diff --git a/ch3.tex b/ch3.tex index 80904af..9cfe886 100644 --- a/ch3.tex +++ b/ch3.tex @@ -3,38 +3,63 @@ qualitative research, methodology varies, but has standard parts: design of the study, sources and their selection, data, the process of analysis, the interpretation, and the approach to validation. Sample selection is recorded and reported - so that others may judge transferability to their own context. Interviews are - the principle technique used by phenomenographical research. Documents - can also be used. Normal conduct of teaching can also provide data that can + so that others may judge transferability to their own context. + The kinds of data in a qualitative study include interviews and documents. + Interviews are + the principle technique used by phenomenographical research. + %Documents can also be used. + Normal conduct of teaching can also provide data that can be used, if in an anonymous, aggregate form. Both deductive and inductive - analysis provide qualitative data. + analysis can be carried out on these data. + The analysis produces a description of the situation under study. + This description may include a narrative, often called a thick and rich description, and also specific attributes, such as categories of findings and relationships among these categories. \section{Design of the Study} - Information learned in tutoring and lecturing undergraduates inspired the research questions. + More specifically, questions asked by the students suggested that they were not learning enough about proof techniques to understand material that appeared later in the curriculum. + So, it seemed useful to discover what their ideas were, about proofs. + We used Bloom's taxonomy of the cognitive domain\cite{bloom1956taxonomy} to subdivide the domain in which we hope to find student ideas. + Correspondingly we created parts of the study: recognition and comprehension were grouped together into one part, application was a part of the study, and the third part included analysis, synthesis, and evaluation. + + We chose a qualitative approach because + we seek to be able to describe the nature of the various + understandings achieved by the students, rather than the relative frequency + with which any particular understanding is obtained. + We chose a phenomenographic approach because it is aimed at identifying and expressing student understandings in a way that transforms these understandings into suggestions how to help them advance their learning. + + We collected data about recognition and comprehension in interviews, and in group help sessions and during tutoring, and also with a written list of questions, and by incorporating observations of computer science classes. + We collected data about application on homeworks, and on practice and actual examinations. + We collected data about analysis in interviews. + We collected data about synthesis on homeworks, and on practice and actual examinations. + - This study is qualitative because - we seek to be able to describe the nature of the various - understandings achieved by the students, rather than the relative frequency - with which any particular understanding is obtained. - - The study was designed to observe undergraduate students as they progressed through the curriculum. + %The study was designed to observe undergraduate students as they progressed through the curriculum. %Changes in conceptualizations of students as they progressed through the curriculum would be interesting if we could detect them. - Consistent with a phenomenographic study, the principle data were interview transcripts. + %Consistent with a phenomenographic study, the principle data were interview transcripts. We conducted over 30 interviews. - Our interview participants were sampled from the students, faculty and graduates of a large public research-oriented university in the northeastern United States. + The conceptualizations of undergraduate computing students were sought. + We incorporated into our design, a method of validation, called triangulation. + To check our findings, we also interviewed faculty and graduate students who had provided courses related to proof. + Our interview participants were sampled from a large public research-oriented university in the northeastern United States. + - We used early interviews to explore students' notions of proof, adapting to the + %this paragraph below is good. + Consistent with a grounded theory approach, we used interviews conducted early in the study to explore students' notions of proof, adapting to the student preference for proof by mathematical induction and incorporating the use of recursive algorithms. + + + We used interviews conducted later in the study to investigate questions that developed from analysis of earlier interviews. + + We used exams to study errors in application of the pumping lemma for regular languages. - We used later interviews to investigate questions that developed from analysis of earlier interviews. + We used homework to observe student attempts at proofs, and to observe student familiarity/facility @@ -69,7 +94,7 @@ The second part was about application: How students attempt to apply proof. The third part was about analysis, synthesis and evaluation: what students do when a situation might be well-addressed by proof. This part informed us about any structure students used as they pursued proof-related activities, and about what students thought was required for an attempted proof to be valid. -This part informed us about the comfort level students have about the use of proof, and the consequences students experience, as a result of their choices about application of proof. +This part also informed us about the comfort level students have about the use of proof, and the consequences students experience, as a result of their choices about application of proof. These parts are summarized in Table \ref{parts}. @@ -94,7 +119,12 @@ This part informed us about the comfort level students have about the use of pro \section{Population Studied} - In a phenomenographic study, it is desirable to sample widely to obtain as broad as possible a view of the multiple ways of experiencing a phenomenon within the population of interest.\cite{marton1997learning} which in turn cites Glaser and Strauss 1967\cite{glaser1968discovery} We studied, by interview, homework and test, undergraduate students who have taken courses involving proof. Typically but not always, these are students majoring in computer science. Some of these undergraduate students are dual majors, in computer science and mathematics. We interviewed graduate students emphasizing those who have been teaching assistants for courses involving proofs. We interviewed faculty who have been taught courses involving proof. We have interviewed former students who have graduated from the department. We have interviewed undergraduates who transferred out of the department. + The population of interest is undergraduate students in the computing disciplines. + In a phenomenographic study, it is desirable to sample widely to obtain as broad as possible a view of the multiple ways of experiencing a phenomenon within the population of interest.\cite{marton1997learning} which in turn cites Glaser and Strauss 1967\cite{glaser1968discovery} We studied undergraduate students who have taken computer science courses involving proof. Typically but not always, these are students majoring in computer science. Some of these undergraduate students are dual majors, in computer science and mathematics. We interviewed graduate students emphasizing those who have been teaching assistants for courses involving proofs. We interviewed faculty who have been taught courses involving proof. We have interviewed former students who have graduated from the department. We have interviewed undergraduates who transferred out of the department. + + The demographics of the interviewed students is representative of the demographics of the computer science/engineering department. + Every student who signed a consent form was requested to schedule an interview, from an interval including 8AM to 9PM. + Every student who scheduled an interview was interviewed. For the benefit of readers wondering to what extent the results might be transferable, demographics of some commonly seen properties of populations are provided: @@ -169,12 +199,12 @@ This part informed us about the comfort level students have about the use of pro - \section{Sample Selection} +% \section{Sample Selection} All participants were volunteers. - Volunteers were sought in all undergraduate classes involving proofs, and also some that did not involve proofs, so that we could sample students at different stages in their undergraduate careers. + Volunteers were sought in all computer science undergraduate classes involving proofs, and also some that did not involve proofs, so that we could sample students at different stages in their undergraduate careers. - Graduate student volunteers were also sought. Most of the graduate student interviews were among teaching assistants in courses that taught and/or used proofs. We also included faculty of courses that involved proofs. + Graduate student volunteers were also sought. Most of the graduate student interviews were among teaching assistants in courses that taught and/or used proofs. We also included faculty of courses that involved proofs. Graduate student and faculty provided another perspective that was used as triangulation, a validation method in qualitative research. % Students from the University of Connecticut who have taken or are taking the relevant courses were offered the opportunity to be interviewed. The undergraduate students @@ -184,7 +214,7 @@ This part informed us about the comfort level students have about the use of pro \subsection{Proofs by Mathematic Induction} - This study is in the first part, about what proof is, but also contributes to the third part, yielding insight into consequences of student use (or not) of proof when the situation warrants. + This part of the study contributes to recognition and comprehension of proof, and also to synthesis, yielding insight into consequences of student use (or not) of proof when the situation warrants. The participants for the study of proof by mathematic induction %We studied students who @@ -198,24 +228,20 @@ This part informed us about the comfort level students have about the use of pro Interviews of eleven students were transcribed for this study. Participants included 2 women and 9 men. Two were international students, a third was a recent immigrant. + %Every student who signed a consent form was requested to schedule an interview, from an interval including 8AM to 9PM. + %Every student who scheduled an interview was interviewed. - \subsection{Domain, Range, Mapping, Relation, Function, Equivalence in Proofs} - This study is in the first part, about what proof is. - - For the study about domain, range, mapping, relation, function and equivalence in proofs, students - taking, or having taken, discrete systems, especially students who - had sought help while taking introductory object oriented programming volunteered - to be interviewed. + \subsection{Purpose of Proof} - This study is in the first part, about why proof appears in the curriculum. + This part of the study contributes the first part; it is about structural relevance. Undergraduate students were sought for this study, because we wanted to know what students thought the purpose was while they were taking the undergraduate subjects. \subsection{Proofs Using the Pumping Lemma for Regular Languages} - This study is in the second part, about how students apply proofs they have been taught. + This part of the study contributes the second part; it is about how students apply proofs they have been taught. The participants for the study of proofs using the pumping lemma for regular languages were forty-two students, of whom thirty-four were men and eight women, @@ -233,14 +259,14 @@ This part informed us about the comfort level students have about the use of pro \subsection{Student Use of Proof for Applicability of Algorithms} - This study is in the third part, about student use of proof. + This part of the study contributes the third part, about student use of proof. The students participating in this part were mainly those having internships or summer jobs. This changed the ratio of domestic to international students, such that a greater proportion were domestic students. Also, the ratio of women to men students was affected, such that a greater proportion were male students. \section{Data Collection} - Our corpus included interview transcripts, homework, practice and real tests, - observations from individual tutoring sessions, and group help sessions. %Interview transcripts were analyzed with thematic analysis. - Homework, practice + Our corpus includes interview transcripts, homework, practice and real tests, + and observations from individual tutoring sessions, and group help sessions. %Interview transcripts were analyzed with thematic analysis. + Homework, and practice and real tests, from several different classes were analyzed for proof attempts. (Incidentally, data from multiple instructors was combined, and no use of information about any specific instructor was used.) Data from individual tutoring sessions and group help sessions were also informative. @@ -256,7 +282,7 @@ This part informed us about the comfort level students have about the use of pro The audio portion of all interviews was collected by electronic recorder and subsequently transferred to a password protected computer. From here the interviews were transcribed, and names were redacted. The redacted interview data were analyzed using the Saturate application. - \subsubsection{Proofs by Mathematic Induction} + \subsubsection{Student Conceptions of What Proofs Is} Interviews were solicited in class by general announcement, and by email. Interviews were conducted in person, using a voice recorder. No further interview script, beyond these following few questions, was used. The interviews @@ -281,15 +307,15 @@ The audio portion of all interviews was collected by electronic recorder and sub Almost every student introduced and described proof by mathematic induction as experienced in their current or recent class. - \subsubsection{Expanded semi-structured interview protocol for domain, range, language, equivalence class in Proofs} - \subsubsection{Expanded semi-structured interview protocol for definitions, language, reasoning in Proofs} + + \subsection{Documents} \subsubsection{Proofs Using the Pumping Lemma for Regular Languages} - The study was carried out on the exam documents. The interpretation was informed + The study was carried out on both real and practice exam documents. The interpretation was informed by the events that occurred in the natural conduct of lectures, help sessions and tutoring. One method of assessing whether students understood the ease of application @@ -304,465 +330,256 @@ The audio portion of all interviews was collected by electronic recorder and sub and the equation itself, from those students engaged in a manipulation with at most superficial understanding. - \section{Phenomenographic Analysis} - - %Describing how analysis was done in detail is really important. - %How do you do phenomenography? - - %Is this the way everything was analyzed? - - - - Marton and Booth\cite[p. 133]{marton1997learning} describe a desirable analysis technique: - - \begin{quote} - [ Apply] the principle of focusing on one aspect of the object and seeking its dimension of variation while holding other aspects frozen.\end{quote} %partial derivative - - Remember to apply both perspectives, - \begin{quote}that pertaining to the individual and that pertaining to the collective. - \end{quote} - -\begin{quote} [E]stablish a perspective with boundaries, within which [one seeks] variation.\end{quote} - - Recalling that Marton and Booth regard the learning objective as a collection of related aspects, with the relationships, we can observe that a component hierarchy can represent the aspects. Recalling that Marton and Booth discuss the depth of understanding, we can observe that one consequence of depth of understanding is the development of a generalization/specialization hierarchy. Marton and Booth contrast situations with phenomena, such that phenomena are understandings and situations serve as relatively concrete examples of phenomena, as used in instruction and assessment. - - We may search for evidence of recognition of aspects; they might be mentioned by learners. Marton and Booth have observed that in different context, different aspects shift between foreground (consciousness) and background. - - \begin{enumerate} - \item search for extracts from data, that might pertain to perspective - \item inspect them in context of own interview - \item inspect them in context of other extracts all interviews on the same theme - \end{enumerate} - - Deductively (in the sense of qualitative analysis), in a reductionist fashion, we may select an aspect of the learning objective and seek mention of it. - - \begin{enumerate} - \item select one aspect of the phenomenon and inspect across all subjects - \item select another aspect - \item whole interview -- to see where these two aspects lie relative to other aspects, and to background - \end{enumerate} - - Holistically, we might attune our investigation to seek evidence that generalization may have occurred. - - \begin{enumerate} - \item all of research problems, one problem at a time, whole transcripts that have particularly interesting ways of handling problem - \end{enumerate} - - Marton and Booth advise persistence ``return to again and again \ldots until there is clarity''. - - Completion may be recognized by the achievement of a result, specifically the ability to identify a number of qualitatively different ways in which phenomenon has been experienced (not forgetting different methods of expression)\cite[p. 133]{marton1997learning}. - - Overlap of the material at the collective level is expected. - - Marton and Booth advise us to ``assume that what people say is logical from their point of view''\cite[p. 134]{marton1997learning}, citing Smedlund\cite{smedslund1970circular} - - - \subsection{Application of Phenomenographic Analysis in this Study} - - We applied phenomenographic analysis, basic inductive analysis and deductive qualitative analysis in this study. - - The analysis for the research question "What do students think proof is for?", which was approached as "Why do you think we teach proof?" exemplifies a phenomenographic approach. One aspect of the phenomenon of proof is its utility' We selected student verbal productions related to the use of proof. We considered them in the context of their own interview, and we compared them to data from other interviews on the same theme. - - The questions are ordered guided by the 1956 version of Bloom's Taxonomy, namely, recognition, comprehension, application, analysis, synthesis, evaluation.\cite{bloom1956taxonomy} - - % % %recognition - - \subsubsection{Phenomenographic Analysis of What Students Think Proofs Are} - - \begin{quote} - Mathematics, as it is currently practiced, is a delicate interplay between monastic contemplation and blowing stuff up with dynamite. (Ellenberg)\cite[page 223]{ellenberg2014not} - - Probably I don't want to keep this, but it's fun at the moment. - \end{quote} - - Some students exhibit an understanding of proof at the "black box" level, i.e., there is understanding of the role of proof, without considering any internal structure. When a proof exists, we can know that the thing the proof proves is true, in the context that applies. We can "use that theorem". Other students, though, do not have this idea consolidated yet. For example, if we consider proof by exhaustion applied to a finite set of cardinality one, we can associate to it, the idea of a test. Students, assigned to test an algorithm for approximating the sine function, knew to invoke their implementation with the value to be tested, but did not check their result, either against the range of the sine function, or by comparison with the provided sine implementation, presenting values over 480 million. - - Moving the the "white box" level, we find a spectrum of variation in student understanding. - The most opaque end of this spectrum has been called "magic incantation". In this conceptualization we find those ideas of statements that are not clear, and use of mathematical symbols that is not understood. - - Ellenberg\cite[page 409]{ellenberg2014not} reports that some mathematicians regard axioms as strings of symbols without meaning, and that this quite formal conceptualization can be contrasted with another conceptualization that axioms are true statements about logical facts. He talks of these conceptualizations being taken by the same individual at different times. - - Our participants seemed to regard axioms as strings of symbols that do mean something, though that meaning the participant ascribed might not be correct (especially as participants did not always know definitions of the entities being related), or the participant might feel unable to ascribe any meaning. We did find participants who appreciated the significance of definitions. They were dual majors in math. - - Another waystation on this dimension of variation is "sequence of statements". A more elaborate idea is "sequence of statements where each next statement is justified by what when before". A yet more complete concept is "finite sequence of statements, starting with the premise and ending with what we want to prove, and justified in each step." A more profound conceptualization was found "finite sequence of statements, starting with axioms and premises, proceeding by logical deduction using (valid) rules of inference to what we wanted to prove, that shows us a consequence of the definitions with which we began, an exploration in which we discover the truth value of what we wanted to show, serving after its creation as an explanation of why the theorem is true". - - A few categories, such as those above, serve to identify a dimension of variation. When our purposes include discovering which points we may want to emphasize, we can examine the categories seeking to identify how they are related and how they differ. - - It can certainly be that having more categories provides more critical aspects. For example, Harel and Sowder\cite{harel1998students} offered extrinsic vs. intrinsic conviction, empirical proof schemes and their most advanced deductive proof schemes as broader categories, and seven useful subcategories of these, yielding six critical aspects that suggest what teachers could usefully vary, to help learners discern items that would advance their knowledge. - - - - - - - % % %comprehension - - \subsubsection{Phenomenographic Analysis of How Students Attempt to Understand Proofs} - Some students are attempting to understand proofs while not recognizing that they are studying a proof. - "Were we studying proofs today?" "No" "Were we discussing certain contexts, and why certain ideas will always be true in those contexts?" "Yes" "Doesn't that seem like proof, then?" "Yes"\\ - "So, you're taking introduction to the theory of computing this semester. Do you seen any use of proofs in that course?" "No"\\ - - - Some students read proofs. - - Some students look up the definitions of terms used in the proofs and some do not. - Some students think (or hope) they can solve problems involving producing proofs, without knowing the definitions of the terms used to pose the problem. - Some students are aware that definitions are given, but "zone out" until examples are given. When examples are given, the students attempt to infer definitions themselves. Some students will compare the definitions they infer with the mathematical community's definitions. Some students do not. - - Some students think that the reading of proofs is normally conducted at the same speed as other reading, such as informal sources of information. - - Of students who read proofs attentively, some try to determine what rule of inference was used in moving from one statement to the next, and some do not. - - Some students notice that lemmas can be proved and then used as building blocks in a larger argument, and some do not. - - Some students can identify the forms of proof learned in discrete systems, when they see them employed in proofs, such as the combination of arguing by contrapositive and modus ponens. Some cannot. - - Some students can identify these forms in an argument if the argument is made about concrete objects, such as cars or specific people. Some of these students have difficulty transferring this ability with concrete objects to application to abstract entities such as sets, algorithms or symbols. - - Some students who achieve with difficulty the ability to recognize the application of rules of inference in one argument about abstract entities, become quicker at recognizing arguments of similar form about other abstract entities, and some continue to achieve with difficulty, as if learning the first argument did not facilitate learning the second argument of the same form. - - Students would attend to diagrammatic representation of proofs, such as a block digram depicting machine descriptions packaged as input for yet other machines to process, but were not observed to employ such diagrams. - - Students have been seen to employ decision tree diagrams. - - Students would attend to algorithm representation of proofs, such as a recursive process that determines a prime factorization, but were not observed to employ such algorithmic descriptions. - - Students answering a list of questions, representing computer science ideas mathematically, in algorithms and in figures found the questions "interesting", "fun", "different" and "non-trivial". - - Except when assigned to do so, students were not observed to attempt to solve simpler problems, such as by imposing partitioning into cases. Except when assigned to do so, students were not observed to attempt to solve more general problems, as is sometimes helpful.\cite[that pin dropping probability problem]{ellenberg2014not} - - % % % structural relevance - - \subsubsection{Phenomenographic Analysis of Reasons for Teaching Proof} - - Excerpts of student transcripts were selected on the basis of being related to this question. A dimension of variation emerged from the data, such that the excerpts seemed readily organized along this dimension. - - \begin{longtable}{|p{7cm}|p{8.5cm}|}\hline - - \endfirsthead - - \multicolumn{2}{c}% - {{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\ - \hline \multicolumn{1}{|c|}{\textbf{Category}} & - \multicolumn{1}{c|}{\textbf{Representative}} \\ \hline - \endhead - - \hline \multicolumn{2}{|r|}{{Continued on next page}} \\ \hline - \endfoot - - \hline \hline - \endlastfoot - -% \begin{tabular}{|p{7cm}|p{8.5cm}|}\hline - Category & Representative\\\hline\hline - - - some students do not see any point to proof& - They teach it to us because they were mathematicians and they like it.\\ - - - & we didn't see ok why do i really have to know the proof of the theorem to do that right? We didn't see the point, because no one taught us the point, so, that's a very important part that was missing.\\\hline - - some students think that it satisfies the curriculum goals, to be able to reproduce a previously taught proof, or follow a procedure to generate a proof, without being personally convinced& - I was able to get a full score, but I don't understand why a proof by induction is convincing\\\hline - - Some students do not see a relationship between a problem and approach& - When I have to prove anything, I always start with proof by mathematic induction, that was the one they taught the most.\\\hline - - Some students are surprised to discover that there is a relation between proof by induction and recursion& - I never noticed that before, but now that you mention it, I see that they are isomorphic.\\\hline - - Some students see the relationship but do not use it& - Professor (redacted) would be really proud of me that I learned to understand proof by induction quite well. \ldots I understand how recursion matches induction, there's a base case, there's a way of proceeding. \ldots I just couldn't figure out how to program the merge-sort algorithm.\\\hline - - some students think the only reason for studying proof is to understand proofs of, for example, resource utilization of known algorithms& - I would never consider writing a proof except on an assignment.\\ - - & I understand the proof of the lower bound on comparison sort. \ldots I understand the proof of the upper bound on searching in a binary search tree. \ldots If I had to prove something about termination on a search tree, I don't know how I would do that.\\ - - - & I know that recursion has the same structure as proof by mathematical induction. \ldots If I had an algorithm with a recursive data structure like a tree, and I had to prove something like termination about it, I'm not sure what approach I would use, it would depend.\\\hline - - Some students see that they could employ proof to explore whether an algorithm can be expected to solve a problem in a given context that includes bounds upon resources that are available for consumption. & mostly design the algorithm first, we had some expectation of what that complexity results would be and then we try to find an approach to prove. -%\end{tabular} - \end{longtable} - - - - - Data were analyzed using a modified version of thematic analysis, which is - in turn a form of basic inductive analysis.\cite{Merriam2002,Merriam2009,braun2006using,fereday2008demonstrating,boyatzis1998transforming} Using thematic analysis, we - read texts, including transcripts, looked for “units of meaning”, and extracted - these phrases. Deductive categorization began with defined categories, and - sorted data into them. Inductive categorization “learned” the categories, in - the sense of machine learning, which is to say, the categories were determined - from the data, as features and relationships found among the data suggested - more and less closely related elements of the data. A check on the development - of categories compared the categories with the collection of units of meaning. - Each category was named by either an actual unit of meaning (obtained during - open coding) or a synonym (developed to capture the essence of the category). - A memo was written to capture the summary meaning of the category. - Next a process called axial coding, found in the literature on grounded theory, - \cite{strauss1990basics,kendall1999axial,glaser2008conceptualization} was applied. This process considered each category in turn as a central - hub; attention focused on pairwise relations between that central category - with each of the others. The strength and character of the posited relationship - between each pair of categories was assessed. On the basis of the relationships - characterized in this exercise, the categories with the strongest interesting relationships - were promoted to main themes. A diagram showing the main - themes and their relationships, qualified by the other, subsidiary themes and - the relationships between the subsidiary and main themes was prepared to - present the findings. Using the process of constant comparison, the structure - of these relationships was reviewed in the light of the meanings of the categories. - A memo was written about each relationship in the diagram, referring - to the meaning of the categories and declaring the meaning of the relationship. - A narrative was written to capture the content of the diagram. Using the - process of constant comparison, the narrative was reviewed to see whether it - captured the sense of the diagram. Units of meaning were compared with the - narrative and their original context, to see whether the narrative seemed to - capture the meaning. The products of the analysis were the narrative and the - diagram. - - %\chapter{Analysis} - - The product of analysis in a phenomenographic study, is a set of categories, and relationships \textit{among} them. - - Marton and Booth\cite[p. 135]{marton1997learning} state "in the late stages of analysis, our researcher [has] a sharply structured object of research, with clearly related faces, rich in meaning. She is able to bring into focus now one aspect, now another; she is able to see how they fit together like pieces of a multidimensional jigsaw puzzle; she is able to turn it around and see it against the background of the different situations that it now transcends." - - Using Marton's overriding categorization of task and objective, we can consider that some students do not know, at least when they are studying CSE2500, that they need to be able to understand some proofs, to be good developers. Therefore, they can logically approach the study of proof in CSE2500 as a task, having some facts that they must memorize. Other students, including those with dual majors in math, wish to improve themselves by improving their ability to couch arguments in mathematical terms and both ascertain facts for themselves and convince others. Students who are aware that there are computer-science related purposes for proof, for example, in the study of algorithms they will be using proofs to understand resource consumption, will recognize the study as having the objective to improve themselves vis-a-vis dealing with proof. - - - Because the relationships are expected to form a partial order, corresponding to set inclusion of subsets of the complete (with respect to the objective of teaching) understanding of the information being taught, we can say relationships \textit{between} categories. - - The set inclusion relationship can be that a deeper understanding includes a more superficial understanding. - It may also be that a deeper understanding qualifies a more superficial understanding, such as being applicable in a restricted domain. Thus, understanding of a liquid as being divisible to any degree can be qualified as to scale such as macroscopic, microscopic and so on. - - In a phenomenographic study, this partial order is referred to as a hierarchical order. - - The objective of teaching, as will be in some parts of this study, the components of proof, may have many parts, called, in phenomenology, internal structure. The granularity of the subdivision of the objective of teaching results in the number of parts in the internal structure. If we let $n$ denote the number of parts obtained with a specific granularity, we see that the number of subsets will be $2^n$, which will be inconveniently large unless the granularity is sufficiently coarse. Thus, we choose a granularity resulting in approximately 4 elements of internal structure. - - Thus, when the teaching objective is, what is a proof, we may limit the granularity, such that the internal structure of a proof is, for example, - \begin{enumerate} - \item that particular statement which is to be proved - \item axioms, premises, suppositions, cases - \item other statements - \item warrants (rules of inference) - \end{enumerate} - - We might choose to pursue finer granularity in some cases, for example, we might pursue "What is a statement?", because instructors have found that not all students arriving in CSE3502 have the same depth of understanding of statement, and some do not have sufficiently deep understanding of "statement" to be able to comprehend a proof. - - - Marton and Booth\cite[p. 22]{marton1997learning} call our attention to learners directing their attention to the sign vs. to the signified. With proofs, Polya \cite{} has mentioned a procedural approach to executing a proof, without understanding, as have Harel and Sowder \cite{harel1998students}, and Tall\cite{tall2001symbols}. Weber and Alcock\cite{weber2004semantic} have observed and described students omitting understanding of warrants in proofs. In each of these cases, the sign is provided, but the signified is at best incompletely understood. - - Analysis can usefully illuminate learning processes, taking note of the temporal domain\cite{marton1997learning}. This has been used by Booth in her analysis of how students understand the process of programming\cite{marton1997learning}. - - [p. 136]\cite{marton1997learning} important to be looking whether conceptualizations appear in a certain case, in a certain period of time (such as, when see proofs again in 3500, 3502, are they recognized as proofs, some no some yes, are they helpful as proofs, or troublesome, some helpful, some not, "never did get that"). - - % % %application - -\subsubsection{Phenomenographic Analysis of How Students Attempt to Apply Proofs (When not assigned)} - -Some students claimed they never constructed proofs when not assigned. - - - - -\subsubsection{Phenomenographic Analysis of Whether students exhibit consequences of inability (such as avoiding recursion)} - -Some students claimed to know how to write recursive algorithms but said they never used them because they did not know when they were applicable. - -% % %analysis - - \subsubsection{Phenomenographic Analysis of How familiar and/or comfortable are students with different (specific) proof techniques: induction, construction, contradiction?} - - When asked about specific proof techniques, some students mentioned proof by mathematical induction "When faced with a proof I start with induction, they taught us that one the most." - - When asked for specific proof techniques other than proof by mathematical induction, students knew the words contradiction and contrapositive, but sometimes could not distinguish between them. - - When asked about proof by construction, some students thought this referred to construction of any proof. - - Some students thought proof by contradiction referred to proving the opposite of something, rather than disproving the opposite of something. - - \subsubsection{Phenomenographic Analysis of Whether students notice structural elements in proofs} + \subsection{Observations from Tutoring and Help Sessions} + These were recorded, at the conclusion of the help session or tutoring session, into notes for manuscripts under preparation at the time. - Some students, in the context of hearing a presentation in an algorithms course, of a proof with a lemma, do not know, by name, what a lemma is. "What's a lemma?" + \section{Method of Analysis} - Some students, in the context of planning to construct a proof, do not choose to divide and conquer the problem, breaking it into component parts, such as cases. "What good does that do, doesn't the proof become longer?" + The phenomenographic approach to analysis has been written about by Marton and Booth\cite{marton1997learning}. + This method works on interview and other data, and aims to produce a set of categories with relationships among them. Moreover, these categories and relations are used to infer critical aspects, which are ideas that are critical for developing to a more advanced conceptualization from a less advanced conceptualization. - Some students describe proofs as a sequence of statements, not commenting on any structure. + The process by which this transformation of data occurs has been further clarified by Marton and Booth\cite[p.103]{marton1997learning}, who have written that an analyst should apply "the principle of focusing on one aspect of the object and seeking its dimension of variation while holding other aspects frozen" is helpful. - \subsubsection{Phenomenographic Analysis of What do students think it takes to make an argument valid?} + One example of applying this principle is the analysis directed to the question of what students think about why proofs are taught in the curriculum. Using the terminology of Marton and Booth, "structural relevance", we consider structural relevance to be an aspect of proof in the curriculum. Students should learn about proof for reasons that are connected with other material in the curriculum. For example, proof by mathematic induction is relevant for understanding the explanation of why context free grammars generate the languages accepted by non-deterministic pushdown automata. We focus on the idea of the students' conceptions of why proof is taught. We look for a dimension of variation: some of the students' ideas about why proof is taught will contain more of the reason underlying the presence of proof in the curriculum. Using this single dimension we can sequence excerpts of student interview transcripts, student utterances, according to how little or much of this reason they recognize. This exercise is provided as an example in Table \ref{exemplar}. - Some students used confused/incorrect forms of rules of inference. + % % % structural relevance + + Excerpts of student transcripts were selected on the basis of being related to this question. A dimension of variation emerged from the data, such that the excerpts seemed readily organized along this dimension. + + \begin{longtable}{|p{7cm}|p{8.5cm}|}\hline + \caption{Phenomenographic Analysis of Reasons for Teaching Proof}\label{exemplar} + \endfirsthead + + \multicolumn{2}{c}% + {{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\ + \hline \multicolumn{1}{|c|}{\textbf{Category}} & + \multicolumn{1}{c|}{\textbf{Representative}} \\ \hline + \endhead + + \hline \multicolumn{2}{|r|}{{Continued on next page}} \\ \hline + \endfoot + + \hline \hline + \endlastfoot + + % \begin{tabular}{|p{7cm}|p{8.5cm}|}\hline + Category & Representative\\\hline\hline + + + some students do not see any point to proof& + They teach it to us because they were mathematicians and they like it.\\ + + + & we didn't see ok why do i really have to know the proof of the theorem to do that right? We didn't see the point, because no one taught us the point, so, that's a very important part that was missing.\\\hline - Some students do not notice that the imposition of a subdivision into cases creates more premises. + some students think that it satisfies the curriculum goals, to be able to reproduce a previously taught proof, or follow a procedure to generate a proof, without being personally convinced& + I was able to get a full score, but I don't understand why a proof by induction is convincing\\\hline + + Some students do not see a relationship between a problem and approach& + When I have to prove anything, I always start with proof by mathematic induction, that was the one they taught the most.\\\hline + + Some students are surprised to discover that there is a relation between proof by induction and recursion& + I never noticed that before, but now that you mention it, I see that they are isomorphic.\\\hline + + Some students see the relationship but do not use it& + Professor (redacted) would be really proud of me that I learned to understand proof by induction quite well. \ldots I understand how recursion matches induction, there's a base case, there's a way of proceeding. \ldots I just couldn't figure out how to program the merge-sort algorithm.\\\hline + + some students think the only reason for studying proof is to understand proofs of, for example, resource utilization of known algorithms& + I would never consider writing a proof except on an assignment.\\ + + & I understand the proof of the lower bound on comparison sort. \ldots I understand the proof of the upper bound on searching in a binary search tree. \ldots If I had to prove something about termination on a search tree, I don't know how I would do that.\\ + + + & I know that recursion has the same structure as proof by mathematical induction. \ldots If I had an algorithm with a recursive data structure like a tree, and I had to prove something like termination about it, I'm not sure what approach I would use, it would depend.\\\hline + + Some students see that they could employ proof to explore whether an algorithm can be expected to solve a problem in a given context that includes bounds upon resources that are available for consumption. & mostly design the algorithm first, we had some expectation of what that complexity results would be and then we try to find an approach to prove. + %\end{tabular} + \end{longtable} + + + Marton and Booth\cite[p. 133]{marton1997learning} note that the phenomenographic method of analysis includes viewing excerpts of student utterances in specific perspectives. They advise "establish a perspective with boundaries, within which one seeks variation", and to remember to apply perspectives "that pertaining to the individual and that pertaining to the collective". So, when we establish a perspective with boundaries, we set a scope, allowing us to admit student text fragments relevant to that scope, filtering out other remarks. When we sequence or categorize the selected utterances, during which we will be comparing data from difference individuals, we must evaluate the utterances within the context of the interview from which they were obtained. For example, one student might be more prone to exaggeration than another. Also, one student may have more mathematical background than another. - Some students do not notice that proof by contradiction introduces (for purposes of contradiction) a premise. - -% % %synthesis + Marton and Booth regard the learning objective as a collection of related aspects, with their relationships; we can observe that a component hierarchy can represent the aspects. Marton and Booth discuss the depth of understanding; we can observe that one consequence of depth of understanding is the development of a generalization/specialization hierarchy. Marton and Booth contrast situations with phenomena, such that phenomena are understandings and situations serve as relatively concrete examples of phenomena, as used in instruction and assessment. -\subsubsection{Phenomenographic Analysis of Whether students incorporate structural elements in proofs} - -Students have asked whether, when using categorization into cases, they must apply the same proof technique in each of the cases. - - Probably needs additional interviews. - - - -% %evaluation - -\subsection{Analysis of Interviews} - -Items excerpted from interviews for analysis should be analyzed in the context of the specific interview and also in the context of the ensemble.\cite{marton1997learning}. - - Data were analyzed using a modified version of thematic analysis, which is + We may search for evidence of recognition of aspects; they might be mentioned by learners. Marton and Booth have observed that in different context, different aspects shift between foreground (consciousness) and background. + Marton and Booth advise us to ``assume that what people say is logical from their point of view''\cite[p. 134]{marton1997learning}, citing Smedlund\cite{smedslund1970circular} + + + + + Marton and Booth \cite[p. 133]{marton1997learning} write that completion may be recognized by the achievement of a result, specifically the ability to identify a number of qualitatively different ways in which phenomenon has been experienced (not forgetting different methods of expression). + + One approach we have taken, besides the single aspect oriented approach exemplified in Table \ref{exemplar}, is to apply basic inductive analysis and deductive qualitative analysis, including axial coding, with the phenomenographic paradigm in mind. + + More specifically, when processing interview data, we transcribe the data, we transfer the transcribed and redacted\footnote{Names of people are removed.} data to a website-based tool named Saturate. We have used both the current Saturate application and the previous version. The previous version has features we prefer to the more recent version. + + We use the Saturate application to select contiguous fragments of text the capture meaning in our judgment. Each selected fragment is labeled. These labels, which are also called codes, can be reused, thus collecting together multiple fragments, as synonymous. A process, called constant comparison, begins at this level of aggregation of the data. A code, representing the synonymous fragments, is chosen, either from among the fragments or not. The fragments sharing a code are compared with one another, to ascertain whether the group with that code is internally cohesive, to such a degree that fragments in any one group are relatively distinct from fragments in other groups. A summary description (called a memo) of each code is written, and fragments are checked for compatibility with the code's description. + + Data were analyzed using a modified version of thematic analysis, which is in turn a form of basic inductive analysis.\cite{Merriam2002,Merriam2009,braun2006using,fereday2008demonstrating,boyatzis1998transforming} Using thematic analysis, we read texts, including transcripts, looked for “units of meaning”, and extracted these phrases. Deductive categorization began with defined categories, and sorted data into them. Inductive categorization “learned” the categories, in the sense of machine learning, which is to say, the categories were determined from the data, as features and relationships found among the data suggested - more and less closely related elements of the data. A check on the development - of categories compared the categories with the collection of units of meaning. - Each category was named by either an actual unit of meaning (obtained during - open coding) or a synonym (developed to capture the essence of the category). + more and less closely related elements of the data. A check on the development + of categories compared the categories with the collection of units of meaning. + Each category was named by either an actual unit of meaning (obtained during + open coding\footnote{Open coding is so-called because it occurs at a time when the analyst is the most open-minded about what the meanings being found in the data might be. \cite[p. 178]{merriam2009qualitative}}) or a synonym (developed to capture the essence of the category). A memo was written to capture the summary meaning of the category. - Next a process called axial coding, found in the literature on grounded theory, - \cite{strauss1990basics,kendall1999axial,glaser2008conceptualization} was applied. This process considered each category in turn as a central - hub; attention focused on pairwise relations between that central category - with each of the others. The strength and character of the posited relationship - between each pair of categories was assessed. On the basis of the relationships - characterized in this exercise, the categories with the strongest interesting relationships - were promoted to main themes. A diagram showing the main - themes and their relationships, qualified by the other, subsidiary themes and - the relationships between the subsidiary and main themes was prepared to - present the findings. Using the process of constant comparison, the structure - of these relationships was reviewed in the light of the meanings of the categories. - A memo was written about each relationship in the diagram, referring - to the meaning of the categories and declaring the meaning of the relationship. - A narrative was written to capture the content of the diagram. Using the - process of constant comparison, the narrative was reviewed to see whether it - captured the sense of the diagram. Units of meaning were compared with the - narrative and their original context, to see whether the narrative seemed to - capture the meaning. The products of the analysis were the narrative and the - diagram. - -% \section{Interview} - Some students remembered taking proofs in high school in geometry. - Some students were taking proofs contemporaneously in philosophy. - Some of the students studying proof in philosophy found them disturbing, expressing a preference for geometrical proofs. - Some students remembered having to furnish proofs of geometrical facts, also facts about prime numbers and sets. - Some students knew that CSE2500 treated proofs because they would be used in later courses. Students did not know why proofs would be used later, and were generally happy to hear some example uses. - Though students were asked whether they made use of proofs spontaneously, none of those interviewed gave an example. - Some students preferred to articulate with code, and some (who were dual computer science / math) sometimes preferred mathematical symbols, depending upon the context. - Some students do wish to convince themselves of things, such as tractable execution times, and correctness. Though students were asked whether they made use of proofs for this purpose, none of those interviewed claimed to do so, rather they mentioned going carefully over their algorithm construction, and considering cases. - - In interviews, the students almost all chose to discuss proofs by mathematical induction. - - \subsubsection{Themes / Categories} - \begin{itemize} - \item Definitions\\ - Students divided into (1)those who found definitions boring, difficult to pay attention to, and undesirable compared to examples, from which they preferred to induce their own definitions, and (2) those who had caught on to the idea that definitions were the carefully crafted building blocks of reasoning. - \item Procedures - Students sometimes learned what was desired in a proof, but learned to produce it by procedure, and were not themselves convinced. - \item Context - Students asked whether the topics for examples and exercises, such as prime numbers, had relevance to programming, with which they had experience, but not unrelated to the topics. - Students did not know the context in which the proofs, or procedure version of proof, was applicable, so, for example, did not apply proof by mathematic induction to recursive algorithms, and did not know how to tell whether recursive algorithms would be applicable. - \item Concrete vs. Abstract - Some students felt quite comfortable with the application of rules of inference to concrete items, but had difficulty transferring application of those rules to mathematical symbols. - \item Symbolization - consistent with Harel and Sowder's 1998 categorization of concepts, we found students who would attempt to write in symbols, but not understand what was denoted, and consequently were uncertain about appropriate operations. Some of these students were glad to see a progression from pseudocode with long variable names to pseudocode with short variable names to mathematical symbolization (formula translation (FORTRAN) in reverse). - \item Applicability of single examples - Some students believed that a few examples constituted a proof. These examples were not generic particular, nor were they transformational, in the sense of Harel and Sowder's 1998 model. - \item Substructure - Students familiar with methods, in the sense of object-oriented programming, and with construction of programs involving multiple method calls, did not always recognize that proofs could be built from multiple lemmas, although they did understand that axioms could be applied. - \item Proofs are used, in computer science, to show resource consumption (complexity class), properties of models of computation, and computability/decidability. No occasion was identified, other than assignment, when undergraduate students recognized they were undertaking proofs. - \item Among graduate students, proofs were undertaken in the context of preparing manuscripts for publication. These were scheduled to be approached after algorithm design, though retroactive adjustment of algorithms did occur for simplifying the proof. - - \end{itemize} - \subsubsection{Relationships} - - \subsection{Analysis of Homework and Tests} - \subsubsection{Proofs} - Proofs submitted on homework and tests were analyzed in several respects. - The overall approach should be valid. For example, students who undertook to prove that the converse was true did not use a valid approach. - The individual statements should each be warranted. - Use of structure, such as lemmas, and care that cases form a partition of the relevant set are gladly noticed. - Proof attempts that lose track of the goal, and proof attempts that assert with insufficient justification, the goal are noted. - \subsubsection{Pumping Lemmas} - We wrote descriptions for each error. Some example descriptions - are in Table II. - - - Table : Some example errors - Let x be empty - $|xy| \leq p, so xy = 0^p$\\ - $|xy| \leq p; let \; x = 0^{p+r}, y = 0^{p+r}, 0 < r < p$\\ - Let’s choose $|xy| = p$\\ - $0^{p+1}0^b1^p \neq 0^{p+1}1^p \therefore xy^2z \not\in \mathcal{L}$ - where $\mathcal{L} = \{0^i1^j, i \neq j\}$\\ - we choose $s = 0^{p+1}1^p$ within $|xy|$\\ - thus $\neq 0^p1^{p+1}$\\ - Let $x = 0^a, y = 0^b1^a$\\ - $x = 0^{p-h}, y = 0^h$\\ - $x = 0^i, y = 0^i, z = 0^i1^j$ - - A handful of students did exhibit their reasoning that for - all segmentations there would exist at least one value of 𝑖 that - would generate a string outside the language. - We categorized the errors as misunderstandings of one or - more of: - 1) ∣𝑥𝑦∣ ≤ 𝑝 permits ∣𝑥𝑦∣ < 𝑝\\ - 2) 𝑥 is the part of the string prior to the cycle\\ - 3) 𝑦 is the part of the string which returns the state of - the automaton to a previously visited state\\ - 4) 𝑧 is the part of the string after the (last) cycle up to - acceptance\\ - 5) 𝑝 − 1 characters is the maximum size of a string - that need not contain a cycle, (strings of length 𝑝 - or greater must reuse a state)\\ - 6) 𝑖 is the number of executions of 𝑦\\ - 7) There must be no segmentation for which pumping - is possible, if pumping cannot occur.\\ - 8) A language is a set of strings.\\ - 9) A language class is a set of languages.\\ - Categories are shown in the chapter on results (labelled table iii).\\ - - - - - \subsection{Analysis of Help Session and Tutoring} - some students, who do know that any statement must and can, be - either true or false, thought implications must be true. - - - -% The study proceeded using prior interview experiences to suggest further investigation. - % Originally asking about proofs and what they were for, we received answers about proof by induction and found out not all students contemplate why the curriculum contains what it does. - % (reference Guzdial on students trusting that whatever curriculum they take, is very likely to qualify them for a job). - %When providing leading questions about why, interview data indicated that not all students apply proof techniques with which they have been successful. - %So, when application of proofs developed as a part of the study. - %Consequently, what clues there may be that prompt recall of proof and proof techniques to application to a problem became part of the study, involving structural relevance. - %Generalization is related to the presence of one situation evolving a response that was learned in the context of a different situation prompted the view of teacher teaching from context of generalization hierarchy present and situation as example and homework situation as another example. - %What opportunities to foster generalization do students notice? - - % \subsection{Within What is a Proof?} - %The study was devoted to proofs, a subject that can be subdivided. - %Part of the study was aimed at the idea of domain, directed at the concept that - %though a variable could identify a scalar, it might also represent a set. - %Part of the study was aimed at the activity of abstraction, because some students - %exhibited the ability to operate at one level of abstraction, not necessarily a - %concrete level, yet the ability to traverse between that level of abstraction and - %a concrete level seemed to be absent. Other students claimed to be able to - %understand concrete examples with ease, but to encounter difficulty when - %short variable names were used within the same logical argument. - - %\subsection{Order of Exploration} - %The order of exploration was data driven, thus the material was sought sometimes - %in reverse order of the curriculum, almost as if seeking bedrock by starting - %at a surface, and working downwards. - - - - - \ No newline at end of file + + + + Then, with a set of codes, we again perform grouping. This time we group codes into categories. Each category is reviewed to check whether the codes contained are relatively cohesive within a category, and relatively distinct from codes in other categories. A memo is written for each category. + + Categories at this point in the analysis are also called initial themes. Themes are used in the process of axial coding. The word axial refers to the hub and spoke placement of the categories, as, one at a time, each category takes its place as a hub in a diagram, with one spoke for each of the other categories, that are ranged around the hub in a circle. Each spoke is labeled with a pairwise relationship between categories. After the relationships have been inferred by considering each pair of categories, the relationships are themselves compared. Participation in multiple strong relationships distinguishes a category, promoting it to a (what's the adjective, something like principle) theme. + Next a process called axial coding, found in the literature on grounded theory, + \cite{strauss1990basics,kendall1999axial,glaser2008conceptualization} was applied. This process considered each category in turn as a central + hub; attention focused on pairwise relations between that central category + with each of the others. The strength and character of the posited relationship + between each pair of categories was assessed. On the basis of the relationships + characterized in this exercise, the categories with the strongest interesting relationships + were promoted to main themes. + + Attending to the phenomenographic paradigm, we seek dimensions of variation. These are delineated by the appearance of multiple categories that are usually related to each other by including more aspects (component parts) of an idea that is a learning objective. + The basic inductive analysis with axial coding described above should make more evident relationships in the data that are of the nature of a dimension of variation. Thus we see that basic inductive analysis with axial coding is compatible with phenomengraphic analysis, in that it can be directed towards achieving the goals of phenomenographic analysis insofar as one or more dimensions of variation can emerge. + + Phenomenographic analysis proceeds beyond the identification of categories and relationships to infer critical aspects. These are differences between related categories, such that discernment by students of the ideas differentiating those two related categories are thought necessary for the students' depth of understanding to develop into the more inclusive category. + + A diagram showing the main + themes and their relationships, qualified by the other, subsidiary themes and + the relationships between the subsidiary and main themes was prepared to + present the findings. Using the process of constant comparison, the structure + of these relationships was reviewed in the light of the meanings of the categories. + A memo was written about each relationship in the diagram, referring + to the meaning of the categories and declaring the meaning of the relationship. + A narrative was written to capture the content of the diagram. Using the + process of constant comparison, the narrative was reviewed to see whether it + captured the sense of the diagram. Units of meaning were compared with the + narrative and their original context, to see whether the narrative seemed to + capture the meaning. The products of the analysis were the narrative and the + diagram. + + + + Using Marton's overriding categorization of task and objective, we can consider that some students do not know, at least when they are studying CSE2500, that they need to be able to understand some proofs, to be good developers. Therefore, they can logically approach the study of proof in CSE2500 as a task, having some facts that they must memorize. Other students, including those with dual majors in math, wish to improve themselves by improving their ability to couch arguments in mathematical terms and both ascertain facts for themselves and convince others. Students who are aware that there are computer-science related purposes for proof, for example, in the study of algorithms they will be using proofs to understand resource consumption, will recognize the study as having the objective to improve themselves vis-a-vis dealing with proof. + + + Because the relationships are expected to form a partial order, corresponding to set inclusion of subsets of the complete (with respect to the objective of teaching) understanding of the information being taught, we can say relationships \textit{between} categories. + + The set inclusion relationship can be that a deeper understanding includes a more superficial understanding. + It may also be that a deeper understanding qualifies a more superficial understanding, such as being applicable in a restricted domain. Thus, understanding of a liquid as being divisible to any degree can be qualified as to scale such as macroscopic, microscopic and so on. + + In a phenomenographic study, this partial order is referred to as a hierarchical order. + + The objective of teaching, as will be in some parts of this study, the components of proof, may have many parts, called, in phenomenology, internal structure. The granularity of the subdivision of the objective of teaching results in the number of parts in the internal structure. If we let $n$ denote the number of parts obtained with a specific granularity, we see that the number of subsets will be $2^n$, which will be inconveniently large unless the granularity is sufficiently coarse. Thus, we choose a granularity resulting in approximately 4 elements of internal structure. + + Thus, when the teaching objective is, what is a proof, we may limit the granularity, such that the internal structure of a proof is, for example, + \begin{enumerate} + \item that particular statement which is to be proved + \item axioms, premises, suppositions, cases + \item other statements + \item warrants (rules of inference) + \end{enumerate} + + We might choose to pursue finer granularity in some cases, for example, we might pursue "What is a statement?", because instructors have found that not all students arriving in CSE3502 have the same depth of understanding of statement, and some do not have sufficiently deep understanding of "statement" to be able to comprehend a proof. + + + Marton and Booth\cite[p. 22]{marton1997learning} call our attention to learners directing their attention to the sign vs. to the signified. With proofs, Polya \cite{} has mentioned a procedural approach to executing a proof, without understanding, as have Harel and Sowder \cite{harel1998students}, and Tall\cite{tall2001symbols}. Weber and Alcock\cite{weber2004semantic} have observed and described students omitting understanding of warrants in proofs. In each of these cases, the sign is provided, but the signified is at best incompletely understood. + + Analysis can usefully illuminate learning processes, taking note of the temporal domain\cite{marton1997learning}. This has been used by Booth in her analysis of how students understand the process of programming\cite{marton1997learning}. + + [p. 136]\cite{marton1997learning} important to be looking whether conceptualizations appear in a certain case, in a certain period of time (such as, when see proofs again in 3500, 3502, are they recognized as proofs, some no some yes, are they helpful as proofs, or troublesome, some helpful, some not, "never did get that"). + + \subsection{Analysis of Interviews} + + Items excerpted from interviews for analysis should be analyzed in the context of the specific interview and also in the context of the ensemble.\cite{marton1997learning}. + + Data were analyzed using a modified version of thematic analysis, which is + in turn a form of basic inductive analysis.\cite{Merriam2002,Merriam2009,braun2006using,fereday2008demonstrating,boyatzis1998transforming} Using thematic analysis, we + read texts, including transcripts, looked for “units of meaning”, and extracted + these phrases. Deductive categorization began with defined categories, and + sorted data into them. Inductive categorization “learned” the categories, in + the sense of machine learning, which is to say, the categories were determined + from the data, as features and relationships found among the data suggested + more and less closely related elements of the data. A check on the development + of categories compared the categories with the collection of units of meaning. + Each category was named by either an actual unit of meaning (obtained during + open coding) or a synonym (developed to capture the essence of the category). + A memo was written to capture the summary meaning of the category. + Next a process called axial coding, found in the literature on grounded theory, + \cite{strauss1990basics,kendall1999axial,glaser2008conceptualization} was applied. This process considered each category in turn as a central + hub; attention focused on pairwise relations between that central category + with each of the others. The strength and character of the posited relationship + between each pair of categories was assessed. On the basis of the relationships + characterized in this exercise, the categories with the strongest interesting relationships + were promoted to main themes. A diagram showing the main + themes and their relationships, qualified by the other, subsidiary themes and + the relationships between the subsidiary and main themes was prepared to + present the findings. Using the process of constant comparison, the structure + of these relationships was reviewed in the light of the meanings of the categories. + A memo was written about each relationship in the diagram, referring + to the meaning of the categories and declaring the meaning of the relationship. + A narrative was written to capture the content of the diagram. Using the + process of constant comparison, the narrative was reviewed to see whether it + captured the sense of the diagram. Units of meaning were compared with the + narrative and their original context, to see whether the narrative seemed to + capture the meaning. The products of the analysis were the narrative and the + diagram. + + + + \subsection{Analysis of Help Session and Tutoring} + some students, who do know that any statement must and can, be + either true or false, thought implications must be true. + + + + % The study proceeded using prior interview experiences to suggest further investigation. + % Originally asking about proofs and what they were for, we received answers about proof by induction and found out not all students contemplate why the curriculum contains what it does. + % (reference Guzdial on students trusting that whatever curriculum they take, is very likely to qualify them for a job). + %When providing leading questions about why, interview data indicated that not all students apply proof techniques with which they have been successful. + %So, when application of proofs developed as a part of the study. + %Consequently, what clues there may be that prompt recall of proof and proof techniques to application to a problem became part of the study, involving structural relevance. + %Generalization is related to the presence of one situation evolving a response that was learned in the context of a different situation prompted the view of teacher teaching from context of generalization hierarchy present and situation as example and homework situation as another example. + %What opportunities to foster generalization do students notice? + + % \subsection{Within What is a Proof?} + %The study was devoted to proofs, a subject that can be subdivided. + %Part of the study was aimed at the idea of domain, directed at the concept that + %though a variable could identify a scalar, it might also represent a set. + %Part of the study was aimed at the activity of abstraction, because some students + %exhibited the ability to operate at one level of abstraction, not necessarily a + %concrete level, yet the ability to traverse between that level of abstraction and + %a concrete level seemed to be absent. Other students claimed to be able to + %understand concrete examples with ease, but to encounter difficulty when + %short variable names were used within the same logical argument. + + %\subsection{Order of Exploration} + %The order of exploration was data driven, thus the material was sought sometimes + %in reverse order of the curriculum, almost as if seeking bedrock by starting + %at a surface, and working downwards. + + + + + + \subsection{Application of Phenomenographic Analysis to Why We Teach Proof} + + The analysis for the research question "What do students think proof is for?", which was approached as "Why do you think we teach proof?" exemplifies a phenomenographic approach. One aspect of the phenomenon of proof is its utility. We set the scope of our perspective to be specific to usefulness. We selected student verbal productions related to the use of proof. We considered them in the context of their own interview, and we compared them to data from other interviews on the same theme. + + We applied phenomenographic analysis by focusing on the aspect of relevance of proof for learning computer science and practicing as a software developer. In this case we had already identified the dimension of variation to be the depth of understanding of why we teach proof. Thus we could select fragments of student utterances and rank them according to depth of understanding. We then presented them in a sequence by rank. + + + + + + + + \section{Method of Presentation of Results} + + + The product of analysis in a phenomenographic study, is a set of categories, and relationships \textit{among} them. + + Marton and Booth\cite[p. 135]{marton1997learning} state "in the late stages of analysis, our researcher [has] a sharply structured object of research, with clearly related faces, rich in meaning. She is able to bring into focus now one aspect, now another; she is able to see how they fit together like pieces of a multidimensional jigsaw puzzle; she is able to turn it around and see it against the background of the different situations that it now transcends." \ No newline at end of file diff --git a/ch4.tex b/ch4.tex index 3ee5c92..d1e49b6 100644 --- a/ch4.tex +++ b/ch4.tex @@ -1,364 +1,282 @@ -\chapter{Results} - -The results of a phenomenographic study comprise a set of categories of description of ways of experiencing (or capability for experiencing \cite[p. 126]{marton1997learning}) a phenomenon, and relations among those categories. - -Booth states\cite{booth2001learning} "The results are communicated as -descriptions of the essential aspects of each category, illustrated by pertinent -extracts from the data". - -Marton and Booth\cite[p. 125]{marton1997learning} give criteria for the quality of a set of descriptive categories. -The collective experience, over all participants in the study should be included. -The individual categories should each stand in clear relation to the phenomenon of the investigation so that each category tells us something distinct about a particular way of experiencing the phenomenon. -The categories have to stand in a logical relationship with one another, a relationship that is frequently hierarchical. -Finally the systems should be parsimonious, which is to say that as few categories should be explicated as is feasible and reasonable, for capturing the critical variation in the data. - -late stages of analysis able to see aspects/facets of research object, -see how they fit together like jigsaw pieces, -see it against background, and communicate it to others. - -There are results for each of the research questions, and some combined results. - -While mainly we are discussing proof in general, it can help to think about one proof at a time. Applying the analytical framework of Marton and Booth\cite[p.43]{marton1997learning} -\begin{table}[placement] -\caption{Outcome space for what is proof, with temporal facet} -\begin{tabular}{|p{4cm}|p{4cm}|p{4cm}|}\hline -Acquiring & Knowing & Making Use of\\\hline\hline -see the steps & remembering the steps & write it out\\\hline -understand the steps & remembering the meaning & produce the meaning\\\hline -understand steps and warrants & understanding the meaning & be able to apply the proof to simple examples\\\hline -analyzed the structure, determine the warrants & understand the relevance & be able to apply the proof in general, know its context of applicability\\\hline -\end{tabular} - -\end{table} -Our goal might be in the lower right, and for some students who do not arrive that far, they might arrive at any of its three neighbors in the chart. - -We are looking for ways of experiencing, for example, one way is, proofs only apply to number facts, vs. proof techniques are separable from number facts and can be used on other domains. - - - -\section{What do students think a proof is?} - - -Some students, when asked what they think a proof is, will report that they think it is a list of true mathematically formulated statements, demonstrating the truth of a mathematically formulated statement. Some students report that a proof has a goal, a statement to be proved true. -Some students know that the identified goal does not have to be known to be true in advance of the first proof. - -Sometimes, though, students have the idea that the proof is an exhibit of their ability to connect known facts, including the goal as a known fact. - - - -Some students, for example some taking philosophy, understand a proof more generally as not having a requirement for a mathematical formulation. Some of these have expressed dislike of such less precisely articulated statements. - -Some students, when prompted, will acknowledge that warrants for these statements are required. Axioms and agreed facts do not require warrants. -Some students, but not all, recognize that premises do not require warrants. -Some students, but not all, recognize that suppositions, as premises, do not require warrants. -Some students, but not all, recognize that cases, as suppositions, do not require warrants. -Some students, but not all, know that progress from one statement to the next, a transformation of a statement, requires a warrant. - -Some of the optional syntactic ornamentation of a proof, such as literal text "Proof:", and "QED" or $\qed$, are used by some students as proxies for the proof. As in the research by Harel and Sowder\cite{harel1998students}, which they describe as "ritual proof", we find in our research that some students claim to recognize a proof when they see these artifacts, and claim they have not seen a proof when they do not see these artifacts. - -Some students are aware that proof, as encountered in class, ought to be a convincing argument. -These students feel that something is wrong when they are not convinced by the proof technique they have learned to execute in a procedural fashion. - -Some students know that proof is convincing others, and also ascertaining for oneself. Of these, some find that proof is convincing for some facts they regard as mathematical, yet do not think proof is applicable to programs as large as those with which they plan to be involved. -Some of these students have not yet acquired the perspective that proving theorems about the number of instruction executions, and/or memory locations needed are both numerical and also applicable to and relevant for software development. - - -\begin{figure}[h] -\centering -\includegraphics[width=0.7\linewidth]{/home/theresesmith/Documents/2015Fall/Research/Thesis/whatThemes} -\caption{Conceptualizations found for what a proof is} -\label{fig:whatThemes} -\end{figure} -\begin{table} - \caption{Critical factors for what a proof is} -\begin{tabular}{|c|c|c|} -\hline Lesser Conceptualization & Better Conceptualization & Critical Factor\\ \hline -\hline List of Known Facts & List of Warranted Facts & Warrants\\ -\hline List of Warranted Facts & Means of Discovery & Tool rather than\\ & & demonstration to teachers\\ \hline -\end{tabular} - \end{table} - -\newpage -\section{How do students approach understanding a proof?} -Those students who felt they understood some proofs approached them by checking whether they felt each line of a proof was true. -Some of these mentioned that a statement should be warranted by previous statements. - -\begin{figure}[h] -\centering -\includegraphics[width=0.7\linewidth]{/home/theresesmith/Documents/2015Fall/Research/Thesis/howThemes} -\caption{Conceptualizations of how to comprehend a proof} -\label{fig:howThemes} -\end{figure} -\begin{table} - \caption{Critical factors for what a proof is} -\begin{tabular}{|c|c|c|} -\hline Lesser Conceptualization & Better Conceptualization & Critical Factor\\ \hline -\hline List of Known Facts & List of Warranted Facts & Warrants\\ -\hline List of Warranted Facts & Means of Discovery & Tool rather than\\ & & demonstration to teachers\\ \hline -\end{tabular} - \end{table} - -\newpage -\section{What do students think a proof is for?} -Some students think that proofs are not applicable to what they do. They think they do not need to know it. Because they do not need to know it, they logically conclude that learning to produce a "proof" procedurally is enough, because, it earns full credit. - -Some students combine the learning about proof with the subject matter that is used to exercise proof techniques; they think proof is for demonstrating facts about numbers. - -Some students claim that they never produce proofs unless assigned to do so in class. - -Let us call the statement to be proved, the target, so as to more clearly articulate the variety of student thinking, by escaping the connotations of "statement to be proved". - -Perhaps not surprisingly in light of the teaching of proof, some students think the purpose of proof is to demonstrate that they can construct a sequence of statements that connects the truth of the premises to the truth of the target. Some of these students regard the truth of the target to be known beforehand. As the purpose of the proof is to exhibit their ability to produce an argument, it is not surprising that students say they never construct proofs unless they are assigned to do so. It is not surprising in this context, that students opt for a procedural approach, learning the parts, for example, of a proof by induction, learning to provide a proof for a base case, and learning to take a premise as a given and a conclusion of an implication to be proved. Some students do not understand why this procedure constructs a proof. Some express an unease -- they wish for the proof procedure to be convincing, to themselves. They are glad when they learn why the procedure does produce a convincing argument. - -Some students recognize proof being used in class, for example in algorithms class and in introduction to the theory of computing. - - - - -Some students felt that proof was for finding out that a mathematical expression was true, or false. Some students knew that some statements could be proved undecidable. - -\begin{figure}[h] -\centering -\includegraphics[width=0.7\linewidth]{/home/theresesmith/Documents/2015Fall/Research/Thesis/whyThemes} -\caption{Conceptualizations about why to study proofs} -\label{fig:whyThemes} -\end{figure} -\begin{table} - \caption{Critical factors for what a proof is} -\begin{tabular}{|c|c|c|} -\hline Lesser Conceptualization & Better Conceptualization & Critical Factor\\ \hline -\hline List of Known Facts & List of Warranted Facts & Warrants\\ -\hline List of Warranted Facts & Means of Discovery & Tool rather than\\ & & demonstration to teachers\\ \hline -\end{tabular} - \end{table} - -\newpage -\section{What do students use proof for, when not assigned?} -Some students claim they never use proofs when not assigned. - -It is not the case that any student, even when prompted, said they chose to carry out a proof without being directed to do so. -This could easily be due to a misunderstanding of the definition of proof. - -\newpage -\section{Do students exhibit any consequence of inability in proof?} -Some students said that they knew how to craft recursive procedures, and enjoyed doing so when assigned problems designated as suitable for recursive implementations. -Some students said they did not employ recursive procedures in situations without a designation that recursive procedures were appropriate. They claimed not to be able to tell when recursive procedures were applicable. - -\newpage -\section{What kind of structure do students notice in proofs?} -Some students think proofs are lists of statements without hierarchical structure. -Some students have asked what lemma means. -Some students knew that lemmas were built for use in larger proofs. -Some students were interested to hear about Dr. Lamport's structure in proofs. - -\newpage -\section{What do students think it takes to make an argument valid?} -Some students, when prompted about rules of inference, felt that when all statement transformations were warranted, an argument was valid. -Some students stated that, when the target of the proof was true, the proof was valid, converse error. - - - - - -We organize our overview of results beginning from an ideal Hilbert-axiomatic style of proof approach, and moving through approximations as they become greater departures from it. -\begin{figure}[h] -\centering -\includegraphics[width=0.7\linewidth]{/home/theresesmith/Documents/2015Fall/Research/Thesis/valid} -\caption{Conceptualizations about validity of proofs} -\label{fig:validityThemes} -\end{figure} -\begin{table} - \caption{Critical factors for what a (valid) proof is} -\begin{tabular}{|c|c|c|} -\hline Lesser Conceptualization & Better Conceptualization & Critical Factor\\ \hline -\hline No Warrants & Some Warrants & Warrants\\ -\hline Some Appropriate Warrants & Fully Warranted & thoroughness\\ \hline -\end{tabular} - \end{table} - -\paragraph{Definition based reasoning} -Some students, and some teaching assistants in their teaching, are not organizing the approach to proof around definitions. -Instead some students and some teaching assistants are focusing on an intuitive approach, involving examples. -Some students use examples to infer definitions. -Some students use single examples as proof. - -Some students are not aware that proofs are illustrated with facts of, for the purposes of the class, less significance than the proof techniques. -Some students are not aware of the relevance of proof to their intended career. -These students do not see any point to learning more than a procedural approach to the proof material, as they believe it to be of no lasting significance to them. - -\paragraph{Generalization and transformation based reasoning} -Some students, and some instructors, do not emphasize that a single presentation can be seen as a representative of a group. For example, Mathematical Association of America\cite{} publishes a proof of the Pythagorean Theorem that uses rectangles to illustrate that, when they are square, the Pythagorean Theorem is being shown to be true, though of course, the rectangles need not always be square. The proof, having been established, does not rely upon the rectangles remaining in a square condition. - + \chapter{Phenomenographic Analysis} + + + + + + + + \subsection{Application of Phenomenographic Analysis in this Study} + + We applied phenomenographic analysis, + + The analysis for the research question "What do students think proof is for?", which was approached as "Why do you think we teach proof?" exemplifies a phenomenographic approach. One aspect of the phenomenon of proof is its utility' We selected student verbal productions related to the use of proof. We considered them in the context of their own interview, and we compared them to data from other interviews on the same theme. + + The questions are ordered guided by the 1956 version of Bloom's Taxonomy, namely, recognition, comprehension, application, analysis, synthesis, evaluation.\cite{bloom1956taxonomy} + + % % %recognition + + \subsubsection{Phenomenographic Analysis of What Students Think Proofs Are} + + \begin{quote} + Mathematics, as it is currently practiced, is a delicate interplay between monastic contemplation and blowing stuff up with dynamite. (Ellenberg)\cite[page 223]{ellenberg2014not} + + Probably I don't want to keep this, but it's fun at the moment. + \end{quote} + + Some students exhibit an understanding of proof at the "black box" level, i.e., there is understanding of the role of proof, without considering any internal structure. When a proof exists, we can know that the thing the proof proves is true, in the context that applies. We can "use that theorem". Other students, though, do not have this idea consolidated yet. For example, if we consider proof by exhaustion applied to a finite set of cardinality one, we can associate to it, the idea of a test. Students, assigned to test an algorithm for approximating the sine function, knew to invoke their implementation with the value to be tested, but did not check their result, either against the range of the sine function, or by comparison with the provided sine implementation, presenting values over 480 million. + + Moving the the "white box" level, we find a spectrum of variation in student understanding. + The most opaque end of this spectrum has been called "magic incantation". In this conceptualization we find those ideas of statements that are not clear, and use of mathematical symbols that is not understood. + + Ellenberg\cite[page 409]{ellenberg2014not} reports that some mathematicians regard axioms as strings of symbols without meaning, and that this quite formal conceptualization can be contrasted with another conceptualization that axioms are true statements about logical facts. He talks of these conceptualizations being taken by the same individual at different times. + + Our participants seemed to regard axioms as strings of symbols that do mean something, though that meaning the participant ascribed might not be correct (especially as participants did not always know definitions of the entities being related), or the participant might feel unable to ascribe any meaning. We did find participants who appreciated the significance of definitions. They were dual majors in math. + + Another waystation on this dimension of variation is "sequence of statements". A more elaborate idea is "sequence of statements where each next statement is justified by what when before". A yet more complete concept is "finite sequence of statements, starting with the premise and ending with what we want to prove, and justified in each step." A more profound conceptualization was found "finite sequence of statements, starting with axioms and premises, proceeding by logical deduction using (valid) rules of inference to what we wanted to prove, that shows us a consequence of the definitions with which we began, an exploration in which we discover the truth value of what we wanted to show, serving after its creation as an explanation of why the theorem is true". + + A few categories, such as those above, serve to identify a dimension of variation. When our purposes include discovering which points we may want to emphasize, we can examine the categories seeking to identify how they are related and how they differ. + + It can certainly be that having more categories provides more critical aspects. For example, Harel and Sowder\cite{harel1998students} offered extrinsic vs. intrinsic conviction, empirical proof schemes and their most advanced deductive proof schemes as broader categories, and seven useful subcategories of these, yielding six critical aspects that suggest what teachers could usefully vary, to help learners discern items that would advance their knowledge. + + + + + + + % % %comprehension + + \subsubsection{Phenomenographic Analysis of How Students Attempt to Understand Proofs} + Some students are attempting to understand proofs while not recognizing that they are studying a proof. + "Were we studying proofs today?" "No" "Were we discussing certain contexts, and why certain ideas will always be true in those contexts?" "Yes" "Doesn't that seem like proof, then?" "Yes"\\ + "So, you're taking introduction to the theory of computing this semester. Do you seen any use of proofs in that course?" "No"\\ + + + Some students read proofs. + + Some students look up the definitions of terms used in the proofs and some do not. + Some students think (or hope) they can solve problems involving producing proofs, without knowing the definitions of the terms used to pose the problem. + Some students are aware that definitions are given, but "zone out" until examples are given. When examples are given, the students attempt to infer definitions themselves. Some students will compare the definitions they infer with the mathematical community's definitions. Some students do not. + + Some students think that the reading of proofs is normally conducted at the same speed as other reading, such as informal sources of information. + + Of students who read proofs attentively, some try to determine what rule of inference was used in moving from one statement to the next, and some do not. + + Some students notice that lemmas can be proved and then used as building blocks in a larger argument, and some do not. + + Some students can identify the forms of proof learned in discrete systems, when they see them employed in proofs, such as the combination of arguing by contrapositive and modus ponens. Some cannot. + + Some students can identify these forms in an argument if the argument is made about concrete objects, such as cars or specific people. Some of these students have difficulty transferring this ability with concrete objects to application to abstract entities such as sets, algorithms or symbols. + + Some students who achieve with difficulty the ability to recognize the application of rules of inference in one argument about abstract entities, become quicker at recognizing arguments of similar form about other abstract entities, and some continue to achieve with difficulty, as if learning the first argument did not facilitate learning the second argument of the same form. + + Students would attend to diagrammatic representation of proofs, such as a block digram depicting machine descriptions packaged as input for yet other machines to process, but were not observed to employ such diagrams. + + Students have been seen to employ decision tree diagrams. + + Students would attend to algorithm representation of proofs, such as a recursive process that determines a prime factorization, but were not observed to employ such algorithmic descriptions. + + Students answering a list of questions, representing computer science ideas mathematically, in algorithms and in figures found the questions "interesting", "fun", "different" and "non-trivial". + + Except when assigned to do so, students were not observed to attempt to solve simpler problems, such as by imposing partitioning into cases. Except when assigned to do so, students were not observed to attempt to solve more general problems, as is sometimes helpful.\cite[that pin dropping probability problem]{ellenberg2014not} + + % % % structural relevance + + \subsubsection{Phenomenographic Analysis of Reasons for Teaching Proof} + + Excerpts of student transcripts were selected on the basis of being related to this question. A dimension of variation emerged from the data, such that the excerpts seemed readily organized along this dimension. + + \begin{longtable}{|p{7cm}|p{8.5cm}|}\hline + + \endfirsthead + + \multicolumn{2}{c}% + {{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\ + \hline \multicolumn{1}{|c|}{\textbf{Category}} & + \multicolumn{1}{c|}{\textbf{Representative}} \\ \hline + \endhead + + \hline \multicolumn{2}{|r|}{{Continued on next page}} \\ \hline + \endfoot + + \hline \hline + \endlastfoot + +% \begin{tabular}{|p{7cm}|p{8.5cm}|}\hline + Category & Representative\\\hline\hline + + + some students do not see any point to proof& + They teach it to us because they were mathematicians and they like it.\\ + + + & we didn't see ok why do i really have to know the proof of the theorem to do that right? We didn't see the point, because no one taught us the point, so, that's a very important part that was missing.\\\hline + some students think that it satisfies the curriculum goals, to be able to reproduce a previously taught proof, or follow a procedure to generate a proof, without being personally convinced& + I was able to get a full score, but I don't understand why a proof by induction is convincing\\\hline + + Some students do not see a relationship between a problem and approach& + When I have to prove anything, I always start with proof by mathematic induction, that was the one they taught the most.\\\hline + + Some students are surprised to discover that there is a relation between proof by induction and recursion& + I never noticed that before, but now that you mention it, I see that they are isomorphic.\\\hline + + Some students see the relationship but do not use it& + Professor (redacted) would be really proud of me that I learned to understand proof by induction quite well. \ldots I understand how recursion matches induction, there's a base case, there's a way of proceeding. \ldots I just couldn't figure out how to program the merge-sort algorithm.\\\hline + + some students think the only reason for studying proof is to understand proofs of, for example, resource utilization of known algorithms& + I would never consider writing a proof except on an assignment.\\ + + & I understand the proof of the lower bound on comparison sort. \ldots I understand the proof of the upper bound on searching in a binary search tree. \ldots If I had to prove something about termination on a search tree, I don't know how I would do that.\\ + + + & I know that recursion has the same structure as proof by mathematical induction. \ldots If I had an algorithm with a recursive data structure like a tree, and I had to prove something like termination about it, I'm not sure what approach I would use, it would depend.\\\hline + + Some students see that they could employ proof to explore whether an algorithm can be expected to solve a problem in a given context that includes bounds upon resources that are available for consumption. & mostly design the algorithm first, we had some expectation of what that complexity results would be and then we try to find an approach to prove. +%\end{tabular} + \end{longtable} + + + + + % % %application + +\subsubsection{Phenomenographic Analysis of How Students Attempt to Apply Proofs (When not assigned)} + +Some students claimed they never constructed proofs when not assigned. + + + + +\subsubsection{Phenomenographic Analysis of Whether students exhibit consequences of inability (such as avoiding recursion)} + +Some students claimed to know how to write recursive algorithms but said they never used them because they did not know when they were applicable. + +% % %analysis + + \subsubsection{Phenomenographic Analysis of How familiar and/or comfortable are students with different (specific) proof techniques: induction, construction, contradiction?} + + When asked about specific proof techniques, some students mentioned proof by mathematical induction "When faced with a proof I start with induction, they taught us that one the most." + + When asked for specific proof techniques other than proof by mathematical induction, students knew the words contradiction and contrapositive, but sometimes could not distinguish between them. + + When asked about proof by construction, some students thought this referred to construction of any proof. + + Some students thought proof by contradiction referred to proving the opposite of something, rather than disproving the opposite of something. + + + \subsubsection{Phenomenographic Analysis of Whether students notice structural elements in proofs} + + Some students, in the context of hearing a presentation in an algorithms course, of a proof with a lemma, do not know, by name, what a lemma is. "What's a lemma?" + + Some students, in the context of planning to construct a proof, do not choose to divide and conquer the problem, breaking it into component parts, such as cases. "What good does that do, doesn't the proof become longer?" + + Some students describe proofs as a sequence of statements, not commenting on any structure. + + \subsubsection{Phenomenographic Analysis of What do students think it takes to make an argument valid?} + + Some students used confused/incorrect forms of rules of inference. + + Some students do not notice that the imposition of a subdivision into cases creates more premises. + + Some students do not notice that proof by contradiction introduces (for purposes of contradiction) a premise. + +% % %synthesis + +\subsubsection{Phenomenographic Analysis of Whether students incorporate structural elements in proofs} + +Students have asked whether, when using categorization into cases, they must apply the same proof technique in each of the cases. + + Probably needs additional interviews. + - -\section{Combined Description} - -There are a couple of ways students work with exercises in proof, that are incomplete. - -Some students reason well with concrete entities, yet are confused with abstractions. These students are not appreciating the value of careful definitions, because they do not use them as tools, or the basis for reasoning. They are more comfortable with examples, because they are operating in a concrete world. - -Some students do not connect the world of concrete objects with the abstract, symbolic representation, but are making use of symbols. Some operations transforming symbolic expressions are performed correctly, but not all. The lack of understanding of the symbols combined with a procedural approach to producing a proof artifact, leaves these students personally unconvinced, and unmotivated to make use of proof when it would be helpful to them. - -Some students do understand application of facts, axioms and rules of inference, and are at home with careful definitions and symbolic concision. Some of these students also study math. - - -\newpage -\section{Diagram of Conceptualizations} - -\begin{figure}[h] -\centering -\includegraphics[width=0.95\linewidth]{./themes} -\caption{Themes from interview data} -\label{fig:themes} -\end{figure} -\newpage -\section{Outcome Space} -The outcomes were not arranged in a single progression. Rather, there were several means, listed below, by which students were able to construct the proof artifact required by the class. The students did not always find the artifact convincing. - -\begin{enumerate} -\item Concrete to Abstract -- generalize the argument, then the entities -\item Hilbert-style axiomatic/definitional proof -\item Abstract operations -- symbols rather than entities, structure of argument -\end{enumerate} - -The concrete to abstract path enabled students to reason with specific cases whose logic made sense to them, then make the step that the logical process itself was an entity that could be reused. The idea that other concrete entities could bear the same relationships, and be subject to the same reasoning constituted a step. The idea that analogies were being made, and that generalization was possible was another step. - -The reasoning by axioms and rules of inference path was known to some students. These students mentioned their appreciation of math, and in some cases their discomfort with philosophy, in connection with symbolization and application of rules of inference. - -One path was operation at the level of symbols, using procedures. This path is distinguished from that involving definitions, because some students, using definitions, were clear about appropriate operations to transform symbolic expressions, but students also sometimes were unsure about denotation and about appropriate operations. - -\section{Critical Factors} -On the path from concrete to structured proofs, called herein "generalize the argument, then the entities", one critical factor is that an argument about one set of concrete entities can be used on another set, having analogous relationships. - -Another critical factor is that, when an argument can be reused, sets of entities that stand in analogous relationships, the relationship can be generalized. When the relationship is generalized, the entities standing in that relationship can be given symbols. - -On the path that starts with symbols, students have not generalized from concrete to abstract entities, rather, they have entered the fray at the level of abstraction of symbols. Thus, a critical factor is to understand the operations appropriate to the symbols, which imbues application of the rules of inference with significance. Another critical factor is that these symbols can represent entities of interest. - - - - - -\section{Earlier Paper Material} - -\subsection{Categories of Experience of Entering Students} -Undergraduate students beginning study of the computing disciplines present -various degrees of preparedness\cite{smith2013categorizing}. Some had no experience, some had had -informal experience, and some had had formal classes. The formal classes -extended from using applications to building applications. Informal experience -ranged from editing configuration files, such as background colors, to full time -jobs extended over multiple summers. -After publishing this paper, we encountered more related information. For -example, consistent with the work of Almstrum\cite{almstrum1996investigating}, we found that, about implications, -some students, who do know that any statement must and can, be -either true or false, thought implications must be true. Some interview participants -enjoyed a modified Moore method\cite{cohen1982modified} geometry class in middle school, -and relished opportunities to create proofs (not yet published). Other students -were not so well prepared. -\subsection{Representation/Symbolization in Pumping Lemmas} -We found that some students may lack facility in notation. For example, in the -application of the pumping lemma, students are expected to understand the -role of $i$, in the context that a string $s$, having component substrings $x$, $y$ and $z$, -can be used to generate other strings, of the form $xy^iz$, where $i$ gives the number -of copies of the substring $y$. Moreover, students are expected to understand that -the subdivision of a string of length $p$, $\sigma_1^a\sigma_2^{p-a}$, where $a \in \{0,1,\ldots,p\}$ -uses $a$ as a parameter, a free variable, not one necessarily bound to a single instance -of a natural number, but a representation of a domain. An excerpt of the errors found on tests is shown in Table . Trigueros et -al. \cite[ p. 3]{jacobs2008developing} have observed that ``students are often unclear about the different -ways letters are used in mathematics''. We have seen this -lack of understanding in a situation in which it was proposed as evidence that -a single example, namely $\sigma_1^a\sigma_2^{p-a}$, formed a proof for a universally quantified -statement. -Some of our results were consistent with the framework described by Harel -and Sowder in 1998\cite{harel1998students}. We found students holding conceptualizations that -Harel and Sowder's 1998 model calls symbolization. Harel and Sowder have -identified another category of conceptualization, that correctly applied transformation -and axiomatic arguments. Some students expressed enthusiasm for -the power that inheres to building arguments with carefully specified component -ideas, in particular how the absence of ambiguity permitted arguments to -extend to great length while remaining valid. Not all of the students had developed -axiomatic conceptualizations of proof. About definitions, we collected -preliminary data on students' conceptualizations of definitions used in proofs. -Some students thought definitions were boring. Some students thought that -they could infer definitions from a few examples. Concerning executive function, -we found that some students do not state the premises clearly, and some -students did not keep track of their goal. About rules of inference, we found -Figure 5.3.1: Some categories / conceptualizations found among students of -introduction to the theory of computing, and published at FIE. -that some students apply invalid approaches to inference. - -\subsection{Abstract Model for Proof by Mathematical Induction and Recursion} -Far from finding agreement that (a) theorems are true as a consequence of -the definitions and the premise, and that (b) proofs serve to show how the -consequence is demonstrated from the premise, axioms and application of -rules of inference, instead we found a variety of notions about proof, including -the well-known procedural interpretation \cite{tall2008transition,weber2004traditional,tall2001symbols}, and the well-known empirical -misconception \cite{harel1998students}. The conceptualization that definitions are not necessarily -of interest compared with the procedures seemed different in kind from the -concept image / concept definition discoveries of R\"osken et al. \cite{rosken2007integrating}. -Interviews with students revealed that some students saw generation of a proof -by mathematic induction as a procedure to be followed, in which they should -produce a base case, and prove it, and should produce an induction step, and -prove that. This was consistent with Weber [?, p. 4-426] who has stated ``in -the studies that I conducted, it was more often the case that undergraduates -applied procedures that were not meaningful to them.'' He went on to give a -quotation from a participant [?, p. 4-426] ``And I prove something and I look at -it, and I thought, well, you know, it's been proved, but I still don't know that I -even agree with it [laughs]. I'm not convinced by my own proof!'' Some of the -students interviewed did not know why this procedure generated a convincing -argument. Polya[?] has written a problem involving all girls being blue-eyed; -a similar problem appears in Sipse \cite{sipser2012introduction} about all horses being the same color. -The purpose of this exercise is to make students aware that the truth of the -inductive step must apply when the base case appears as the premise. In some -cases, this point was not clear to the students. -Students' conceptualizations of proof by mathematical induction can support -their choice to apply recursive algorithms. One student reported success at -both mathematical induction and recursive algorithm application without ever -noticing any connection. This student opined that having learned recursion -with figures, and proof by mathematical induction without figures, that no -occasion for the information to spontaneously connect occurred. Students reporting -ability to implement assigned problems recursively, but not the ability -to understand proof by mathematical induction also reported that ability to -write recursive programs did not result in recognition of when recursive solutions -might be applicable in general. Students reporting ability to implement -assigned problems recursively, and also the ability to prove using mathematical -induction also reported preferring to implement recursive solutions in -problems as they arose. -Our work on students' choices of algorithmic approaches was consistent with -work by other researchers in computer science education\cite{booth1997phenomenography} on conceptualizations -of algorithms. Our work served to unify that of mathematician educators -with computer science educators, by providing a plausible explanation why -the conceptualizations of recursive algorithms that were found, might exist. -\subsection{Proofs by Induction} -Table 5.3.3: The Outcome Space for Proofs by Induction -Category Description -1Following procedure The method is learned, without understanding -2Understands base case The idea that a base case is proved by an existence -proof, often with a specific example -3Understands implication The idea that an implication is proved by -assuming the premise is not used -4Does not understand connection -Sees the implication and proves it well, but -does not anchor the succession to a base -case -5Does understand the argument -Understands the argument -6Knows why recursion -works -Can tailor the argument to explain recursive -algorithms -7Appreciates data structures -supporting recursion -Can see the benefit to algorithm from recursive -data structure -\subsection{Pumping Lemmas} -TABLE III. CATEGORIES\\ -understand inequality\\ -formulate correctly\\ -distinguish between particular and generic particular\\ -correctly apply universal quantifier\\ -recognize string as member of language set - +% %evaluation + + % \section{Interview} + Some students remembered taking proofs in high school in geometry. + Some students were taking proofs contemporaneously in philosophy. + Some of the students studying proof in philosophy found them disturbing, expressing a preference for geometrical proofs. + Some students remembered having to furnish proofs of geometrical facts, also facts about prime numbers and sets. + Some students knew that CSE2500 treated proofs because they would be used in later courses. Students did not know why proofs would be used later, and were generally happy to hear some example uses. + Though students were asked whether they made use of proofs spontaneously, none of those interviewed gave an example. + Some students preferred to articulate with code, and some (who were dual computer science / math) sometimes preferred mathematical symbols, depending upon the context. + Some students do wish to convince themselves of things, such as tractable execution times, and correctness. Though students were asked whether they made use of proofs for this purpose, none of those interviewed claimed to do so, rather they mentioned going carefully over their algorithm construction, and considering cases. + + In interviews, the students almost all chose to discuss proofs by mathematical induction. + + \subsubsection{Themes / Categories} + \begin{itemize} + \item Definitions\\ + Students divided into (1)those who found definitions boring, difficult to pay attention to, and undesirable compared to examples, from which they preferred to induce their own definitions, and (2) those who had caught on to the idea that definitions were the carefully crafted building blocks of reasoning. + \item Procedures + Students sometimes learned what was desired in a proof, but learned to produce it by procedure, and were not themselves convinced. + \item Context + Students asked whether the topics for examples and exercises, such as prime numbers, had relevance to programming, with which they had experience, but not unrelated to the topics. + Students did not know the context in which the proofs, or procedure version of proof, was applicable, so, for example, did not apply proof by mathematic induction to recursive algorithms, and did not know how to tell whether recursive algorithms would be applicable. + \item Concrete vs. Abstract + Some students felt quite comfortable with the application of rules of inference to concrete items, but had difficulty transferring application of those rules to mathematical symbols. + \item Symbolization + consistent with Harel and Sowder's 1998 categorization of concepts, we found students who would attempt to write in symbols, but not understand what was denoted, and consequently were uncertain about appropriate operations. Some of these students were glad to see a progression from pseudocode with long variable names to pseudocode with short variable names to mathematical symbolization (formula translation (FORTRAN) in reverse). + \item Applicability of single examples + Some students believed that a few examples constituted a proof. These examples were not generic particular, nor were they transformational, in the sense of Harel and Sowder's 1998 model. + \item Substructure + Students familiar with methods, in the sense of object-oriented programming, and with construction of programs involving multiple method calls, did not always recognize that proofs could be built from multiple lemmas, although they did understand that axioms could be applied. + \item Proofs are used, in computer science, to show resource consumption (complexity class), properties of models of computation, and computability/decidability. No occasion was identified, other than assignment, when undergraduate students recognized they were undertaking proofs. + \item Among graduate students, proofs were undertaken in the context of preparing manuscripts for publication. These were scheduled to be approached after algorithm design, though retroactive adjustment of algorithms did occur for simplifying the proof. + + \end{itemize} + \subsubsection{Relationships} + + \subsection{Analysis of Homework and Tests} + \subsubsection{Proofs} + Proofs submitted on homework and tests were analyzed in several respects. + The overall approach should be valid. For example, students who undertook to prove that the converse was true did not use a valid approach. + The individual statements should each be warranted. + Use of structure, such as lemmas, and care that cases form a partition of the relevant set are gladly noticed. + Proof attempts that lose track of the goal, and proof attempts that assert with insufficient justification, the goal are noted. + \subsubsection{Pumping Lemmas} + We wrote descriptions for each error. Some example descriptions + are in Table II. + + + Table : Some example errors + Let x be empty + $|xy| \leq p, so xy = 0^p$\\ + $|xy| \leq p; let \; x = 0^{p+r}, y = 0^{p+r}, 0 < r < p$\\ + Let’s choose $|xy| = p$\\ + $0^{p+1}0^b1^p \neq 0^{p+1}1^p \therefore xy^2z \not\in \mathcal{L}$ + where $\mathcal{L} = \{0^i1^j, i \neq j\}$\\ + we choose $s = 0^{p+1}1^p$ within $|xy|$\\ + thus $\neq 0^p1^{p+1}$\\ + Let $x = 0^a, y = 0^b1^a$\\ + $x = 0^{p-h}, y = 0^h$\\ + $x = 0^i, y = 0^i, z = 0^i1^j$ + + A handful of students did exhibit their reasoning that for + all segmentations there would exist at least one value of $i$ that + would generate a string outside the language. + We categorized the errors as misunderstandings of one or + more of: + + \cite[get some page reference]{sipser2012introduction} + 1) ∣𝑥𝑦∣ ≤ 𝑝 permits ∣𝑥𝑦∣ < 𝑝\\ + 2) 𝑥 is the part of the string prior to the cycle\\ + 3) 𝑦 is the part of the string which returns the state of + the automaton to a previously visited state\\ + 4) 𝑧 is the part of the string after the (last) cycle up to + acceptance\\ + 5) 𝑝 − 1 characters is the maximum size of a string + that need not contain a cycle, (strings of length 𝑝 + or greater must reuse a state)\\ + 6) 𝑖 is the number of executions of 𝑦\\ + 7) There must be no segmentation for which pumping + is possible, if pumping cannot occur.\\ + 8) A language is a set of strings.\\ + 9) A language class is a set of languages.\\ + Categories are shown in the chapter on results (labelled table iii).\\ + + + \ No newline at end of file diff --git a/ch5.tex b/ch5.tex index 39fcc33..3ee5c92 100644 --- a/ch5.tex +++ b/ch5.tex @@ -1,804 +1,10 @@ -\chapter{Interpretation/Discussion} - -\section{Interpretation} -Here I want to put the ideas about definitions and abstraction. Without abstraction -definitions are more cumbersome to remember and operate with. This -discourages use of the axiomatic proof conceptions, because they are based on -definitions. -What about Valiant? His establishing of definitions in circuits in the mind by -conjunctions, and by disjunctions, of ideas. Without abstraction for definitions, -this is more cumbersome. - -Is intuition helping or opposing our educational objectives? Can we get help from it? - -\subsection{Productive and Counterproductive Beliefs} -What do they ``know'', and what do they ``know that isn't so''. -Some will be conscious, some will be unconscious. -\subsection{Productive and Counterproductive Momentum} -What are they trying to learn? Is it aligned with the departmental curriculum? the course goals? -\subsection{Published papers} -Three papers in this area were published: -\begin{itemize} - -\item CCSCNE: Categorizing the School Experience of Entering Computing -Students \cite{smith2013categorizing} -\item FIE: Mathematization in Teaching Pumping Lemmas \cite{smith2013mathematization} -\item Koli Calling: Computer Science Students’ Concepts of Proof by Induction\cite{smith2014computer} - -\end{itemize} -\paragraph{Categories of Experience of Entering Students} -Undergraduate students beginning study of the computing disciplines present -various degrees of preparedness\cite{smith2013categorizing}. Some had no experience, some had had -informal experience, and some had had formal classes. The formal classes -extended from using applications to building applications. Informal experience -ranged from editing configuration files, such as background colors, to full time -jobs extended over multiple summers. -After publishing this paper, we encountered more related information. For -example, consistent with the work of Almstrum\cite{almstrum1996investigating}, we found that, about implications, -some students, who do know that any statement must and can, be -either true or false, thought implications must be true. Some interview participants -enjoyed a modified Moore method\cite{cohen1982modified} geometry class in middle school, -and relished opportunities to create proofs (not yet published). Other students -were not so well prepared. - - - - -\paragraph{Representation/Symbolization in Pumping Lemmas} -We found that some students may lack facility in notation. For example, in the -application of the pumping lemma, students are expected to understand the -role of $i$, in the context that a string $s$, having component substrings $x$, $y$ and $z$, -can be used to generate other strings, of the form $xy^iz$, where $i$ gives the number -of copies of the substring $y$. Moreover, students are expected to understand that -the subdivision of a string of length $p$, $\sigma_1^a\sigma_2^{p-a}$, where $a \in \{0,1,\ldots,p\}$ -uses $a$ as a parameter, a free variable, not one necessarily bound to a single instance -of a natural number, but a representation of a domain. An excerpt of the errors found on tests is shown in Table . Trigueros et -al. \cite[ p. 3]{jacobs2008developing} have observed that ``students are often unclear about the different -ways letters are used in mathematics''. We have seen this -lack of understanding in a situation in which it was proposed as evidence that -a single example, namely $\sigma_1^a\sigma_2^{p-a}$, formed a proof for a universally quantified -statement. -Some of our results were consistent with the framework described by Harel -and Sowder in 1998\cite{harel1998students}. We found students holding conceptualizations that -Harel and Sowder's 1998 model calls symbolization. Harel and Sowder have -identified another category of conceptualization, that correctly applied transformation -and axiomatic arguments. Some students expressed enthusiasm for -the power that inheres to building arguments with carefully specified component -ideas, in particular how the absence of ambiguity permitted arguments to -extend to great length while remaining valid. Not all of the students had developed -axiomatic conceptualizations of proof. About definitions, we collected -preliminary data on students' conceptualizations of definitions used in proofs. -Some students thought definitions were boring. Some students thought that -they could infer definitions from a few examples. Concerning executive function, -we found that some students do not state the premises clearly, and some -students did not keep track of their goal. About rules of inference, we found -Figure 5.3.1: Some categories / conceptualizations found among students of -introduction to the theory of computing, and published at FIE. -that some students apply invalid approaches to inference. - -We have found students holding conceptualizations -that Harel and Sowder's 1998 model\cite{harel1998students} calls symbolization: We have found -that some students may lack facility in notation. For example, in the application -of the pumping lemma, students are expected to understand the role of $i$, -in the context that a string $s$, having component substrings $x$, $y$ and $z$, can be -used to generate other strings, of the form $xy^iz$, where $i$ gives the number of -copies of the substring $y$. Moreover, students are expected to understand that -the subdivision of a string of length $p$, expressed as $\sigma_1^a\sigma_2^{p-a}$, where $a \in \{0,1,\ldots,p\}$ -uses $a$ as a parameter, a free variable, not one necessarily bound to a single instance -of a natural number, but a representation of a domain. -Trigueros et al.\cite[p. 3]{jacobs2008developing} have observed that ``students are often unclear about the different way letters are used in mathematics.'' - We saw this -lack of understanding in a situation in which it was proposed as evidence that -a single example, namely $\sigma_1^a\sigma_2^{p-a}$, formed a proof for a universally quantified -statement. An excerpt of the errors found on tests is shown in Table . - -Table : Some example errors\\ -Let x be empty\\ -$|xy| \leq p, so xy = 0^p$\\ -$|xy| \leq p; let x = 0^{p+r}, y = 0^{p+r}, 0 < r < p$\\ -Let’s choose $|xy| = p$\\ -$0^{p+1}0^b1^p \neq 0^{p+1}1^p \therefore xy^2z \not\in \mathcal{L}$\\ -where $\mathcal{L} = \{0^i1^j, i \neq j\}$\\ -we choose $s = 0^{p+1}1^p$ within $|xy|$\\ -thus $\neq 0^p1^{p+1}$\\ -Let $x = 0^a, y = 0^b1^a$\\ -$x = 0^{p-h}, y = 0^h$\\ -$x = 0^i, y = 0^i, z = 0^i1^j$\\ -Figure 5.3.3: Some categories / conceptualizations found among students of -introduction to the theory of computing, and published at FIE. - -Some of our results were consistent with the framework described by Harel -and Sowder in 1998[?]. We found students holding conceptualizations that -Harel and Sowder's 1998 model calls symbolization. Harel and Sowder have -identified another category of conceptualization, that correctly applied transformation -and axiomatic arguments. Some students expressed enthusiasm for -the power that inheres to building arguments with carefully specified component -ideas, in particular how the absence of ambiguity permitted arguments to -extend to great length while remaining valid. Not all of the students had developed -axiomatic conceptualizations of proof. About definitions, we collected -preliminary data on students' conceptualizations of definitions used in proofs. -Some students thought definitions were boring. Some students thought that -they could infer definitions from a few examples. Concerning executive function, we found that some students do not state the premises clearly, and some -students did not keep track of their goal. About rules of inference, we found -that some students apply invalid approaches to inference. -\paragraph{Abstract Model for Proof by Mathematical Induction and Recursion} -Far from finding agreement that (a) theorems are true as a consequence of -the definitions and the premise, and that (b) proofs serve to show how the -consequence is demonstrated from the premise, axioms and application of -rules of inference, instead we found a variety of notions about proof, including -the well-known procedural interpretation \cite{tall2008transition,weber2004traditional,tall2001symbols}, and the well-known empirical -misconception \cite{harel1998students}. The conceptualization that definitions are not necessarily -of interest compared with the procedures seemed different in kind from the -concept image / concept definition discoveries of R\"osken et al. \cite{rosken2007integrating}. -Interviews with students revealed that some students saw generation of a proof -by mathematic induction as a procedure to be followed, in which they should -produce a base case, and prove it, and should produce an induction step, and -prove that. This was consistent with Weber [?, p. 4-426] who has stated ``in -the studies that I conducted, it was more often the case that undergraduates -applied procedures that were not meaningful to them.'' He went on to give a -quotation from a participant [?, p. 4-426] ``And I prove something and I look at -it, and I thought, well, you know, it's been proved, but I still don't know that I -even agree with it [laughs]. I'm not convinced by my own proof!'' Some of the -students interviewed did not know why this procedure generated a convincing -argument. Polya[?] has written a problem involving all girls being blue-eyed; -a similar problem appears in Sipser \cite{sipser2012introduction} about all horses being the same color. -The purpose of this exercise is to make students aware that the truth of the -inductive step must apply when the base case appears as the premise. In some -cases, this point was not clear to the students. -Students' conceptualizations of proof by mathematical induction can support -their choice to apply recursive algorithms. One student reported success at -both mathematical induction and recursive algorithm application without ever -noticing any connection. This student opined that having learned recursion -with figures, and proof by mathematical induction without figures, that no -occasion for the information to spontaneously connect occurred. Students reporting -ability to implement assigned problems recursively, but not the ability -to understand proof by mathematical induction also reported that ability to -write recursive programs did not result in recognition of when recursive solutions -might be applicable in general. Students reporting ability to implement -assigned problems recursively, and also the ability to prove using mathematical -induction also reported preferring to implement recursive solutions in -problems as they arose. - - (Say something about how this is consistent with the procedural conceptualization, bifurcation in Tall's writing.) -Our work on students' choices of algorithmic approaches was consistent with -work by other researchers in computer science education\cite{booth1997phenomenography} on conceptualizations -of algorithms. Our work served to unify that of mathematician educators -with computer science educators, by providing a plausible explanation why -the conceptualizations of recursive algorithms that were found, might exist. -\begin{figure} -\centering -%\includegraphics[width=0.7\linewidth]{./} -\caption{Categories of Student Conceptualizations of Proof by Induction -that Recursion Works} -\end{figure} - -Figure 5.3.2: -\begin{table}[h] -\caption{The Outcome Space for Proofs by Induction} -\begin{tabular}{|p{.2cm}|p{6cm}|p{6cm}|} -\hline & Category & Description \\ -\hline 1 & Following procedure & The method is learned, without understanding \\ -\hline 2 & Understands base case & The idea that a base case is proved by an existence -proof, often with a specific example\\ -\hline 3 & Understands implication & The idea that an implication is proved by -assuming the premise is not used\\ -\hline 4 & Does not understand connection & Sees the implication and proves it well, but -does not anchor the succession to a base -case \\ -\hline 5 & Does understand the argument & Understands the argument \\ -\hline 6 & Knows why recursion works & Can tailor the argument to explain recursive -algorithms \\ -\hline 7 & Appreciates data structures supporting recursion & Can see the benefit to algorithm from recursive data structure \\ -\hline -\end{tabular} -\end{table} - - - -\subsection{Helping Students Discern Abstraction} -Recall that variation theory holds that students cannot discern a thing unless -contrast is provided. Pang has pointed out that [], for persons aware of only -one language, ``speaking'' and ``speaking their language'' are conflated. Only -when the existence of a second language is known, does the idea of speaking -become separated from the idea of speaking a specific language. -(Here is a specialization (Hofstadter), formation of a new conjunct (Valiant), see Besold 2015) -Abstraction is important in computer science, and is worthy of investigation. -Inquiry into students' conceptualizations of formalization using symbols, symbolization, -has shown similar results among students of mathematics and of -computer science [?, ?]. Student populations contain the conceptualization that -proofs ought to be expressed using symbols, and some proof attempts show -that not all students are able to formalize meaningfully. Mathematics and computer -science pedagogies differ on the recommended style of variable names in -symbolization. In mathematics, there is a preference for single letter variable -names, and in computer science it is recognized that longer variable names assist -readers in understanding. In mathematics the use of single variable names -is preferred because it is thought to contribute to cultivating students' ability -to learn abstraction. If, in computer science education, we apply variation -Table 5.3.5: The Outcome Space for Proofs by Induction\\ -Category Description\\ - 1 Following procedure The method is learned, without understanding\\ - 2 Understands base case The idea that a base case is proved by an existence -proof, often with a specific example\\ -3 Understands implication The idea that an implication is proved by -assuming the premise is not used\\ -4 Does not understand connection -Sees the implication and proves it well, but -does not anchor the succession to a base -case\\ -5 Does understand the argument -Understands the argument\\ -6 Knows why recursion -works\\ -Can tailor the argument to explain recursive -algorithms\\ -7 Appreciates data structures -supporting recursion\\ -Can see the benefit to algorithm from recursive -data structure\\ -theory, we gain confidence in the idea that students may discern the process -of abstraction as we vary the names of the variables. We could imagine deriving -code from a requirement about a specific class, and using corresponding -variable names, and we could show the process of promoting the code into a -more general class in the inheritance hierarchy, changing the variable names to -correspond to the more general domain of objects. Thus we can borrow from -the approach used by mathematics education, but make it more explicit, taking -advantage of computer science's explicit treatment of inheritance hierarchies in -object oriented code. Seeking evidence of students' conception of abstraction, -we could examine overridden methods to see whether variable names in more -and less general implementations bear that relation to one another. - -\subsection {Algebra} -In middle or high school algebra students became familiar with the use of letters in equations, -and solving equations which resulted in individual values, or no value, being attached to the letters. - -Ideally the ability to understand expressions, to formulate pre- and post conditions would be acquired. -As we have seen that this occurs sometimes, but does not always occur, -there may be benefit to some students to review this idea. -We might choose to emphasize abstraction in this process. - -\subsection{Geometry} -In high school geometry, formal proofs of geometric properties are covered. -Students are exposed to a form for argument, and are given examples of use of rules of inference to perform logical deduction. -We have seen that sometimes this process is appreciated in enough generality to be recognized -as an example of argumentation. -We have seen as well, that some students found this process entirely specific to geometry, -doubting that it had broader application. - -\subsection{Seeing a Broader Context} - -It may be that some students do not see a separation between the activity of formalization on the one hand, -and the application area of finding solutions to equations on the other. -It may be that some students do not see a separation between the activity of deducing using logic on the one hand, and the application area of learning geometric facts on the other. -It seems consistent with neglecting opportunities for abstraction, that these separations are not always seen, -Speciation, an idea which uses abstraction, providing a hierarchy of properties animals and plants might have, was not recognized early or universally. So, it is -not surprising that abstraction, which involves choice about which details to defer, and which to regard as significant, is not always obvious. -In the machine learning perspective, features can be learned. -What do I want to say, it takes some effort to recognize features? -There might be a way to formulate choice of features such that some better efficiency is gained by thinking of the features in that order vs. another order. -(Such as, we never have to think about some features for some parts of the tree.) -If we think about knowledge being organized in neural networks, such that abstraction has a physical manifestation, we can see that ideas between which there is little distance (by some measure, neurons, glia?) in the tree will more frequently elicit one another by linkages at the metabolic level. -At the neural level, modifications for efficiency are constantly taking place (Do we have this from Kandel and Squire?). -We might wish to exploit this in the way we teach, to exhibit the abstraction deliberately, to minimize the amount of neural connection remodeling that would occur in the process of providing an efficient neural connection remodeling that would occur in the process of providing an efficient neural representation. -Being able to learn by analogy testifies to the utility of having a neural representation that corresponds to abstraction. -Students who are working without hierarchical organization of concepts are at a disadvantage. -\subsection{Mathematics tests in high school that involve proving} -What can we learn from students of computer science who excelled in reasoning to this level? - - -%\chapter{Discussion} - -\section{Discussion} - \subsection{Importance} -Importance goes here, rather than in analysis\\ - -Programmers/developers who produce and/or verify software that is used in safety critical applications, such as medical equipment, self-driving cars, and defense-related equipment should be able to know that their software functions correctly. - -Programmers/developers who produce and/or verify software that is expected to perform work, such as search, efficiently, should be able to know that their algorithms are efficient. - -Computer science is the expected background preparation for people working in these careers. -Proof is the method that is used to ascertain, and to convince, that these goals have been achieved. -\subsection{Interpretation of Results} -As in mathematics, some students learn as procedure that which we would prefer that they understand. -Some procedural learning is insufficiently accompanied by an understanding as to which contexts to which it applies, and has become in some cases what Whitehead calls "inert knowledge". - -\subsection{More Discussion} -\paragraph{Helping Students Discern Abstraction} -Recall that variation theory holds that students cannot discern a thing unless -contrast is provided. Pang has pointed out that [], for persons aware of only -one language, ``speaking'' and ``speaking their language'' are conflated. Only -when the existence of a second language is known, does the idea of speaking -become separated from the idea of speaking a specific language. -(Here is a specialization (Hofstadter), formation of a new conjunct (Valiant), see Besold 2015) -Abstraction is important in computer science, and is worthy of investigation. -Inquiry into students' conceptualizations of formalization using symbols, symbolization, -has shown similar results among students of mathematics and of -computer science [?, ?]. Student populations contain the conceptualization that -proofs ought to be expressed using symbols, and some proof attempts show -that not all students are able to formalize meaningfully. Mathematics and computer -science pedagogies differ on the recommended style of variable names in -symbolization. In mathematics, there is a preference for single letter variable -names, and in computer science it is recognized that longer variable names assist -readers in understanding. In mathematics the use of single variable names -is preferred because it is thought to contribute to cultivating students' ability -to learn abstraction. If, in computer science education, we apply variation -Table 5.3.5: The Outcome Space for Proofs by Induction\\ -Category Description\\ - 1 Following procedure The method is learned, without understanding\\ - 2 Understands base case The idea that a base case is proved by an existence -proof, often with a specific example\\ -3 Understands implication The idea that an implication is proved by -assuming the premise is not used\\ -4 Does not understand connection -Sees the implication and proves it well, but -does not anchor the succession to a base -case\\ -5 Does understand the argument -Understands the argument\\ -6 Knows why recursion -works\\ -Can tailor the argument to explain recursive -algorithms\\ -7 Appreciates data structures -supporting recursion\\ -Can see the benefit to algorithm from recursive -data structure\\ -theory, we gain confidence in the idea that students may discern the process -of abstraction as we vary the names of the variables. We could imagine deriving -code from a requirement about a specific class, and using corresponding -variable names, and we could show the process of promoting the code into a -more general class in the inheritance hierarchy, changing the variable names to -correspond to the more general domain of objects. Thus we can borrow from -the approach used by mathematics education, but make it more explicit, taking -advantage of computer science's explicit treatment of inheritance hierarchies in -object oriented code. Seeking evidence of students' conception of abstraction, -we could examine overridden methods to see whether variable names in more -and less general implementations bear that relation to one another. -\paragraph{Helping Students Discern Abstraction} -\paragraph {Algebra} -In middle or high school algebra students became familiar with the use of letters in equations, -and solving equations which resulted in individual values, or no value, being attached to the letters. - -Ideally the ability to understand expressions, to formulate pre- and post conditions would be acquired. -As we have seen that this occurs sometimes, but does not always occur, -there may be benefit to some students to review this idea. -We might choose to emphasize abstraction in this process. - -\paragraph{Geometry} -In high school geometry, formal proofs of geometric properties are covered. -Students are exposed to a form for argument, and are given examples of use of rules of inference to perform logical deduction. -We have seen that sometimes this process is appreciated in enough generality to be recognized -as an example of argumentation. -We have seen as well, that some students found this process entirely specific to geometry, -doubting that it had broader application. - -\paragraph{Seeing a Broader Context} - -It may be that some students do not see a separation between the activity of formalization on the one hand, -and the application area of finding solutions to equations on the other. -It may be that some students do not see a separation between the activity of deducing using logic on the one hand, and the application area of learning geometric facts on the other. -It seems consistent with neglecting opportunities for abstraction, that these separations are not always seen, -Speciation, an idea which uses abstraction, providing a hierarchy of properties animals and plants might have, was not recognized early or universally. So, it is -not surprising that abstraction, which involves choice about which details to defer, and which to regard as significant, is not always obvious. -In the machine learning perspective, features can be learned. -What do I want to say, it takes some effort to recognize features? -There might be a way to formulate choice of features such that some better efficiency is gained by thinking of the features in that order vs. another order. -(Such as, we never have to think about some features for some parts of the tree.) -If we think about knowledge being organized in neural networks, such that abstraction has a physical manifestation, we can see that ideas between which there is little distance (by some measure, neurons, glia?) in the tree will more frequently elicit one another by linkages at the metabolic level. -At the neural level, modifications for efficiency are constantly taking place (Do we have this from Kandel and Squire?). -We might wish to exploit this in the way we teach, to exhibit the abstraction deliberately, to minimize the amount of neural connection remodeling that would occur in the process of providing an efficient neural connection remodeling that would occur in the process of providing an efficient neural representation. -Being able to learn by analogy testifies to the utility of having a neural representation that corresponds to abstraction. -Students who are working without hierarchical organization of concepts are at a disadvantage. -\paragraph{Mathematics tests in high school that involve proving} -What can we learn from students of computer science who excelled in reasoning to this level? - - -\section{Previously Published Work} - -Three papers in this area have been published to date: -\begin{itemize} - -\item CCSCNE: Categorizing the School Experience of Entering Computing -Students -\item FIE: Mathematization in Teaching Pumping Lemmas -\item Koli Calling: Computer Science Students' Concepts of Proof by Induction - -\end{itemize} -\subsection{Categories of Experience of Entering Students} -Undergraduate students beginning study of the computing disciplines present -a various degrees of preparedness.\cite{reilly2014examination} Some interview participants enjoyed -a modified Moore method\cite{cohen1982modified} geometry class in middle school, and relished -opportunities to create proofs (not yet published). Other students are not so -well prepared. -After publishing this paper, more information relating to its topic has been -encountered. For example, consistent with the work of Almstrum \cite{almstrum1996investigating}, we have -found that, about implications, some students, who do know that any statement -must and can, be either true or false, think implications must be true. -\subsection{Categories of Experience of Entering Students} -Undergraduate students beginning study of the computing disciplines present -various degrees of preparedness\cite{smith2013categorizing}. Some had no experience, some had had -informal experience, and some had had formal classes. The formal classes -extended from using applications to building applications. Informal experience -ranged from editing configuration files, such as background colors, to full time -jobs extended over multiple summers. -After publishing this paper, we encountered more related information. For -example, consistent with the work of Almstrum\cite{almstrum1996investigating}, we found that, about implications, -some students, who do know that any statement must and can, be -either true or false, thought implications must be true. Some interview participants -enjoyed a modified Moore method\cite{cohen1982modified} geometry class in middle school, -and relished opportunities to create proofs (not yet published). Other students -were not so well prepared. -\subsection{Representation/Symbolization in Pumping Lemmas} -Some of our results to date are consistent with the framework described by -Harel and Sowder in 1998\cite{harel1998students}. We have found students holding conceptualizations -that Harel and Sowder's 1998 model\cite{harel1998students} calls symbolization: We have found -that some students may lack facility in notation. For example, in the application -of the pumping lemma, students are expected to understand the role of $i$, -in the context that a string $s$, having component substrings $x$, $y$ and $z$, can be -used to generate other strings, of the form $xy^iz$, where $i$ gives the number of -copies of the substring $y$. Moreover, students are expected to understand that -the subdivision of a string of length $p$, expressed as $\sigma_1^a\sigma_2^{p-a}$, where $a \in \{0,1,\ldots,p\}$ -uses $a$ as a parameter, a free variable, not one necessarily bound to a single instance -of a natural number, but a representation of a domain. We have seen this -lack of understanding in a situation in which it was proposed as evidence that -a single example, namely $\sigma_1^a\sigma_2^{p-a}$, formed a proof for a universally quantified -statement. An excerpt of the errors found on tests is shown in Table . - -Table : Some example errors\\ -Let x be empty\\ -$|xy| \leq p, so xy = 0^p$\\ -$|xy| \leq p; let x = 0^{p+r}, y = 0^{p+r}, 0 < r < p$\\ -Let's choose $|xy| = p$\\ -$0^{p+1}0^b1^p \neq 0^{p+1}1^p \therefore xy^2z \not\in \mathcal{L}$\\ -where $\mathcal{L} = \{0^i1^j, i \neq j\}$\\ -we choose $s = 0^{p+1}1^p$ within $|xy|$\\ -thus $\neq 0^p1^{p+1}$\\ -Let $x = 0^a, y = 0^b1^a$\\ -$x = 0^{p-h}, y = 0^h$\\ -$x = 0^i, y = 0^i, z = 0^i1^j$ - -Figure: Some categories / conceptualizations found among students of -introduction to the theory of computing, and published at FIE. - -Harel and Sowder identified a category of conceptualization that correctly -applied transformation and axiomatic arguments. Some students expressed -enthusiasm for the power that inheres to building arguments with carefully -specified component ideas, in particular how the absence of ambiguity permitted -arguments to extend to great length while remaining valid. Not all of the -students had developed axiomatic conceptualizations of proof. About definitions, -we have collected preliminary data on students' conceptualizations of -definitions used in proofs. Some students think definitions are boring. Some -students think that they can infer definitions from a few examples. Concerning -executive function, we have found that some students do not state the -premises clearly, and some students do not keep track of their goal. About -rules of inference, we have found that some students apply invalid approaches -to inference. -\subsection{Representation/Symbolization in Pumping Lemmas} -We found that some students may lack facility in notation. For example, in the -application of the pumping lemma, students are expected to understand the -role of $i$, in the context that a string $s$, having component substrings $x$, $y$ and $z$, -can be used to generate other strings, of the form $xy^iz$, where $i$ gives the number -of copies of the substring $y$. Moreover, students are expected to understand that -the subdivision of a string of length $p$, $\sigma_1^a\sigma_2^{p-a}$, where $a \in \{0,1,\ldots,p\}$ -uses $a$ as a parameter, a free variable, not one necessarily bound to a single instance -of a natural number, but a representation of a domain. An excerpt of the errors found on tests is shown in Table . Trigueros et -al. \cite[ p. 3]{jacobs2008developing} have observed that ``students are often unclear about the different -ways letters are used in mathematics''. We have seen this -lack of understanding in a situation in which it was proposed as evidence that -a single example, namely $\sigma_1^a\sigma_2^{p-a}$, formed a proof for a universally quantified -statement. -Some of our results were consistent with the framework described by Harel -and Sowder in 1998\cite{harel1998students}. We found students holding conceptualizations that -Harel and Sowder's 1998 model calls symbolization. Harel and Sowder have -identified another category of conceptualization, that correctly applied transformation -and axiomatic arguments. Some students expressed enthusiasm for -the power that inheres to building arguments with carefully specified component -ideas, in particular how the absence of ambiguity permitted arguments to -extend to great length while remaining valid. Not all of the students had developed -axiomatic conceptualizations of proof. About definitions, we collected -preliminary data on students' conceptualizations of definitions used in proofs. -Some students thought definitions were boring. Some students thought that -they could infer definitions from a few examples. Concerning executive function, -we found that some students do not state the premises clearly, and some -students did not keep track of their goal. About rules of inference, we found -Figure 5.3.1: Some categories / conceptualizations found among students of -introduction to the theory of computing, and published at FIE. -that some students apply invalid approaches to inference. -\subsection{Abstract Model for Proof by Mathematical Induction and Recursion} -Far from finding agreement that (a) theorems are true as a consequence of -the definitions and the premise, and that (b) proofs serve to show how the -consequence is demonstrated from the premise, axioms and application of -rules of inference, instead we found a variety of notions about proof, including -the well-known procedural interpretation \cite{tall2008transition,weber2004traditional,tall2001symbols}, and the well-known empirical -misconception \cite{harel1998students}. The conceptualization that definitions are not necessarily -of interest compared with the procedures seemed different in kind from the -concept image / concept definition discoveries of R\"osken et al. \cite{rosken2007integrating}. -Interviews with students revealed that some students saw generation of a proof -by mathematic induction as a procedure to be followed, in which they should -produce a base case, and prove it, and should produce an induction step, and -prove that. This was consistent with Weber [?, p. 4-426] who has stated ``in -the studies that I conducted, it was more often the case that undergraduates -applied procedures that were not meaningful to them.'' He went on to give a -quotation from a participant [?, p. 4-426] ``And I prove something and I look at -it, and I thought, well, you know, it's been proved, but I still don't know that I -even agree with it [laughs]. I'm not convinced by my own proof!'' Some of the -students interviewed did not know why this procedure generated a convincing -argument. Polya[?] has written a problem involving all girls being blue-eyed; -a similar problem appears in Sipse \cite{sipser2012introduction} about all horses being the same color. -The purpose of this exercise is to make students aware that the truth of the -inductive step must apply when the base case appears as the premise. In some -cases, this point was not clear to the students. -Students' conceptualizations of proof by mathematical induction can support -their choice to apply recursive algorithms. One student reported success at -both mathematical induction and recursive algorithm application without ever -noticing any connection. This student opined that having learned recursion -with figures, and proof by mathematical induction without figures, that no -occasion for the information to spontaneously connect occurred. Students reporting -ability to implement assigned problems recursively, but not the ability -to understand proof by mathematical induction also reported that ability to -write recursive programs did not result in recognition of when recursive solutions -might be applicable in general. Students reporting ability to implement -assigned problems recursively, and also the ability to prove using mathematical -induction also reported preferring to implement recursive solutions in -problems as they arose. -Our work on students' choices of algorithmic approaches was consistent with -work by other researchers in computer science education\cite{booth1997phenomenography} on conceptualizations -of algorithms. Our work served to unify that of mathematician educators -with computer science educators, by providing a plausible explanation why -the conceptualizations of recursive algorithms that were found, might exist. -\subsection{Proofs by Induction} -Table 5.3.3: The Outcome Space for Proofs by Induction -Category Description -1Following procedure The method is learned, without understanding -2Understands base case The idea that a base case is proved by an existence -proof, often with a specific example -3Understands implication The idea that an implication is proved by -assuming the premise is not used -4Does not understand connection -Sees the implication and proves it well, but -does not anchor the succession to a base -case -5Does understand the argument -Understands the argument -6Knows why recursion -works -Can tailor the argument to explain recursive -algorithms -7Appreciates data structures -supporting recursion -Can see the benefit to algorithm from recursive -data structure -\subsection{Abstract Model for Proof by Mathematical Induction and Recursion} -Interviews with students revealed that some students see generation of a proof -by mathematic induction as a procedure to be followed, in which they produce -a base case, and prove it, and produce an induction step, and prove that. Some -of the students interviewed did not know why this procedure generated a -convincing argument. Moore, as reported in Polya[] noted that some students -of mathematics formed the same conceptualization, that there is a procedure, -but it does not necessarily produce a convincing argument. Polya[] wrote -a problem involving all girls being blue-eyed; a similar problem appears in -Sipser\cite{sipser2012introduction} about all horses being the same color. The purpose of this exercise is -to make students aware that the truth of the inductive step must apply when -the base case appears as the premise. In some cases, this point was not clear to -the students. -Students' conceptualizations of proof by mathematical induction can support -their choosing to apply recursive algorithms. One student reported success at -both mathematical induction and recursive algorithm application without ever -noticing any connection. This student opined that having learned recursion -with figures, and proof by mathematical induction without figures, that no -occasion for the information to spontaneously connect occurred. Students reporting -ability to implement assigned problems recursively, but not the ability -to understand proof by mathematical induction also reported that ability to -write recursive programs did not result in recognition of when recursive solutions -might be applicable in general. Students reporting ability to implement -assigned problems recursively, and also the ability to prove using mathematical -induction also reported preferring to implement recursive solutions in -problems as they arose. -Our work on students' choices of algorithmic approaches is consistent with -work by other researchers in computer science education\cite{} on conceptualizations -of algorithms. Our work served to unify that of mathematician educators -with that of computer science educators, by providing a plausible explanation why -the conceptualizations of recursive algorithms that were found, might exist. - -Figure 4.0.2: Conceptualizations of proof by induction and recursion, published -in Koli Calling +\chapter{Results} -Index Element of Model -\begin{enumerate} +The results of a phenomenographic study comprise a set of categories of description of ways of experiencing (or capability for experiencing \cite[p. 126]{marton1997learning}) a phenomenon, and relations among those categories. -\item Some students begin learning proof by mathematical induction as if it were -a procedure. -\item Some students learn two parts of this proof technique without seeing any -connection between the two. -\item Some students do not find the procedure to be a convincing argument. -\item Some students would not employ proof by mathematical induction to explore -whether a recursive algorithm would apply to a given problem. -\item Some students understand both proof by mathematical induction and also -recursion and had never noticed any similarity. - -\end{enumerate} - -\subsection{Pumping Lemmas} -TABLE III. CATEGORIES\\ -understand inequality\\ -formulate correctly\\ -distinguish between particular and generic particular\\ -correctly apply universal quantifier\\ -recognize string as member of language set - - -\section{Results of Combined Investigations} -There are some categories that are shared among the several contexts. -\section{Categories} -Categories found in one or more investigations - -Categories\\ -Definition of proof as convincing (to mathematicians) argument is not -always understood\\ -Definitions in general are not always recognized as significant building -blocks in arguments\\ -The idea of a false statement sometimes becomes troublesome when -negation is being learned.\\ -In particular, accepting that an implication may be false, can be troublesome. -Notation is sometimes difficult.\\ -Ideas presented relying on notation are not always connected with -ideas presented relying on figures.\\ -Warrants are not always recognized.\\ -Students do not always traverse levels of abstraction effectively.\\ -The applicability of valid argument forms to contexts of interest is not -always appreciated. -\section{Critical Factors} -To determine critical factors, we can convert negative categories into achievement -levels. -\begin{tabular}{p{3cm}p{3cm}} -Categories & Achievement Levels\\ -The idea of a false statement -sometimes becomes troublesome -when negation is being -learned.&\\ -&True and false make sense, and -we can make arguments using -them.\\ -Definition of proof as convincing -argument is not always understood&\\ -Warrants are not always recognized.&\\ -&Proof can sometimes be obtained -through a series of warranted assertions.\\ -Definitions in general are not always -recognized as significant -building blocks in arguments&\\ -&Using agreed definitions and -valid rules of inference we can -sometimes explore the consequences -of definitions.\\ -Notation is sometimes difficult.&\\ -&Notation helps.\\ -Ideas presented relying on notation -are not always connected -with ideas presented relying on -figures.&\\ -&We might wish to help students -traverse multiple rendering of -ideas.\\ -Students do not always traverse -levels of abstraction effectively.&\\ -&We might wish to help students -traverse multiple levels of abstraction.\\ -The applicability of valid argument -forms to contexts of interest -is not always appreciated.&\\ -&We might wish to give exercise -with authentic (career related) -examples\\ - -\end{tabular} - -Using the achievement levels we can infer critical factors. -\begin{tabular}{p{3cm}p{3cm}} -Achievement Levels& Critical Factors\\ -True and false make sense, and -we can make arguments using -them.&\\ -& True and false apply to assertions.\\ -Proof can sometimes be obtained -through a series of warranted assertions. -& Proof is exploration and discovery.\\ -Using agreed definitions and -valid rules of inference we can -sometimes explore the consequences -of definitions.&\\ -& Efficiency but also abstraction -are aided by notation.\\ -Notation helps.&\\ -& Notation is one representation -and there are others. Ideas appear -in multiple guises.\\ -We might wish to help students -traverse multiple rendering of -ideas.&\\ -& When notation allows for multiple -interpretations, abstraction -above those multiple interpretations -has been achieved.\\ -We might wish to help students -traverse multiple levels of abstraction.&\\ -&Multiple levels of abstraction are -relevant at the same time.\\ -We might wish to give exercise -with authentic (career related) -examples.&\\ -& Authentic applications show the -use of this knowledge.\\ -\end{tabular} - - -\subsection{Abstraction} -Literature reports \cite{} students of CS have trouble with abstraction. -Taking abstraction to be the ability to select some details to ignore, -and thereby find a simpler model of an entity, we can transform the ideal -knowledge transfer experience into on disabled by a lack of ability to see this dimension. -The multiple-inheritance hierarchy that could be used to organize -definitions and relationships of ideas is less able. More entities will be -grouped together than effective use of the multiple inheritance hierarchy -would consider equivalent. -Another useful concept that students have been seen to underappreciate is the significance of careful definitions. -Abstraction hierarchies allow for efficiency in definitions. -A new entity can be defined as a specialization of an existing entity, and its differences -make up the new definition material. -In the absence of this multiple inheritance hierarchy, every definition in its full length -is attached to its entity. -Tie in with Mazur. For students holding the same granularity of refinement of -concepts, conversations would be easier, because there would be fewer disconnects as one participant expressed a thought on one degree of refinement far from that of another student. -If the ideas implying the refinement of the definition inheritance graph, being different from one -discussant to the next, are rare, and/or the meaning of the sentence does not depend upon it, these exchanges are not too disruptive or distressing. -On the other hand when two sets of refinement are very different, -and the meaning of the exchange depends upon the refinement in the speaker, that the hearer does not have, -then some degree of failure of communication will ensue. -Absence of abstraction converts tree of topics into sequence of topics. -Tree of proof examples (say of application of proof technique) into sequence of examples. -Might detract from recognizing what is a related example. -Would detract from plausible inference technique of ``related problem seen before''. -We have a goal for student programming that they should strive for segments of programs -(e.g., method implementations) to be small. One way of accomplishing this is to use abstraction, -such as combining instructions into a method, and calling the method. -If students have difficulty with abstraction, they might -have difficulty with choosing groups of instructions to represent a method. -Correspondingly, if they practice grouping instruction into methods, and using those methods, they would -be gaining practice relevant to using abstraction. -One way to cultivate abstraction is to pose a question of which one of several examples is different. -When several things are examples of one abstract idea and one is not, identifying the one that is different involves noticing the abstraction. -These questions could be instantiated using blocks of code. - -\subsection{Definitions} -Without abstraction the burdensomeness of definition is increased. This could contribute to the reluctance of students to embrace definitions. -\subsection{Symbolization} -Use of symbols is a kind of abstraction. -Symbolization is the syntax for simple, clear definitions as Gries\cite[p.205]{gries2012science}(Science of Programming) recommends for construction of programs. -Students will be hindered at this program derivation/development style if symbolization is a not-yet-acquired skill. -Program development/derivation should begin with a formulation of the requirements. -Students may arrive with some programming experience that is of a more intuitive, less -mathematically disciplined sort. -We have to ask how we desire to cultivate the abilities of such students. -Vygotsky discussed language acquisition by children, in which some children will have begun to invent -some terms for items in their environment, and will need to be guided to abandon neologisms for the naming generally agreed in their environment. -Kuhn discussed the reluctance of scientists who have been rewarded for operating in one -perspective on nature to adopt a different perspective. -Instructor may encounter a similar reluctance on the part of students -to adapt a scientifically/mathematically disciplined approach to programming, -especially if the students have experienced some success in their earlier work. -To win over such students, -demonstration of superior outcomes on problems, especially on problems that seem insoluble otherwise, -are more frequently convincing. -Happily, Professor Gries has provided such examples. -By showing superior relative efficacy of these approaches in an activity the students recognize as -desirable, instructors could motivate the students to learn symbolization. -\subsection{Structure} -sequence vs. sequence that has come about from combining parts. Refer to Leslie Lamport's structure for proofs. Combine with Gries' proofs for deriving code. The purpose for getting through Goguen and Malcolm is that it applies to imperative programs. - - - - - -The results of a phenomenographic study comprise a set of categories of description of ways of experiencing (or capability for experiencing [p. 126]) a phenomenon, and relations among those categories. +Booth states\cite{booth2001learning} "The results are communicated as +descriptions of the essential aspects of each category, illustrated by pertinent +extracts from the data". Marton and Booth\cite[p. 125]{marton1997learning} give criteria for the quality of a set of descriptive categories. The collective experience, over all participants in the study should be included. @@ -830,18 +36,8 @@ We are looking for ways of experiencing, for example, one way is, proofs only ap - \section{What do students think a proof is?} -One participant opined "Some students think proof is a magical incantation." - -Some students do not recognize a proof when they are looking at one. When asked "Were we doing proofs today?", such a student said "No." When asked, "Did we not show that certain things must always be the case when certain contextual requirements were met?", the answer was "Yes, oh, I guess that's a proof." When asked, "Why didn't it appear to be a proof?", the answer was "It didn't start by saying 'Proof:', and it didn't end by saying 'QED', or that box." - -Some students say that a proof is a sequence of statements, starting with one distinguished statement, called a premise and ending with a distinguished statement, what we wanted to prove. - -Some students say "What's a lemma?". Some students cannot, when asked for credit, identify the use of structure, such as a lemma, in a proof. - -Some students know that proof ought to be a convincing argument. Some of these are not convinced by a proof by induction. When assured that they would enjoy a certain proof in introduction to the theory of computation, because it was a proof by induction, voices in a class claimed, "We never 'got' that". Some students, when asked what they think a proof is, will report that they think it is a list of true mathematically formulated statements, demonstrating the truth of a mathematically formulated statement. Some students report that a proof has a goal, a statement to be proved true. Some students know that the identified goal does not have to be known to be true in advance of the first proof. @@ -1046,114 +242,123 @@ On the path that starts with symbols, students have not generalized from concret - - - - - +\section{Earlier Paper Material} +\subsection{Categories of Experience of Entering Students} +Undergraduate students beginning study of the computing disciplines present +various degrees of preparedness\cite{smith2013categorizing}. Some had no experience, some had had +informal experience, and some had had formal classes. The formal classes +extended from using applications to building applications. Informal experience +ranged from editing configuration files, such as background colors, to full time +jobs extended over multiple summers. +After publishing this paper, we encountered more related information. For +example, consistent with the work of Almstrum\cite{almstrum1996investigating}, we found that, about implications, +some students, who do know that any statement must and can, be +either true or false, thought implications must be true. Some interview participants +enjoyed a modified Moore method\cite{cohen1982modified} geometry class in middle school, +and relished opportunities to create proofs (not yet published). Other students +were not so well prepared. +\subsection{Representation/Symbolization in Pumping Lemmas} +We found that some students may lack facility in notation. For example, in the +application of the pumping lemma, students are expected to understand the +role of $i$, in the context that a string $s$, having component substrings $x$, $y$ and $z$, +can be used to generate other strings, of the form $xy^iz$, where $i$ gives the number +of copies of the substring $y$. Moreover, students are expected to understand that +the subdivision of a string of length $p$, $\sigma_1^a\sigma_2^{p-a}$, where $a \in \{0,1,\ldots,p\}$ +uses $a$ as a parameter, a free variable, not one necessarily bound to a single instance +of a natural number, but a representation of a domain. An excerpt of the errors found on tests is shown in Table . Trigueros et +al. \cite[ p. 3]{jacobs2008developing} have observed that ``students are often unclear about the different +ways letters are used in mathematics''. We have seen this +lack of understanding in a situation in which it was proposed as evidence that +a single example, namely $\sigma_1^a\sigma_2^{p-a}$, formed a proof for a universally quantified +statement. +Some of our results were consistent with the framework described by Harel +and Sowder in 1998\cite{harel1998students}. We found students holding conceptualizations that +Harel and Sowder's 1998 model calls symbolization. Harel and Sowder have +identified another category of conceptualization, that correctly applied transformation +and axiomatic arguments. Some students expressed enthusiasm for +the power that inheres to building arguments with carefully specified component +ideas, in particular how the absence of ambiguity permitted arguments to +extend to great length while remaining valid. Not all of the students had developed +axiomatic conceptualizations of proof. About definitions, we collected +preliminary data on students' conceptualizations of definitions used in proofs. +Some students thought definitions were boring. Some students thought that +they could infer definitions from a few examples. Concerning executive function, +we found that some students do not state the premises clearly, and some +students did not keep track of their goal. About rules of inference, we found +Figure 5.3.1: Some categories / conceptualizations found among students of +introduction to the theory of computing, and published at FIE. +that some students apply invalid approaches to inference. -\section{More Discussion} -\paragraph{Helping Students Discern Abstraction} -Recall that variation theory holds that students cannot discern a thing unless -contrast is provided. Pang has pointed out that [], for persons aware of only -one language, ``speaking'' and ``speaking their language'' are conflated. Only -when the existence of a second language is known, does the idea of speaking -become separated from the idea of speaking a specific language. -(Here is a specialization (Hofstadter), formation of a new conjunct (Valiant), see Besold 2015) -Abstraction is important in computer science, and is worthy of investigation. -Inquiry into students' conceptualizations of formalization using symbols, symbolization, -has shown similar results among students of mathematics and of -computer science [?, ?]. Student populations contain the conceptualization that -proofs ought to be expressed using symbols, and some proof attempts show -that not all students are able to formalize meaningfully. Mathematics and computer -science pedagogies differ on the recommended style of variable names in -symbolization. In mathematics, there is a preference for single letter variable -names, and in computer science it is recognized that longer variable names assist -readers in understanding. In mathematics the use of single variable names -is preferred because it is thought to contribute to cultivating students' ability -to learn abstraction. If, in computer science education, we apply variation -Table 5.3.5: The Outcome Space for Proofs by Induction\\ -Category Description\\ - 1 Following procedure The method is learned, without understanding\\ - 2 Understands base case The idea that a base case is proved by an existence -proof, often with a specific example\\ -3 Understands implication The idea that an implication is proved by -assuming the premise is not used\\ -4 Does not understand connection +\subsection{Abstract Model for Proof by Mathematical Induction and Recursion} +Far from finding agreement that (a) theorems are true as a consequence of +the definitions and the premise, and that (b) proofs serve to show how the +consequence is demonstrated from the premise, axioms and application of +rules of inference, instead we found a variety of notions about proof, including +the well-known procedural interpretation \cite{tall2008transition,weber2004traditional,tall2001symbols}, and the well-known empirical +misconception \cite{harel1998students}. The conceptualization that definitions are not necessarily +of interest compared with the procedures seemed different in kind from the +concept image / concept definition discoveries of R\"osken et al. \cite{rosken2007integrating}. +Interviews with students revealed that some students saw generation of a proof +by mathematic induction as a procedure to be followed, in which they should +produce a base case, and prove it, and should produce an induction step, and +prove that. This was consistent with Weber [?, p. 4-426] who has stated ``in +the studies that I conducted, it was more often the case that undergraduates +applied procedures that were not meaningful to them.'' He went on to give a +quotation from a participant [?, p. 4-426] ``And I prove something and I look at +it, and I thought, well, you know, it's been proved, but I still don't know that I +even agree with it [laughs]. I'm not convinced by my own proof!'' Some of the +students interviewed did not know why this procedure generated a convincing +argument. Polya[?] has written a problem involving all girls being blue-eyed; +a similar problem appears in Sipse \cite{sipser2012introduction} about all horses being the same color. +The purpose of this exercise is to make students aware that the truth of the +inductive step must apply when the base case appears as the premise. In some +cases, this point was not clear to the students. +Students' conceptualizations of proof by mathematical induction can support +their choice to apply recursive algorithms. One student reported success at +both mathematical induction and recursive algorithm application without ever +noticing any connection. This student opined that having learned recursion +with figures, and proof by mathematical induction without figures, that no +occasion for the information to spontaneously connect occurred. Students reporting +ability to implement assigned problems recursively, but not the ability +to understand proof by mathematical induction also reported that ability to +write recursive programs did not result in recognition of when recursive solutions +might be applicable in general. Students reporting ability to implement +assigned problems recursively, and also the ability to prove using mathematical +induction also reported preferring to implement recursive solutions in +problems as they arose. +Our work on students' choices of algorithmic approaches was consistent with +work by other researchers in computer science education\cite{booth1997phenomenography} on conceptualizations +of algorithms. Our work served to unify that of mathematician educators +with computer science educators, by providing a plausible explanation why +the conceptualizations of recursive algorithms that were found, might exist. +\subsection{Proofs by Induction} +Table 5.3.3: The Outcome Space for Proofs by Induction +Category Description +1Following procedure The method is learned, without understanding +2Understands base case The idea that a base case is proved by an existence +proof, often with a specific example +3Understands implication The idea that an implication is proved by +assuming the premise is not used +4Does not understand connection Sees the implication and proves it well, but does not anchor the succession to a base -case\\ -5 Does understand the argument -Understands the argument\\ -6 Knows why recursion -works\\ +case +5Does understand the argument +Understands the argument +6Knows why recursion +works Can tailor the argument to explain recursive -algorithms\\ -7 Appreciates data structures -supporting recursion\\ +algorithms +7Appreciates data structures +supporting recursion Can see the benefit to algorithm from recursive -data structure\\ -theory, we gain confidence in the idea that students may discern the process -of abstraction as we vary the names of the variables. We could imagine deriving -code from a requirement about a specific class, and using corresponding -variable names, and we could show the process of promoting the code into a -more general class in the inheritance hierarchy, changing the variable names to -correspond to the more general domain of objects. Thus we can borrow from -the approach used by mathematics education, but make it more explicit, taking -advantage of computer science's explicit treatment of inheritance hierarchies in -object oriented code. Seeking evidence of students' conception of abstraction, -we could examine overridden methods to see whether variable names in more -and less general implementations bear that relation to one another. -\paragraph{Helping Students Discern Abstraction} -\paragraph {Algebra} -In middle or high school algebra students became familiar with the use of letters in equations, -and solving equations which resulted in individual values, or no value, being attached to the letters. - -Ideally the ability to understand expressions, to formulate pre- and post conditions would be acquired. -As we have seen that this occurs sometimes, but does not always occur, -there may be benefit to some students to review this idea. -We might choose to emphasize abstraction in this process. - -\paragraph{Geometry} -In high school geometry, formal proofs of geometric properties are covered. -Students are exposed to a form for argument, and are given examples of use of rules of inference to perform logical deduction. -We have seen that sometimes this process is appreciated in enough generality to be recognized -as an example of argumentation. -We have seen as well, that some students found this process entirely specific to geometry, -doubting that it had broader application. - -\paragraph{Seeing a Broader Context} - -It may be that some students do not see a separation between the activity of formalization on the one hand, -and the application area of finding solutions to equations on the other. -It may be that some students do not see a separation between the activity of deducing using logic on the one hand, and the application area of learning geometric facts on the other. -It seems consistent with neglecting opportunities for abstraction, that these separations are not always seen, -Speciation, an idea which uses abstraction, providing a hierarchy of properties animals and plants might have, was not recognized early or universally. So, it is -not surprising that abstraction, which involves choice about which details to defer, and which to regard as significant, is not always obvious. -In the machine learning perspective, features can be learned. -What do I want to say, it takes some effort to recognize features? -There might be a way to formulate choice of features such that some better efficiency is gained by thinking of the features in that order vs. another order. -(Such as, we never have to think about some features for some parts of the tree.) -If we think about knowledge being organized in neural networks, such that abstraction has a physical manifestation, we can see that ideas between which there is little distance (by some measure, neurons, glia?) in the tree will more frequently elicit one another by linkages at the metabolic level. -At the neural level, modifications for efficiency are constantly taking place (Do we have this from Kandel and Squire?). -We might wish to exploit this in the way we teach, to exhibit the abstraction deliberately, to minimize the amount of neural connection remodeling that would occur in the process of providing an efficient neural connection remodeling that would occur in the process of providing an efficient neural representation. -Being able to learn by analogy testifies to the utility of having a neural representation that corresponds to abstraction. -Students who are working without hierarchical organization of concepts are at a disadvantage. -\paragraph{Mathematics tests in high school that involve proving} -What can we learn from students of computer science who excelled in reasoning to this level? - - - - - - - - - - - - - - +data structure +\subsection{Pumping Lemmas} +TABLE III. CATEGORIES\\ +understand inequality\\ +formulate correctly\\ +distinguish between particular and generic particular\\ +correctly apply universal quantifier\\ +recognize string as member of language set diff --git a/ch6.tex b/ch6.tex index 1223214..39fcc33 100644 --- a/ch6.tex +++ b/ch6.tex @@ -1,228 +1,1159 @@ -\chapter{Validity and Reliability} -strategies for trustworthy, valid, reliable -what about generalizability? %(e.g., to people with ASD) -\section{Interviews} -software programs that were used, to manage, organize data\\ -analyze as we go\\ -inductive and comparative\\ -precisely how the analysis was done\\ -thorough explanation of any strategies, such as discourse analysis -\section{Documents} -\section{Validity and Reliability in Proofs Using the Pumping Lemma -for Regular Languages} -Some support for the validity of the results comes from seeing several variations -of each proposed error type. We found in our data multiple versions of -unwarranted restrictions: choosing $x$ to be empty or choosing the length of $xy$ -to be $p$, and others. We found in literature warnings against attempts to prove -statements with universal qualifiers true by means of showing the existence -of examples \cite{devlin2012mathematical,Franklin}. These warnings suggest these errors have occurred before. -In textbooks \cite{epp2010discrete,rosen2003} we find single counterexamples for showing such a -statement false, and the method of exhaustion for showing a finite universal -statement to be true. Proof by contradiction for the purpose of showing such a -statement true, i.e., that any particular tentative counterexample contained an -inherent contradiction, is not itself universally accepted, due to not necessarily -being constructive \cite[p. 2]{bridges2007did}. We also found in our data, several versions -of misunderstanding inequalities. We found support in literature for errors of -misunderstanding how to work with inequalities, by students of this level\cite{Mattuck}. -\section{Proof by Induction} -We were encouraged by the overlap in description among interview participants. -The interviews were certainly not the same, but common elements, -specifically that there is a form to proofs by induction, appeared. Some students -referred to this form as steps, others as a procedure, or framework. Moreover, -degrees of understanding filled in a spectrum, from joyful deep understanding -to admissions of not understanding why the steps of proof by induction prove -anything, and conceptions in between. These included a supposition why a -proof of an induction step would, in combination with an established base case, -constitute a proof by induction, that its originator characterized as ``weird''. -\section{Domain, Range, Mapping, Relation, Function, Equivalence -Relation in Proofs} -\section{Definitions, Language, Reasoning in Proofs} -\section{Equivalence Class, Generic Particular, Abstraction in Proofs} -Using an analogy, I claim, is saying there is a set of relations among things $a_i$ that -we agree upon, furthermore, I might wish to teach that there is a corresponding -set of relations among things $b_i$. I might wish to say, use the relation we agree -upon for municipalities provide addresses for homes that can be used for -surface mail, and I might wish to teach that there is a corresponding provision -of addresses for items a computer programmer might wish to use for storage -and recall. We can note that addresses make use of a hierarchy of place names, -countries, states, cities, streets, street numbers, apartment numbers. We can -note that structured data types can correspondingly make use of instances and -fields and indices that can be arranged in a tree. -In the absence of abstraction, the surface mail address hierarchy might not pose -much more difficulty, but the data structure might, because the fields therein -are more subject to change than municipalities. In the absence of abstraction, -the comparison between one hierarchical arrangement with another would be -more difficult, because it is the structure of the abstraction itself, namely, the -choices of features regarded as significant throughout the tree, that is to be -recalled and used as a scaffold for the new information. -``An alternative pathway towards abstraction involves recognizing an analogy -between two structures in different domains, which then focuses one's attention -on the abstract structure they share. This new abstraction then becomes a -’concrete’ concept that one can study'' \cite [p 449]{}. -\section{Vertical Integration and Explanation} -It is accepted that, in discrete math, it is helpful to work problems. We may -inquire, what is it about working problems that helps? We can, at the neurophysiology -level, expect that long term potentiation of synapses, for the -synapses collocated with the long term memory for the concepts in the problem, -is being carried out, as the thinking about the problem occurs. We can -recall that a sense of reward, as might be gained from success at a problem, -or pleasantness in a problem statement, helps consolidate the memory for the -ideas that have been gained. We can recall that depression resulting from -avoidance of sadness at failure to solve a problem reduces the ability of the -hippocampus to support the formation of long term memory. We can, at the -cognitive neuroscience level, expect that the opportunity for like structures to -be recognized occurs, and analogies, are made, and the abstraction hierarchy of -concepts related to the problem is remodeled, extended, to more closely mirror -the mathematical definitions being used. We can, at the phenomenography -level, suppose that fine distinctions between concepts will be more likely to be -noted, because the mental structures that support these are forming. We can, -at the computer science education level, consider how to bring activity into -lecture, that poses the analogies and distinctions we wish the student to gain, -by varying the examples of the concepts such that a representative example -is contrasted with a non-example, in an ambience that fosters curiosity and -rewards progress. -\section{ Validation} -We draw a connection between epistemology and validation. Epistemology -is why we believe what we believe. Thus, it makes sense to apply our epistemological -framework to explain why we believe what we believe about our -results and interpretation. -Our epistemology is informed by the work of others over a wide range of disciplines: -Computer science education, mathematics education, and education -generally, especially phenomenography. Cognitive aspects, including memory -and attention, are shared by and form a bridge between phenomenography -and cognitive neuroscience. Neuroscience provides interesting relevant information. -Ira Black, MD, in [?, ?, [p 40] ``A satisfactory mechanistic description of any -well-framed cognitive process requires that we simultaneously explain it at -multiple levels of analysis. Different levels provide complementary insights -to characterization and causality that are unobtainable from any single line of -analysis.'' -\subsection{Validation at the Level of Computer Science Education} -Interviews with computer science educators, both instructors and teaching -assistants have provided a diversity of viewpoints, and generally support the -interpretations we have given. For example, one instructor, when asked what -students thought proofs were, said ``some kind of magic incantation'', and -teaching assistants have said ``students really struggle with this''. -\subsection{Validation at the Level of Mathematics Education} -The literature of mathematics education includes work on students' learning -about proof. Our work with computer science students has had the benefit -of some students who are dual majors of math and computer science, which -has allowed us to trace the similarities and differences of these cohorts of students. -The significance of definitions, the necessity and utility of proof, the role -played by interest in forming procedures and functions, the difference between -functional and procedural programming have differed in these three cohorts, -in so far as we have been able to examine. We did not explore the interest in -developing procedures/functions or procedural or functional programming in -mathematics majors who were not also computer science majors. - - - -\subsection{ Validation at the Level of Phenomenography/Variation Theory} -Variation theory supports our observation that comparing and contrasting fine -distinctions in material being taught aids the process of learning. We used -the difference between assignment and equality testing, manifest in the java -expression of ``=='' vs. ``=''. We compared a software procedure representation -with a mathematical formulation (the latter using only ``=''), for comprehensibility -by students. This helped us to see that barriers to student understanding -exist, for some students of computer science, at the level of formulation. It also -helped us see that the barrier between the internalization and interiorization -of Harel and Sowder might be less of a barrier in students of computer science -who are routinely conscious of the need to analyze procedures. -\subsection{Validation at the Level of Cognitive Neuroscience} -Many students have expressed an interest in learning from examples, and researchers studying students' acquisition of the ability to prove have observed a category for concepts called ``perceptual'' where students mistakenly believe or hope, that examples constitute a proof (of universality). Valiant points out two cases where examples are very effective in learning: -when on may employ elimination when the concept to be learned is a conjunction and an exemplar exhibits a variable in negative form, it is clear that that variable is not needed for set membership. -When a concept is a disjunction any variable that appears in positive form in a negative example is shown to be insufficient to guarantee membership. - -Valaint [p. 171] Humans do not argue readily from the contrapositive. -P.C. Wason 1983 Realism and rationality and the selection task. Thinking and Reasoning: Psychological Approaches Evans, ed., Routledge - - - -Valiant, in Circuits of the Mind [] that an ``important class that is not currently -known to be learnable is disjunctive normal form (or DNF for short)\ldots This -appears to be a most natural generalization of simple conjunctions from the -viewpoint of modeling human concepts. It can express the idea that examples -of a concept fall into a number of somewhat distinct categories, each corresponding -to noe of the conjunction. When discussing inductive learning we -have a \textit{hierarchical} context in mind. If we wish to learn DNF formulae, but do not -have an algorithm for learning these direction, we can nevertheless attempt to -learn these in stages. For example, to learn $x_1x_2 \land x_2x_3$ we could first learn the -simple conjunctions $x_1x_2$ and $x_2x_3$ separately in some fashion. Having learned -these we can learn the DNF when learning hierarchical in this way more is -required of the teacher or environment than in the simplest case of learning -by example. Somehow the subconcept $x_1x_2$ must be learned separately in supervised -or unsupervised mode. In the former case, for example, a teacher -may have to teach the name of this subconcept in unsupervised memorization -mode and then identify positive or negative examples of it so that it is learned -in supervised mode inductively. Alternatively, this subconcept may be learned -in unsupervised mode either by memorization or be correlational learning. `` -Valiant has written\cite{this is in a separate pdf} that the hippocampus is likely to be the location where the allocation of new memory locations is carried out. - -Valiant has observe that, given a set of concepts that can be hierarchically related, in the absence of hierarchy, when instead the concepts are flattened out, it is more unwieldy to make analogies, such as $A^B$ is analogous to $C$. -This is supportive of our interpretation of student data, in which we suggest that students who -find abstraction challenging, will in turn find remembering and using definitions more challenging, and will be at a disadvantage in terms of advancing to definition-based axiomatic reasoning. - - - - -\subsection{ Validation at the Level of Neurophysiology} -(Says who?) suggests that memory for events that are observed through one sensory modality are stored near the nervous tissue that process the input for that modality. -This is supportive of our interpretation of student data, in which we suggest that students who - learn proof by mathematic induction, represented symbolically, and recursive algorithms, represented pictorially, do not always ``see'' the analogy right away, because the memory traces are not activated at the same time. -When considering the two topics at the same time in discussion, both ideas are recalled, corresponding to metabolic activity in the memory (possibly tow different, separate, regions) facilitating formation of connection between the two ideas. -From Chapter 18 Migration in the Hippocampus, of Cellular Migration and Formation of Neuronal Connections: Comprehensive Developmental Neuroscience Vol 2, 2013, Elsevier ``the proliferative subgranular zone located at the border between the granular cell layer and the hilus, which serves as the major site for persistent neurogenesis in the adult hippocampus'' ``one of the more unique aspects of hippocampas development is the formation of the dentate gyrus, which involves formation of a specialized neural stem cell niche.'' ``hippocampus harbors neural circuitry essential for learning and memory {Lisman 1999} relating hippocampal circuitry to function neuron'' ``Cajal-Retzius cells \ldots somewhat penetrate the boundary but tend to avoid invading into olfactory cortex of olfactory bulb Bielle et al. 2005'', maybe -``signalling pathways \ldots regulate the redial glia-guided migration in the neocortex'' p.335 (not only during development) -0.339 nice graphic for germinative layers (was that supposed to be p.339?) -persistent neurogenesis in adult hippocampus Altman and Bayer 1990 Migration and Distribution percursors Comp Neurol subgranular zone Li et al. 2009 neurogenic zone Li and Pleasure 2005 Morphogensis of Dentat gyrus - - -% % % % % % % % % % % % % % % % % % % % % % -% % % % % % % % % % % % % % % % % % -\section{Validity and Reliability} -We checked for internal consistency and reinforcement, and for external compatibility -of our findings with existing educational literature in computer science -and in mathematics. We noted the phenomenological work of Gian-Carlo -Rota \cite{rota1997phenomenology} who has reported that memory for mathematical proof and its elements -is noticeably improved when a proof is deemed to be beautiful. We were encouraged -by the overlap in description among interview participants. In the -literature of mathematics education, we found researchers [?] reporting quite -similar conceptions of proof by mathematical induction in students of mathematics. -In the literature of computer science education we found research \cite{booth1997phenomenography} -on a different topic, but with similar results. Booth reported categories of -conceptions of recursion similar to our categories of conception of proof by -mathematical induction. -\section{Researcher Bias and Assumptions} -Researcher Bias and Assumptions -strategies for trustworthy, valid, reliable -what about generalizability? %(e.g., to people with ASD) -\subsection{Proofs Using the Pumping Lemma for Regular Languages} -The author believes diagrams aid the abstraction process. The researchers -tend to believe that students want to learn, and will try to comprehend and to -become able to apply the material, and that the limitations temporarily present -in the student can be overcome by explanation and practice. +\chapter{Interpretation/Discussion} + +\section{Interpretation} +Here I want to put the ideas about definitions and abstraction. Without abstraction +definitions are more cumbersome to remember and operate with. This +discourages use of the axiomatic proof conceptions, because they are based on +definitions. +What about Valiant? His establishing of definitions in circuits in the mind by +conjunctions, and by disjunctions, of ideas. Without abstraction for definitions, +this is more cumbersome. + +Is intuition helping or opposing our educational objectives? Can we get help from it? + +\subsection{Productive and Counterproductive Beliefs} +What do they ``know'', and what do they ``know that isn't so''. +Some will be conscious, some will be unconscious. +\subsection{Productive and Counterproductive Momentum} +What are they trying to learn? Is it aligned with the departmental curriculum? the course goals? +\subsection{Published papers} +Three papers in this area were published: +\begin{itemize} + +\item CCSCNE: Categorizing the School Experience of Entering Computing +Students \cite{smith2013categorizing} +\item FIE: Mathematization in Teaching Pumping Lemmas \cite{smith2013mathematization} +\item Koli Calling: Computer Science Students’ Concepts of Proof by Induction\cite{smith2014computer} + +\end{itemize} +\paragraph{Categories of Experience of Entering Students} +Undergraduate students beginning study of the computing disciplines present +various degrees of preparedness\cite{smith2013categorizing}. Some had no experience, some had had +informal experience, and some had had formal classes. The formal classes +extended from using applications to building applications. Informal experience +ranged from editing configuration files, such as background colors, to full time +jobs extended over multiple summers. +After publishing this paper, we encountered more related information. For +example, consistent with the work of Almstrum\cite{almstrum1996investigating}, we found that, about implications, +some students, who do know that any statement must and can, be +either true or false, thought implications must be true. Some interview participants +enjoyed a modified Moore method\cite{cohen1982modified} geometry class in middle school, +and relished opportunities to create proofs (not yet published). Other students +were not so well prepared. + + + + +\paragraph{Representation/Symbolization in Pumping Lemmas} +We found that some students may lack facility in notation. For example, in the +application of the pumping lemma, students are expected to understand the +role of $i$, in the context that a string $s$, having component substrings $x$, $y$ and $z$, +can be used to generate other strings, of the form $xy^iz$, where $i$ gives the number +of copies of the substring $y$. Moreover, students are expected to understand that +the subdivision of a string of length $p$, $\sigma_1^a\sigma_2^{p-a}$, where $a \in \{0,1,\ldots,p\}$ +uses $a$ as a parameter, a free variable, not one necessarily bound to a single instance +of a natural number, but a representation of a domain. An excerpt of the errors found on tests is shown in Table . Trigueros et +al. \cite[ p. 3]{jacobs2008developing} have observed that ``students are often unclear about the different +ways letters are used in mathematics''. We have seen this +lack of understanding in a situation in which it was proposed as evidence that +a single example, namely $\sigma_1^a\sigma_2^{p-a}$, formed a proof for a universally quantified +statement. +Some of our results were consistent with the framework described by Harel +and Sowder in 1998\cite{harel1998students}. We found students holding conceptualizations that +Harel and Sowder's 1998 model calls symbolization. Harel and Sowder have +identified another category of conceptualization, that correctly applied transformation +and axiomatic arguments. Some students expressed enthusiasm for +the power that inheres to building arguments with carefully specified component +ideas, in particular how the absence of ambiguity permitted arguments to +extend to great length while remaining valid. Not all of the students had developed +axiomatic conceptualizations of proof. About definitions, we collected +preliminary data on students' conceptualizations of definitions used in proofs. +Some students thought definitions were boring. Some students thought that +they could infer definitions from a few examples. Concerning executive function, +we found that some students do not state the premises clearly, and some +students did not keep track of their goal. About rules of inference, we found +Figure 5.3.1: Some categories / conceptualizations found among students of +introduction to the theory of computing, and published at FIE. +that some students apply invalid approaches to inference. + +We have found students holding conceptualizations +that Harel and Sowder's 1998 model\cite{harel1998students} calls symbolization: We have found +that some students may lack facility in notation. For example, in the application +of the pumping lemma, students are expected to understand the role of $i$, +in the context that a string $s$, having component substrings $x$, $y$ and $z$, can be +used to generate other strings, of the form $xy^iz$, where $i$ gives the number of +copies of the substring $y$. Moreover, students are expected to understand that +the subdivision of a string of length $p$, expressed as $\sigma_1^a\sigma_2^{p-a}$, where $a \in \{0,1,\ldots,p\}$ +uses $a$ as a parameter, a free variable, not one necessarily bound to a single instance +of a natural number, but a representation of a domain. +Trigueros et al.\cite[p. 3]{jacobs2008developing} have observed that ``students are often unclear about the different way letters are used in mathematics.'' + We saw this +lack of understanding in a situation in which it was proposed as evidence that +a single example, namely $\sigma_1^a\sigma_2^{p-a}$, formed a proof for a universally quantified +statement. An excerpt of the errors found on tests is shown in Table . + +Table : Some example errors\\ +Let x be empty\\ +$|xy| \leq p, so xy = 0^p$\\ +$|xy| \leq p; let x = 0^{p+r}, y = 0^{p+r}, 0 < r < p$\\ +Let’s choose $|xy| = p$\\ +$0^{p+1}0^b1^p \neq 0^{p+1}1^p \therefore xy^2z \not\in \mathcal{L}$\\ +where $\mathcal{L} = \{0^i1^j, i \neq j\}$\\ +we choose $s = 0^{p+1}1^p$ within $|xy|$\\ +thus $\neq 0^p1^{p+1}$\\ +Let $x = 0^a, y = 0^b1^a$\\ +$x = 0^{p-h}, y = 0^h$\\ +$x = 0^i, y = 0^i, z = 0^i1^j$\\ +Figure 5.3.3: Some categories / conceptualizations found among students of +introduction to the theory of computing, and published at FIE. + +Some of our results were consistent with the framework described by Harel +and Sowder in 1998[?]. We found students holding conceptualizations that +Harel and Sowder's 1998 model calls symbolization. Harel and Sowder have +identified another category of conceptualization, that correctly applied transformation +and axiomatic arguments. Some students expressed enthusiasm for +the power that inheres to building arguments with carefully specified component +ideas, in particular how the absence of ambiguity permitted arguments to +extend to great length while remaining valid. Not all of the students had developed +axiomatic conceptualizations of proof. About definitions, we collected +preliminary data on students' conceptualizations of definitions used in proofs. +Some students thought definitions were boring. Some students thought that +they could infer definitions from a few examples. Concerning executive function, we found that some students do not state the premises clearly, and some +students did not keep track of their goal. About rules of inference, we found +that some students apply invalid approaches to inference. +\paragraph{Abstract Model for Proof by Mathematical Induction and Recursion} +Far from finding agreement that (a) theorems are true as a consequence of +the definitions and the premise, and that (b) proofs serve to show how the +consequence is demonstrated from the premise, axioms and application of +rules of inference, instead we found a variety of notions about proof, including +the well-known procedural interpretation \cite{tall2008transition,weber2004traditional,tall2001symbols}, and the well-known empirical +misconception \cite{harel1998students}. The conceptualization that definitions are not necessarily +of interest compared with the procedures seemed different in kind from the +concept image / concept definition discoveries of R\"osken et al. \cite{rosken2007integrating}. +Interviews with students revealed that some students saw generation of a proof +by mathematic induction as a procedure to be followed, in which they should +produce a base case, and prove it, and should produce an induction step, and +prove that. This was consistent with Weber [?, p. 4-426] who has stated ``in +the studies that I conducted, it was more often the case that undergraduates +applied procedures that were not meaningful to them.'' He went on to give a +quotation from a participant [?, p. 4-426] ``And I prove something and I look at +it, and I thought, well, you know, it's been proved, but I still don't know that I +even agree with it [laughs]. I'm not convinced by my own proof!'' Some of the +students interviewed did not know why this procedure generated a convincing +argument. Polya[?] has written a problem involving all girls being blue-eyed; +a similar problem appears in Sipser \cite{sipser2012introduction} about all horses being the same color. +The purpose of this exercise is to make students aware that the truth of the +inductive step must apply when the base case appears as the premise. In some +cases, this point was not clear to the students. +Students' conceptualizations of proof by mathematical induction can support +their choice to apply recursive algorithms. One student reported success at +both mathematical induction and recursive algorithm application without ever +noticing any connection. This student opined that having learned recursion +with figures, and proof by mathematical induction without figures, that no +occasion for the information to spontaneously connect occurred. Students reporting +ability to implement assigned problems recursively, but not the ability +to understand proof by mathematical induction also reported that ability to +write recursive programs did not result in recognition of when recursive solutions +might be applicable in general. Students reporting ability to implement +assigned problems recursively, and also the ability to prove using mathematical +induction also reported preferring to implement recursive solutions in +problems as they arose. + + (Say something about how this is consistent with the procedural conceptualization, bifurcation in Tall's writing.) +Our work on students' choices of algorithmic approaches was consistent with +work by other researchers in computer science education\cite{booth1997phenomenography} on conceptualizations +of algorithms. Our work served to unify that of mathematician educators +with computer science educators, by providing a plausible explanation why +the conceptualizations of recursive algorithms that were found, might exist. +\begin{figure} +\centering +%\includegraphics[width=0.7\linewidth]{./} +\caption{Categories of Student Conceptualizations of Proof by Induction +that Recursion Works} +\end{figure} + +Figure 5.3.2: +\begin{table}[h] +\caption{The Outcome Space for Proofs by Induction} +\begin{tabular}{|p{.2cm}|p{6cm}|p{6cm}|} +\hline & Category & Description \\ +\hline 1 & Following procedure & The method is learned, without understanding \\ +\hline 2 & Understands base case & The idea that a base case is proved by an existence +proof, often with a specific example\\ +\hline 3 & Understands implication & The idea that an implication is proved by +assuming the premise is not used\\ +\hline 4 & Does not understand connection & Sees the implication and proves it well, but +does not anchor the succession to a base +case \\ +\hline 5 & Does understand the argument & Understands the argument \\ +\hline 6 & Knows why recursion works & Can tailor the argument to explain recursive +algorithms \\ +\hline 7 & Appreciates data structures supporting recursion & Can see the benefit to algorithm from recursive data structure \\ +\hline +\end{tabular} +\end{table} + + + +\subsection{Helping Students Discern Abstraction} +Recall that variation theory holds that students cannot discern a thing unless +contrast is provided. Pang has pointed out that [], for persons aware of only +one language, ``speaking'' and ``speaking their language'' are conflated. Only +when the existence of a second language is known, does the idea of speaking +become separated from the idea of speaking a specific language. +(Here is a specialization (Hofstadter), formation of a new conjunct (Valiant), see Besold 2015) +Abstraction is important in computer science, and is worthy of investigation. +Inquiry into students' conceptualizations of formalization using symbols, symbolization, +has shown similar results among students of mathematics and of +computer science [?, ?]. Student populations contain the conceptualization that +proofs ought to be expressed using symbols, and some proof attempts show +that not all students are able to formalize meaningfully. Mathematics and computer +science pedagogies differ on the recommended style of variable names in +symbolization. In mathematics, there is a preference for single letter variable +names, and in computer science it is recognized that longer variable names assist +readers in understanding. In mathematics the use of single variable names +is preferred because it is thought to contribute to cultivating students' ability +to learn abstraction. If, in computer science education, we apply variation +Table 5.3.5: The Outcome Space for Proofs by Induction\\ +Category Description\\ + 1 Following procedure The method is learned, without understanding\\ + 2 Understands base case The idea that a base case is proved by an existence +proof, often with a specific example\\ +3 Understands implication The idea that an implication is proved by +assuming the premise is not used\\ +4 Does not understand connection +Sees the implication and proves it well, but +does not anchor the succession to a base +case\\ +5 Does understand the argument +Understands the argument\\ +6 Knows why recursion +works\\ +Can tailor the argument to explain recursive +algorithms\\ +7 Appreciates data structures +supporting recursion\\ +Can see the benefit to algorithm from recursive +data structure\\ +theory, we gain confidence in the idea that students may discern the process +of abstraction as we vary the names of the variables. We could imagine deriving +code from a requirement about a specific class, and using corresponding +variable names, and we could show the process of promoting the code into a +more general class in the inheritance hierarchy, changing the variable names to +correspond to the more general domain of objects. Thus we can borrow from +the approach used by mathematics education, but make it more explicit, taking +advantage of computer science's explicit treatment of inheritance hierarchies in +object oriented code. Seeking evidence of students' conception of abstraction, +we could examine overridden methods to see whether variable names in more +and less general implementations bear that relation to one another. + +\subsection {Algebra} +In middle or high school algebra students became familiar with the use of letters in equations, +and solving equations which resulted in individual values, or no value, being attached to the letters. + +Ideally the ability to understand expressions, to formulate pre- and post conditions would be acquired. +As we have seen that this occurs sometimes, but does not always occur, +there may be benefit to some students to review this idea. +We might choose to emphasize abstraction in this process. + +\subsection{Geometry} +In high school geometry, formal proofs of geometric properties are covered. +Students are exposed to a form for argument, and are given examples of use of rules of inference to perform logical deduction. +We have seen that sometimes this process is appreciated in enough generality to be recognized +as an example of argumentation. +We have seen as well, that some students found this process entirely specific to geometry, +doubting that it had broader application. + +\subsection{Seeing a Broader Context} + +It may be that some students do not see a separation between the activity of formalization on the one hand, +and the application area of finding solutions to equations on the other. +It may be that some students do not see a separation between the activity of deducing using logic on the one hand, and the application area of learning geometric facts on the other. +It seems consistent with neglecting opportunities for abstraction, that these separations are not always seen, +Speciation, an idea which uses abstraction, providing a hierarchy of properties animals and plants might have, was not recognized early or universally. So, it is +not surprising that abstraction, which involves choice about which details to defer, and which to regard as significant, is not always obvious. +In the machine learning perspective, features can be learned. +What do I want to say, it takes some effort to recognize features? +There might be a way to formulate choice of features such that some better efficiency is gained by thinking of the features in that order vs. another order. +(Such as, we never have to think about some features for some parts of the tree.) +If we think about knowledge being organized in neural networks, such that abstraction has a physical manifestation, we can see that ideas between which there is little distance (by some measure, neurons, glia?) in the tree will more frequently elicit one another by linkages at the metabolic level. +At the neural level, modifications for efficiency are constantly taking place (Do we have this from Kandel and Squire?). +We might wish to exploit this in the way we teach, to exhibit the abstraction deliberately, to minimize the amount of neural connection remodeling that would occur in the process of providing an efficient neural connection remodeling that would occur in the process of providing an efficient neural representation. +Being able to learn by analogy testifies to the utility of having a neural representation that corresponds to abstraction. +Students who are working without hierarchical organization of concepts are at a disadvantage. +\subsection{Mathematics tests in high school that involve proving} +What can we learn from students of computer science who excelled in reasoning to this level? + + +%\chapter{Discussion} + +\section{Discussion} + \subsection{Importance} +Importance goes here, rather than in analysis\\ + +Programmers/developers who produce and/or verify software that is used in safety critical applications, such as medical equipment, self-driving cars, and defense-related equipment should be able to know that their software functions correctly. + +Programmers/developers who produce and/or verify software that is expected to perform work, such as search, efficiently, should be able to know that their algorithms are efficient. + +Computer science is the expected background preparation for people working in these careers. +Proof is the method that is used to ascertain, and to convince, that these goals have been achieved. +\subsection{Interpretation of Results} +As in mathematics, some students learn as procedure that which we would prefer that they understand. +Some procedural learning is insufficiently accompanied by an understanding as to which contexts to which it applies, and has become in some cases what Whitehead calls "inert knowledge". + +\subsection{More Discussion} +\paragraph{Helping Students Discern Abstraction} +Recall that variation theory holds that students cannot discern a thing unless +contrast is provided. Pang has pointed out that [], for persons aware of only +one language, ``speaking'' and ``speaking their language'' are conflated. Only +when the existence of a second language is known, does the idea of speaking +become separated from the idea of speaking a specific language. +(Here is a specialization (Hofstadter), formation of a new conjunct (Valiant), see Besold 2015) +Abstraction is important in computer science, and is worthy of investigation. +Inquiry into students' conceptualizations of formalization using symbols, symbolization, +has shown similar results among students of mathematics and of +computer science [?, ?]. Student populations contain the conceptualization that +proofs ought to be expressed using symbols, and some proof attempts show +that not all students are able to formalize meaningfully. Mathematics and computer +science pedagogies differ on the recommended style of variable names in +symbolization. In mathematics, there is a preference for single letter variable +names, and in computer science it is recognized that longer variable names assist +readers in understanding. In mathematics the use of single variable names +is preferred because it is thought to contribute to cultivating students' ability +to learn abstraction. If, in computer science education, we apply variation +Table 5.3.5: The Outcome Space for Proofs by Induction\\ +Category Description\\ + 1 Following procedure The method is learned, without understanding\\ + 2 Understands base case The idea that a base case is proved by an existence +proof, often with a specific example\\ +3 Understands implication The idea that an implication is proved by +assuming the premise is not used\\ +4 Does not understand connection +Sees the implication and proves it well, but +does not anchor the succession to a base +case\\ +5 Does understand the argument +Understands the argument\\ +6 Knows why recursion +works\\ +Can tailor the argument to explain recursive +algorithms\\ +7 Appreciates data structures +supporting recursion\\ +Can see the benefit to algorithm from recursive +data structure\\ +theory, we gain confidence in the idea that students may discern the process +of abstraction as we vary the names of the variables. We could imagine deriving +code from a requirement about a specific class, and using corresponding +variable names, and we could show the process of promoting the code into a +more general class in the inheritance hierarchy, changing the variable names to +correspond to the more general domain of objects. Thus we can borrow from +the approach used by mathematics education, but make it more explicit, taking +advantage of computer science's explicit treatment of inheritance hierarchies in +object oriented code. Seeking evidence of students' conception of abstraction, +we could examine overridden methods to see whether variable names in more +and less general implementations bear that relation to one another. +\paragraph{Helping Students Discern Abstraction} +\paragraph {Algebra} +In middle or high school algebra students became familiar with the use of letters in equations, +and solving equations which resulted in individual values, or no value, being attached to the letters. + +Ideally the ability to understand expressions, to formulate pre- and post conditions would be acquired. +As we have seen that this occurs sometimes, but does not always occur, +there may be benefit to some students to review this idea. +We might choose to emphasize abstraction in this process. + +\paragraph{Geometry} +In high school geometry, formal proofs of geometric properties are covered. +Students are exposed to a form for argument, and are given examples of use of rules of inference to perform logical deduction. +We have seen that sometimes this process is appreciated in enough generality to be recognized +as an example of argumentation. +We have seen as well, that some students found this process entirely specific to geometry, +doubting that it had broader application. + +\paragraph{Seeing a Broader Context} + +It may be that some students do not see a separation between the activity of formalization on the one hand, +and the application area of finding solutions to equations on the other. +It may be that some students do not see a separation between the activity of deducing using logic on the one hand, and the application area of learning geometric facts on the other. +It seems consistent with neglecting opportunities for abstraction, that these separations are not always seen, +Speciation, an idea which uses abstraction, providing a hierarchy of properties animals and plants might have, was not recognized early or universally. So, it is +not surprising that abstraction, which involves choice about which details to defer, and which to regard as significant, is not always obvious. +In the machine learning perspective, features can be learned. +What do I want to say, it takes some effort to recognize features? +There might be a way to formulate choice of features such that some better efficiency is gained by thinking of the features in that order vs. another order. +(Such as, we never have to think about some features for some parts of the tree.) +If we think about knowledge being organized in neural networks, such that abstraction has a physical manifestation, we can see that ideas between which there is little distance (by some measure, neurons, glia?) in the tree will more frequently elicit one another by linkages at the metabolic level. +At the neural level, modifications for efficiency are constantly taking place (Do we have this from Kandel and Squire?). +We might wish to exploit this in the way we teach, to exhibit the abstraction deliberately, to minimize the amount of neural connection remodeling that would occur in the process of providing an efficient neural connection remodeling that would occur in the process of providing an efficient neural representation. +Being able to learn by analogy testifies to the utility of having a neural representation that corresponds to abstraction. +Students who are working without hierarchical organization of concepts are at a disadvantage. +\paragraph{Mathematics tests in high school that involve proving} +What can we learn from students of computer science who excelled in reasoning to this level? + + +\section{Previously Published Work} + +Three papers in this area have been published to date: +\begin{itemize} + +\item CCSCNE: Categorizing the School Experience of Entering Computing +Students +\item FIE: Mathematization in Teaching Pumping Lemmas +\item Koli Calling: Computer Science Students' Concepts of Proof by Induction + +\end{itemize} +\subsection{Categories of Experience of Entering Students} +Undergraduate students beginning study of the computing disciplines present +a various degrees of preparedness.\cite{reilly2014examination} Some interview participants enjoyed +a modified Moore method\cite{cohen1982modified} geometry class in middle school, and relished +opportunities to create proofs (not yet published). Other students are not so +well prepared. +After publishing this paper, more information relating to its topic has been +encountered. For example, consistent with the work of Almstrum \cite{almstrum1996investigating}, we have +found that, about implications, some students, who do know that any statement +must and can, be either true or false, think implications must be true. +\subsection{Categories of Experience of Entering Students} +Undergraduate students beginning study of the computing disciplines present +various degrees of preparedness\cite{smith2013categorizing}. Some had no experience, some had had +informal experience, and some had had formal classes. The formal classes +extended from using applications to building applications. Informal experience +ranged from editing configuration files, such as background colors, to full time +jobs extended over multiple summers. +After publishing this paper, we encountered more related information. For +example, consistent with the work of Almstrum\cite{almstrum1996investigating}, we found that, about implications, +some students, who do know that any statement must and can, be +either true or false, thought implications must be true. Some interview participants +enjoyed a modified Moore method\cite{cohen1982modified} geometry class in middle school, +and relished opportunities to create proofs (not yet published). Other students +were not so well prepared. +\subsection{Representation/Symbolization in Pumping Lemmas} +Some of our results to date are consistent with the framework described by +Harel and Sowder in 1998\cite{harel1998students}. We have found students holding conceptualizations +that Harel and Sowder's 1998 model\cite{harel1998students} calls symbolization: We have found +that some students may lack facility in notation. For example, in the application +of the pumping lemma, students are expected to understand the role of $i$, +in the context that a string $s$, having component substrings $x$, $y$ and $z$, can be +used to generate other strings, of the form $xy^iz$, where $i$ gives the number of +copies of the substring $y$. Moreover, students are expected to understand that +the subdivision of a string of length $p$, expressed as $\sigma_1^a\sigma_2^{p-a}$, where $a \in \{0,1,\ldots,p\}$ +uses $a$ as a parameter, a free variable, not one necessarily bound to a single instance +of a natural number, but a representation of a domain. We have seen this +lack of understanding in a situation in which it was proposed as evidence that +a single example, namely $\sigma_1^a\sigma_2^{p-a}$, formed a proof for a universally quantified +statement. An excerpt of the errors found on tests is shown in Table . + +Table : Some example errors\\ +Let x be empty\\ +$|xy| \leq p, so xy = 0^p$\\ +$|xy| \leq p; let x = 0^{p+r}, y = 0^{p+r}, 0 < r < p$\\ +Let's choose $|xy| = p$\\ +$0^{p+1}0^b1^p \neq 0^{p+1}1^p \therefore xy^2z \not\in \mathcal{L}$\\ +where $\mathcal{L} = \{0^i1^j, i \neq j\}$\\ +we choose $s = 0^{p+1}1^p$ within $|xy|$\\ +thus $\neq 0^p1^{p+1}$\\ +Let $x = 0^a, y = 0^b1^a$\\ +$x = 0^{p-h}, y = 0^h$\\ +$x = 0^i, y = 0^i, z = 0^i1^j$ + +Figure: Some categories / conceptualizations found among students of +introduction to the theory of computing, and published at FIE. + +Harel and Sowder identified a category of conceptualization that correctly +applied transformation and axiomatic arguments. Some students expressed +enthusiasm for the power that inheres to building arguments with carefully +specified component ideas, in particular how the absence of ambiguity permitted +arguments to extend to great length while remaining valid. Not all of the +students had developed axiomatic conceptualizations of proof. About definitions, +we have collected preliminary data on students' conceptualizations of +definitions used in proofs. Some students think definitions are boring. Some +students think that they can infer definitions from a few examples. Concerning +executive function, we have found that some students do not state the +premises clearly, and some students do not keep track of their goal. About +rules of inference, we have found that some students apply invalid approaches +to inference. +\subsection{Representation/Symbolization in Pumping Lemmas} +We found that some students may lack facility in notation. For example, in the +application of the pumping lemma, students are expected to understand the +role of $i$, in the context that a string $s$, having component substrings $x$, $y$ and $z$, +can be used to generate other strings, of the form $xy^iz$, where $i$ gives the number +of copies of the substring $y$. Moreover, students are expected to understand that +the subdivision of a string of length $p$, $\sigma_1^a\sigma_2^{p-a}$, where $a \in \{0,1,\ldots,p\}$ +uses $a$ as a parameter, a free variable, not one necessarily bound to a single instance +of a natural number, but a representation of a domain. An excerpt of the errors found on tests is shown in Table . Trigueros et +al. \cite[ p. 3]{jacobs2008developing} have observed that ``students are often unclear about the different +ways letters are used in mathematics''. We have seen this +lack of understanding in a situation in which it was proposed as evidence that +a single example, namely $\sigma_1^a\sigma_2^{p-a}$, formed a proof for a universally quantified +statement. +Some of our results were consistent with the framework described by Harel +and Sowder in 1998\cite{harel1998students}. We found students holding conceptualizations that +Harel and Sowder's 1998 model calls symbolization. Harel and Sowder have +identified another category of conceptualization, that correctly applied transformation +and axiomatic arguments. Some students expressed enthusiasm for +the power that inheres to building arguments with carefully specified component +ideas, in particular how the absence of ambiguity permitted arguments to +extend to great length while remaining valid. Not all of the students had developed +axiomatic conceptualizations of proof. About definitions, we collected +preliminary data on students' conceptualizations of definitions used in proofs. +Some students thought definitions were boring. Some students thought that +they could infer definitions from a few examples. Concerning executive function, +we found that some students do not state the premises clearly, and some +students did not keep track of their goal. About rules of inference, we found +Figure 5.3.1: Some categories / conceptualizations found among students of +introduction to the theory of computing, and published at FIE. +that some students apply invalid approaches to inference. +\subsection{Abstract Model for Proof by Mathematical Induction and Recursion} +Far from finding agreement that (a) theorems are true as a consequence of +the definitions and the premise, and that (b) proofs serve to show how the +consequence is demonstrated from the premise, axioms and application of +rules of inference, instead we found a variety of notions about proof, including +the well-known procedural interpretation \cite{tall2008transition,weber2004traditional,tall2001symbols}, and the well-known empirical +misconception \cite{harel1998students}. The conceptualization that definitions are not necessarily +of interest compared with the procedures seemed different in kind from the +concept image / concept definition discoveries of R\"osken et al. \cite{rosken2007integrating}. +Interviews with students revealed that some students saw generation of a proof +by mathematic induction as a procedure to be followed, in which they should +produce a base case, and prove it, and should produce an induction step, and +prove that. This was consistent with Weber [?, p. 4-426] who has stated ``in +the studies that I conducted, it was more often the case that undergraduates +applied procedures that were not meaningful to them.'' He went on to give a +quotation from a participant [?, p. 4-426] ``And I prove something and I look at +it, and I thought, well, you know, it's been proved, but I still don't know that I +even agree with it [laughs]. I'm not convinced by my own proof!'' Some of the +students interviewed did not know why this procedure generated a convincing +argument. Polya[?] has written a problem involving all girls being blue-eyed; +a similar problem appears in Sipse \cite{sipser2012introduction} about all horses being the same color. +The purpose of this exercise is to make students aware that the truth of the +inductive step must apply when the base case appears as the premise. In some +cases, this point was not clear to the students. +Students' conceptualizations of proof by mathematical induction can support +their choice to apply recursive algorithms. One student reported success at +both mathematical induction and recursive algorithm application without ever +noticing any connection. This student opined that having learned recursion +with figures, and proof by mathematical induction without figures, that no +occasion for the information to spontaneously connect occurred. Students reporting +ability to implement assigned problems recursively, but not the ability +to understand proof by mathematical induction also reported that ability to +write recursive programs did not result in recognition of when recursive solutions +might be applicable in general. Students reporting ability to implement +assigned problems recursively, and also the ability to prove using mathematical +induction also reported preferring to implement recursive solutions in +problems as they arose. +Our work on students' choices of algorithmic approaches was consistent with +work by other researchers in computer science education\cite{booth1997phenomenography} on conceptualizations +of algorithms. Our work served to unify that of mathematician educators +with computer science educators, by providing a plausible explanation why +the conceptualizations of recursive algorithms that were found, might exist. \subsection{Proofs by Induction} -\subsection{Domain, Range, Mapping, Relation, Function, Equivalence Relation -in Proofs} -\subsection{Definitions, Language, Reasoning in Proofs} -\subsection{Equivalence Class, Generic Particular, Abstraction in Proofs} +Table 5.3.3: The Outcome Space for Proofs by Induction +Category Description +1Following procedure The method is learned, without understanding +2Understands base case The idea that a base case is proved by an existence +proof, often with a specific example +3Understands implication The idea that an implication is proved by +assuming the premise is not used +4Does not understand connection +Sees the implication and proves it well, but +does not anchor the succession to a base +case +5Does understand the argument +Understands the argument +6Knows why recursion +works +Can tailor the argument to explain recursive +algorithms +7Appreciates data structures +supporting recursion +Can see the benefit to algorithm from recursive +data structure +\subsection{Abstract Model for Proof by Mathematical Induction and Recursion} +Interviews with students revealed that some students see generation of a proof +by mathematic induction as a procedure to be followed, in which they produce +a base case, and prove it, and produce an induction step, and prove that. Some +of the students interviewed did not know why this procedure generated a +convincing argument. Moore, as reported in Polya[] noted that some students +of mathematics formed the same conceptualization, that there is a procedure, +but it does not necessarily produce a convincing argument. Polya[] wrote +a problem involving all girls being blue-eyed; a similar problem appears in +Sipser\cite{sipser2012introduction} about all horses being the same color. The purpose of this exercise is +to make students aware that the truth of the inductive step must apply when +the base case appears as the premise. In some cases, this point was not clear to +the students. +Students' conceptualizations of proof by mathematical induction can support +their choosing to apply recursive algorithms. One student reported success at +both mathematical induction and recursive algorithm application without ever +noticing any connection. This student opined that having learned recursion +with figures, and proof by mathematical induction without figures, that no +occasion for the information to spontaneously connect occurred. Students reporting +ability to implement assigned problems recursively, but not the ability +to understand proof by mathematical induction also reported that ability to +write recursive programs did not result in recognition of when recursive solutions +might be applicable in general. Students reporting ability to implement +assigned problems recursively, and also the ability to prove using mathematical +induction also reported preferring to implement recursive solutions in +problems as they arose. +Our work on students' choices of algorithmic approaches is consistent with +work by other researchers in computer science education\cite{} on conceptualizations +of algorithms. Our work served to unify that of mathematician educators +with that of computer science educators, by providing a plausible explanation why +the conceptualizations of recursive algorithms that were found, might exist. + +Figure 4.0.2: Conceptualizations of proof by induction and recursion, published +in Koli Calling + +Index Element of Model +\begin{enumerate} + +\item Some students begin learning proof by mathematical induction as if it were +a procedure. +\item Some students learn two parts of this proof technique without seeing any +connection between the two. +\item Some students do not find the procedure to be a convincing argument. +\item Some students would not employ proof by mathematical induction to explore +whether a recursive algorithm would apply to a given problem. +\item Some students understand both proof by mathematical induction and also +recursion and had never noticed any similarity. + +\end{enumerate} + +\subsection{Pumping Lemmas} +TABLE III. CATEGORIES\\ +understand inequality\\ +formulate correctly\\ +distinguish between particular and generic particular\\ +correctly apply universal quantifier\\ +recognize string as member of language set + + +\section{Results of Combined Investigations} +There are some categories that are shared among the several contexts. +\section{Categories} +Categories found in one or more investigations + +Categories\\ +Definition of proof as convincing (to mathematicians) argument is not +always understood\\ +Definitions in general are not always recognized as significant building +blocks in arguments\\ +The idea of a false statement sometimes becomes troublesome when +negation is being learned.\\ +In particular, accepting that an implication may be false, can be troublesome. +Notation is sometimes difficult.\\ +Ideas presented relying on notation are not always connected with +ideas presented relying on figures.\\ +Warrants are not always recognized.\\ +Students do not always traverse levels of abstraction effectively.\\ +The applicability of valid argument forms to contexts of interest is not +always appreciated. +\section{Critical Factors} +To determine critical factors, we can convert negative categories into achievement +levels. +\begin{tabular}{p{3cm}p{3cm}} +Categories & Achievement Levels\\ +The idea of a false statement +sometimes becomes troublesome +when negation is being +learned.&\\ +&True and false make sense, and +we can make arguments using +them.\\ +Definition of proof as convincing +argument is not always understood&\\ +Warrants are not always recognized.&\\ +&Proof can sometimes be obtained +through a series of warranted assertions.\\ +Definitions in general are not always +recognized as significant +building blocks in arguments&\\ +&Using agreed definitions and +valid rules of inference we can +sometimes explore the consequences +of definitions.\\ +Notation is sometimes difficult.&\\ +&Notation helps.\\ +Ideas presented relying on notation +are not always connected +with ideas presented relying on +figures.&\\ +&We might wish to help students +traverse multiple rendering of +ideas.\\ +Students do not always traverse +levels of abstraction effectively.&\\ +&We might wish to help students +traverse multiple levels of abstraction.\\ +The applicability of valid argument +forms to contexts of interest +is not always appreciated.&\\ +&We might wish to give exercise +with authentic (career related) +examples\\ + +\end{tabular} + +Using the achievement levels we can infer critical factors. +\begin{tabular}{p{3cm}p{3cm}} +Achievement Levels& Critical Factors\\ +True and false make sense, and +we can make arguments using +them.&\\ +& True and false apply to assertions.\\ +Proof can sometimes be obtained +through a series of warranted assertions. +& Proof is exploration and discovery.\\ +Using agreed definitions and +valid rules of inference we can +sometimes explore the consequences +of definitions.&\\ +& Efficiency but also abstraction +are aided by notation.\\ +Notation helps.&\\ +& Notation is one representation +and there are others. Ideas appear +in multiple guises.\\ +We might wish to help students +traverse multiple rendering of +ideas.&\\ +& When notation allows for multiple +interpretations, abstraction +above those multiple interpretations +has been achieved.\\ +We might wish to help students +traverse multiple levels of abstraction.&\\ +&Multiple levels of abstraction are +relevant at the same time.\\ +We might wish to give exercise +with authentic (career related) +examples.&\\ +& Authentic applications show the +use of this knowledge.\\ +\end{tabular} + + +\subsection{Abstraction} +Literature reports \cite{} students of CS have trouble with abstraction. +Taking abstraction to be the ability to select some details to ignore, +and thereby find a simpler model of an entity, we can transform the ideal +knowledge transfer experience into on disabled by a lack of ability to see this dimension. +The multiple-inheritance hierarchy that could be used to organize +definitions and relationships of ideas is less able. More entities will be +grouped together than effective use of the multiple inheritance hierarchy +would consider equivalent. +Another useful concept that students have been seen to underappreciate is the significance of careful definitions. +Abstraction hierarchies allow for efficiency in definitions. +A new entity can be defined as a specialization of an existing entity, and its differences +make up the new definition material. +In the absence of this multiple inheritance hierarchy, every definition in its full length +is attached to its entity. +Tie in with Mazur. For students holding the same granularity of refinement of +concepts, conversations would be easier, because there would be fewer disconnects as one participant expressed a thought on one degree of refinement far from that of another student. +If the ideas implying the refinement of the definition inheritance graph, being different from one +discussant to the next, are rare, and/or the meaning of the sentence does not depend upon it, these exchanges are not too disruptive or distressing. +On the other hand when two sets of refinement are very different, +and the meaning of the exchange depends upon the refinement in the speaker, that the hearer does not have, +then some degree of failure of communication will ensue. +Absence of abstraction converts tree of topics into sequence of topics. +Tree of proof examples (say of application of proof technique) into sequence of examples. +Might detract from recognizing what is a related example. +Would detract from plausible inference technique of ``related problem seen before''. +We have a goal for student programming that they should strive for segments of programs +(e.g., method implementations) to be small. One way of accomplishing this is to use abstraction, +such as combining instructions into a method, and calling the method. +If students have difficulty with abstraction, they might +have difficulty with choosing groups of instructions to represent a method. +Correspondingly, if they practice grouping instruction into methods, and using those methods, they would +be gaining practice relevant to using abstraction. +One way to cultivate abstraction is to pose a question of which one of several examples is different. +When several things are examples of one abstract idea and one is not, identifying the one that is different involves noticing the abstraction. +These questions could be instantiated using blocks of code. + +\subsection{Definitions} +Without abstraction the burdensomeness of definition is increased. This could contribute to the reluctance of students to embrace definitions. +\subsection{Symbolization} +Use of symbols is a kind of abstraction. +Symbolization is the syntax for simple, clear definitions as Gries\cite[p.205]{gries2012science}(Science of Programming) recommends for construction of programs. +Students will be hindered at this program derivation/development style if symbolization is a not-yet-acquired skill. +Program development/derivation should begin with a formulation of the requirements. +Students may arrive with some programming experience that is of a more intuitive, less +mathematically disciplined sort. +We have to ask how we desire to cultivate the abilities of such students. +Vygotsky discussed language acquisition by children, in which some children will have begun to invent +some terms for items in their environment, and will need to be guided to abandon neologisms for the naming generally agreed in their environment. +Kuhn discussed the reluctance of scientists who have been rewarded for operating in one +perspective on nature to adopt a different perspective. +Instructor may encounter a similar reluctance on the part of students +to adapt a scientifically/mathematically disciplined approach to programming, +especially if the students have experienced some success in their earlier work. +To win over such students, +demonstration of superior outcomes on problems, especially on problems that seem insoluble otherwise, +are more frequently convincing. +Happily, Professor Gries has provided such examples. +By showing superior relative efficacy of these approaches in an activity the students recognize as +desirable, instructors could motivate the students to learn symbolization. +\subsection{Structure} +sequence vs. sequence that has come about from combining parts. Refer to Leslie Lamport's structure for proofs. Combine with Gries' proofs for deriving code. The purpose for getting through Goguen and Malcolm is that it applies to imperative programs. + + + + + +The results of a phenomenographic study comprise a set of categories of description of ways of experiencing (or capability for experiencing [p. 126]) a phenomenon, and relations among those categories. + +Marton and Booth\cite[p. 125]{marton1997learning} give criteria for the quality of a set of descriptive categories. +The collective experience, over all participants in the study should be included. +The individual categories should each stand in clear relation to the phenomenon of the investigation so that each category tells us something distinct about a particular way of experiencing the phenomenon. +The categories have to stand in a logical relationship with one another, a relationship that is frequently hierarchical. +Finally the systems should be parsimonious, which is to say that as few categories should be explicated as is feasible and reasonable, for capturing the critical variation in the data. + +late stages of analysis able to see aspects/facets of research object, +see how they fit together like jigsaw pieces, +see it against background, and communicate it to others. + +There are results for each of the research questions, and some combined results. + +While mainly we are discussing proof in general, it can help to think about one proof at a time. Applying the analytical framework of Marton and Booth\cite[p.43]{marton1997learning} +\begin{table}[placement] +\caption{Outcome space for what is proof, with temporal facet} +\begin{tabular}{|p{4cm}|p{4cm}|p{4cm}|}\hline +Acquiring & Knowing & Making Use of\\\hline\hline +see the steps & remembering the steps & write it out\\\hline +understand the steps & remembering the meaning & produce the meaning\\\hline +understand steps and warrants & understanding the meaning & be able to apply the proof to simple examples\\\hline +analyzed the structure, determine the warrants & understand the relevance & be able to apply the proof in general, know its context of applicability\\\hline +\end{tabular} + +\end{table} +Our goal might be in the lower right, and for some students who do not arrive that far, they might arrive at any of its three neighbors in the chart. + +We are looking for ways of experiencing, for example, one way is, proofs only apply to number facts, vs. proof techniques are separable from number facts and can be used on other domains. + + + + +\section{What do students think a proof is?} + +One participant opined "Some students think proof is a magical incantation." + +Some students do not recognize a proof when they are looking at one. When asked "Were we doing proofs today?", such a student said "No." When asked, "Did we not show that certain things must always be the case when certain contextual requirements were met?", the answer was "Yes, oh, I guess that's a proof." When asked, "Why didn't it appear to be a proof?", the answer was "It didn't start by saying 'Proof:', and it didn't end by saying 'QED', or that box." + +Some students say that a proof is a sequence of statements, starting with one distinguished statement, called a premise and ending with a distinguished statement, what we wanted to prove. + +Some students say "What's a lemma?". Some students cannot, when asked for credit, identify the use of structure, such as a lemma, in a proof. + +Some students know that proof ought to be a convincing argument. Some of these are not convinced by a proof by induction. When assured that they would enjoy a certain proof in introduction to the theory of computation, because it was a proof by induction, voices in a class claimed, "We never 'got' that". + +Some students, when asked what they think a proof is, will report that they think it is a list of true mathematically formulated statements, demonstrating the truth of a mathematically formulated statement. Some students report that a proof has a goal, a statement to be proved true. +Some students know that the identified goal does not have to be known to be true in advance of the first proof. + +Sometimes, though, students have the idea that the proof is an exhibit of their ability to connect known facts, including the goal as a known fact. + + + +Some students, for example some taking philosophy, understand a proof more generally as not having a requirement for a mathematical formulation. Some of these have expressed dislike of such less precisely articulated statements. + +Some students, when prompted, will acknowledge that warrants for these statements are required. Axioms and agreed facts do not require warrants. +Some students, but not all, recognize that premises do not require warrants. +Some students, but not all, recognize that suppositions, as premises, do not require warrants. +Some students, but not all, recognize that cases, as suppositions, do not require warrants. +Some students, but not all, know that progress from one statement to the next, a transformation of a statement, requires a warrant. + +Some of the optional syntactic ornamentation of a proof, such as literal text "Proof:", and "QED" or $\qed$, are used by some students as proxies for the proof. As in the research by Harel and Sowder\cite{harel1998students}, which they describe as "ritual proof", we find in our research that some students claim to recognize a proof when they see these artifacts, and claim they have not seen a proof when they do not see these artifacts. + +Some students are aware that proof, as encountered in class, ought to be a convincing argument. +These students feel that something is wrong when they are not convinced by the proof technique they have learned to execute in a procedural fashion. + +Some students know that proof is convincing others, and also ascertaining for oneself. Of these, some find that proof is convincing for some facts they regard as mathematical, yet do not think proof is applicable to programs as large as those with which they plan to be involved. +Some of these students have not yet acquired the perspective that proving theorems about the number of instruction executions, and/or memory locations needed are both numerical and also applicable to and relevant for software development. + + +\begin{figure}[h] +\centering +\includegraphics[width=0.7\linewidth]{/home/theresesmith/Documents/2015Fall/Research/Thesis/whatThemes} +\caption{Conceptualizations found for what a proof is} +\label{fig:whatThemes} +\end{figure} +\begin{table} + \caption{Critical factors for what a proof is} +\begin{tabular}{|c|c|c|} +\hline Lesser Conceptualization & Better Conceptualization & Critical Factor\\ \hline +\hline List of Known Facts & List of Warranted Facts & Warrants\\ +\hline List of Warranted Facts & Means of Discovery & Tool rather than\\ & & demonstration to teachers\\ \hline +\end{tabular} + \end{table} + +\newpage +\section{How do students approach understanding a proof?} +Those students who felt they understood some proofs approached them by checking whether they felt each line of a proof was true. +Some of these mentioned that a statement should be warranted by previous statements. + +\begin{figure}[h] +\centering +\includegraphics[width=0.7\linewidth]{/home/theresesmith/Documents/2015Fall/Research/Thesis/howThemes} +\caption{Conceptualizations of how to comprehend a proof} +\label{fig:howThemes} +\end{figure} +\begin{table} + \caption{Critical factors for what a proof is} +\begin{tabular}{|c|c|c|} +\hline Lesser Conceptualization & Better Conceptualization & Critical Factor\\ \hline +\hline List of Known Facts & List of Warranted Facts & Warrants\\ +\hline List of Warranted Facts & Means of Discovery & Tool rather than\\ & & demonstration to teachers\\ \hline +\end{tabular} + \end{table} + +\newpage +\section{What do students think a proof is for?} +Some students think that proofs are not applicable to what they do. They think they do not need to know it. Because they do not need to know it, they logically conclude that learning to produce a "proof" procedurally is enough, because, it earns full credit. + +Some students combine the learning about proof with the subject matter that is used to exercise proof techniques; they think proof is for demonstrating facts about numbers. + +Some students claim that they never produce proofs unless assigned to do so in class. + +Let us call the statement to be proved, the target, so as to more clearly articulate the variety of student thinking, by escaping the connotations of "statement to be proved". + +Perhaps not surprisingly in light of the teaching of proof, some students think the purpose of proof is to demonstrate that they can construct a sequence of statements that connects the truth of the premises to the truth of the target. Some of these students regard the truth of the target to be known beforehand. As the purpose of the proof is to exhibit their ability to produce an argument, it is not surprising that students say they never construct proofs unless they are assigned to do so. It is not surprising in this context, that students opt for a procedural approach, learning the parts, for example, of a proof by induction, learning to provide a proof for a base case, and learning to take a premise as a given and a conclusion of an implication to be proved. Some students do not understand why this procedure constructs a proof. Some express an unease -- they wish for the proof procedure to be convincing, to themselves. They are glad when they learn why the procedure does produce a convincing argument. + +Some students recognize proof being used in class, for example in algorithms class and in introduction to the theory of computing. + + + + +Some students felt that proof was for finding out that a mathematical expression was true, or false. Some students knew that some statements could be proved undecidable. + +\begin{figure}[h] +\centering +\includegraphics[width=0.7\linewidth]{/home/theresesmith/Documents/2015Fall/Research/Thesis/whyThemes} +\caption{Conceptualizations about why to study proofs} +\label{fig:whyThemes} +\end{figure} +\begin{table} + \caption{Critical factors for what a proof is} +\begin{tabular}{|c|c|c|} +\hline Lesser Conceptualization & Better Conceptualization & Critical Factor\\ \hline +\hline List of Known Facts & List of Warranted Facts & Warrants\\ +\hline List of Warranted Facts & Means of Discovery & Tool rather than\\ & & demonstration to teachers\\ \hline +\end{tabular} + \end{table} + +\newpage +\section{What do students use proof for, when not assigned?} +Some students claim they never use proofs when not assigned. + +It is not the case that any student, even when prompted, said they chose to carry out a proof without being directed to do so. +This could easily be due to a misunderstanding of the definition of proof. + +\newpage +\section{Do students exhibit any consequence of inability in proof?} +Some students said that they knew how to craft recursive procedures, and enjoyed doing so when assigned problems designated as suitable for recursive implementations. +Some students said they did not employ recursive procedures in situations without a designation that recursive procedures were appropriate. They claimed not to be able to tell when recursive procedures were applicable. + +\newpage +\section{What kind of structure do students notice in proofs?} +Some students think proofs are lists of statements without hierarchical structure. +Some students have asked what lemma means. +Some students knew that lemmas were built for use in larger proofs. +Some students were interested to hear about Dr. Lamport's structure in proofs. + +\newpage +\section{What do students think it takes to make an argument valid?} +Some students, when prompted about rules of inference, felt that when all statement transformations were warranted, an argument was valid. +Some students stated that, when the target of the proof was true, the proof was valid, converse error. + + + + + +We organize our overview of results beginning from an ideal Hilbert-axiomatic style of proof approach, and moving through approximations as they become greater departures from it. +\begin{figure}[h] +\centering +\includegraphics[width=0.7\linewidth]{/home/theresesmith/Documents/2015Fall/Research/Thesis/valid} +\caption{Conceptualizations about validity of proofs} +\label{fig:validityThemes} +\end{figure} +\begin{table} + \caption{Critical factors for what a (valid) proof is} +\begin{tabular}{|c|c|c|} +\hline Lesser Conceptualization & Better Conceptualization & Critical Factor\\ \hline +\hline No Warrants & Some Warrants & Warrants\\ +\hline Some Appropriate Warrants & Fully Warranted & thoroughness\\ \hline +\end{tabular} + \end{table} + +\paragraph{Definition based reasoning} +Some students, and some teaching assistants in their teaching, are not organizing the approach to proof around definitions. +Instead some students and some teaching assistants are focusing on an intuitive approach, involving examples. +Some students use examples to infer definitions. +Some students use single examples as proof. + +Some students are not aware that proofs are illustrated with facts of, for the purposes of the class, less significance than the proof techniques. +Some students are not aware of the relevance of proof to their intended career. +These students do not see any point to learning more than a procedural approach to the proof material, as they believe it to be of no lasting significance to them. + +\paragraph{Generalization and transformation based reasoning} +Some students, and some instructors, do not emphasize that a single presentation can be seen as a representative of a group. For example, Mathematical Association of America\cite{} publishes a proof of the Pythagorean Theorem that uses rectangles to illustrate that, when they are square, the Pythagorean Theorem is being shown to be true, though of course, the rectangles need not always be square. The proof, having been established, does not rely upon the rectangles remaining in a square condition. + + + + + + + +\section{Combined Description} + +There are a couple of ways students work with exercises in proof, that are incomplete. + +Some students reason well with concrete entities, yet are confused with abstractions. These students are not appreciating the value of careful definitions, because they do not use them as tools, or the basis for reasoning. They are more comfortable with examples, because they are operating in a concrete world. + +Some students do not connect the world of concrete objects with the abstract, symbolic representation, but are making use of symbols. Some operations transforming symbolic expressions are performed correctly, but not all. The lack of understanding of the symbols combined with a procedural approach to producing a proof artifact, leaves these students personally unconvinced, and unmotivated to make use of proof when it would be helpful to them. + +Some students do understand application of facts, axioms and rules of inference, and are at home with careful definitions and symbolic concision. Some of these students also study math. + + +\newpage +\section{Diagram of Conceptualizations} + +\begin{figure}[h] +\centering +\includegraphics[width=0.95\linewidth]{./themes} +\caption{Themes from interview data} +\label{fig:themes} +\end{figure} +\newpage +\section{Outcome Space} +The outcomes were not arranged in a single progression. Rather, there were several means, listed below, by which students were able to construct the proof artifact required by the class. The students did not always find the artifact convincing. + +\begin{enumerate} +\item Concrete to Abstract -- generalize the argument, then the entities +\item Hilbert-style axiomatic/definitional proof +\item Abstract operations -- symbols rather than entities, structure of argument +\end{enumerate} + +The concrete to abstract path enabled students to reason with specific cases whose logic made sense to them, then make the step that the logical process itself was an entity that could be reused. The idea that other concrete entities could bear the same relationships, and be subject to the same reasoning constituted a step. The idea that analogies were being made, and that generalization was possible was another step. + +The reasoning by axioms and rules of inference path was known to some students. These students mentioned their appreciation of math, and in some cases their discomfort with philosophy, in connection with symbolization and application of rules of inference. + +One path was operation at the level of symbols, using procedures. This path is distinguished from that involving definitions, because some students, using definitions, were clear about appropriate operations to transform symbolic expressions, but students also sometimes were unsure about denotation and about appropriate operations. + +\section{Critical Factors} +On the path from concrete to structured proofs, called herein "generalize the argument, then the entities", one critical factor is that an argument about one set of concrete entities can be used on another set, having analogous relationships. + +Another critical factor is that, when an argument can be reused, sets of entities that stand in analogous relationships, the relationship can be generalized. When the relationship is generalized, the entities standing in that relationship can be given symbols. + +On the path that starts with symbols, students have not generalized from concrete to abstract entities, rather, they have entered the fray at the level of abstraction of symbols. Thus, a critical factor is to understand the operations appropriate to the symbols, which imbues application of the rules of inference with significance. Another critical factor is that these symbols can represent entities of interest. + + + + + + + + + + + + +\section{More Discussion} +\paragraph{Helping Students Discern Abstraction} +Recall that variation theory holds that students cannot discern a thing unless +contrast is provided. Pang has pointed out that [], for persons aware of only +one language, ``speaking'' and ``speaking their language'' are conflated. Only +when the existence of a second language is known, does the idea of speaking +become separated from the idea of speaking a specific language. +(Here is a specialization (Hofstadter), formation of a new conjunct (Valiant), see Besold 2015) +Abstraction is important in computer science, and is worthy of investigation. +Inquiry into students' conceptualizations of formalization using symbols, symbolization, +has shown similar results among students of mathematics and of +computer science [?, ?]. Student populations contain the conceptualization that +proofs ought to be expressed using symbols, and some proof attempts show +that not all students are able to formalize meaningfully. Mathematics and computer +science pedagogies differ on the recommended style of variable names in +symbolization. In mathematics, there is a preference for single letter variable +names, and in computer science it is recognized that longer variable names assist +readers in understanding. In mathematics the use of single variable names +is preferred because it is thought to contribute to cultivating students' ability +to learn abstraction. If, in computer science education, we apply variation +Table 5.3.5: The Outcome Space for Proofs by Induction\\ +Category Description\\ + 1 Following procedure The method is learned, without understanding\\ + 2 Understands base case The idea that a base case is proved by an existence +proof, often with a specific example\\ +3 Understands implication The idea that an implication is proved by +assuming the premise is not used\\ +4 Does not understand connection +Sees the implication and proves it well, but +does not anchor the succession to a base +case\\ +5 Does understand the argument +Understands the argument\\ +6 Knows why recursion +works\\ +Can tailor the argument to explain recursive +algorithms\\ +7 Appreciates data structures +supporting recursion\\ +Can see the benefit to algorithm from recursive +data structure\\ +theory, we gain confidence in the idea that students may discern the process +of abstraction as we vary the names of the variables. We could imagine deriving +code from a requirement about a specific class, and using corresponding +variable names, and we could show the process of promoting the code into a +more general class in the inheritance hierarchy, changing the variable names to +correspond to the more general domain of objects. Thus we can borrow from +the approach used by mathematics education, but make it more explicit, taking +advantage of computer science's explicit treatment of inheritance hierarchies in +object oriented code. Seeking evidence of students' conception of abstraction, +we could examine overridden methods to see whether variable names in more +and less general implementations bear that relation to one another. +\paragraph{Helping Students Discern Abstraction} +\paragraph {Algebra} +In middle or high school algebra students became familiar with the use of letters in equations, +and solving equations which resulted in individual values, or no value, being attached to the letters. + +Ideally the ability to understand expressions, to formulate pre- and post conditions would be acquired. +As we have seen that this occurs sometimes, but does not always occur, +there may be benefit to some students to review this idea. +We might choose to emphasize abstraction in this process. + +\paragraph{Geometry} +In high school geometry, formal proofs of geometric properties are covered. +Students are exposed to a form for argument, and are given examples of use of rules of inference to perform logical deduction. +We have seen that sometimes this process is appreciated in enough generality to be recognized +as an example of argumentation. +We have seen as well, that some students found this process entirely specific to geometry, +doubting that it had broader application. + +\paragraph{Seeing a Broader Context} + +It may be that some students do not see a separation between the activity of formalization on the one hand, +and the application area of finding solutions to equations on the other. +It may be that some students do not see a separation between the activity of deducing using logic on the one hand, and the application area of learning geometric facts on the other. +It seems consistent with neglecting opportunities for abstraction, that these separations are not always seen, +Speciation, an idea which uses abstraction, providing a hierarchy of properties animals and plants might have, was not recognized early or universally. So, it is +not surprising that abstraction, which involves choice about which details to defer, and which to regard as significant, is not always obvious. +In the machine learning perspective, features can be learned. +What do I want to say, it takes some effort to recognize features? +There might be a way to formulate choice of features such that some better efficiency is gained by thinking of the features in that order vs. another order. +(Such as, we never have to think about some features for some parts of the tree.) +If we think about knowledge being organized in neural networks, such that abstraction has a physical manifestation, we can see that ideas between which there is little distance (by some measure, neurons, glia?) in the tree will more frequently elicit one another by linkages at the metabolic level. +At the neural level, modifications for efficiency are constantly taking place (Do we have this from Kandel and Squire?). +We might wish to exploit this in the way we teach, to exhibit the abstraction deliberately, to minimize the amount of neural connection remodeling that would occur in the process of providing an efficient neural connection remodeling that would occur in the process of providing an efficient neural representation. +Being able to learn by analogy testifies to the utility of having a neural representation that corresponds to abstraction. +Students who are working without hierarchical organization of concepts are at a disadvantage. +\paragraph{Mathematics tests in high school that involve proving} +What can we learn from students of computer science who excelled in reasoning to this level? + + + + -\subsection{Assessing Validity} -The results have bridged papers in computer science education, by Professor Booth\cite{booth1997phenomenography}?\cite{}, and mathematics education, by \cite{}. -The results have been consistent with results of others in mathematics education. -% % % % % % % % % % % % % % % % % % % % % % % % % % % -% % % % % % % % % % % % % % % % % % % % % % % % % % % % -\section{Validity} -By choosing a varied population we hoped to obtain transferable results. -We compared our results with existing publications. -We performed a little triangulation by using multiple views into the student population: interviews and tests. We also compared the results from this with information obtained in tutoring and larger help sessions. We consulted faculty, who had experience with teaching this material, and who had experience with students who were supposed to have learned this material in prerequisites. -% % % % % % % % % % % % % % % % % % % % % % % -\subsection{Assessing Validity} + + -The results have bridged papers in computer science education, by Professor Booth\cite{}\cite{booth1997phenomenography}?, and mathematics education, by \cite{}. -The results have been consistent with results of others in mathematics education. diff --git a/ch7.tex b/ch7.tex index d12746b..ee96a3d 100644 --- a/ch7.tex +++ b/ch7.tex @@ -1,1152 +1,228 @@ -\chapter{Related Work} -Maybe I want to put this in the order of definition, relationships, specialization, symbolization, analogy, generalization/abstraction -\section{Goal Definition} -Schoenfeld gives this expression of a goal for instructors teaching proof: He states that mathematics literature includes ``knowledge and perspectives of world-class mathematicians vs. more or less ordinary PhDs''\cite[p. 74]{schoenfeld1998reflections}. Do we have such characterization for computer scientists? If so, what are these characteristics that are relevant for computer science students, and what projection do these have into students' conceptualizations? - - \section{Methods} -Archavi et al.\cite[p. 4]{arcavi1998teaching} report using microgenetic analysis, which they describe as having roots in both cognitive science and ethnography. -``Schoenfeld, Smith and Arcavi\cite{schoenfeld1993learning} describe it as striving 'for explanations that are both locally and globally consistent, accounting for as much observed detail as possible and not contradicting any other related explanations'''. -Schoenfeld is, among other categorizations, a cognitive scientist, and uses this knowledge to inform his teaching.Archavi et al.\cite[p. 4]{arcavi1998teaching} . - -Research on teaching and learning about proof in mathematics education has -produced an extensive literature. Only a small sampling is mentioned below. -Mathematics educators, including Keith Weber[?], Harel and Sowder in -1998[?], and David Tall[?] have studied students' learning of proof in the mathematics -curriculum. Leron, in 1983, [?] has described the structural method -for proof construction, attributing it to recent ideas from computer science. -Lamport, in 1995, [?] in work on proof construction, has given one approach -that computer science students might find compatible with their background. -Velleman, in 2006, has written software and a textbook [?] about proving with -a structured approach. Weber has reported the success of several approaches -to pedagogy [?]. -Barnard [?] has commented upon students negating statements with quantifiers. -Edwards and Ward [?, p. 223] have discussed the role of definitions for undergraduate -mathematics courses, stating ``the enculturation of college mathematics -students into the field of mathematics includes their acceptance and -understanding of the role of mathematical definitions''. Bills and Tall [?] have -distinguished student understanding of definitions that is sufficient that the -student can use them in proofs. -Harel and Sowder [?] and Harel and Brown [?] have conducted qualitative -research on mathematics students' conceptualization of proofs. They have developed -three main categories, each with several subcategories. Evidence from -our studies is consistent with the presence of these categories of conceptualizations -in the population of CS(E) students. -Tall[?, ?] has also categorized mathematics students' understanding of proof. -He has studied the development of cognitive abilities used in proof, starting, -as did Piaget,[?] with abilities believed present at birth. -Yang and Lin have modeled reading comprehension.[?] -Leron[?] has written about encouraging students to attend to proof structure -by teaching with generic proofs (proofs that use a generic particular). -Mejia-Ramos et al.[?] have built a model for proof comprehension. They have -observed that students who are assessed on appreciation of structural and -other appropriate features of a proof, rather than on rote reproduction, are -more likely to develop a deeper understanding of proof. -Knipping and Reid[41] have examined proof in mathematics education. -Weber[?, ?, ?, ?, ?, ?, ?, ?, ?] has investigated students' approaches to and difficulties -with proof. When studying student proof attempts in group theory, Weber -has found that some typical students' inabilities to construct proofs arise despite -having adequate factual and procedural knowledge, the ability to apply that -knowledge in a productive manner was lacking. [?] More specifically applying -the knowledge was seen to include selecting among facts, guided by knowledge -of which were important, for those most likely to be useful. [?] Alcock -and Weber,[?] have studied students' understanding of warrants, the support -for the use of a particular inference. Weber has published a framework for describing -the processes that undergraduate students use to construct proofs. [?] -Almstrum[?] has investigated the understanding of undergraduate computer -science students of problems related to logic, compared to problems only -weakly related to logic, and has shown that some students have trouble with -the notion of truth or falsity. -Healy and Hoyles[?] have reported on algebra students' preferences for the -content of convincing arguments, and their distinction between preferences -for ascertaining vs. preferences about what was likely to be well-received on -assessments. -{\.I}mamo{\u g}lu[?, ?] has studied the conceptualizations of proof of students who -were preparing to become mathematics and science teachers, in their freshman -and senior years. -Knuth has applied qualitative research to the conceptualizations of proof by -high school mathematics teachers [?, ?]. -Because our work with proof also has explored the consequences for the student -in terms of algorithm choice, including recursive algorithms with proof -by mathematical induction, the work of Booth[?], who has used phenomenography -to develop a model of students' understanding of recursive algorithms -is related. -Zhang and Wildemuth[?] have described qualitative analysis of content. -\section{ Proofs Using the Pumping Lemma for Regular Languages} -Mattuck[36] states ``analysis replaces the equalities of calculus with inequalities: -certainty with uncertainty. This represents for students a step up in -maturity.''[page xiii] and ``these are things which I find that many of my students -don't seem to know, or don't know explicitly. They subtract inequalities -\ldots ``. -\subsection{Quantifiers} -In 2010 Pillay [44] asserted that ``there has been no research into the actual -learning difficulties experienced by students with the different topics'' in formal -languages and automata theory. Of the pumping lemmas, Pillay states ``A -majority of the students made logical errors when proving that a language -is regular and using the Pumping Lemma to show that a language is nonregular. -These could be attributed to a lack of problem-solving skills and an -understanding of the Pumping Lemma.'' Devlin[18] observes that quantifiers -can appear daunting to the uninitiated, and that statements containing multiple -quantifiers can be difficult to understand. -\subsection{ Symbols} -H\"uttel and N{\o}rmark[45] described a successful method for improving both -student activity level in the course and final grades, which combines peer -assessment with creation of notes that can be used during the exam. (``The -incentive was that their answers to text (CHECK) questions would be available for them -to use at the written exam. No other textual aids would be allowed at the -exam.''[p. 4]) The better performance on the exam is welcome; whether it is -due to having notes compared to closed book, or having performed the review -might not be certain. -According to Arnoux and Finkel[46], it is not unusual for students to acquire -mathematical knowledge without attaching meaning to it, and leaving them -unable to solve some problems. They go on to report that Paivio proved -that ``double coding (verbal and visual)'' facilitated remembering. They also -report that different parts of the brain are used to process verbal and visual -information, and therefore more of the brain is involved when both verbal and -pictorial communication is used. They prefer multi-modal representations. -Xing[47] writes about aiding students comprehension of proofs being aided by -graphs. She reports ``students feel that Pumping Lemma(PL) is so abstract to -grasp that using it to prove that a language is non-regular is a daunting task.'' -She shows a graphically laid out proof that a given language is not regular. This -graph has the advantage over a traditional proof, i.e., a sequence of statements, -that the dependencies of states on axioms or intermediate results are plainly -shown by graph edges. -Simon et al.[43] ask ``Is it possible that students plug and chug in computing, not -really understanding the concepts as we would like them to?'' and go on to say -``We posit that the need exists for computing instructors to design assessments -more directly targeting understanding, not just doing, computing. And, of -course, to adopt teaching approaches that support student development of -these skills.'' -Mazur[25] developed peer instruction to address students' propensity to practice -a plug-and-chug approach to problems. This approach has been applied -to computer science teaching, including theory of computation, by several researchers -including Simon, Zingaro, Porter, Bailey-Lee and others[48, 49, 50, -51, 27]. -\subsection{Teaching Pumping Lemmas} -In 2003 Weidmann[39] wrote a dissertation on teaching Automata Theory to -students at the college level. She found that past performance in prerequisite -theory courses was a statistically significant indicator for success in their college -level course. She described a theoretical framework called ``pedagogical -positivism'', a stance between logical positivism and constructivism, allowing -the notion of a teaching method best suited to a group of students to learn Automata -Theory. She interviewed a teacher with ``several'' years of experience -teaching this course (p. 5), who ``admitted that she did not have a better way -to teach abstract thinking other than repeated exposure'' (p. 98). -In chapter 5, Discussion, Conclusions and Implications, of this dissertation[39], -the suggestion ``Instead of simply providing the solution to a problem in class, -or stating the intuitive leap that makes the problem easy to solve, the students -should be exposed to the iterative thought process that lead to the intuition -that created the solution.''(p. 201) appears. One suggestion is ``Learning objectives -should be set to focus on familiarity with formalisms and rigorous -mathematical notations” (p. 224) and another suggestion is “Include programming -projects as part of the required coursework''(p. 224). The combination -of these brings to mind the suggestion of Harel and Papert[40]: ``constructing -personally designed pieces of instructional software'', and the thought that the -students might dwell more effectively on the notion of abstraction as they tried -to teach someone else about it. -\section{ Proof by Induction} -Kinnunen and Simon [7] describe an example applying phenomenography to -computing education research, listing several recent examples, and also providing -a detailed description of a mainly data- but also theory-driven refinement -of categories. -Berglund, Eckerdal and Thun\'e [16, 3, 4] have applied phenomenography to -computing education research, obtaining classifications by judicious grouping -of student conceptions derived from interview data. Eckerdal et al. [4] describe -how the results using phenomenography showed additional insights beyond -other methods. -Jones and Herbst [6] considered which theoretical frameworks might be most -useful for studying student teacher interactions in the context of learning about -proofs. Bussey et al. [2] illustrated student teacher interactions in the space of -learning, and the objects of learning, in variation theory, modified from the -model of Rundgren and Tibell [13]. -Reid and Petocz [12] used phenomenography to study students' conceptions -of statistics. Their purposes included to ``enable teachers to develop curricula -that focus on enhancing the student learning environment and guiding -student conceptions of statistics.'' They asked students to describe how they -understood statistics and then organised student responses into a hierarchy of -conceptions. They used interviews to understand individual students, and the -group of interviews to show the variations they found. They found the students with the most superficial understanding to be carrying out steps without -knowing their meaning. -Krantz [8] describes proof by induction, giving several examples in this book -of proof techniques for computer science. -\section{Domain, Range, Mapping, Relation, Function, Equivalence Relation in Proofs} -Marilyn Carlson, \cite{carlson1998cross}shows that we can easily expect too much from our students in terms of what they understand of functions. This has significance for what we think are adequate examples to use for proof by mathematic induction, for example. -\section{ Definitions, Language, Reasoning in Proofs} -Weber, Alcock, Knuth -\subsection{Procedural vs. Understanding} -Is it that Tall and bifurcation are about learning the procedural vs. understanding approach to dealing with proof? -There are indications\cite[p. 18]{loewenberg2003mathematical} that mathematics teachers in grade school and high school who were mathematics majors themselves learned a procedural approach to mathematics and "lacked an understanding of the meanings of the computational procedures or of the solutions. Their knowledge was often fragmented, and they did not integrate ideasthat could have been connected (e.g., whole-number division, fractions, decimals, or division in algebraic expressions.)" -\subsection{Recall of Relevant Information vs. Inert Knowledge} -Bransford et al.\cite[p. 296]{bransford2000designs} attempt to address the problem identified by as inert knowledge (in the sense of Whitehead\cite{whitehead1959aims}). They situated class activity in a problem solving environment, and they showed\cite{van1992jasper} that this instruction had better results for students' ability to transfer skills to new word problems than traditional instruction. - -Lehrer et al. \cite[p. 334]{lehrer2000inter} found that "at least in some circumstances, giving children models may be less helpful than fostering their propensity to construct, evaluate, and revise models of their own to solve problems that they consider personally meaningful." - -\section{Abstraction} -Lesh and Doerr\cite{lesh2000symbolizing} used model-eliciting to engage students in the creation of a combination of meaningful descriptions, explanations and procedures. These models were recognized as tools to be shared and reused, consequently the idea of generalization was implicit. Lesh and Doerr argue that by illuminating the idea of a tool that may be reused, they divide the problem of teaching generalization into two parts: making a general model, and discerning its domain of applicability. - - -Kemmerer\cite{kemmerer} - -According to Huang et al.\cite{huang2015highest}, a large body of research suggests that an abstract cognitive processing style produces greater creativity. Empirically, decades of work have shown that both abstract thinking and creativity are consistently linked to right-hemispheric activation in the brain (e.g., Fink et al., 1996\cite{fink1996brain} and Mihov et al., 2010\cite{Mihov2010442}). -Miron-Spektor et al.\cite{Miron-Spektor20111065} have shown that observing anger communicated through sarcasm enhances complex thinking and solving of creative problems. - -(We want to know, do they understand deduction, abstraction, where are they on van Hiele levels ) - -According to Gray and Tall \cite[p. 117]{gray1994duality}, Hiebert and Lefevre observed ``a connected web \ldots a network in which the linking relationships are as prominent as the discrete pieces of information \ldots a unit of conceptual knowledge cannot be an isolated piece of information; -by definition is it part of conceptual knowledge only if the holder recognizes its relationship to other pieces of information ``\cite[p. 3-4]{hiebert2013conceptual} -conceptual knowledge is harder to assess than other kinds of knowledge. - -\section{Diagrams in Proof} -Gibson\cite{gibson1998students} examined students' use of diagrams in proofs, and found that diagrams helped link students' ideas to mathematization, namely, to representation in symbols, and also to support variation, in the sense of the critical difference between what Harel and Sowder\cite{harel1998students} call perceptual and transformational conceptualizations. +\chapter{Validity and Reliability} +strategies for trustworthy, valid, reliable +what about generalizability? %(e.g., to people with ASD) +\section{Interviews} +software programs that were used, to manage, organize data\\ +analyze as we go\\ +inductive and comparative\\ +precisely how the analysis was done\\ +thorough explanation of any strategies, such as discourse analysis +\section{Documents} +\section{Validity and Reliability in Proofs Using the Pumping Lemma +for Regular Languages} +Some support for the validity of the results comes from seeing several variations +of each proposed error type. We found in our data multiple versions of +unwarranted restrictions: choosing $x$ to be empty or choosing the length of $xy$ +to be $p$, and others. We found in literature warnings against attempts to prove +statements with universal qualifiers true by means of showing the existence +of examples \cite{devlin2012mathematical,Franklin}. These warnings suggest these errors have occurred before. +In textbooks \cite{epp2010discrete,rosen2003} we find single counterexamples for showing such a +statement false, and the method of exhaustion for showing a finite universal +statement to be true. Proof by contradiction for the purpose of showing such a +statement true, i.e., that any particular tentative counterexample contained an +inherent contradiction, is not itself universally accepted, due to not necessarily +being constructive \cite[p. 2]{bridges2007did}. We also found in our data, several versions +of misunderstanding inequalities. We found support in literature for errors of +misunderstanding how to work with inequalities, by students of this level\cite{Mattuck}. +\section{Proof by Induction} +We were encouraged by the overlap in description among interview participants. +The interviews were certainly not the same, but common elements, +specifically that there is a form to proofs by induction, appeared. Some students +referred to this form as steps, others as a procedure, or framework. Moreover, +degrees of understanding filled in a spectrum, from joyful deep understanding +to admissions of not understanding why the steps of proof by induction prove +anything, and conceptions in between. These included a supposition why a +proof of an induction step would, in combination with an established base case, +constitute a proof by induction, that its originator characterized as ``weird''. +\section{Domain, Range, Mapping, Relation, Function, Equivalence +Relation in Proofs} +\section{Definitions, Language, Reasoning in Proofs} \section{Equivalence Class, Generic Particular, Abstraction in Proofs} -\section{Computer Science Education} -Thun\'e and Eckerdahl\cite{thune2009variation} have applied variation theory has been applied to the teaching of computer science. - - -\section{Educational Psychology} -\subsection{What do students need to construct?} - -Archavi et al.\cite[p. 13]{arcavi1998teaching} ``I'd like to have you doing some mathematics and I will do everything I can --- including using grading --- as a device for having you do that.'' - -\section{Phenomenography, Variation Theory} - -Marton and Booth\cite{marton1997learning} have written - -{\aa}kerlind \cite{aakerlind2012variation} has written on how - -Runesson\cite{runesson2005beyond} has applied variation theory to math - -\section{Constructivism} - - - -Vygotsky in Language and Thought said we do as individuals build up thoughts and then becoming socialized with shared language, some accommodation would need to be enforced onto the child. [p.17] the psychological problem is to become convinced that always, necessarily a given picture has to appear as one of a multiple of possible graphs of the same category (i.e. only as a representative of a class \ldots must be grasped not in a final fixed state but rather \textit{in construction} the point moving) - -Vygotsky\cite[p. 49]{vygotsky1978mind} noted that "one child selected a picture of an onion to recall the word 'dinner'. When asked why she chose the picture, she gave the perfectly satisfactory answer, 'Because i eat an onion'. However, she was unable to recall the word 'dinner' during the experiment. This example shows that the ability to form elementary associations is not sufficient to ensure that the associative relation will fulfill the \textit{instrumental} function necessary to produce recall." - -\subsection{What do students need to construct?} - -Archavi et al.\cite[p. 10]{arcavi1998teaching} ``To be successful, students must know both the appropriate heuristics and the mathematics required to solve the problem. -\subsection{Intuition} -Students have some knowledge constructed already, and it is not all conscious. - -overconfidence --- counter by ``search for reasons it might be wrong'' Koriat et al., 1980\\ -confidence --- doesn't correlate with correctness\\ -be as to conform Tweney Doherty Mynatt 1981\\ -``renouce several of his funcamental beliefs with regard to reality'' [p. 39]\\ -$$ Combine desire to gain points having taken the place of desire to learn, with propensity to learn how to take tests rather than how to believe, and obtain ``How we answer the tests, or, what we really think'' $$ -Piaget (Piaget-Beth 1966 [p. 195] ) ontogenetic construction of evidence a new domain integrates former domain as subdomain\\ -(n heast?) intuition tends to survive even when contradicted by systematic formal instruction [p. 47]\\ -Polanyi 1969 [p. 143-144]\\ -\ldots in the structure of tacit knowledge we have found a mechanism which can produce discoveries by steps we cannot specify this mechanism may then account for scientific intuition \ldots not the supreme immediate knowledge called intuition by Leibnitz or Spinoza or Husserl, but a work-a-day skill for scientific guessing with a chance of guessing right.\\ -Polanyi sees a deep analogy between integrative capacity -$$ there we have ``unsupervised'' specialization network formation during consolidation, perhaps $$ ``Where to turn for a logic by which such tacit powers can achieve and uphold true conclusions'' Polanyi 1967 [p. 137]\\ -$$I'm thinking about conscious / unconscious$$ -$$ When we do something consciously we can be checking, when we do something unconsciously, we might not be $$ -Fischbein [p.59] ``Inferential affirmatory intuition may have an inductive or deductive structures. After one has found that a certain number of elements (objects, substances, individual, mathematical entities, etc.) have certain properties in common one tends \textit{intuitively} to generalize and to affirm that the \textit{whole} category of elements possesses that property. This is not a mere logical operation. -The generalization appears more of less suddenly with a feeling of confidence. -This is a fundamental source of hypotheses in science. -According to Poincar\'e ``generalization by induction copied, so to speak, from the proedures of experimental sciences'' is one of the basic categories of intuition (Poincar\'e 1920 [p. 20]).\\ -\cite[p. 67]{fischbein1987intuition}(check) ``One morning walking on the bluff, the idea came to me with just the same characteristics of brevity, suddenness and immediate certainty that the arithmetical transformation of indeterminate ternary indefinite forms were identical to those of the non-Euclidean geometry Poincar\'e 1913 [p. 388]''\\ -David Tall mathematician and psychological analyst, moment of insight ``never felt that he made 'conjectures'; what he say were 'truths' evidenced by strong resonances in his mind. Even though they often later proved to be false, at the time he felt much emotion vested in their truth \ldots intense intuitive certainties. Yet at the same time his contact with them often seemed tenuous and trasient; initially he had to write them down, even though they might be imperfect, before they vanished like ghosts in the night (Tall 1980 [p. 33])\\ -$$ being unconscious seems to go with consolidation. Either unconscious because attending to something else like walking on a bluff, or asleep, being unconscious is relevant to these integration occurring and then moving into consciousness $$ - -$$''Really wants to know'' implies an openness to change the pre-determined ideas, and ``complyig with requriements'' does not imply readiness to revise $$\\ -\cite[p. 68]{Fischbein} citing Feller ``experienced player absorbs a complicated situation at a glance and is unable to account rationally for his intuition''\\ -$$ it (the part of the brain doing the reasoning) is functioning without conscious oversight (the neurons that did that when the capability was new have been deemed extraneous and removed)$$) -$$Brooks -- the first system is done carefully with all consciousness like the beginning chessplayer in Feller, the second system has some unwarranted conviction and the third system has mostly warranted $$\\ -\cite[p. 69]{Fischbein} Felix Kelin (1898) trained intuition\\ -Suppes 1966 train the intuition for finding an writing mathematical proofs''\\ -\cite[p. 72]{Fischbein} categorical syllogism type AAA seems easiest for which, EAE, AII 65\%, EIO -(These are categorical syllogism types. See www.philosophypages.com/lg/e07a.htm) -\cite[p. 77]{Fischbein} -AAA and modus ponens come earliest and are class inclusion, maybe $\bar{p} \rightarrow q$ never develop\\ -re \cite[p. 81]{Fischbein}, $$ people have data stored to supply their intuition and it our be wrong education involves opening it up to conscious inspection, fixing it, and restoring the rapid unconscious operation $$\\ -\cite[p. 106]{Fischbein} citing \cite[p. 228]{Wertheimer 1961} ``These thoughts did not come in any verbal formulation. I very rarely think in words at all. T thought comes, and I may try to express it in words afterwards''. Einstein\\ -\cite[p. 119]{Fischbein} A total fusion of the generality of a principle and a particular directly graspable (in this case figural) expression of it. It is this kind of fusion which is the essence of intuition.\\ -\cite[p. 120]{Fischbein} specific, directly convincing example and the general principle derived through similarity and proportionality from the particular case.\\ -\cite[p. 129]{Fischbein} Analogy frequently intervenes in mathematical reasoning, Polya writes about great analogies \\ -138 1985 software by David Tall\\ -\cite[p. 144]{Fischbein}ways in which people process concepts Smith Meliss 1981\\ -\cite[p. 147]{Fischbein} ``For many students the concepts of parallelogram, square and rectangle are not organized hierarchically. They represent classes of quadrilaterals of the same generality''. -$$ some programmers before unified method were guilty of writing code this way, with wasteful effect. Moreover, Liskov Substitutability Principle was enunciated to help people know how to populate hierarchies when they were learning to do so. Students who are not organizing their concepts hierarchically are similarly disadvantaged. As I believe this hierarchical situation is consolidated during sleep or relaxation, ti becomes a research question$$ -\cite[p. 159]{Fischbein} analogies already similar diagram post concept\\ -\cite[p. 165]{Fischbein} diagram relies on intervening structure (conceptual structure) else it does not communicate\\ -stability of intuition, Ajzen? 1983 epistemic freezing\\ -\cite[p. 214]{Fischbein} concept of intuitive loading --- have to know students first before knowing how to teach them - - -Intuition in Science and Mathematics An Educational Approach Efraim Fischbein 1987 Reidel Publish -Westcott combines theoretical analysis with experimental findings. -Andrea di Sessa building a theory of intuition -Bergson 1952 essence of lining changing phenomena -Kant intellectual (does not exist) and sensible intuition -1980 [p. 268] - -Poincar\'e useful -Hahn 1956 source of misconception -to use or to eliminate? -Berne professional quality work without awareness says Westcott 1968 [p.42-46] -immediacy --- is that because crosses into conscious unconscious\\ -consistency, brevity of expression $\rightarrow$ bearty -if the result of the brain's consolidation of knowledge, info, into uncouscious knowledge, usable, available for recall -going into intuition, what you taught us for the test, or what we really think -Fischbein p. 9 ``One may not be aware of the existence of such an explicit representation but it continues to act tacitly and to influence ways of reasoning. - -Seymour Papert 1980 apparently says something about intuition - -Brouwer, Weil, Kline 1980 [pp 306-327] - -The sum of the angles of a triangle is equal to two right angles'' -Connect bewteen intuitino and reasoning -\subsection{Social Constructivism} -Archavi et al.\cite[p. 6]{arcavi1998teaching} ``Students' mathematical activity takes place in an inherently social milieu.'' -\subsection{Tall: Set-befores and Met-befores} -McGowan and Tall \cite[p. 172]{ (2010 Jour. Math. Behav.)} ``If learning defaults to the goal of learning how, it can be successful. However, if it is accompanied by a lack of conceptual meaning so that mistakes occur, it can become fragile and more likely to fail in the longer term. At this stage the problems may proliferate as the student becomes confused as to which rule to use, where to use it, and how to interpret it. - -Tall and Mejia-Ramos \cite[p. 138]{2010, Explanation and Proof in Mathmatics, Springer} ``Here proof develops through generalized arithmetic and algebraic manipulation'', -different kinds of warrants for truth $$ so assess student by asking what kind of warrant$$ see Pinto and Tall (1999 and 2002) build on met-befores. -\begin{figure}[tbph] -\centering -\includegraphics[width=0.7\linewidth]{chp7p1} -\caption{How proof develops, Tall Mejia-Ramos} -\label{fig:chp7p1} -\end{figure} - -\subsection{Harel and Sowder} - - \cite[p. 237]{harel1998students}Rather than gradually refining students' conception of what constitutes evidence and justification in mathematics, we impose on them proof methods and implication rules that in many cases are utterly extraneous to what convinces them. -Editors Schoenfeld et al.\cite{kaput1998research} describe that Harel and Sowder\cite{harel1998students} ``characterize students' cognitive schemes of proof''. - -The subdivisions in the 1998 version of categories of conceptualizations \cite{harel1998students}, specifically intuitive -- axiomatic, structural and axiomatizing, -matter much in computer science, because intuitive -- axiomatic could be thought to be less used in computer science than in math, program's content could be less intuitive than Euclidean geometry, more subject to checking by assertion checking or debugger examination. - - \cite[p. 268]{harel1998students} contextual proof scheme: --students have learned to work in a context, e.g., $\mathbb{R}^n$, and so, interpret statements that have greater generality as restricted to be in the context they have learned ``he or shee has not yet abstracted the concept \ldots beyond this specific context''. Compare this with Pang's (is it Pong?) observation that for students who know only one language, ``speaking'' and ``speaking that language'' are concepts that are undifferentiated. - - \cite[p. 274]{harel1998students} ``An important distinction between the structured proof scheme and the intuitive proof scheme is the ability to separate the abstract statements of mathematics (e.g., $1+1=2$) from their corresponding quantitative observations (e.g., 1 apple + 1 apple = 2 apples) or the axiomatically -- based observations from their corresponding visual phenomena \ldots ``, ``axiomatic proof scheme is epistemologically an extension of transformational proof scheme. One might mistakenly think of the axiomatic proof scheme is the ability to reason formally \ldots ``. - - \subsection{Pirie and Kieren Model of Mathematical Understanding} - \cite{meel1998honors} - Verify these are due to Pirie and Kieren rather than to Meel. - - \paragraph{Primitive Knowing} - This is brought by the student, and is also known as intuitive knowledge, situated knowledge, prior knowledge and informal knowledge. - \paragraph{Image Making} - any mental image not necessarily pictorial - \paragraph{Image Having} - mental picture / objects, concept image, frame, knowledge representation structure, students' alternative frameworks - \paragraph{Property Noticing} - unselfconscious knowing, can notice distinctions combinations connections between mental objects - \paragraph{Formalizing} - abstract (this is a verb) common qualities from classes of images, classlike mental objects built from noticed properties, description of these class-like mental objects results in production of full mathematical definitions - \paragraph{Observing} - ability to consider one's own formal thinking, organize personal thought processes, recognize ramifications, - \paragraph{Structuring} - axiomatic system, conceive proofs of properties associated with a concept - \paragraph{Inventising} - create new questions, develop new concepts - \paragraph{folding back} - reorganizing lower level understanding to accommodate new information - - -\subsection{van Hiele Levels} -Abstraction is before deduction -\subsection{Performance Levels} -Baranchik and Cherkas\cite{baranchik1998supplementary} found three levels of understanding in a population taking algebra exams: -\begin{enumerate} -\item Early skills --- arithmetic and elementary algebra -\item Later Skills --- subsequent algebra and a variety of skills involving methematical abstraction, and -\item Formalism --- either devising a solution strategy or reformulating a problem into a standard form that permits a solution using early or later skills -\end{enumerate} -\subsection{Student Centered} -Carlson\cite{carlson1998cross} has concluded that ``\ldots an individual's view of the function concept evolves over a period of many years and requires an effort of 'sense making' to understand an orchestrate individual function components to work in concert.'' -\subsection{Use of Diagrams} -Gibson\cite{kaput1998research} states ``Diagrams aided students' thinking by corresponding more closely to the part of their understanding with which they were operating at the time and by reducing the burden that proving placed on their thinking.'' -\cite[p. 205]{kaput1998research} The nature of internal representations, however, is unclear because they are not observable.: -$$ nature of internal representations can be broad, and we can perhaps influence the nature of internal representations, which are ultimately neural nets, by how we teach, and nature of internal representations is such that some, e.g., perceptual, are nto as helpful as others, e.g., transformational. Get the superior colliculus involved, see the motion Ties in with variation theory. Also visualization parts of brain.(B17?) $$ - -Winn, B, (get the citation from Gibson article in Kaput RCME 1998)( Charts, graphs and diagrams in Ed. materials Psych Illus Basic Research Vol 1 Springer 1987 pp. 152-198) has a spectrum for internal representations from pictures to works, the word end is called abstract. - -Zimmerman, Visual thinking in Calculus Visualization Teaching Learning Math 1991 -Gibson\cite[p. 132]{kaput1998research} ``There is no doubt that diagrams play a heuristic role in motivating and understanding proofs'' -Tall 1991 Intuition and rigor, role of visualization in teaching learning mathematics -Gibson\cite[p. 288]{kaput1998research} ``When student used visual language I inferred that they were operating with the visual part of their understanding'' -Gibson\cite[p. 289]{kaput1998research} ``Students indicated that diagrams helped them understand information by appealing to their natural thinking. They said that diagrams seemed to coincide with the way their 'minds work' and that information represented visually seemed easier or clearer than verbal/symbolic representations.'' -more concrete than verbal/symbolic -Gibson\cite[p. 290]{kaput1998research} ``used it to help me see what would be happening'' -$$ executive parts of brain is engaging visual parts of brain$$ -easier than holding the mental image is look at the drawn image -Gibson\cite[p. 291]{kaput1998research}''When I read the definitions you can't think about the whole thing at once, but when you have a picture you can'' -Gibson\cite[p. 294]{kaput1998research}''Because students did not usually think of their criteria in terms of formal definitions, their ability to decide whether their criteria had been met was hindered when they worked with information represented in only verbal/symbolic form.'' -``They could obtain ideas more readily from diagrams than they could from verbal/symbolic representations'' -Gibson\cite[p. 297]{kaput1998research} Why always keep the picture in your mind when you can have it on the paper, allowing you to focus more on how to get to the end of the proof instead of always having to recall the picture in each individual step?'' -$$visual rather than mirror area is possible$$ -Gibson\cite[p. 298]{kaput1998research} ``students sometimes used diagrams to help them express their ideas'' symbolically -$$compare proofs without words$$ -Gibson\cite[p. 298]{kaput1998research} ``diagrams helped Laura write out her ideas by helping her connect her ideas to verbal/symbolic representations of these ideas'' -Gibson\cite[p. 299]{kaput1998research}''you need to down load that picture on here so that you can touch it and then allow your brain to think about the words you need to say'' -visualization does not always help, Gibson quoted some sources -Gibson\cite[p. 302]{kaput1998research} ``when attempting to solve unfamiliar problems, students can benefit from using diagrams'' -Moore\cite[p. 262]{moore1994making} ``The students' ability to use the definitions in the proofs depended on their knowledge of the formal definitions, which in turn depended on their informal concept images. The students often needed to develop their concept images through examples, diagrams, graphs and others means before they could understand the formal verbal or symbolic definitions'' %[p. 262], Moore R Making the transition to formal proof Ed Studeies in Math 1994. -Gibson\cite[p. 303]{kaput1998research}''That students would operate in this manner (with the visual part of their concept images) and that such behavior might be of benefit is reasonable when one considers the nature of the concepts in the proofs together with the students' experiences as visual beings and the physiology of their brains''. - - -\section{Cognitive Science} -Archavi et al.\cite[p. 6]{arcavi1998teaching} Mathematics requires abstraction, and problems should inspire generalization and specialization. - - - -\subsection{Intrinsic Reward} -Archavi et al.\cite[p. 9]{arcavi1998teaching} Good problems are ``\ldots non-routine and interesting mathematical tasks, which students want and like to solve, and for which they lack readily accessible means to achieve a solution''. -\subsection{What do students need to construct?} - -Archavi et al.\cite[p. 13]{arcavi1998teaching} `` There were occasions later in the course in which the whole-class discussion also dealt with issues of mathematical elegance and aesthetics.'' - - -Leslie Valiant\cite[p. 103]{valiant2000circuits} points out that representations, for models of cognition, are not all equally learnable. -(in polynomially many steps, p. 104) -Easily learnable representations (of concepts) ``include Boolean conjunctions (e.g., $x_1 \land x_5 \land \bar{x_y}$) and Boolean disjunctions (e.g., $x_1 \lor \bar{x_3} \lor x_8$) \ldots An important class that is not currently learnable is disjunctive normal form (or DNF for short)'', (e.g., $x_1\bar{x_2}x_3 \lor x_1x_2 \lor x_2x_4x_7$), describes a concept whose membership can be attained in one of three ways, in two of which $x_2$ must b true, but in the other of which $x_2$ may be false, so long as $x_1$ and $x_3$ are true. -He goes on to observe these may be learned in stages, stating ``more is required of the teacher or environment than in the simplest case of learning by example'' [p. 104] -He uses the idea later clarified by Marton and Pang\cite{} stating ``a teacher may have to teach the name of this subconcept and then identify positive and negative examples of it'' [p. 104]. -`` In this context, learning theory can be thought of as defining the granularity with which learning can proceed without intervention \ldots the largest chunks of information that can be learned feasibly without their having to be broken up into smaller chunks'' [p. 104] -(Combine this with the approximately 7 chunks in short term memory?) - -Generalization and analogy are directly addressed in mathematics teaching by assigning students to ``search for connections and extensions of problems''\cite{santos1998instructional}. -\subsection{Analogical Reasoning} -Gentner and Smith\cite{gentner2012analogical} define analogical reasoning as "the ability to perceive and use relational similarity between two situations and events", and have stated that analogical reasoning is fundamental to human cognition. -They state that \cite[p. 131]{gentner2012analogical} ``Analogy is often the most effective way for people to learn a new relational abstraction; this makes it highly valuable in education.'' -Because we wish to obtain the value inherent in reasoning by analogy, we note that it depends upon recognition of relationships, and abstraction, to compare relationships at a level divested of some specifics. -Abstraction, for students of computer science, has been observed to be difficult to learn\cite{or2004cognitive} in that context. -Nevertheless, application of proverbs, such as "Don't cry wolf.", is routinely expected in education of children\cite{lutzer1988comprehension}. -Or-Bach and Lavy show empirical data and provide insight into the difficulties of computer science students who have trouble extracting common features from a problem statement that emphasizes differences, and promoting those to a more general class, while maintaining the differences in the more specific classes. -The relationship from one class to a related class in an inheritance hierarchy, motivated as it has been by code reuse, is more stereotyped than the relationships in proverbs, which are not restricted to generalization/specialization. So, we should be careful about generalizing the difficulty students of computer science have with abstraction. -Gentner and Smith go on to say\cite[p. 131]{gentner2012analogical} that analogical reasoning is characterized by retrieval, in which a current topic in working memory may remind a person of a prior analogous situation in long term memory; mapping, which involves aligning the representations and projecting inferences from one analog to another; and evaluation, which judges the success of the alignment of the representations and inferences. -Thus we see that the relationships are key in analogical reasoning, compared with being stereotyped in establishing inheritance hierarchies. -Gentner and Smith\cite[p. 133]{gentner2012analogical} remark that "Another benefit of analogy is \textit{abstraction}: that is, we may derive a more general understanding based on abstracting the common relational pattern." and "analogies can also call attention to certain differences between the analogs." -Though we might wish to have people readily retrieve knowledge that would, by analogy, be helpful to solving a current problem, Gick and Holyoak\cite{gick1980analogical} showed that people do not always retrieve the knowledge they have, rendering it, at least temporarily and for this purpose, what Alfred North Whitehead called "inert knowledge"\cite{whitehead1959aims}. -Gentner and Toupin \cite{gentner1986systematicity} have observed, however that, older children (and not younger children) benefited from systematicity: a summary statement of the structure of the relationships. There is a shift that can occur from focussing on objects to focusing on relatiohships, called a "relational shift", which has been the subject of research\cite{gentner1988metaphor,rattermann1998more,bulloch2009makes}. -Dunbar\cite{dunbar2000scientists} outlines three important strategies that scientists use: attention to unexpected findings, analogic reasoning, and distributed reasoning. Dunbar states\cite[p. 54]{dunbar2000scientists} "our analyses suggest that analogy is a very powerful way of filling in gaps in current knowledge and suggesting experimental strategies that scientists should use" and "If scientific reasoning is viewed as a search in a problem space, then analogy allows the scientist to leap to different parts of the space rather than slowly searching through it until they find a solution". - -Day and Gentner\cite{day2007nonintentional} showed in -Day and Gentner\cite[p. 41]{day2007nonintentional} -"Gentner and Medina proposed that -schemas and other abstractions are often derived via a -process of repeated analogizing over instances (see also -Cheng \& Holyoak, 1985)." - -\textbf{This, schemas and abstractions, earned a double question mark. Maybe that means it should be reported in more length.} -Day and Gentner\cite[p. 41]{day2007nonintentional}"The goal of this research was to investigate an important open question: Can a single prior instance influence how a new episode is understood, and if so, does it do -so by using a structurally sensitive mapping process, as -in analogy?" -Day and Gentner\cite[p. 42]{day2007nonintentional}"The results are consistent with the claim that individuals -may use a single prior instance as a source for nonintentional inference based on structural commonalities. The -pattern of inferences is what would be expected if participants were structurally aligning the two representations -and drawing inferences about the target from relationally -similar aspects of the base. Participants' responses that the -inferred information had actually been stated in the target -story suggest that these inferences were not deliberately -considered and evaluated, but rather were spontaneously -incorporated into the target representations as they were -being created." - -\subsection{Generalization} -Generalization is thought to result when multiple instances of analogies, sharing the same structure of relationships, have been considered. *who was I reading before kowatari?" - -Ball states\cite[p. 38]{loewenberg2003mathematical} "Generalization involves searching for patterns, structures, and relationships in data or mathematical symbols. These patterns, structure, and relationships transcend the particulars of the data or symbols and point to more--general conclusions that can be made about all data or symbols in a particular class. Hypothesizing and testing generalizations about observations or data is a critical part of problem solving." - -She continues \cite[p. 38]{loewenberg2003mathematical} "In one of the simpler common exercises designed to develop young students' capabilities to generalize, students are presented with a series of numbers and are asked to predict what the next number in the series will be. \ldots Representational practice play an important role in generalizing. For example, being able to represent an odd number as $2k+1$ shows the general structure of an odd number. \ldots Representing the structure using symbolic notation premits a direct view of the general form." - -Lesh and Doerr\cite[p. 379]{lesh2000symbolizing} encourage students to construct models, that may include "a combination of spoken words, written symbols, pictures or diagrams, or references to other models or real-life experiences \ldots in any case, the representation tends to organize and simplify the situation so that additional information can be noticed, or so that attention can be direct toward underlying patterns and regularities, which may, in turn, drive changes in conceptions." - -Bowers\cite[p 390]{bower2000postscript} summarizes ideas on generalizing saying: "Bransford et al.\cite{bransford2000designs} describe several studies to support the claim that 'people's representations of problems and experiences have strong effects on the degree to which they will transfer their knowledge to new settings'. Similarly, Lesh and Doerr\cite{lesh2000symbolizing} argue that the models students produce when engaging in model-eliciting problems are not just solutions to the problem at hand, but instead stand as more generalized conceptual tools that can be 'shared and reused in other situations' ". - -Huth et al.\cite{huth2012continuous} - A continuous semantic space describes the representation - - % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % - - - $$ One reason we delve into this is that we want to know something about to what degree different factors that might possibly assist learning are significant. For example, how important is it to note the beauty of a proof, and what is the significance of the order in which a proof is presented (for example, lemmas first), and how quickly might we expect students to grasp a hierarchy of abstractions. We learn possibly surprising things, such as, if we provide a fragrance during learning a proof with ideas about spatial locations, and we provide that same fragrance during the early part of sleep, the memory will be consolidated more effectively, as demonstrated by subsequent recall during an awake state.$$ - - Cerf et al. \cite{cerf2014studying} opine "early neuron activity observed here could represent the early stages of the formation of a thought or recollection, the states at which we may not yet be fully aware of the content of the thought" - - \subsection{Cognitive Neuroscience} - Cognitive neuroscience provides evidence for believing that suspense, and concern for characters, is useful in helping students selectively attend to, and remember at an abstracted level, the material they are seeing. - For example, - Bezdek et al. \cite{Bezdek2015338} have measured brain responses corresponding to attention, and have shown that attention is modulated by the emotional flow of a narrative as it unfolds over time, and that suspense is associated with increased central processing (of the visual field) and decreased peripheral processing. Moreover they have reason to believe that this attention does produce downstream consequences, reflecting encoding of content at a level abstracted from visual features. They have brain metabolism imaging showing decreases in activity that has been associated with mind-wandering\cite{Christoff20098719}. - Kosslyn and K\"onig \cite[p. 56--57]{kosslyn1992wet} describe some layout of lower level functions in the brain, "The ventral (object-properties-encoding) system in the temporal lobes not only registers key properties of shapes, but also encodes color and texture; this information is matched to that of objects stored in visual memory. This temporal-lobe memory stores information in a visual code, which cannot be accessed by input from other sensory modalities. \ldots The outputs from the \ldots encoding systems come together at an \textit{associative memory} (which relies on tissue in various places in the brain) \ldots once the appropriate information is accessed, one knows the name of the object, categories to which it belongs, sounds it makes, and so forth. \ldots there is ample evidence that the frontal lobe plays a critical role in this process" - Kosslyn and K\"onig \cite[p. 78]{kosslyn1992wet} "For example, if a caterpillar is still against a twig, it may be very difficult to notice. But if it is moving, we may identify it immediately. Computing motion relations appears to be qualitatively distinct from computing the organization of portions of static images, and distinct regions of the brain apparently encode motion (particularly areas MT and MST). There appears to be a distinct \textit{motion relations subsystem}. The motion relations subsystem extracts key aspects of motion fields, and operations concurrently with the preprocessing subsystem." - $$ knowing whether two (simultaneously presented) parts of a proposed presentation will require attention from two systems able to operate concurrently, or will make conflicting demands upon one attention window is useful$$. - - - Kosslyn and K\"onig \cite[p. 102]{kosslyn1992wet} "the frontal lobe clearly has a role in setting up plans. More important, the frontal lobe clearly has a role in formulating and testing hypotheses."$$needed for creating proofs$$. - - Kosslyn and K\"onig \cite[p. 104]{kosslyn1992wet}"a subsystem that \textit{engages} attention. \ldots the thalamus. The thalamus is a kind of switching station, connecting many parts of the cortex." - - Kosslyn and K\"onig \cite[p. 112]{kosslyn1992wet} "Visual object agnosia is often divided into two types Patients who are diagnosed as having \textit{apperceptive} agnosia have difficulty putting together visual information to form an integrated perception on an object. Some such patients describe the world as fragmented or chaotic. These patients cannot determine whether two objects are the same or different, let alone identify an object they see. However, even these patients are not blind. Patients who are diagnosed as having \textit{associative agnosia}, in contrast, have difficulty associating the perceptual input with previously stored information. Patients who have 'pure' associative agnosia can discriminate between and properly compare shapes, even though they cannot identify the shapes \ldots could tell whether objects were the same or different, but not what they were. In short, a problem appreciating the shape of an object is apperceptive; a problem identifying the object while being able to distinguish its shape is associative." - $$ it would be nice to diagnose student difficulties to this degree, especially if an treatment could be associated with the diagnosis. Moreover, there could be a strong analogy here. Students could recognize two applications of rules of inference as being the same without being able to identify which rule of inference it is. This suggests a clicker question about rule applications, as to whether they are the same or not, or, choose the rule application that is different. $$. - - Kosslyn and K\"onig \cite[p. 114]{kosslyn1992wet} "Prosopagnosia is particularly puzzling because at least some of these patients apparently recognize faces unconsciously, and never become aware that they have done so; patients with prosopagnosia showed changes in the electrical properties of their skin (i.e., increased electrodemal skin conductance responses) when they were viewing familiar faces, compared to unfamiliar ones, even when they claimed to have no idea whom they were looking at. The stimulus must be matching a stored memory of a face in the pattern activiation subsystem (at least to some degree), but processing sops early -- and the persons is never aware that a match was made. - - Kosslyn and K\"onig \cite[p. 118]{kosslyn1992wet} There are processes that activate stored visual information to generate visual mental images. - - Posner for systems involved in attention - - Kosslyn and K\"onig \cite[p. 124]{kosslyn1992wet} "one of the most intriguing aspects of the neglect syndrome is that it appears to affect consciousness itself." - - Kosslyn and K\"onig \cite[p. 132]{kosslyn1992wet} "Patricia Goldman-Rakic and her colleagues showed that the frontal lobes contain a short-term spatial memory." - - Kosslyn and K\"onig \cite[p. 136]{kosslyn1992wet} "the (mental) image is not like a picture; it is facing almost as soon as it is generated."(Fink, Pinker, Farah, Chambers, Reisberg) - - Kosslyn and K\"onig \cite[p. 147]{kosslyn1992wet} "In either case, instructions somehow must be provided to the attention shifting subsystems; if the image is truly novel, the instructions cannot be previously stored. Thus, it is of interest that there are rich connections from the frontal lobes (which presumably direct the process), not only to parts of the parietal lobes known to be involved in attention, but also to the subcortical structures involved in shifting attention." Goldman-Rakic, posner, 1987, 1988, petersen 1990 - - $$Kosslyn et al. have inferred an organization of processing in the brain, and we as instructors could think about how we are deploying the learning task over this architecture. We could think about what path information has follow, to be learned to the extent that it can be called into future use. That is, while some instruction could produce inert knowledge, the goal is usable knowledge.$$ - - Kosslyn and K\"onig \cite[p. 152--153]{kosslyn1992wet} "By inferring that images occur in the same visual buffer that is used in perception, we expect that the properties of the buffer that affect perception also should affect imagery. If so, then the inability to maintain patterns in images for very long may be another consequence of this common mechanism. That is, in perception one does not want an image to linger; one wants to 'clear the buffer' every time the eyes move. Indeed, if images did not fade rapidly, they would smear and become overlaid. \ldots Thus, patterns in the visual buffer are transient in both perception and imagery." - - Kosslyn and K\"onig \cite[p. 154]{kosslyn1992wet} "A symbolic use of imagery involves the same subsystems used in other kinds of imagery reasoning; one must generate the image and retain it long enough to operate on it and 'see' the results. But there is one critical difference between this kind of symbolic imagery and the other sorts: One now must decide how to convert abstract material to particular patterns in the image. For example, one could visualize relative intelligence not only as dots on a line, but also as a set of circles" - - Kosslyn and K\"onig \cite[p. 163]{kosslyn1992wet} both hemispheres can generate images (posterior of brain), instructions come from frontal lobes. Left hemisphere naturally, right hemisphere with training. Kosslyn and K\"onig \cite[p. 164]{kosslyn1992wet} both hemispheres can generate images, but in different ways. - - Kosslyn and K\"onig \cite[p. 165]{kosslyn1992wet} image transformations require contributions from both hemispheres - - Kosslyn and K\"onig \cite[p. 168]{kosslyn1992wet} "Learning to read these (already familiar) words is learning to use an additional route into associative memory to access the information that is stored with words. The problem is, that we initially learned to access the relevant memories on the basis of hearing sounds, \ldots How are we able to use a pattern of lines to access these memories?" - - $$subvocalization is an inferior way to accomplish this$$ - - - - - - - Kosslyn and K\"onig \cite[p. 193]{kosslyn1992wet} "With repeated use, the entire pattern of lines becomes an entry in the pattern activation subsystem and the pattern recognized there is associated with a word in associative memory" - - $$repetition of relationships between items allows us to claim a pattern, a pattern suggests the thought of generalization, the symbolic representation of the pattern makes generalization more efficiently notated, maybe increases awareness of the possibility of generalization, and the ability to generalize is used in transfer of knowledge to further applications, counteracting inert knowledge$$ - - Kosslyn and K\"onig \cite[p. 204]{kosslyn1992wet} "the information that allows one to understand a word must be in associative memory, and possibly is stored only in the left hemisphere. (The left hemisphere is typically the site of the most language processing, and hence it would not be surprising if representations of word meaning were stored primarily in this hemisphere.) \ldots the left hemisphere associative memory \ldots presumably implements the word shape/sound associations." - - Kosslyn and K\"onig \cite[p. 207]{kosslyn1992wet} writing about a form of dyslexia that has provided insight, describe ideas of Coltheart\cite{schmalz2015getting} "He hypothesized that only words that name visible objects or properties are stored in the right hemisphere, \ldots " - - Kosslyn and K\"onig \cite[p. 214]{kosslyn1992wet} "Sounds are represented initially in a cortical structure in the inner part of the superior (upper) temporal lobes. Heschl's gyrus \ldots is the auditory analog to area V1; indeed, this structure is often called A1 (to indicate that it is the first cortical auditory area). As in vision, the raw auditory input is organized and represented in the auditory buffer prior to high-level processing. Ulrich Neisser posited that such a buffer serves as an 'echoic memory';" rather than being organized as visual input, where spatially adjacent cells have spatially adjacent receptive fields, instead it is organized so that spatially adjacent cells receive input at different sound pitches/frequencies. - - Kosslyn and K\"onig \cite[p. 215--216]{kosslyn1992wet} "Recall that in vision there are two subcortical pathways from the eyes to the brain, the geniculo-striate and the tecto-pulvinar; the tecto-pulvinar pathway (eye, superior colliculus, pulvinar, cortext) draws attention to potential regions of interest. The inferior colliculus projects to the deep and superior colliculus play a similar role in audition. \ldots the auditory receptive fields of neurons in the superior colliculus shift with changes in eye position, allowing the auditory and visual maps to remain aligned. (see, for example Maddox et al.,\cite{maddox2014directing}) Hence, one tupically pays attention to a single object, registering its appearance and sounds at the same time." - - Kosslyn and K\"onig \cite[p. 219--220]{kosslyn1992wet} "visual preprocessing subsystem becomes 'tuned' by experience to encode useful visual patterns \ldots the separate auditory areas are also connected with reciprocal connections \ldots the reasoning we used to infer than experience tunes the visual preprocessing subsystem also leads us to expect that stimulus properties that distinguish between words will be noted downstream, and feedback will reinforce the encoding of those properties in the auditory preprocessing subsystem" - - Kosslyn and K\"onig \cite[p. 220--221]{kosslyn1992wet} "\textit {categorical perception} \ldots lies at the heart of what is accomplished by the auditory preprocessing subsystem \ldots the same categories are extracted when a word is spoken by different people -- and these categories help one to understand the words spoken under different circumstances." - - $$ - Kandel talks about removing some extraneous (i.e., those that might be redundant, maybe less often on) connections, do I think this is having the effect of categorization/generalization? - Choosing consciously to de-emphasize consideration of details at a high level, and unconsciously removing less-often exercised distinctions could have a similar effect. - $$ - Kosslyn and K\"onig \cite[p. 225]{kosslyn1992wet} "We are led to infer than the representations of the sounds of individual words depend on temporal--parietal cortex" - - Kosslyn and K\"onig \cite[p. 227]{kosslyn1992wet} "If we assume that unimodal memories are stored in the subsystem that encodes them", which is done by Squire (1987)\cite{squire1987memory}(check this, there is a google book) - - - - David and Squire reviewed protein synthesis connected with memory formation\cite{davis1984protein} " Evidence from learning curves, examination of short-term retention, and posttraining drug injection indicate that initial acquisition is not dependent on such synthesis, but it appears that protein synthesis, during or shortly after training, is an essential step in the formation of long-term memory." - - Kosslyn and K\"onig \cite[p. 230]{kosslyn1992wet} "it is possible that semantic information is organized along these lines, with appearance-based meanings segregated from use-based meanings". - - Kosslyn and K\"onig \cite[p. 343]{kosslyn1992wet} "We inferred in Chapters 3 and 4 that unimodal visual information is probably stored in the inferior temporal lobe (in the object-properties-encoding subsystem), and we inferred in Chapter 6 that unimodal auditory information is probably stored in temporal-parietal cortex." - - Kosslyn and K\"onig \cite[p. 344]{kosslyn1992wet} "information in associative memory can be activated by input from any perceptual modality". Associations can be established between representations in perceptual memory and in associative memory. "memory formation subsystems rely on anatomical structures \ldots the principal members of this set being the \textit{hippocampus} (and related cortex), the \textit{limbic thalamus}, and the \textit{basal forebrain}." - - Kosslyn and K\"onig \cite[p. 345]{kosslyn1992wet} "the hippocampus receives input from a number of other structures (the septum and the hypothalamus, via the fornix; the anterior thalamic nucleus and the subcallosal area, via the cingulum; and the amygdala). \ldots The hypothalamus appears to be involved in motivation, and the amygdala appears to have a role in emotion; clearly both factors affect what we remember. \ldots The hippocampus plays a critical role in the storage of new perceptual representations. \ldots also plays a critical role in storing associations between representations. (Mishkin and Appenzeller 1987, Squire 1987)" Long term potentiation is a phenomenon of hippocampal cells in neural microcircuits involved in storing associations between representations. - - Kosslyn and K\"onig \cite[p. 346]{kosslyn1992wet} "many of the thalamic nuclei appear to be involved in attentional processes (Posner and others) \ldots The basal forebrain \ldots in turn issues a signal that new representations and/or associations should be stored.(Mishkin and Appenzeller 1987) This is a biochemical signal, consisting of the release of \textit{acetylcholine}." - - Kosslyn and K\"onig \cite[p. 347]{kosslyn1992wet} "we shall decompose the memory formation subsystem into two more precisely characterized subsystems, \ldots these subsystems are involved in initiating the learning sequence, and in changing selected connections strengths in particular neural networks, respectively." - - Kosslyn and K\"onig \cite[p. 347]{kosslyn1992wet} "the \textit{striatum} plays a critical role in skill acquisition \ldots the striatum receives information from cortical perceptual areas". - - Kosslyn and K\"onig \cite[p. 349--350]{kosslyn1992wet} "If novel stimuli are perceived, \ldots when a match is not obtained, this information is sent to the frontal lobes and provides input to the print-now sybsystem; recall that perceptual encoding subsystems have anatomical projects into the frontal lobe (Goldman-Rakic, 1987) Thus, the perceptual encoding subsystems send outputs to the associative memory \ldots Structural changes are initiated that will allow the systems later to reconstitute the pattern of activation evoked by the novel stimulus" - - Kosslyn and K\"onig \cite[p. 351]{kosslyn1992wet} "The key to storing new information in memory is the ability to change the 'strengths' of connections among neurons in just the right way. This \ldots has been documented in actual neural networks (Kandel and Schwartz, 1985, Shepherd 1988)" - - Kosslyn and K\"onig \cite[p. 353]{kosslyn1992wet} "These changes (in synaptic connections) apparently begin in the hippocampus \ldots this phenomenon is called long term potentiation. \ldots it can last hours, days, weeks, or even longer -- depending on \ldots as well as various properties of the stimulus." - - $$ It is the accumulation of surface area from vesicles delivering neurotransmitter at the presynaptic cell in the synaptic cleft that creates microenvironments that are more effective (NMDA activated/molecule of neurotransmitter delivered) in activating the postsynatic neuron, since the corresponding membrane that is used to restore the number of vesicles is taken from smooth area.$$ - - Kosslyn and K\"onig \cite[p. 358--359]{kosslyn1992wet} "The idea that associative memory hooks back into unimodal perceptual representations is consistent with a range of clinical findings. \ldots Squire suggests that the perceptual systems that encode information may actually store much of it.(Squire 87) \ldots We speculate that associative memory depends in part on the superior, posterior temporal lobe, if only because patients with lesions in this area often appear to have disrupted associations." - - Kosslyn and K\"onig \cite[p. 361]{kosslyn1992wet} "When an object is later perceived and input enters associative memory, the relevant associations are activated." - - Kosslyn and K\"onig \cite[p. 370--371]{kosslyn1992wet} "We often store new information even if it has no obvious relevance to any goal or problem at hand. This is called \textit{incidental} memory.\ldots if one pays attention to a stimulus, it is likely to be stored 'automatically,' with no decision to do so. \ldots one tries to memorize the information, and is more likely to remembers it than if the effort were not made. This sort of memory is called \textit{intentional} memory. \ldots the longer the information is attended to, the more likely it is that the memory formation subsystems eventually will store it in memory \ldots if the property lookup subsystems access more information about the to-be-remembered material, memory will be improved. \ldots called a \textit{depth of processing} effect" - - Kosslyn and K\"onig \cite[p. 372]{kosslyn1992wet} "one can store information relatively effectively by inventing distinctive \textit{retrieval cues}. \ldots Allan Paivio reviews a large amount of evidence that we remember information better when we use a 'dual code' (visual and verbal) than when we store it in only a single way (Paivio 1971)" - - Kosslyn and K\"onig \cite[p. 373]{kosslyn1992wet} "when one becomes an expert in any domain, one often cannot report how one performs the task. Much, if not most, of the information in memory cannot be directly accessed and communicated" - - $$ it is not in conscious memory anymore$$ - Lisman and Sternberg\cite{lisman2013habit} talk about Habit and nonhabit systems for unconscious and conscious behavior: Implications for multitasking. - - Peter Graf and Daniel Schacter implicit memory - - Kosslyn and K\"onig \cite[p. 375]{kosslyn1992wet} "There is considerable evidence that priming tasks and explicit memory tasks rely on distinct processing subsystems \ldots certain drugs impair both recall and recognition in explicit memory tasks, but do not affect the magnitude of priming." - - Kosslyn and K\"onig \cite[p. 376]{kosslyn1992wet} In priming, "The words apparently were processed to some level within the system, even when the patient was fully unconscious." - - Kosslyn and K\"onig \cite[p. 377]{kosslyn1992wet} "priming has two components, only one of which is perceptual" There is an advantage to having the priming and the recall be in the same sensory modality, only for the right hemisphere. " Graf and Schacter also showed that implicit memory for associations between words is modality-specific. - - $$ to make the most of the priming effect, the students should have practice writing responses to stimuli, like test questions. For memorization type test questions, that are answered by handwriting, note-taking in class, of the same item, should help.$$ - - Kosslyn and K\"onig \cite[p. 380]{kosslyn1992wet} "Mishkin suggests that a subcortical structure called the \textit{substantia nigra} is critically involved in the 'reinforcement' process that strengthens connections between perceptual states and responses." - Stimulus/response learning, striatum, dopamine. Is this what is used in flash cards? This kind of learning is restricted, [p. 383]"the response can only be evoked by the appropriate stimulus" - - Kosslyn and K\"onig \cite[p. 387]{kosslyn1992wet} "material in \textit{working memory} is used to aid reasoning processes (Baddeley 1986). Reasoning processes only can operate on information in short-term memory, but relatively little information can be stored in short-term memory." - $$ so lemmas are good$$ - - Kosslyn and K\"onig \cite[p. 387-388]{kosslyn1992wet} " We interpret the short-term memory structures Patricia Goldman-Rakic reports in the frontal lobe (Goldman-Rakic 1987, 1988) as extension of the perceptual encoding subsystems, which serve to make perceptual information immediately available to the decision processes; according to this view, it is debatable whether one want to conceive of the information as actually being stored -- as opposed to monitored -- in the frontal lobe. - Working memory, then, corresponds to the activated information in the long-term memories, the information in short-term memories, and the decision processes that manage which information is activated in the long-term memories and retained in the short-term memories(Kosslyn 1991)" - - Kosslyn and K\"onig \cite[p. 398]{kosslyn1992wet} "recall that Goldman-Rakic provides evidence that perceptual information ultimately projects to the frontal lobe(Goldman-Rakic 1987, 1988)" - - Kosslyn and K\"onig \cite[p. 398--399]{kosslyn1992wet}explicit memory formation uses acetylcholine, modifying the strengths of connections in the appropriate networks, implemented in part in the hippocampus and related cortex - - - - - - - - Squire and Dede\cite{} - Martin\cite{martin2015grapes}"some of the latest functional - neuroimaging findings on the organization of object concepts - in the human brain. I argue that these data provide strong - support for viewing concepts as the products of highly inter- - active neural circuits grounded in the action, perception, and - emotion systems. The nodes of these circuits are defined by - regions representing specific object properties (e.g., form, color, and motion) and thus are property-specific, rather than - strictly modality-specific. How these circuits are modified - by external and internal environmental demands, the distinction between representational content and format, and the - grounding of abstract social concepts are also discussed." - - $$we have modality specific long term, and we have association area for other than single modality, we have consolidation and reconsolidation, do we have spatial arrays of concepts$$ - \subsection{Brain Imaging} - Bachmann\cite{bachmann2015brain} discusses "brain-imaging markers of neural correlates of consciousness" - $$when we are learning a skill we can be conscious of exercising that skill, and later, we can become less conscious of specifically how we exercise that skill. Is the part of the brain that is conscious moving? Is the location of the memory moving? Could be both. Don't some motor skills move to the cerebellum, and aren't there problems with this for dyslexics? What's going on when something suddenly appears in consciousness? I think the metabolic activity brings the consciousness to the place where the memory is. But in these kind of answers popping into consciousness (Poincar\'e?) isn't it a new synthesis that pops into consciousness?$$ - - Luck and LeClerc \cite{luck2014potentiation} wrote about the Potentiation of Associative Memory by Emotions: An Event-Related FMRI Study, - - Brain imaging provides evidence for believing that creativity, (generating novel ideas, such as proofs) can be improved by training and takes time corresponding to reorganizing intercortical interactions\cite{kowatari2009neural}. (Look here kowatari more, there is something about predominance of right prefrontal over left.) - According to Huang et al.\cite{huang2015highest}, a large body of research suggests that an abstract cognitive processing style produces greater creativity. Empirically, decades of work have shown that both abstract thinking and creativity are consistently linked to right-hemispheric activation in the brain (e.g., Fink et al., 1996\cite{fink1996brain} and Mihov et al., 2010\cite{Mihov2010442}) - . - - Squire tells us \cite{squire2015conscious} "findings suggest that fMRI activity in the medial temporal lobe reflects processes related to the formation of long-term memory" - - Chambers et al.~\cite[p. 1045]{chambers2003developmental}, citing Gurden et al, Mulder et al., Robinson and Kolb, and Hyman and Melenka~\cite{gurden1999integrity,mulder1997short,hyman2001addiction} state "Dopamine transmission in nucleus accumbens and prefrontal cortex regions projecting to the nucleus accumbens has been implicated in mechanisms of learning and plasticity, including changes in long-term potentiation and morphology of neuronal dendritic trees." - - \subsection{Brain structure} - - \begin{quote} - Who can trace out the secret threads by which our concepts are united?\\ - Hermann von Helmholtz - \end{quote}\cite{kounios2015eureka} - - Takeuchi et al. \cite{takeuchi2010training} Training of working memory impacts structural connectivity - - Melby-Lerv{\aa}g \cite{melby2013working} "current findings cast doubt on both the clinical relevance of working memory training programs and their utility as methods of enhancing cognitive functioning in typically developing children and healthy adults. " - - Rutishauser et al.\cite[p. 104-105]{fried2014single} medial temporal lobe "lesion subjects showed reduced (but above chance) recognition memory performance but, strikingly, has a complete lack of performance improvement for items shown together with task-irrelevant novel attributes. In contrast, normal control subjects had a substantial gain in memory performance for these items (Kishiyama et al., 2004\cite{kishiyama2004restorff}). Together, this indicates that the 'von Restorff effect' (Wallace, 1965\cite{wallace1965review}; Kinsbourne \& George, 1974\cite{kinsbourne1974mechanism}; Hunt, 1995\cite{hunt1995subtlety}; Parker et al., 1998\cite{parker1998restorff}) is driven by neuronal mechanisms that reside in the MTL. \ldots Novelty responses are thus a sensitive measure to quantify learning and plasticity. \ldots We identified a subpopulation of single neurons in the hippocampus and the amygdala that showed striking differences in their spiking response. \ldots the same cell would indicate the novelty of a stimulus regardless of which category the stimulus was from. \ldots These cells \ldots also show changes in firing rate as a function of repeated presentation even when subjects are only passively viewing stimuli without an explicit memory task (Perdreira et al., 2010\cite{pedreira2010responses})" - - Rutishauser et al.\cite[p. 106]{rutishauser2014single} "it seems that there are at least two distinct classes of novelty-sensitive single neurons in the human MTL: abstract and visually tuned. The first class of untuned general novelty detectors could serve to signal the significance of stimuli during the acquisition of new memories (Lisman \& Otmakova, 2001\cite{lisman2001storage}). It has been suggested that such neurons trigger dopaminergic release through projections to the ventral tegmental area (Lisman \& Grace, 2005\cite{lisman2005hippocampal})" - - Mormann et al.\cite[p. 131--133]{mormann2014visual} reports "Another seminal study described neurons representing a specific semantic concept in the human MTL (Quian Quiroga et al., 2005). \ldots A problem in determining the precise turning curves of semantic neurons in the MTL is to find a suitable parameterization of the stimulus space \ldots compounded in human studies by the short duration of any one experiment in any one patient." - - Ouchi et al.\cite{ouchi2013reduced} "Adult neurogenesis is known to be important in hippocampus-dependent memory" - - Ojemann\cite[p. 262]{ojemann2014human} has demonstrated that lateral temporal neurons are involved in recent memory, making use of attention to increase the success of consolidation of memory of items competing with distractions. "The evolutionary changes in the brain orgainzation for recent memory involve an expansion of the cortical component with relative conservation of the medial temporal -- hippocampal component. \ldots A substantial literature has established the importance of the temporal lobe in learning. \ldots neurons with changes during \ldots memory were significantly more likely to have associative learning changes." - - Kowatari et al.\cite{kowatari2009neural} state "In the experts, creativity was quantitatively correlated with the degree of dominance of the right prefrontal cortex over that of the left, \ldots Our results supported the hypothesis that training increases creativity via reorganized intercortical interactions." - - Waisman et al.\cite{waisman2014brain} observe that "various studies demonstrate that when complexity of the (arithmetic) problems rises, more brain areas simultaneously support the solving process", citing Zamarian et al.\cite{zamarian2009neuroscience}. - - Waisman et al.\cite{waisman2014brain} investigated "cortical activity associated with solving problems that require translation between symbolic and graphical representations". - - Waisman et al.\cite[p.691]{waisman2014brain} stated that the "posterior parietal cortex is know to be activated when mental representations are manipulated (Zacks 2008)\cite{zacks2008neuroimaging}. - - Deng et al.\cite{deng2010new} report that "Neurons born in the subventricular zone(SVZ) differentiate and integrate into the local neural network as granule cells of the dentate gyrus." - - Knutson et al.\cite{knutson2001anticipation} state that a region in the nucleus accumbens codes for expected positive incentive value, and - \cite[p. 4]{knutson2001anticipation} it is "an apparently lateralized response of the right nucleus accumbens." - - Chambers et al.~\cite{chambers2003developmental} explained that "Adolescent neurodevelopment occurs in brain regions associated with motivation, impulsivity, and addiction. Adolescent impulsivity and/or novelty seeking as a transitional trait behavior can be explained in part by maturational changes in frontal cortical and subcortical monoaminergic systems. These developmental processes may advantagiously promote learning drives" - - $$ these systems that we are trying to use to shepherd information from the place it begins (sensory to MTL) to the right prefrontal cortex, from where it can be retrieved for creative application to problem solving, are developing$$ - - \begin{figure} - \centering - \includegraphics[width=0.7\linewidth]{./l73f1.jpeg} - \caption{In this figure reprinted from Chambers\cite{chambers2003developmental}, (I plan to do something, either redraw or ask permission) we see how sensory input, new information, is delivered to hippocampus. Other sources have shown us that new neurons are created in or near hippocampus that are a response to new information arriving and related to memory for the new information. We have seen how presence of dopamine, from striatum and VTA assist the consolidation of new memory into longer term memory. We have seen that for monomodal information, the long term memory is stored in cortex near where input is provided by that modality (visual cortex, auditory cortex) and for multimodal information, the long term memory is stored in association cortex. We have seen that information stored in association cortex is more readily retrieved, as any of the associated modalities can help retrieve it. We have seen that this consolidation requires protein and is facilitated by sleep, (I forget which of REM or slow wave sleep.) We have seen how reconsolidation can occur, assisted by nucleus accumbens, and can result in information accessible to the prefrontal cortex, on the right side. We have seen how anticipation of positive reward activates the nucleus accumbens on the right side. Could it be that anticipation of positive reward occurs in REM sleep, I wonder.} - \label{fig:poster23} - \end{figure} - - Anderson et al.~\cite[p. 53]{anderson2011cognitive} state "There is some reason to suspect that the angular gyrus (ANG) may also be engaged to serve the metacognitive activities of monitoring and reflecting." (on non-routine problem solving). They go on to say "Regions close to the right ANG have been found to play a variety of metacognitive functions, citing a review by Decety \& Lamm\cite{decety2007role}. - - Anderson et al.~\cite[p. 54]{anderson2011cognitive} state "Another region that is potentially involved in metacognition is Brodmann Area 10 or frontopolar cortex (FPC), particularly its lateral portion, (citing Fletcher and Henson\cite{fletcher2001frontal}). A number of converging lines of research suggest that this region of the brain may be critical in the ability to extend knowledge." - Anderson et al.~\cite[p. 58]{anderson2011cognitive} state "The left ANG is often distinguished from the right in many theories including the triple code, but the pattern of ANG effects in this experiment is basically the same in the two hemispheres." - Anderson et al.~\cite[p. 62]{anderson2011cognitive} state "In every case (brain areas related to metacognition), the patterns are roughly bilaterally symmetric." - - - \subsection{Brain function} - - - - R. Quian Quiroga\cite{quiroga2012concept} opines that concept cells are the building blocks of declarative memory functions. - - Suthana et al.\cite{suthana2012memory} report on Memory enhancement and deep-brain stimulation of the entorhinal area. - - Imamoglu (fix the accents) et al.\cite{imamoglu2012changes} discuss changes in functional connectivity support conscious object recognition. - - Murayama and Kitagami\cite{murayama2014consolidation}dopaminergic memory consolidation effect can result from extrinsic reward. - - $$ - so, give a little quiz at the end of lecture, covering the main points, and hand out tickets in exchange for handing in quizzes. Then, tickets can be handed in with homework to count for points. So they are paid at the time they are thinking about quiz contents, and that is expected to help with consolidation of the material on the quiz.$$ - - Born and Wilhelm\cite{born2012system} discuss System consolidation of memory during sleep - - Diekelmann et al.\cite{diekelmann2012offline} describe that Offline consolidation of memory varies with time in slow wave sleep and can be accelerated by cuing memory reactivations. - - Taylor et al.\cite{tayler2013reactivation} describe Reactivation of neural ensembles during the retrieval of recent and remote memory. - - Cowansage et al.\cite{cowansage2014direct} describe the Direct reactivation of a coherent neocortical memory of context - - Lustenberger et al.\cite{lustenberger2012triangular} discuss a triangular relationship between sleep spindle activity, general cognitive ability and the efficiency of declarative learning. - - Roux and Uhlhaas\cite{roux2014working} consider working memory and neural oscillations, questioning whether alpha--gamma versus theta--gamma codes for distinct WM information. - - Walker and Stickgold\cite{walker2014sleep} consider Sleep, memory and plasticity. - - Tonnoni and Cirelli\cite{tononi2014sleep} discuss Sleep and the price of plasticity: from synaptic and cellular homeostasis to memory consolidation and integration. - - Rutishauser et al.\cite[p. 107]{rutishauser2014single} "Neurons coded this information very reliably: The decoder (a function of a population of neurons) could tell (for correct trials) whether the stimulus was new or old on a single trial basis with an accuracy of 75\% \ldots (using) a single previously identified novelty/familiarity neuron. Performance increased to 93\% if six neurons were considered. \ldots for 75\% of error trials, the decoder predicted what the correct response would have been (but which was not given by the subject). \ldots a simple decoder outperform(s) the patient \ldots The neurons have better memory than the patient demonstrated behaviorally," - - Rutishauser et al.\cite[p. 111]{rutishauser2014single} "Many factors modulate the probability that a memory for a stimulus will be formed. Examples include attention, motivation, and saliency of the stimulus (Paller \& Wagner, 2002\cite{paller2002observing}). Structurally, the modification of synaptic circuits by plasticity mechanisms is thought to underlie memory formation (Martin et al., 2000\cite{martin2000synaptic}) - - Rutishauser et al.\cite[p. 112]{rutishauser2014single} "findings from animal literature indicate that theta oscillations and the timing of neuronal activity relative to the ongoing theta oscillations have a strong influence on plasticity as well as learning, suggesting the possibility that the two are functionally linked by theta oscillations. \ldots The power of theta oscillations measured on the scalp \ldots can be predictive of whether a memory is formed or not (Klimesch et al., 1996\cite{klimesch1996theta}; Sederberg et al., 2003\cite{sederberg2003theta}) \ldots what is relevant for whether a memory was formed or not is whether the preferred phase of a particular neuron was followed faithfully and not the absolute phase \ldots what was predictive (of whether a memory was formed or not) was whether the spikes that were fired were phase locked to ongoing theta or not." - - Rutishauser et al.\cite[p. 113]{rutishauser2014single} "the spike field coherence at the time of learning was already indicative of whether a memory was later strong or weak." - - Mormann et al.\cite[p. 140]{mormann2014visual} state "The process of encoding episodic memories consists of associating pieces of semantic information (what happened where with whom involved and so on) in a defined temporal order. Lesion studies in humans have shown that structures in the MTL are essential for the encoding of episodic memories (Squire et al., 2004; Squire et al., 2007; Squire, 2009; Milner et al. 1968, Scoville \& Millner 1957). Representations of semantic information at the single unit level are frequently found in these very structures and thus might provide a unique opportunity to investigate how our brain links pieces of semantic information together into episodic memories (Quiroga, 2012). \ldots Episodic memories are always context dependent whereas semantic memories are context invariant and can emerge via generalization of recurring context-dependent experiences (Buzsaki, 2005)." - - Mormann et al.\cite[p. 142--143]{mormann2014visual} state "Earlier notions that the amygdala might be specialized to elicit or mediate fear responses (LeDoux, 1996) have been supplemented by more abstract accounts whereby the amygdala processes ambiguity or unpredictability in the environment (Herry et al., 2007) and mediates an organism's vigilance and arousal (Davis and Whalen, 2001). \ldots In humans, neuroimaging studies of the amygdala argue for a broad role in processing stimuli that are strongly rewarding or punishing (Sander et al., 2003; Ohman et al., 2007) \ldots In human fMRI memory studies, increased blood flow in the amygdala during encoding was found to correlate with improved memory formation (e.g, Canli et al., 2000). In addition, the phase locking of human amygdala neurons to ongoing theta oscillations was found to predict memory formation (Rutishauser et al., 2011) \ldots there might be a distinction between the hippocampus and the amygdala in terms of the extent to which unconscious information reaches those areas. If indeed unconscious information can reach the amygdala but not the hippocampus and surrounding structures" - - Paz and Pare\cite{paz2013physiological}" in emotionally - arousing conditions, whether positively or negatively valenced, - the amygdala allows incoming information to be processed - more efficiently in distributed cerebral networks." - - $$ This is just corroborating that infectious enthusiasm for a subject, on the part of the instructor, helps students learn.$$ - - Schwabe et al. \cite{schwabe2014reconsolidation} there is reconsolidation - - Patel et al.\cite[p. 205]{patel2014human} "Functions such as reward processing, motivations, and learning have since been attribted to basal ganglia circuits." - - Patel et al.\cite[p. 207]{patel2014human} "Examined impact of emotional valance and cACC responsitveness to complext attnetion tasks \dots Examined reward properties of dopaminergic neurons using virtual financial reward" - - Patel et al.\cite[p. 208--209]{patel2014human} "when there is a differentce in expected and actual outcome -- a prediction error signal -- midbrain dopaminergic neurons rapidly fire at the onset of the unexpected reward. \ldots This feature is thought to drive reward-based learning and adaptive behavior. \ldots It is thought that phasic dopaminergic activity is the neural substrate of this type (classical conditioning) of learning (Schultz, 1998) \ldots dopamine release in the striatum" - - $$ So, deliver unexpected rewards. Maybe some clicker questions have greater value.$$ - - Patel et al.\cite[p. 210]{patel2014human} "the ventral striatum has become a focal point in studies of reinforcement learning, \ldots phasically active neurons, thought to be the medium spiny neurons that make up about 95\% of striatal neurons, are more relevant to strengthiening functional circuits diring instrumental conditioning and procedural learning (Graybiel 2008, Jog 1999)" anticipation of rewards, including secondary rewards (monetary) - - - Patel et al.\cite[p. 212]{patel2014human} "NAcc activity reliably encodes the anticipation of reward proportional to reward magnitude" - - Patel et al.\cite[p. 213--214]{patel2014human} "each subregion of the ACC is predomintely involved with distinct roles, such as motivation and cognition (Paus, 2001). Notably these areas are not homogeneous and demonstrate some overlap in function. \ldots The portion of the ACC implicated in reward processing and cognition is the dACC. \ldots The vmPFC is strongly connected to the limbic system, which is a group of brain structures that are implicated in several functions such as emotion, motivation, and memory." - - Ojemann\cite[p. 268]{ojemann2014human} states "there are also 'subconscious' or 'implicit' memory processes, processes of which the subject is unaware that change performance. One such process is repetition priming, a shortening of reaction time with repeated presentation of the same item. Changes in lateral temporal cortical single neuron activity related to this, \ldots that study included recordings throughout lateral temporal cortex, neurons with those significant implicit memory changes were all in superior temporal gyrus or superior portion of middle gyrus, significantly more superior and posterior than neurons with recent memory processes \ldots implicit memory then involves neural networks in posterior superior temporal cortex, largely separate from those for recent memory" - - Rutishauser et al.\cite[p. 348]{rutishauser2014next} state "Different timescales of memory formation have been described at the neuronal level." - - Rutishauser et al.\cite[p. 348]{rutishauser2014next} "It has remained very difficult to directly link the mechanisms of synaptic plasticity to memories" - %$$ Hebbian by deposition of membrane, increasing the folds per volume, because membrane harvesting for vesicle retrieval is from flat$$ - - Rutishauser et al.\cite[p. 348]{rutishauser2014next} "We are able to recall memories that were established years or even decades ago, but the processes by which such remote call(sic) works remain largely unknown." - %$$ it is one cell activating another, and glial cells bringing in blood to the region in response to activated neuron. Remember, it was Wilder Penfield stimulating neurons that brought back remote memories$$ - Rutishauser et al.\cite[p. 348]{rutishauser2014next} "emotions are likely to play a key role in memory formation (Cahill et al. 1995, Fanselow \& Gale 2004, Phelps, 2004) - - - Rutishauser et al.\cite[p. 351]{rutishauser2014next} "There is ample evidence that multiple tasks show sleep-dependent enhancement (Stickgold, 2005)" - - There is a journal Neurobiology of Learning and Memory.(single neuron book p 182) - - Patel et al.\cite[p. 211]{patel2014human} "Knutson and colleagues further demonstrated that NAcc activation increases proportional to the magnitude of the anticipated monetary reward (Knutson et al, 2001) - - According to Kowatari et al.\cite[p.1679]{kowatari2009neural}, Carlsson et al.\cite{carlsson2000neurobiology} "reported that both hemispheres were involved in highly creative subjects." Kowatari et al. \cite[p.1679]{kowatari2009neural} "found that professional training reorganized brain activation patterns, which was correlated with increased creativity." - Kowatari et al. \cite[p.1682]{kowatari2009neural} "In the expert group, a right and left hemispheric difference was obvious; only the right prefrontal cortex (PFC) and parietal cortex (PC) were activated in the expert group, whereas in the novice group, bilateral PFC and PC were activated." - Kowatari et al. \cite[p.1682]{kowatari2009neural} "This results indicated that the direct or indirect interaction between the right and left PFC might contribute to producing highly original designs in the expert group." - - Kowatari et al. \cite[p.1683]{kowatari2009neural} "Based on these observations, we postulated that \ldots brain regions that are involved in yielding high creativity indices shifted from the PC to the PFC." - - Deng et al.\cite[p. 341]{deng2010new} report that "the adult born dentate granule cells (DGCs) exhibit stronger synaptic plasticity than mature DGCs, as indicated by their lower threshold for the induction of long term potentiation (LTP) and their higher LTP amplitude. and, citing Toni et al.\cite{toni2008neurons}, say "structural modification of dendritic spines and axonal boutons continues to occur as the just-born DGCs become older" Toni et al.\cite{toni2008neurons} report that neurons born in the adult dentate gyrus form functional synapses with target cells. - - Deng et al.\cite[p. 343]{deng2010new} report, citing Kee et al. \cite{kee2007preferential} that "learning by a mouse when a set of adult-born DGC was at least 4-6 weeks of age led to preferential activation of these cells during memory retrieval" during the learned task "when the DGCs were 10 weeks old." - - Trouche et al. \cite{trouche2009recruitment} state that recruitment of adult-generated neurons into functional hippocampal networks contributes to updating and strengthening of spatial memory. - - Deng et al.\cite[p. 343]{deng2010new} surmise that finding suggest that, compared with their mature counterparts, adult-born DGCs may be specifically activated by an animal's experiences and thus can make unique contributions to learning and memory. - - Deng et al.\cite[p. 344]{deng2010new} opine that neurogenesis allows plasticity to be mostly localized to newborn immature DGCs, preserving the information that is represented by mature DGCs. %dentate granule cells - - Deng et al.\cite[p. 348]{deng2010new} report, citing Kitamura et al.\cite{kitamura2009adult} "a recent study in mice suggested that adult neurogenesis facilitated memory reorganization that led to a gradual reduction of the hippocampus-dependence of memories and the permanent storage of these memories in extra-hippocampal regions. - - Diekelmann and Born\cite{diekelmann2010memory} state that "Sleep has been identified as a state that optimizes the consolidation of newly acquired information in memory, depending on the specific conditions of learning and timing of sleep. Consolidation during sleep promotes both quantitative and qualitative changes of memory representations. Through specific patterns of neuromodulator activity and electric field potential oscillations, slow-wave sleep (SWS) and rapid eye movement (REM) sleep support system consolidation and synaptic consolidation, respectively. During SWS, slow oscillations, spindles and ripples -- at minimum cholinergic activity -- coordinated the re-activation and redistribution of hippocampus-dependent memories to neocortical sites, whereas during REM sleep, local increases in plasticity-related immediate-early gene activity -- at high cholinergic and theta activity -- might favour the subsequent synaptic consolidations of memories in the cortex." - - Diekelmann and Born\cite{diekelmann2010memory} state that, among its functions, "sleep's role in the establishment of memories seems to be particularly important". promoting primarily the consolidation of memory. - They tell us that "Consolidation refers to a process that transforms new and initially labile memories encoded in the awake state into more stable representations that become integrated into the network of pre-existing long-term memories.", involving active re-processing of 'fresh' memories within the neuronal networks that were used for encoding them. - - Diekelmann and Born\cite{diekelmann2010memory} , citing Stickgold et al., and Walker et al.\cite{stickgold2000visual.walker2003dissociable} state that "For optimal benefit on procedural memory consolidation, sleep does not need to occur immediately but should happen on the same day as initial training." - - Chambers et al.~\cite[p. 1044]{chambers2003developmental}, when discussing events activating loops within primary motivation circuity, state "These events may also facilitate mechanisms of neuroplasticity among nucleus accumbens neurons, and their afferents." - - Chambers et al.~\cite[p. 1045]{chambers2003developmental}, citing Masterman et al.~\cite{masterman1997frontal}, stated "dopamine release into the nucleus accumbens is associated with motivational stimuli, subjective reward, premotor cognition(thought), and learning of new behaviors" - - Chambers et al.~\cite[p. 1045]{chambers2003developmental}, citing Waelit et al.~\cite{waelti2001dopamine}, stated "Rewards delivered in intermittent, random, or unexpected fashions have greater capacity over repeated trials to maintain dopamine cell firing and reward-conditioned behavior." - - Chambers et al.~\cite[p. 1045]{chambers2003developmental} state that "well-learned motivated behaviors or habits performed under expected contingencies become less dependent on nucleus accumbens dopamine release." - - Chambers et al.~\cite[p. 1045]{chambers2003developmental}, citing Yates~\cite{yates1990theories} "In adolescence, the prefrontal cortex has not yet maximized a variety of cognitive functions \ldots Measures of prefrontal cortex function, including working memory, complex problem solving, abstract thinking, and sustained logical thinking, improve markedly during adolescence. - - Wittmann et al.~\cite{wittmann2005reward}show that activation of dopaminergic midbrain is associated with enhanced hippocampus-dependent long term memory formation. - - Wittmann et al.~\cite{wittmann2005reward} state that reward anticipation reliably elicits a dopaminergic response. They hypothesize that the known improved dopamine driven synaptic plasticity and long-term potentiation is associated with better memory consolidation in the hippocampus. Their findings are consistent with they hypothesis that activation of dopaminergic midbrain regions enhances hippocampus-dependent memory formation, possibly by enhancing consolidation. They have shown that activity of the ventral tegmental area and medial substantia nigra accompanied hippocampal activity related to memory formation, in that both structures were activated by novelty and in relation to subsequent free recall performance. The areas that respond to signals related to reward are the dopaminergic areas when the response is reward prediction and the mesial frontal cortex after learning the contingency between the predicting stimulus and the reward; there is a shift with learning.\cite{knutson2003region}. Their findings support their hypothesis that the hippocampus is a major site for the neuromodulatory influence of reward on long-term memory formation. - - Lee et al.~\cite{lee2007strategic} reported on neuroanatomical correlates of converting between symbolic algebra and a pictorial representation. They found that for conversion in either direction, the active areas include the both left frontal gyri, intraparietal sulci bilaterally, which are linked to working memory and quantitative processing. They also found that using the symbolic method activated the posterior superior parietal lobules and the precuneus, in contrast to the pictorial method. They conclude that the symbolic and pictorial strategies impose different attentional demands. - - Lee et al.~\cite{lee2007strategic}, citing Anderson et al.~\cite{anderson2003information}, observe that algebraic transformation is subserved by the left posterior parietal region and the left dorsal lateral prefrontal cortex. Lee et al.~\cite{lee2007strategic}, citing Sohn et al.~\cite{sohn2004behavioral}, who found that "anterior prefrontal activation - was greater in the story condition and posterior parietal - activation was greater in the equation condition". - - $$ then creativity is found in the same or nearby location to the storytelling way, rather than the symbolic representation way$$ - - Lee et al.~\cite[p. 167]{lee2007strategic} report the symbolic method was associated with activation in the left precuneus and bilateral posterior superior parietal lobules. This finding suggest the symbolic condition recruited attentional processes more extensively than did the model method. Also activated were various loci in the visual processing area and in the basal ganglia. The model condition did not activate any areas beyond those activated by the symbolic ". According to Lee et al.~\cite[p. 167]{lee2007strategic}, both the precuneus and the posterior superior parietal lobules are associated with attentional processes. - $$so, maybe it takes more attention to work with symbols. After all, they are more concise$$ - - Lee et al.~\cite[p. 168]{lee2007strategic} citing Owen et al.~\cite{owen2005n} say that dorsolateral prefrontal, overlapping middle frontal is involved in reorganizing material into pre-existing knowledge structures. - - $$ thinking about proof might use this $$ - - Lee et al.~\cite[p. 169]{lee2007strategic} found that "occipital areas were activated in the algebraic condition. This suggests participants spent more time viewing the questions in the algebraic conditions.\ldots activation in the posterior superior parietal lobules might be related" - - Wittmann et al.~\cite{wittmann2007anticipation} found that the "hippocampus differed from the response profile of SN/VTA in responding to expected and 'unexpected' novelty". Their results demonstrate parallels between the processing of novelty and reward in the SN/VTA. Their fMRI analysis revealed that cues predicting novel images elicited significantly higher SN/VTA activation than cues predicting familiar stimuli.[p. 198] - Wittmann et al.~\cite[p. 200]{wittmann2007anticipation} "Irrespective of whether the dopaminergic midbrain drives the hippocampus or vice versa, coactivation of the hippocampus and SN/VTA could be associated with increased dopaminergic input to the hippocampus during anticipation. This, in turn, could induce a state that enhances learning for upcoming novel stimuli" - - Keller and Menon~\cite{keller2009gender} studied brain activation during mathematical cognition, and compared men and women. They found that the same brain areas were used: right intr-parietal sulcus areas and angular gyrus regions, ventral stream of right lingual and parahippocampal gyri. "Females had greater regional density and greater regional volume where males showed greater fMRI activation. \ldots Our findings provide evidence for gender differences in the functional and structural organization of the right hemisphere brain areas involved in mathematical cognition. Together with the lack of behavioral differences, our results point to more efficient use of neural processing resources in females." - Keller and Menon~\cite[p. 348]{keller2009gender}stated "Gender differences were all localized to the right posterior regions of the brain." - - Diekelmann et al.~\cite[p. 116]{diekelmann2010memory} observes that "a great number of studies indicate that sleep supports consolidation of memory in all major memory systems \ldots There is growing evidence that explicit encoding, even in procedural tasks, involves a dialogue between the prefrontal cortex and the hippocampus, citing \cite{schendan2003fmri}, which also integrates intentional and motivational aspects \ldots Sleep changes memory representations quantitatively and qualitatively. \ldots a strengthening of associations \ldots qualitative changes in memory representations." - - $Isay$ In order to encourage the prefrontal cortex involvement, we should be explicit about motivation for choosing one inference rule over another as the demonstration / pedagogical proofs are exhibited.$$ - - Diekelmann et al.~\cite[p. 116]{diekelmann2010memory} observes that "subjects learned single relations between different objects which, unknown to the subject, relied upon an embedded hierarchy, citing \cite{ellenbogen2007human}. When learning was followed by sleep, subjects at a re-test were better at inferring the relationship between the most distant object, which had not been learned before. Likewise, after sleep subjects more easily solved a logical calculus problem that they were unable to solve before sleep or after corresponding intervals of wakefulness citing \cite{wagner2004sleep}. Of note, sleep facilitated the gain of insight into the problem only if adequate encoding of the task was ensured before sleep." - $$ need the definition of adequate encoding. Does this use of encoding refer to placing the representation into, say, hippocampus?$$ - Diekelmann et al.~\cite[p. 116]{diekelmann2010memory} "sleep can re-organize newly encoded memory representations, enabling generation of new assoications and the extraction of invariant features" - $$ here is generalization, abstraction, we want cs students to learn$$ - Diekelmann et al.~\cite[p. 116]{diekelmann2010memory} "from complex stimuli, and therby easing novel inferences and insights. Re-organization of memory representations during sleep also promotes the transformation of implicit into explicit knowledge \ldots procedural and declarative memory systems interact during sleep-dependent consolidation." - They also observe that once implicit memory has become explicit, subjects no longer showed improvement in implicit procedural skill. - $$I'm connecting explicit with conscious. Once the subject is conscious of (interiorized after internalized, using Harel and Sowder's 1998 scheme, thinking about what you are doing as you do it) the procedure, they carry it out consciously, which could be more slowly. Maybe eventually they will become unconscious of how they carry out this skill, maybe the slowing process of conscious execution will drop away$$ - - (what has Stickgold been learning about sleep, memory, lately?) - - Diekelmann et al.~\cite[p. 117]{diekelmann2010memory} "It is assumed that the re-activations during system consolidation stimulate the redistribution of hippocampal memories to neocortical storage sites" - - (probably there is more recent on redistribution) - Diekelmann et al.~\cite[p. 117]{diekelmann2010memory} "In addition to system consolidation, consolidation involves strengthening of memory representation at the synaptic level." - - $$Maybe this is so that when the memories are moved farther out from the hippocampus, there is a stronger trail to it$$ - - Diekelmann et al.~\cite[p. 121--122]{diekelmann2010memory} In the active system consolidation view, "It is assumed that in the waking brain events are initially encoded in parallel in neocortical networks and in the hippocampus. During subsequent periods of SWS the newly acquired memory traces are repeatedly re-activated and thereby become gradually redistributed such that connections within the neocortex are strengthened, forming more persistent memory representations. Re-activation of the new representations gradually adapt them to pre-existing neocortical 'knowledge networks', thereby promoting the extraction of invariant repeating features and qualitative changes in the memory representations." - - Diekelmann et al.~\cite[p. 122]{diekelmann2010memory}, citing Hasselmo et al.\cite{} suggest that "acetylcholine serves as a switch between modes of brain activity, from encoding during wakefulness to consolidation during SWS" - - Diekelmann et al.~\cite[p. 122]{diekelmann2010memory}, citing Wagner and Born \cite{wagner2008memory} observes that glucocorticoids (cortisol in humans) block the hippocampal information flow to the neocortex, and if the level of glucocorticoids is artificially increased during SWS, the consolidation of declarative memories is blocked. - - $Isay>$ so, we need to find out whether students who play video games on the way to retiring are decreasing or increasing their cortisol in the process.$$ - - - There is evidence that some games help reduce cortisol "The impact of playing computer games on cortisol concentration of saliva before and after the game showed that the amount of saliva plasma after playing the game has dropped significantly." \cite{aliyari2015effects}, casual video games decrease stress\cite{russoniello2009effectiveness} and there is evidence that Tetris in particular reduces stress\cite{mercer2015stress} and there is evidence that excessive use of violent video games by some young men \cite{eickhoff2015excessive} can negatively impact their cognitive effectiveness. Maass et al. conducted a study on 117 university students; Maass et al. state "The more time spent on media the poorer cognitive performance is. This association has mainly been found for general-audience, violent, and action-loaded contents but not for educational contents. \ldots A significant univariate difference was found for high- vs. low-arousing contents in general (independent of type of media), the high-arousing content leading to poorer ability to concentrate after media use. The expected mediating and moderating effects are not supported. The study yields evidence that short-term mechanisms might play a role in explaining the negative correlations between media use and cognitive performance." \cite{maass2015does} - - $Isay>$ might wish to advise students who play video games on the way to retiring to, close to retiring, use those that reduce stress$$ - - Diekelmann et al.~\cite[p. 122]{diekelmann2010memory} report "The concept of a redistribution of memories during sleep has been corroborated by human brain imaging studies (82,83,149,159,158) Interestingly, in these studies, hippocampus-dependent memories were particularly redistributed to medial prefrontal cortex regions (82,83,122)." - - $$ if it is the case that when students. trying to understand some math in computer science, but opting to memorize, given the time available (Is this what David Tall's bifurcation is about?) form memories that are not hippocampus-dependent (procedural memories are less hippocampus dependent)? If so, is it then also the case that they are less likely to be conveyed to the medial prefrontal cortex? Is the intervention then that instructors provide better explicit descriptions, give exercises to write something in code, which we hope is a bridge from internalized/implicit to interiorized/explicit, declarative, and avail ourselves of emotional support by evaluating the beauty, to aid in recruiting hippocampus-dependent memory formation? $$ - - Diekelmann et al.~\cite[p. 122]{diekelmann2010memory} state that "These regions not only have a key role in the recall and binding of these memories once the are stored for the long term, citing Frankland and Bontempi \cite{frankland2005organization}, \ldots prefrontal-hippocampal system might provide a selection mechanism that determines which memory enters sleep-dependent consolidation." - - Diekelmann et al.~\cite[p. 123]{diekelmann2010memory} state that the REM time interval upregulation of genes related to plasticity is dependent upon the learning experience in prior wakefulness, and is localized to the brain regions involved in prior learning, citing Ribeiro \cite{ribeiro2007novel,ribeiro2002induction}. - - \paragraph{Steps} - \begin{enumerate} - \item the input is a demonstration of a proof that explicitly describes the motivation for choosing each successive inference rule after another, and explicitly describes the motivation for choosing each lemma, i.e., why was that part of the proof handled separately, (e.g., we know we can, and we know it will be useful, but something about how we suspect it will be useful) Moreover, because we know emotional content is helpful in consolidation of memory during sleep, we remark upon such beauty as we may find in the proof. - Note that it is important to be, in the unfolding of time, orderly. - \item sensory input appears in sensory acquisition (accompany with pleasant scent\cite{born2012system}, and relevance to future plans\cite{born2012system}) - \item sensory input conveyed to medial temporal lobe, where it is sometimes seen with single neuron instrumentation - \item unexpected novelty stimulates dopamine, differently from expected novelty (as in exploration) - \item reward stimulates dopamine, or, if prediction of reward is learned, the stimulus role is transferred to the prediction - \item dopamine can come from SN/VTA - \item dopaminergic midbrain is associated with enhanced hippocampus-dependent long-term memory formation \cite{wittmann2005reward}, i.e., rewards help form long term memory, and it occurs in some sleep phase, maybe REM or slow wave - \item LTP is divided into early and late, and dopamine contributes to late~\cite{wittmann2005reward} - \item dopamine is also used for long term depression, which is also learning - \item new neurons are generated~\cite{deng2010new} related to new hippocampus memories - - \item explicit (vs. only implicit) learning favors access to sleep-dependent consolidation\cite[p. 115]{diekelmann2010memory} - \item motivational tagging of memories, might signal behavioral effort and relevance and mediate preferential consolidation\cite[p. 116]{diekelmann2010memory} - \item if video games are played just before retiring, they should probably not be violent. Conversely, if violent video games are played, something else, such as Tetris, should be used after the violent ones, prior to retiring. - \item sleep deprivation is expected to be detrimental to learning - \item sleep occurring 3 hours after learning was more effective than sleep delayed by more than 10 hours.\cite{gais2006sleep,talamini2008sleep,walker2003dissociable} - \item use same scent as in class, during SWS - \item slow oscillations typically seen in slow wave sleep (earlier part of sleep) have a causal role in the consolidation of hippocampus-dependent memories~\cite[p. 119]{diekelmann2010memory} - \item ripples typically seen in slow wave sleep (earlier part of sleep) have a causal role in the consolidation of memories~\cite[p. 119]{diekelmann2010memory} - \item it is not a particular sleep stage per se that mediate memory consolidation, but rather the neurophysiological mechanisms associated with those sleep stages~\cite[p. 116]{diekelmann2010memory} - \item re-activation of encoded memories occurs during slow wave sleep, in the order the remembered material was experienced Maquet\cite{maquet2000experience} cited in Diekelmann\cite[p. 117]{diekelmann2010memory} - \item theta oscillations, associated with REM sleep, have been found specifically over the right prefrontal cortex to be correlated with the consolidation of emotional memories\cite{nishida2009rem} - \item practice retrieval\cite{Bridge01082015}, better after sleep, can disrupt decoding if before sleep (check with Bridge 2015) - \item once attention to an item of knowledge has been rewarded, subsequent attention to that item is involuntary \cite{sali2014role} - \item consolidation occurs, differentiated by number of modalities, could be inert knowledge, as Whitehead~\cite{whitehead1959aims}, or retrievable, preferably. The number of related modalities, the more easily retrieved. - \item Retrieval allows for reconsolidation.\cite{sandrini2015modulating} (check it)\cite{schwabe2014reconsolidation}(check it) \cite{forcato2013role}(check it) - \cite{walker2003dissociable} (check it) - \item dopamine, nucleus accumbens helps reconsolidation occur with connection to right pre-frontal cortex \cite{knutson2001anticipation} and this is helped by anticipation of positive reward - \item right pre-frontal cortex is what experts use to be creative \cite{kowatari2009neural}, so we want to shepherd our taught material here, so that it is readily retrievable for inventing proofs - \end{enumerate} - - Chou et al.~\cite[p. 726]{chou2011sex} observes "the analytic brain for mathematical and logical cognition comprises the inferior frontal gyrus, parietal cortex and supramarginal gyrus", citing Dehaene et al, 1998k Goel et al., 1998, Zago et al., 2001. They measured, using fractional anisotropy (FA), microstructure of white matter that differed significantly in several areas, between men and women. - $$Of these, at least the bilateral precuneus has been identified as of interest in mental activity related to mathematical proofs.$$ - They state ~\cite[p. 731]{chou2011sex} "the interaction analysis of dispositional measures by sex demonstrated that FA of the WM \ldots underlying occipital gyrus and postcentral gyrus was negatively associated with systematic quotient (SQ)in females.\ldots males exhibited larger FA in the WM of hippocampus whereas females showed larger FA in WM of parahippocampal gyrus \ldots females typically hold an advantage in tasks related to declarative memory, in which the parahippocampal gyrus has been implicated, such as in the retrieval and recognition of longterm \ldots memories." - - Lisman et al.\cite{lisman2011neohebbian} report that "For novel information and motivational events such as rewards this signal at hippocampal CA1 synapses is mediated by the neuromodulator, dopamine." They summarize a consequence of the Hebb framework "if cell A represented object A and cell B represented object B, the co-occurrence of the two object would, by the Hebb rule, strengthen the synaptic linkage between these cells. This link would subsequently be evident when only object A was presented because it would lead to the firing of cell B, thus bringing object B to mind by association." - - Lisman et al.\cite[p. 537]{lisman2011neohebbian} state "Two types of experiments demonstrate that dopamine can strengthen the synaptic potentiation produced by learning itself." - %PPT pedunculopontine tegmentum - Lisman et al.\cite[p. 540]{lisman2011neohebbian} state "Thus, novelty, reward stimuli and aversive stimuli are all able to activate the dopamine system \ldots in humans. - - Lisman et al.\cite[p. 540]{lisman2011neohebbian} state "This reward-related memory enhancement was associated with a coactivation of SN/VTA, striatum, and hippocampus, as detected by fMRI (\cite{adcock2006reward,wittmann2005reward}) Memory enhancement after long retention intervals (e.g. 24h) has been consistently found (\cite{krebs2009personality,wittmann2011behavioral}). Moreover, the enhancement was greater at late timepoints than at early intervals (i.e. 3 wks vs 20 min)(47) - - $$ using novelty, reward (or punishment, i.e., if there are bad test grades, maybe they can be used), make these memories.$$ - - Lisman et al.\cite[p. 540]{lisman2011neohebbian}point out that it is necessary to be able to encode and consolidate after a single exposure. - - \begin{figure} - \centering - \includegraphics[width=0.7\linewidth]{./lismanNeoHebbian} - \caption{connectivity in medial temporal lobs and hippocampus ventral tegmentum loop processing info about object and spatial context. Allows perirhinal cortical info about novelity to general dopamine response} - \label{fig:lismanNeoHebbian} - \end{figure} - Lisman et al.\cite[p. 542]{lisman2011neohebbian} summarize a model, citing Frey and Morris~\cite{frey1997synaptic} "weak stimulation induces on ly early long term potentiation (LTP). By contrast, stronger stimulation produces the dompamine-dependent protein synthesis that allows late LTP.\ldots the memory for event sthat occur before or after the dopamine release would depend not only on their own properties, but also on whether they fell within the penumbra of a dopamine-releasing stimulus." - - $$Music that gives "chills" gives dopamine, so if we could play something like the prelude to the StarWars IV before class, students would be primed. Some national anthems might be chill inducing, such as the French, Marseillaise (lyrics?) How long is this penumbra? It could be different for each person. for rodents, 1/2 hour$$ - - Lisman et al.\cite[p. 542]{lisman2011neohebbian} "novel photographs of natural scenes ('strong events', such as those one would expect to see in the magazine National Geographic)" had a penumbra of at least 5 minutes. - - $$so, have some slides with nat geo pictures, at least every 5 minutes, new every time$$ - - Lisman et al.\cite[p. 542]{lisman2011neohebbian} "Cholinergic\cite{sarter2005unraveling} and noradrenergic\cite{frey2008synaptic} projections to Medial temporal lobes can also modulate Long term potentiation and long-term memory. \ldots Reward-related SN/VTA activation improves memory for the rewarded stimulus but not for the non-rewarded stimuli given in close temporal proximity \cite{wittmann2011behavioral} implying either a very short or a very stimulus-specific penumbra. This is at odds with the observation that novelty-related activation of the SN/VTA has a long (ca 30 min) penumbra that affect memory for unrelated information (e.g., exposure to novel scenes can affect memory for words)\cite{fenker2008novel} One possible resolution is that the duration or stimulus-specificity of the penumbra depends on the type of motivational event that triggers dopamine release." - - Lisman et al.\cite[p. 544]{lisman2011neohebbian} "the ability to recollect newly acquired information could be intrinsically rewarding. In fact, the study of human learning has revealed an interesting puzzle; long-term retention is not helped by simple re-exposure to recently learned material but is greatly helped by retesting even when subjects already know the answer \cite{karpicke2008critical}. One interesting possibility is that retesting provides an opportunity to generate intrinsic reward signals, thereby enhancing long-term persistence of newly learned material." - - Mains et al.~\cite{Mains01072015}"Embedding three (clicker) questions within a - 30 min lecture increased students' knowledge - immediately after the lecture and 2 weeks later. We - hypothesise that this increase was due to forced - information retrieval by students during the learning - process, a form of the testing effect." - - Bridge and Voss \cite{Bridge01082015} "Cueing with actively retrieved objects facilitated memory of associated objects, which was associated with unique patterns of viewing behavior during study and enhanced ERP correlates of retrieval during test, relative to other reminder cues that were not actively retrieved. Active short-term retrieval therefore enhanced binding of retrieved elements with others, thus creating powerful memory cues for entire episodes." - - $$students may make their own flash cards to practice retrieval, but it appears a more effective strategy would have multiple differing cues$$ - - Rottschy e al.\cite[p. 830]{rottschy2012modelling} define "working memory subsumes the capability to memorize, retrieve and utilize information for a limited period of time". - - Rottschy e al.\cite[p. 836]{rottschy2012modelling} "experiments using non-verbal material showed significantly higher convergence in the left (pre-)motor area and bilateral dorsal pre motor cortex." - - Rottschy e al.\cite[p. 843]{rottschy2012modelling} "selective attention system (Shulman et al.) \ldots is right-dominant and \ldots includes the temporo-parietal junction. \ldots apparent overlap between a distributed central executive for working memory, (and) the attention system" - - $$we can get their attention, for example by music that gives chills, and it will bring blood circulation to those areas, so, working memory will be supplied with circulation$$ - - - - Wittman et al.\cite{wittmann2011behavioral}"Recent functional imaging studies link reward-related activation of the midbrain substantia nigra –ventral tegmental area - (SN/VTA), the site of origin of ascending dopaminergic projections, with improved long-term episodic memory. Here, - we investigated in two behavioral experiments how (1) the contingency between item properties and reward, (2) the magnitude of reward, (3) the uncertainty of outcomes, and (4) the contextual availability of reward affect long-term memory. - We show that episodic memory is enhanced only when rewards are specifically predicted by the semantic identity of the - stimuli and changes nonlinearly with increasing reward magnitude. These effects are specific to reward and do not occur in - relation to outcome uncertainty alone. These behavioral specifications are relevant for the functional interpretation of how - reward-related activation of the SN/VTA, and more generally dopaminergic neuromodulation, contribute to long-term - memory." - - Abraham et al.\cite[p. 1906]{abraham2012creativity} investigates a specific aspect of creativity that they call conceptual expansion. They found that this activity selectively involved the anterior inferior frontal gyrus, the temporal poles and the lateral frontopolar cortex. These findings "go against \ldots dominance of the right hemisphere during creating thinking, and indicate \ldots anterior cingulate cortex \ldots (for) abstract facets of cognitive control." - - Abraham et al.\cite[p. 1907]{abraham2012creativity} explain that conceptual expansion "refers to the ability to widen the conceptual structures of acquired concepts, a process that is especially critical in the formulation of novel ideas", citing Ward~\cite{ward1994structured}. - Abraham et al.\cite[p. 1910 -- 1911]{abraham2012creativity} found that the regions involved were left anterior inferior frontal gyrus, lateral frontopolar cortex, temporal poles, posterior regions in the inferior frontal gyrus, the middle frontal gyrus, the anterior cingulate cortex, the dorsomedial prefrontal cortex and the inferior parietal lobule. The "activation pattern is strongly lateralized to the left hemisphere." For working memory, "the overall brain activation pattern as a function of working memory was stronger in the right hemisphere" - - \begin{figure} - \centering - \includegraphics[width=0.7\linewidth]{./frontopolar-cortex} - \caption{left frontal polar cortex playing a particularly relevant role in concept expansion, is thought to mediate cognitive control at the most abstract level of information processing. The left dorsolateral pre frontal cortex, and the dorsomedial prefrontal cortex (BA8/9 and 8) also showed stronger brain activity} - \label{fig:frontopolar-cortex} - \end{figure} - - \begin{figure} - \centering - \includegraphics[width=0.7\linewidth]{./inferiorfrontalgyrus} - \caption{Conceptual expension was associated with greater brain activity in left anterior inferior frontal gyrus (right is pictured)} - \label{fig:inferiorfrontalgyrus} - \end{figure} - - \begin{figure} - \centering - \includegraphics[width=0.9\linewidth]{./gr1} - \caption{The inferior frontal gyrus, temporal poles and frontopolar cortex are involved in coceptual expansion. The roles of the anterior cingulate cortex and the dorsolateralprefrontal cortex were found to be most responsive in conceptual expansion, and active in divergent thinking.} - \label{fig:gr1} - \end{figure} - - Abraham et al.\cite[p. 1912]{abraham2012creativity} In a measure intended to isolate cognitive control processes, results showed the dorsolateral prefrontal cortex and superior parietal lobule bilaterally, and "only the right dorsolateral prefrontal cortex and anterior cingulate were found to be involved." - - $$we are seeing that being creative calls upon left side structures, and when we get specifically to being creative with symbolic and diagrammatic representations, we may call upon also the right sides. $$ - - Abraham et al.\cite[p. 1913]{abraham2012creativity} "The roles of the anterior cingulate cortex and the dorsolateral pre-frontal cortex are particularly noteworthy given the patterns of activation in these regions in the current study. Not only were they found to be more activated during divergent thinking compared to working memory, more importantly, they were also found to be most responsive as a function of conceptual expansion. \ldots the posterior aspect of the dorsomedial pre frontal cortex was also activated as a function of conceptual expansion. \ldots As this region has been discussed with reference to concepts that are central to hypothetical reasoning, such as constructive processes in cognition~\cite{abraham2008thinking} which involve flexible recombination of representations from memory~\cite{schacter1998cognitive} and evaluative judgment~\cite{zysset2003functional}, the dorsomedial prefrontal cortex may prove to be highly relevant structure for select aspects of creative thinking. \ldots speaks against the ubiquitous idea the right brain is more 'creative' than the left. \ldots in the current study, we have explored the deliberate problem solving mode of creating thinking under time constraints. There is, however, another vast dimension of creative thinking where idea generation occurs spontaneously, effortlessly, and/or in a state of defocused attention~\cite{|} In fact, creating idea generation is far less likely to result from deliberate cogitation during real everyday problem solving, but instead, it occur spontaneously and unpredictably. This unconscious non-deliberate spectrum of creating thinking \ldots is less amenable to well-controlled investigation. " - - Born and Wilhelm~\cite[p. 192]{born2012system} "Experimental evidence for these three central implication is provided: It has been shown that reactivation of memories during slow-wave sleep(SWS) plays a causal role for consolidation, that sleep and specifically SWS consolidates preferentially memories with relevance for future plans, and that sleep produces qualitative change in memory representations such that extraction of explicit and conscious knowledge from implicitly learned materials is facilitated." - - $$In procedural memory we don't need to know why, there might not be any, (for example, remember the melody) but for some things they are accompanied by why. Is declarative everything other than procedural? What about implicit vs. explicit? These both are compatible with "why". There are times when we can use knowing why to save on what would otherwise need to be remembered. Does it have a name? Is it named in David Tall's article with bi-furcation in the name?$$ - - Born and Wilhelm~\cite[p. 195]{born2012system} "Via the olfactory system odour stimulation acquires immediate access to the hippocampus. \ldots we found that the odour when re-exposed during SWS after learning induced a distinct activation of the left hippocampus, i.e. the odour served as a cue that reactivated the new memories for the card locations encoded in the left hippocampus, thereby enhancing these memories \ldots hippocampal networks are particularly sensitive in SWS to inputs capable of reactivating memories." - - Born and Wilhelm~\cite[p. 197]{born2012system} "explicit encoding favours access to sleep-dependent memory consolidation (\cite{robertson2004awareness}). Involvement of the prefrontal-hippocampal system underlying explicit encoding has been proposed as prerequisite for consolidation to occur during sleep(\cite{marshall2007contribution}). \ldots emotionality of the encoded events can increase the memory benefit from sleep (\cite{kuriyama2004sleep,wagner2006brief})." - - Wagner et al. \cite{wagner2006brief} investigated memory after a four hour interval of sleep. "Sleep following learning compared with wakefulness enhanced memory for emotional texts after 4 years (p = .001). No such - enhancement was observed for neutral texts (p = .571)." - - Born and Wilhelm~\cite[p. 197]{born2012system} "Processing of anticipatory aspects of behaviour such as expaectancies and plans is particularly linked to executive functions of the prefrontal cortex that regulates activation of memory representations during anticipated retrieval and accommodates specifically the intentional and prospective aspects of a memory representation~\cite{polyn2008memory} \ldots prefrontal tagging of memories explicitly encoded under control of the prefrontal-hippocampal system could be decisive for the selectivity in off-line memory consolidation" - - Born and Wilhelm~\cite[p. 199]{born2012system} "there is convergent evidence \ldots that the system consolidation process during sleep supports the extraction of invariant and repeating features in newly encoded memories, and in this way, the conversion of implicit into an explicit and conscious form of memory \ldots more than twice as many subjects of the sleep group gained insight into the hidden structure as compared with the wake control group \ldots subjects who had slept after \ldots training were distinctly more able to deliberately generate the sequence underlying \ldots than the subjects who had stayed awake" - - Born and Wilhelm~\cite[p. 201]{born2012system} "sleep appears to prime the transformation of implicitly encoded information into explicit knowledge, i.e., something that is not conscious before sleep enters consciousness through sleep". - - Dudai~\cite[p. 229]{dudai2012restless} defines declarative memory as that which requires conscious awareness for retrieval (facts, events), and nondeclarative can be retrieved in absence of conscious awareness (habit, skill). - - Dudai~\cite[p. 231]{dudai2012restless} reminds "university students can improve their memory bye practicing self-testing, because retrieval practice is a powerful mnemonic enhancer", citing Karpicke and Roediger~\cite{karpicke2008critical}. - - Forcato et al.~\cite[p. 1]{forcato2013role} observe "the reconsolidation process alllows new information to be integrated into the background of the original memory; second it strengthens the original memory. \ldots at least one labilization-reconsolidation process strengthens a memory via evaluation 5 days after its re-stabilization. \ldots this effect is not triggered by retrieval only. \ldots repeated labilization-reconsolidation processes made the memory more resistant to interference during re-stabilzation." - - Forcato et al.~\cite[p. 1]{forcato2013role} "reconsolidation does not represent recapitulation of initiali consolidation, but rather, if refers to the functional role of this process: to stabilize memories." - - Forcato et al.~\cite[p. 2]{forcato2013role} "when the reminder only included contextual cues (context reminder), the memory was evoked but not labilized." - - A reminder has the effect of labilization which allows reconsolidation. "We found that just one labilization-reconsolidation process was enough to strengthen a memory that was evaluated 5 days following its re-stabilization. \ldots Memory persistence is increased by repeated triggering of labilization-reconsolidation." - - Wirebring et al. \cite{}"Repeated testing is known to produce superior long-term retention of the to-be-learned material compared with repeated encoding and - other learning techniques, much because it fosters repeated memory retrieval. This study demonstrates that repeated memory retrieval - might strengthen memory by inducing more differentiated or elaborated memory representations in the parietal cortex, and at the same - time reducing demands on prefrontal-cortex-mediated cognitive control processes during retrieval. The findings contrast with recent - demonstrations that repeated encoding induces less differentiated or elaborated memory representations. Together, this study suggests - a potential neurocognitive explanation of why repeated retrieval is more beneficial for long-term retention than repeated encoding, a - phenomenon known as the testing effect." - - \subsection{Educational Neuroscience} - - \section{Systems Biology} - be sure to get the Kandel, Dudai 2014 review - - \section{Physiologically Informed Constructivism} - Research has shown the progression of an idea, from a short term representation in the medial temporal lobe, to a consolidated memory in an area related to the single sensory modality with which that idea was received, to a representation in association cortex when multiple sensory modalities are involved, to a reconfiguration in right prefrontal cortex when an idea is used creatively. - How we as instructors shepherd these representations in the minds of our students can be suggested by cognitive neuroscience. One example is a deeper understanding of the utility, for memory and attention, to intrinsically rewarding learning activities. Another example is the application of multiple sensory modalities, such as representations in figures, text and mathematical symbols, and also pseudocode. Yet another is the deliberate construction of pseudo-monetary rewards, of unpredictable value, in the immediate aftermath of - instruction of a new idea. (This derives its effect from the dopamine response to the monetary reward, which is augmented by not knowing the amount.) - - We can expect information stored in the cortex near a single sensory modality to be retrieved when a similar situation recurs. When multiple modalities are involved there are more ways to arrive at increased metabolism reactivating this memory, bringing it into consciousness. - - Music is rewarding. The areas of the brain that are rewarded by memory are the same ones that help us remember. (Blood Zatorre salimpoor) We can reward the students / administer dopamine to help them remember. We have from Blood and Zatorre~\cite{blood2001intensely} that music so intensely pleasurable that it creates "chills" is correlated with activity in nucleus accumbens, which we also know, from (whom?) advances consolidation of memory representations in medial temporal cortex into longer term memory. %Here I am thinking how REM sleep is seen to be helpful to consolidation of memory, and I am thinking about the "subject matter" of REM sleep, and wondering whether a method more time-efficient than sleep could be developed - - $$ bagpipe music generates chills$$ - - We want the students to construct an understanding of the material, for example, the correctness or resource utilization of an algorithm, and to be able to creatively apply the information they have to construct a proof of it. - - The material first arrives in the student's awareness through some sensory modality, and we wish the information to be remembered, not only as inert knowledge, but as information that enters consciousness in response to a situation in which it is usefully employed in a proof. - - Lisman and Grace\cite{lisman2005hippocampal} describes the hippocampal--ventral-tegmental area loop, controlling the entry of information into long term memory, facilitated by dopamine(DA). One stimulus for production of DA is the arrival of an unexpected reward \cite{schultz2000neuronal}, another is novelty\cite{kafkas2015striatal} (check this). Instructors can provide unexpected rewards with, for example, increased points for clicker questions. Other ways to provoke DA release, in the part of the brain that produces memories, is with music.\cite{salimpoor2015predictions} (check that this is the right article). Lisman and Grace\cite{lisman2005hippocampal} describe reactions that they state "provide a basis for the dopaminergic modulation of early long term potentiation (LTP)", which is a change to the signaling behavior of neurons in the brain. They continue "It follows that novelty itself should enhance LTP." Moreover they claim\cite[p. 707]{lisman2005hippocampal} "There is reasonable evidence from animal experiments that DA enhances learning, as would be expected from its enhancement of LTP." and \cite[p. 708]{lisman2005hippocampal}"The spiny cells in the accumbens are a likely site for combining novelty signals and goal-dependent motivational signals." and \cite[p. 709]{lisman2005hippocampal} "reasonable working hypothesis that \ldots combines novelty signals with information about saliency and goals \ldots thereby enhance the entry of the information into memory." \cite[p. 709]{lisman2005hippocampal} "without dopamine, late LTP does not occur and early LTP decays within about an hour." - - Wittmann et al.\cite{wittmann2005reward}"Long=term potentiation in the hippocampus can be enhanced and prolonged by dopaminergic inputs from midbrain structures such as the substantia nigra. This improved synaptic plasticity is hypothesize to be associated with better memory consolidation in the hippocampus." They say that reward anticipation reliably elicits a dopaminergic response. They hypothesize that "activation of dopaminergic midbrain regions enhances hippocampus-dependent memory formation, possibly by enhancing consolidation." Wittmann et al.\cite[p. 464--465]{wittmann2005reward}"supports our hypothesis that the hippocampus is a major site for the neuromodulatory influence of reward on long-term memory formation \ldots supports the hypothesis that dopaminergic neuromodulation enhances hippocampus-dependent memory formation \ldots It is likely that a greater proportion of subsequently recallable items will undergo hippocampus-dependent consolidation than of subsequently recognizable items. \ldots these results provide evidence for a relationship between activation of dopaminergic areas and hippocampus-dependent long-term memory formation." - - Here I want to diagram a sequence of activities mapped onto the brain areas, showing the progression of memory from the medial temporal lobes with new dentate granular cells, followed by consolidation in one sensory cortext or association cortext followed by reconsolidation into prefrontal and parietal followed by prefrontal from where it can be used creatively in the construction of proofs. It would be good if I could cite references for each milestone, and associate with conceptualizations, such as those in Harel and Sowder and Tall. - - % % % % % % % % % % % % % % % % - - \subsection{Intuition} - help and obstacle\\ along with obstacles arising from intuition there exist epistemological obstacles Bachelard 1938 Brosseau 1983 preventing acquisiton of new knowledge. There exist didactic obstacles. - epistemological obstacles - \subsection{Met-befores} - Tall\\ - McGowan Tall\cite{Metaphor or met-before 2010 Jour Math Behavior}, The idea met-before (formerly met afore) emphasize that a metaphor relates new knowledge (the 'target') in terms of existing knowledge (the 'source') developed from previous experience, so that new ideas can be related to familiar knowledge already in the grasp of the learner. - - Small example of a met-before: Integrated circuits are available at many levels of integhration, from singletons of transistors to tens of millions. - One simple circuit function ins the counter. The number of cluck edges since the last reset, modulo the number given by $2^n$, where $n$ is the number of bits in the counter, is represented digitally, that is, by a voltage level whose domain has been partitioned into that representing 1 and that representing 0. A counter may reset immediately (so to speak) upon the reset signal. This kind is called asynchronous reset. The synchronous reset kind resets after a clock edge occurs during a reset input. - Having, for the purposes of this example, set the context this way, now consider this problem, from Santos-Trigueros\cite{[p. 74]{}} - ``Nine counters with digits from 1 through 9 are placed on a table.'' - Had we not just been discussing counters in another setting, this sentence might have been understood as intended, more quickly. That is, the context can serve as a distraction, to be overcome. It can help the instructor to realize that students arrive in class, not only lacking wished-for preparation, but also bringing unhelpful contexts. - \subsection{Harel and Sowder} - \subsection{van Hiele Levels} - \subsection{Student Centered} - something about students' perspectives are not always well-matched to their needs - - Students might have in mind material they would like to learn, and there may be also a lack of appreciation for material in required courses - Students may have a rate at which they would like to learn --- points at which they would like to pause and integrate new material with things they already know. - \subsection{Attention} - Lindquist \cite{lindquist2013mind} describes Mind wandering during lectures: Observations of the prevalence and correlates of attentional lapses, and their relationships with task characteristics and memory. - - Brosch et al.\cite{brosch2013impact} examined the impact of emotion on perception, attention, memory, and decision-making. - - Sali et al.\cite{sali2014role} show that "Previously rewarded stimuli involuntarily capture attention." - - \subsection{Memory} - Ojemann\cite[p. 257]{ojemann2014human} " \textit{episodic, explicit} or \textit{declarative} memory, memory for the specific event (specific name, word) that occurs at encoding " - - Ojemann\cite[p. 257]{ojemann2014human} "Our memory paradigm requires storage over a short period and thus a measure of \textit{recent} or \textit{short-term} memory. In the functional neuroimaging literature memory for this duration is often referred to as \textit{working} memory. - - Ojemann\cite[p. 257--258]{ojemann2014human} "Lateral temporal cortex \ldots models of brain stubstrate for recent verbal memory \ldots during the encoding phase." - - \subsection{Social Constructivism} - Attempts at communication, as in conversations about material, are regarded - as helpful to learning. - - McGowan Tall 2013 Jour Math Behav - ``One student wrote that she knew her answer was correct (it was actually incorrect) because the other members of her group agreed with her. These students consistently evaluated both the numerical expression 'minus a number squared' and a quadratic function with a negative-valued input incorrectly throughout the remaining twelve weeks of the semester. This cautions us to realize that cooperative leaning amongst students who are failing to make sense of the mathematics may reinforce their problematic conceptions rather than reconstruct them'' [p. 533] - - Though it is easy to assume that communication between practitioners is carried out verbally, there are examples of proofs without words \cite{nelson1993proofs}. - - D\"orfler\cite[p. 129]{dorfler2000means} proposes that by discussion, abstraction can be promoted "the abstract terms might serve the purpose of talking about a variety of concrete, even physical experiences such as describing observations \ldots This abstracted manner of talking then acquires some independence from the experiences and experiential phenomena referred to, so that the abstract objects gain discursive existence. First, the abstract description lends 'meaning' to the experiences. Later, the abstract objects derive their 'meaning' from their taken-to-be representations or applications." - - D\"orfler\cite[p. 129]{dorfler2000means} summarizes "our experiences with material objects are schematized in an image schema that is projected metaphorically to terms such as \textit{abstract object}. \ldots At times, these image schemas might even be exteriorized by symbolic expressions that, in turn, can serve as generic terms. \ldots A cognitively oriented explanation for the failure to be inducted into the mathematical discourse is, therefore, the lack of image schemata on which to base the discursive extension \ldots this lack might results from an absence of pertinent experiences \ldots there can be differences in the abilities of each individual to schematize his or her experiences and to make metaphoric use of words." - - - - \subsection{Beliefs about Diagrams} %after social constructivism, because people attempt to communicate with diagrams - helpful, hindering, post-conceptual (i.e., they refer to existing concepts, maybe do not convey new ones), - Hilbert's ``who does not'' with a, b, c\\ - perceptual to transformative\\ - animation - \subsection{Semiotics} - Because sybmolization supports generalization\cite{loewenberg2003mathematical} and operations in mathematics\cite{schoenfeld1998reflections}, and because symbols are used also in efficient communication with others, symbolization is a skill our students need. - - Van Oers\cite[p. 133--134]{van2000appropriation} emphasizes the role of the adoption of symbol use, as students learn mathematics, stating "Mathematics as a discipline is now generally conceived of as an activity in which constructive representation, with the help of symbols, plays a decisive role" citing Bishop, Freudenthal and Kaput. - - Van Oers\cite[p. 133--134]{van2000appropriation} summarizes: "According to Freudentahl, mathematics is basically an activity of mathematizing: that is, organizing a (concrete empirical or abstract mental) domain, representing it with the hop of symbols \ldots experimenting with symbolic means". Freudenthal\cite[p. 10]{freudenthal1973mathematics} describes some of the history of symbol use: "Another algebraic idea is symbolism, the use of signs which do not belong to everyday language, to indicate variables. 'Think of a number' is how the problems are introduced in old narrative algebra. In Diaphantus' work the word 'number' becomes more and more a computation symbol. This continues in Indian and Arabian mathematics. The \textit{cossists} of the late Middle Ages has a whole system of symbols for the unknown and its powers \ldots " - - Van Oers\cite[p. 135]{van2000appropriation} states that van Hiele "emphasized the importance of symbols in mathematics for grasping the mathematical meaning. \ldots appropriating the meaning of symbols is primarily a communicative process" requiring a mutual understanding of the meaning. - - Van Oers\cite[p. 136]{van2000appropriation} states: "Symbols are indispensable as means for coding the results of thinking. More importantly, however, symbols also function as ways of organizing in the course of thinking". He goes on to report " According to empirical investigations of the development of mathematical thinking in pupils, the failure of meaningful appropriation of mathematical symbols has turned out to be one of the main problems in mathematics learning.", citing Hughes, 1986 and Walkerdine, 1988, also Miles and Miles 1992. - - Van Oers\cite[p. 170]{van2000appropriation} states: "The analysis of symbolizing in a mathematical context has led us to the realization that symbol use is intrinsically related to meaning, negotiation of meaning, and communication." - - Nemirovsky and Monk described symbolizing\cite[p. 177--178]{nemirovskymonk} "Conceiving of symbolizing as the creation of a space in which the absent is made present and ready at hand elicits at least two major issues: (a) the nature of such a space, and (b) the ways in which the absent is made present and ready at hand". - They observe \cite[p. 178]{nemirovskymonk} "our play as children is a crucial activity through which each one of us has practiced and learned to symbolize", citing Piaget 1962, Slade and Wolf 1994 and Winnicott, 1971/1992. - Nemirovsky and Monk give an example \cite[p. 204]{nemirovskymonk} "Lin shifted her attention from being immersed in creating something \ldots to reflecting on it as a particular manner of doing things". They go on to say, "Symbolizing is making possible the sudden and unanticipated encounter with past experience that can radically transform the 'here and now' of the symbolizer. - - $$ sudden, unanticipated, coming into consciousness is a relevant idea $$. - - Nemirovsky and Monk state\cite[p. 212]{nemirovskymonk} "Insights that 'come' or 'happen' to us is a way of saying that we often experience what we become as surprising and unexpected." - - Bransford et al.\cite{bransford2000designs} mention "a common problem of expertise, namely, that things become so intuitively obvious that one forgets the difficulties that novices have in grasping new ideas". - - $$ consider that procedures to which we have become so accustomed that we do not need conscious attention (e.g., shifting gears for a car or bike) could be represented more efficiently in neurons, see Kandel and Squire on Memory and Attention\cite{squire2000memory} and Squire \cite{squire2015conscious}$$ \ No newline at end of file +Using an analogy, I claim, is saying there is a set of relations among things $a_i$ that +we agree upon, furthermore, I might wish to teach that there is a corresponding +set of relations among things $b_i$. I might wish to say, use the relation we agree +upon for municipalities provide addresses for homes that can be used for +surface mail, and I might wish to teach that there is a corresponding provision +of addresses for items a computer programmer might wish to use for storage +and recall. We can note that addresses make use of a hierarchy of place names, +countries, states, cities, streets, street numbers, apartment numbers. We can +note that structured data types can correspondingly make use of instances and +fields and indices that can be arranged in a tree. +In the absence of abstraction, the surface mail address hierarchy might not pose +much more difficulty, but the data structure might, because the fields therein +are more subject to change than municipalities. In the absence of abstraction, +the comparison between one hierarchical arrangement with another would be +more difficult, because it is the structure of the abstraction itself, namely, the +choices of features regarded as significant throughout the tree, that is to be +recalled and used as a scaffold for the new information. +``An alternative pathway towards abstraction involves recognizing an analogy +between two structures in different domains, which then focuses one's attention +on the abstract structure they share. This new abstraction then becomes a +’concrete’ concept that one can study'' \cite [p 449]{}. +\section{Vertical Integration and Explanation} +It is accepted that, in discrete math, it is helpful to work problems. We may +inquire, what is it about working problems that helps? We can, at the neurophysiology +level, expect that long term potentiation of synapses, for the +synapses collocated with the long term memory for the concepts in the problem, +is being carried out, as the thinking about the problem occurs. We can +recall that a sense of reward, as might be gained from success at a problem, +or pleasantness in a problem statement, helps consolidate the memory for the +ideas that have been gained. We can recall that depression resulting from +avoidance of sadness at failure to solve a problem reduces the ability of the +hippocampus to support the formation of long term memory. We can, at the +cognitive neuroscience level, expect that the opportunity for like structures to +be recognized occurs, and analogies, are made, and the abstraction hierarchy of +concepts related to the problem is remodeled, extended, to more closely mirror +the mathematical definitions being used. We can, at the phenomenography +level, suppose that fine distinctions between concepts will be more likely to be +noted, because the mental structures that support these are forming. We can, +at the computer science education level, consider how to bring activity into +lecture, that poses the analogies and distinctions we wish the student to gain, +by varying the examples of the concepts such that a representative example +is contrasted with a non-example, in an ambience that fosters curiosity and +rewards progress. +\section{ Validation} +We draw a connection between epistemology and validation. Epistemology +is why we believe what we believe. Thus, it makes sense to apply our epistemological +framework to explain why we believe what we believe about our +results and interpretation. +Our epistemology is informed by the work of others over a wide range of disciplines: +Computer science education, mathematics education, and education +generally, especially phenomenography. Cognitive aspects, including memory +and attention, are shared by and form a bridge between phenomenography +and cognitive neuroscience. Neuroscience provides interesting relevant information. +Ira Black, MD, in [?, ?, [p 40] ``A satisfactory mechanistic description of any +well-framed cognitive process requires that we simultaneously explain it at +multiple levels of analysis. Different levels provide complementary insights +to characterization and causality that are unobtainable from any single line of +analysis.'' +\subsection{Validation at the Level of Computer Science Education} +Interviews with computer science educators, both instructors and teaching +assistants have provided a diversity of viewpoints, and generally support the +interpretations we have given. For example, one instructor, when asked what +students thought proofs were, said ``some kind of magic incantation'', and +teaching assistants have said ``students really struggle with this''. +\subsection{Validation at the Level of Mathematics Education} +The literature of mathematics education includes work on students' learning +about proof. Our work with computer science students has had the benefit +of some students who are dual majors of math and computer science, which +has allowed us to trace the similarities and differences of these cohorts of students. +The significance of definitions, the necessity and utility of proof, the role +played by interest in forming procedures and functions, the difference between +functional and procedural programming have differed in these three cohorts, +in so far as we have been able to examine. We did not explore the interest in +developing procedures/functions or procedural or functional programming in +mathematics majors who were not also computer science majors. + + + +\subsection{ Validation at the Level of Phenomenography/Variation Theory} +Variation theory supports our observation that comparing and contrasting fine +distinctions in material being taught aids the process of learning. We used +the difference between assignment and equality testing, manifest in the java +expression of ``=='' vs. ``=''. We compared a software procedure representation +with a mathematical formulation (the latter using only ``=''), for comprehensibility +by students. This helped us to see that barriers to student understanding +exist, for some students of computer science, at the level of formulation. It also +helped us see that the barrier between the internalization and interiorization +of Harel and Sowder might be less of a barrier in students of computer science +who are routinely conscious of the need to analyze procedures. +\subsection{Validation at the Level of Cognitive Neuroscience} +Many students have expressed an interest in learning from examples, and researchers studying students' acquisition of the ability to prove have observed a category for concepts called ``perceptual'' where students mistakenly believe or hope, that examples constitute a proof (of universality). Valiant points out two cases where examples are very effective in learning: +when on may employ elimination when the concept to be learned is a conjunction and an exemplar exhibits a variable in negative form, it is clear that that variable is not needed for set membership. +When a concept is a disjunction any variable that appears in positive form in a negative example is shown to be insufficient to guarantee membership. + +Valaint [p. 171] Humans do not argue readily from the contrapositive. +P.C. Wason 1983 Realism and rationality and the selection task. Thinking and Reasoning: Psychological Approaches Evans, ed., Routledge + + + +Valiant, in Circuits of the Mind [] that an ``important class that is not currently +known to be learnable is disjunctive normal form (or DNF for short)\ldots This +appears to be a most natural generalization of simple conjunctions from the +viewpoint of modeling human concepts. It can express the idea that examples +of a concept fall into a number of somewhat distinct categories, each corresponding +to noe of the conjunction. When discussing inductive learning we +have a \textit{hierarchical} context in mind. If we wish to learn DNF formulae, but do not +have an algorithm for learning these direction, we can nevertheless attempt to +learn these in stages. For example, to learn $x_1x_2 \land x_2x_3$ we could first learn the +simple conjunctions $x_1x_2$ and $x_2x_3$ separately in some fashion. Having learned +these we can learn the DNF when learning hierarchical in this way more is +required of the teacher or environment than in the simplest case of learning +by example. Somehow the subconcept $x_1x_2$ must be learned separately in supervised +or unsupervised mode. In the former case, for example, a teacher +may have to teach the name of this subconcept in unsupervised memorization +mode and then identify positive or negative examples of it so that it is learned +in supervised mode inductively. Alternatively, this subconcept may be learned +in unsupervised mode either by memorization or be correlational learning. `` +Valiant has written\cite{this is in a separate pdf} that the hippocampus is likely to be the location where the allocation of new memory locations is carried out. + +Valiant has observe that, given a set of concepts that can be hierarchically related, in the absence of hierarchy, when instead the concepts are flattened out, it is more unwieldy to make analogies, such as $A^B$ is analogous to $C$. +This is supportive of our interpretation of student data, in which we suggest that students who +find abstraction challenging, will in turn find remembering and using definitions more challenging, and will be at a disadvantage in terms of advancing to definition-based axiomatic reasoning. + + + + +\subsection{ Validation at the Level of Neurophysiology} +(Says who?) suggests that memory for events that are observed through one sensory modality are stored near the nervous tissue that process the input for that modality. +This is supportive of our interpretation of student data, in which we suggest that students who + learn proof by mathematic induction, represented symbolically, and recursive algorithms, represented pictorially, do not always ``see'' the analogy right away, because the memory traces are not activated at the same time. +When considering the two topics at the same time in discussion, both ideas are recalled, corresponding to metabolic activity in the memory (possibly tow different, separate, regions) facilitating formation of connection between the two ideas. +From Chapter 18 Migration in the Hippocampus, of Cellular Migration and Formation of Neuronal Connections: Comprehensive Developmental Neuroscience Vol 2, 2013, Elsevier ``the proliferative subgranular zone located at the border between the granular cell layer and the hilus, which serves as the major site for persistent neurogenesis in the adult hippocampus'' ``one of the more unique aspects of hippocampas development is the formation of the dentate gyrus, which involves formation of a specialized neural stem cell niche.'' ``hippocampus harbors neural circuitry essential for learning and memory {Lisman 1999} relating hippocampal circuitry to function neuron'' ``Cajal-Retzius cells \ldots somewhat penetrate the boundary but tend to avoid invading into olfactory cortex of olfactory bulb Bielle et al. 2005'', maybe +``signalling pathways \ldots regulate the redial glia-guided migration in the neocortex'' p.335 (not only during development) +0.339 nice graphic for germinative layers (was that supposed to be p.339?) +persistent neurogenesis in adult hippocampus Altman and Bayer 1990 Migration and Distribution percursors Comp Neurol subgranular zone Li et al. 2009 neurogenic zone Li and Pleasure 2005 Morphogensis of Dentat gyrus + + +% % % % % % % % % % % % % % % % % % % % % % +% % % % % % % % % % % % % % % % % % +\section{Validity and Reliability} +We checked for internal consistency and reinforcement, and for external compatibility +of our findings with existing educational literature in computer science +and in mathematics. We noted the phenomenological work of Gian-Carlo +Rota \cite{rota1997phenomenology} who has reported that memory for mathematical proof and its elements +is noticeably improved when a proof is deemed to be beautiful. We were encouraged +by the overlap in description among interview participants. In the +literature of mathematics education, we found researchers [?] reporting quite +similar conceptions of proof by mathematical induction in students of mathematics. +In the literature of computer science education we found research \cite{booth1997phenomenography} +on a different topic, but with similar results. Booth reported categories of +conceptions of recursion similar to our categories of conception of proof by +mathematical induction. +\section{Researcher Bias and Assumptions} +Researcher Bias and Assumptions +strategies for trustworthy, valid, reliable +what about generalizability? %(e.g., to people with ASD) +\subsection{Proofs Using the Pumping Lemma for Regular Languages} +The author believes diagrams aid the abstraction process. The researchers +tend to believe that students want to learn, and will try to comprehend and to +become able to apply the material, and that the limitations temporarily present +in the student can be overcome by explanation and practice. +\subsection{Proofs by Induction} +\subsection{Domain, Range, Mapping, Relation, Function, Equivalence Relation +in Proofs} +\subsection{Definitions, Language, Reasoning in Proofs} +\subsection{Equivalence Class, Generic Particular, Abstraction in Proofs} + +\subsection{Assessing Validity} + +The results have bridged papers in computer science education, by Professor Booth\cite{booth1997phenomenography}?\cite{}, and mathematics education, by \cite{}. + +The results have been consistent with results of others in mathematics education. + +% % % % % % % % % % % % % % % % % % % % % % % % % % % +% % % % % % % % % % % % % % % % % % % % % % % % % % % % +\section{Validity} +By choosing a varied population we hoped to obtain transferable results. + +We compared our results with existing publications. + +We performed a little triangulation by using multiple views into the student population: interviews and tests. We also compared the results from this with information obtained in tutoring and larger help sessions. We consulted faculty, who had experience with teaching this material, and who had experience with students who were supposed to have learned this material in prerequisites. + +% % % % % % % % % % % % % % % % % % % % % % % +\subsection{Assessing Validity} + +The results have bridged papers in computer science education, by Professor Booth\cite{}\cite{booth1997phenomenography}?, and mathematics education, by \cite{}. + +The results have been consistent with results of others in mathematics education. diff --git a/ch8.tex b/ch8.tex index f734585..0945d42 100644 --- a/ch8.tex +++ b/ch8.tex @@ -1,124 +1,1152 @@ -\chapter{Conclusion} -\begin{quote}Harel and Sowder \cite[p. 277?] {harel1998students}by their natures, teaching experiments and interview studies do not give definitive conclusions. They can, however, offer indications of the state of affairs and a framework in which to interpret other work.\end{quote} - - - -Are CS students' conceptualizations more like Harel and Sowder, or more like Tall? -Are the several schemes (Pirie Kieren, etc. complementary? reconcilable? Is one more likely than another based on cognitive neuroscience of language? (proofs are in a language after all)) - - - -This research suggests that suitable question for a larger study -\section{ Recognizing an Endpoint} -A qualitative study is thought to be finished when an internally consistent -narrative, compatible with the data, both situating the data and explaining -them, has been produced. -For our research questions, a model, accompanied by a narrative combining -the information obtained from inquiry about these topics will complete the -work. Data from our extended student body, that provide a persuasive model -containing categories of conceptualizations, and that are closely enough related -that some insight about concepts differentiating adjacent categories can -be inferred, are thought sufficient to generate this narrative. The proposed -differentiating concepts are thought to have the potential to become material -for a larger survey, thereby providing a starting point for new work. -I expect to find a model similar to that of Harel and Sowder 1998[?], but -modified because of the different emphases on material in computer science -compared to mathematics. First, students of computer science should be very -familiar with the idea of consciously constructing, examining and evaluating a -process, from their study of algorithms. Because of this, the category internalization -might be subsumed by the category interiorization. -From empirical data, we know that there are students of computer science -who think that proofs might be irrelevant to their career; it would be hard to -imagine a mathematics student who thought so. CS students who do not think -proof is part of their career might be relatively content with conceptualizations -corresponding to outside sources of conviction. We found computer science -students whose conception of proof includes that a single example is sufficient -for proving a universally quantified statement. We found computer science -students whose conception of proof is that definitions are barely interesting, -and who find demonstrations based on definitions unconvincing. Because our -findings were not quantitative, we could not compare the population of categories. -Nevertheless, the relationships between categories, and the resulting -critical factors, might be different, especially in the area of Harel and Sowder's -internalization and interiorization. -Because the scope is broader, involving proof for deciding whether or not an -algorithm is suitable for a problem, I expect we will find more categories, -related to algorithms and their applicability. -The product of a phenomenographical investigation is categories of conceptualization -and critical aspects that distinguish one category from the previous. -One hopes that by identifying critical aspects, suggestions about what to emphasize -when teaching, and what to seek in assessments are also clarified. This -investigation is intended to develop insight into students understandings of -proofs, that are the meanings they have fashioned for themselves, based on how -they have interpreted what they have heard or read. By examining some of -these understandings, we might find directions in which to improve our teaching. -Moreover, observations about the conceptualizations of students early in -the curriculum can forewarn instructors, helping them recognize the preparation -of incoming students. Perhaps we could use this to prepare remediation -materials. -For example, we can use UML diagrams and ``trie'' data structures to emphasize -definitions for families of concepts. We can choose groups of examples, -and non-examples of proofs whose correctness turns on the qualification that -distinguishes a subclass from its immediate superclass. -Beyond this, one may hope that qualitative research suggests worthwhile questions -for larger scale investigations. - -Application of findings about students -of mathematics to students of computer science is fraught by differences -in the preparation and interests related to algorithms. One likely difference is -motivation: students of mathematics know that proof is the principal means -of discourse in their community, but students of computer science might not -be aware of the importance of proof to their work. Not all differences favor -students of mathematics. In particular, the categories internalization and -interiorization of Harel and Sowder’s 1998 model\cite{harel1998students} are apt to be, in students -interested in algorithms, more closely related, than in students of mathematics. -There may be a difference regarding abstraction. Both mathematics and -computer science deal in abstraction, and students in both disciplines struggle -with it. \cite{mason1989mathematical,hazzan2003students}. In mathematics, following Vi\`ete, \cite{viete2006analytic}, single letter variable names -are used. These are thought to support the learning of abstraction, for example, Gray and Tall \cite[p. 121]{gray1994duality} observe ``we want to encompass the growing compressibility of knowledge characteristic of successful mathematicians. Here, not only is a single symbol viewed in a flexible way '' - and -in computer science abstraction, one way to exhibit abstraction is UML diagrams. Because the -``trie'' structure and International Standards Organization ISO standard 11179 -are computer science approaches to management of definitions, it could be that -computer science students would be more accessible to noticing the desirability -of concept definitions over concept images (see R\"osken and Rolka, \cite{rosken2007integrating} and -Rasslan and Tall \cite{rasslan2002definitions}). It would be interesting to know whether any of several -approaches reported by Weber [?] could be used, perhaps in modified form, for -instruction of students of computer science. The Action Process Object Schema -approach of Dubinsky \cite{dubinsky2002apos} sounds compatible with computer science students' -interests. An approach due to Leron and Dubinsky uses computer programming -\cite{leron1983structuring}, another \cite{leron1995abstract} is directed more to learning group theory than to learning proof -construction. Also specific to students concerned with algorithms, we may -wish to extend the notion of social constructivism from that of Piaget \cite{}, [?] and -of Vygotsky, [?] where it was necessarily a person with whom the learner was -communicating, and therefore with whom it was necessary to share a basis for -communication, to include a compiler and runtime execution environment, as -students of computing disciplines must also comply with rules (e.g., syntax) -used in these systems. Recalling the work of Papert and Harel\cite{harel1991constructionism}, we might -call this constructivism with constructionism. Constructionism is an approach -to learning in which the person learns through design and programming. -A cluster of related problems exists, which includes what students conceptualizations -are, about some elements of proof they should understand: -\begin{itemize} -\item what internal representations do students use? -\item Is there a gamut of internal representations, and does that help with abstraction? -\item mathematization, which is the ability to represent problems in mathematical -notation -\item interiorization, which is the ability to examine and discuss the process of -creating proof -\item comprehension of simple proofs, which is the ability to see that, and why, -an argument is convincing -\item proof analysis, which includes the ability to analyze simple proofs to -recognize structure -\item problem recognition, which is the ability to see that a problem is one that -matches a known solution technique -\item transformational approach, which is considering the consequences of -varying features of the problem -\item axiomatic approach, which is the exploration of the consequences of -definitions -\item construction of valid arguments, which is to synthesize deductions with -component parts, including warrants -\end{itemize} - -\section{Application of Findings} - -\section{ Perspective on Future Directions} \ No newline at end of file +\chapter{Related Work} +Maybe I want to put this in the order of definition, relationships, specialization, symbolization, analogy, generalization/abstraction +\section{Goal Definition} +Schoenfeld gives this expression of a goal for instructors teaching proof: He states that mathematics literature includes ``knowledge and perspectives of world-class mathematicians vs. more or less ordinary PhDs''\cite[p. 74]{schoenfeld1998reflections}. Do we have such characterization for computer scientists? If so, what are these characteristics that are relevant for computer science students, and what projection do these have into students' conceptualizations? + + \section{Methods} +Archavi et al.\cite[p. 4]{arcavi1998teaching} report using microgenetic analysis, which they describe as having roots in both cognitive science and ethnography. +``Schoenfeld, Smith and Arcavi\cite{schoenfeld1993learning} describe it as striving 'for explanations that are both locally and globally consistent, accounting for as much observed detail as possible and not contradicting any other related explanations'''. +Schoenfeld is, among other categorizations, a cognitive scientist, and uses this knowledge to inform his teaching.Archavi et al.\cite[p. 4]{arcavi1998teaching} . + +Research on teaching and learning about proof in mathematics education has +produced an extensive literature. Only a small sampling is mentioned below. +Mathematics educators, including Keith Weber[?], Harel and Sowder in +1998[?], and David Tall[?] have studied students' learning of proof in the mathematics +curriculum. Leron, in 1983, [?] has described the structural method +for proof construction, attributing it to recent ideas from computer science. +Lamport, in 1995, [?] in work on proof construction, has given one approach +that computer science students might find compatible with their background. +Velleman, in 2006, has written software and a textbook [?] about proving with +a structured approach. Weber has reported the success of several approaches +to pedagogy [?]. +Barnard [?] has commented upon students negating statements with quantifiers. +Edwards and Ward [?, p. 223] have discussed the role of definitions for undergraduate +mathematics courses, stating ``the enculturation of college mathematics +students into the field of mathematics includes their acceptance and +understanding of the role of mathematical definitions''. Bills and Tall [?] have +distinguished student understanding of definitions that is sufficient that the +student can use them in proofs. +Harel and Sowder [?] and Harel and Brown [?] have conducted qualitative +research on mathematics students' conceptualization of proofs. They have developed +three main categories, each with several subcategories. Evidence from +our studies is consistent with the presence of these categories of conceptualizations +in the population of CS(E) students. +Tall[?, ?] has also categorized mathematics students' understanding of proof. +He has studied the development of cognitive abilities used in proof, starting, +as did Piaget,[?] with abilities believed present at birth. +Yang and Lin have modeled reading comprehension.[?] +Leron[?] has written about encouraging students to attend to proof structure +by teaching with generic proofs (proofs that use a generic particular). +Mejia-Ramos et al.[?] have built a model for proof comprehension. They have +observed that students who are assessed on appreciation of structural and +other appropriate features of a proof, rather than on rote reproduction, are +more likely to develop a deeper understanding of proof. +Knipping and Reid[41] have examined proof in mathematics education. +Weber[?, ?, ?, ?, ?, ?, ?, ?, ?] has investigated students' approaches to and difficulties +with proof. When studying student proof attempts in group theory, Weber +has found that some typical students' inabilities to construct proofs arise despite +having adequate factual and procedural knowledge, the ability to apply that +knowledge in a productive manner was lacking. [?] More specifically applying +the knowledge was seen to include selecting among facts, guided by knowledge +of which were important, for those most likely to be useful. [?] Alcock +and Weber,[?] have studied students' understanding of warrants, the support +for the use of a particular inference. Weber has published a framework for describing +the processes that undergraduate students use to construct proofs. [?] +Almstrum[?] has investigated the understanding of undergraduate computer +science students of problems related to logic, compared to problems only +weakly related to logic, and has shown that some students have trouble with +the notion of truth or falsity. +Healy and Hoyles[?] have reported on algebra students' preferences for the +content of convincing arguments, and their distinction between preferences +for ascertaining vs. preferences about what was likely to be well-received on +assessments. +{\.I}mamo{\u g}lu[?, ?] has studied the conceptualizations of proof of students who +were preparing to become mathematics and science teachers, in their freshman +and senior years. +Knuth has applied qualitative research to the conceptualizations of proof by +high school mathematics teachers [?, ?]. +Because our work with proof also has explored the consequences for the student +in terms of algorithm choice, including recursive algorithms with proof +by mathematical induction, the work of Booth[?], who has used phenomenography +to develop a model of students' understanding of recursive algorithms +is related. +Zhang and Wildemuth[?] have described qualitative analysis of content. +\section{ Proofs Using the Pumping Lemma for Regular Languages} +Mattuck[36] states ``analysis replaces the equalities of calculus with inequalities: +certainty with uncertainty. This represents for students a step up in +maturity.''[page xiii] and ``these are things which I find that many of my students +don't seem to know, or don't know explicitly. They subtract inequalities +\ldots ``. +\subsection{Quantifiers} +In 2010 Pillay [44] asserted that ``there has been no research into the actual +learning difficulties experienced by students with the different topics'' in formal +languages and automata theory. Of the pumping lemmas, Pillay states ``A +majority of the students made logical errors when proving that a language +is regular and using the Pumping Lemma to show that a language is nonregular. +These could be attributed to a lack of problem-solving skills and an +understanding of the Pumping Lemma.'' Devlin[18] observes that quantifiers +can appear daunting to the uninitiated, and that statements containing multiple +quantifiers can be difficult to understand. +\subsection{ Symbols} +H\"uttel and N{\o}rmark[45] described a successful method for improving both +student activity level in the course and final grades, which combines peer +assessment with creation of notes that can be used during the exam. (``The +incentive was that their answers to text (CHECK) questions would be available for them +to use at the written exam. No other textual aids would be allowed at the +exam.''[p. 4]) The better performance on the exam is welcome; whether it is +due to having notes compared to closed book, or having performed the review +might not be certain. +According to Arnoux and Finkel[46], it is not unusual for students to acquire +mathematical knowledge without attaching meaning to it, and leaving them +unable to solve some problems. They go on to report that Paivio proved +that ``double coding (verbal and visual)'' facilitated remembering. They also +report that different parts of the brain are used to process verbal and visual +information, and therefore more of the brain is involved when both verbal and +pictorial communication is used. They prefer multi-modal representations. +Xing[47] writes about aiding students comprehension of proofs being aided by +graphs. She reports ``students feel that Pumping Lemma(PL) is so abstract to +grasp that using it to prove that a language is non-regular is a daunting task.'' +She shows a graphically laid out proof that a given language is not regular. This +graph has the advantage over a traditional proof, i.e., a sequence of statements, +that the dependencies of states on axioms or intermediate results are plainly +shown by graph edges. +Simon et al.[43] ask ``Is it possible that students plug and chug in computing, not +really understanding the concepts as we would like them to?'' and go on to say +``We posit that the need exists for computing instructors to design assessments +more directly targeting understanding, not just doing, computing. And, of +course, to adopt teaching approaches that support student development of +these skills.'' +Mazur[25] developed peer instruction to address students' propensity to practice +a plug-and-chug approach to problems. This approach has been applied +to computer science teaching, including theory of computation, by several researchers +including Simon, Zingaro, Porter, Bailey-Lee and others[48, 49, 50, +51, 27]. +\subsection{Teaching Pumping Lemmas} +In 2003 Weidmann[39] wrote a dissertation on teaching Automata Theory to +students at the college level. She found that past performance in prerequisite +theory courses was a statistically significant indicator for success in their college +level course. She described a theoretical framework called ``pedagogical +positivism'', a stance between logical positivism and constructivism, allowing +the notion of a teaching method best suited to a group of students to learn Automata +Theory. She interviewed a teacher with ``several'' years of experience +teaching this course (p. 5), who ``admitted that she did not have a better way +to teach abstract thinking other than repeated exposure'' (p. 98). +In chapter 5, Discussion, Conclusions and Implications, of this dissertation[39], +the suggestion ``Instead of simply providing the solution to a problem in class, +or stating the intuitive leap that makes the problem easy to solve, the students +should be exposed to the iterative thought process that lead to the intuition +that created the solution.''(p. 201) appears. One suggestion is ``Learning objectives +should be set to focus on familiarity with formalisms and rigorous +mathematical notations” (p. 224) and another suggestion is “Include programming +projects as part of the required coursework''(p. 224). The combination +of these brings to mind the suggestion of Harel and Papert[40]: ``constructing +personally designed pieces of instructional software'', and the thought that the +students might dwell more effectively on the notion of abstraction as they tried +to teach someone else about it. +\section{ Proof by Induction} +Kinnunen and Simon [7] describe an example applying phenomenography to +computing education research, listing several recent examples, and also providing +a detailed description of a mainly data- but also theory-driven refinement +of categories. +Berglund, Eckerdal and Thun\'e [16, 3, 4] have applied phenomenography to +computing education research, obtaining classifications by judicious grouping +of student conceptions derived from interview data. Eckerdal et al. [4] describe +how the results using phenomenography showed additional insights beyond +other methods. +Jones and Herbst [6] considered which theoretical frameworks might be most +useful for studying student teacher interactions in the context of learning about +proofs. Bussey et al. [2] illustrated student teacher interactions in the space of +learning, and the objects of learning, in variation theory, modified from the +model of Rundgren and Tibell [13]. +Reid and Petocz [12] used phenomenography to study students' conceptions +of statistics. Their purposes included to ``enable teachers to develop curricula +that focus on enhancing the student learning environment and guiding +student conceptions of statistics.'' They asked students to describe how they +understood statistics and then organised student responses into a hierarchy of +conceptions. They used interviews to understand individual students, and the +group of interviews to show the variations they found. They found the students with the most superficial understanding to be carrying out steps without +knowing their meaning. +Krantz [8] describes proof by induction, giving several examples in this book +of proof techniques for computer science. +\section{Domain, Range, Mapping, Relation, Function, Equivalence Relation in Proofs} +Marilyn Carlson, \cite{carlson1998cross}shows that we can easily expect too much from our students in terms of what they understand of functions. This has significance for what we think are adequate examples to use for proof by mathematic induction, for example. +\section{ Definitions, Language, Reasoning in Proofs} +Weber, Alcock, Knuth +\subsection{Procedural vs. Understanding} +Is it that Tall and bifurcation are about learning the procedural vs. understanding approach to dealing with proof? +There are indications\cite[p. 18]{loewenberg2003mathematical} that mathematics teachers in grade school and high school who were mathematics majors themselves learned a procedural approach to mathematics and "lacked an understanding of the meanings of the computational procedures or of the solutions. Their knowledge was often fragmented, and they did not integrate ideasthat could have been connected (e.g., whole-number division, fractions, decimals, or division in algebraic expressions.)" +\subsection{Recall of Relevant Information vs. Inert Knowledge} +Bransford et al.\cite[p. 296]{bransford2000designs} attempt to address the problem identified by as inert knowledge (in the sense of Whitehead\cite{whitehead1959aims}). They situated class activity in a problem solving environment, and they showed\cite{van1992jasper} that this instruction had better results for students' ability to transfer skills to new word problems than traditional instruction. + +Lehrer et al. \cite[p. 334]{lehrer2000inter} found that "at least in some circumstances, giving children models may be less helpful than fostering their propensity to construct, evaluate, and revise models of their own to solve problems that they consider personally meaningful." + +\section{Abstraction} +Lesh and Doerr\cite{lesh2000symbolizing} used model-eliciting to engage students in the creation of a combination of meaningful descriptions, explanations and procedures. These models were recognized as tools to be shared and reused, consequently the idea of generalization was implicit. Lesh and Doerr argue that by illuminating the idea of a tool that may be reused, they divide the problem of teaching generalization into two parts: making a general model, and discerning its domain of applicability. + + +Kemmerer\cite{kemmerer} + +According to Huang et al.\cite{huang2015highest}, a large body of research suggests that an abstract cognitive processing style produces greater creativity. Empirically, decades of work have shown that both abstract thinking and creativity are consistently linked to right-hemispheric activation in the brain (e.g., Fink et al., 1996\cite{fink1996brain} and Mihov et al., 2010\cite{Mihov2010442}). +Miron-Spektor et al.\cite{Miron-Spektor20111065} have shown that observing anger communicated through sarcasm enhances complex thinking and solving of creative problems. + +(We want to know, do they understand deduction, abstraction, where are they on van Hiele levels ) + +According to Gray and Tall \cite[p. 117]{gray1994duality}, Hiebert and Lefevre observed ``a connected web \ldots a network in which the linking relationships are as prominent as the discrete pieces of information \ldots a unit of conceptual knowledge cannot be an isolated piece of information; +by definition is it part of conceptual knowledge only if the holder recognizes its relationship to other pieces of information ``\cite[p. 3-4]{hiebert2013conceptual} +conceptual knowledge is harder to assess than other kinds of knowledge. + +\section{Diagrams in Proof} +Gibson\cite{gibson1998students} examined students' use of diagrams in proofs, and found that diagrams helped link students' ideas to mathematization, namely, to representation in symbols, and also to support variation, in the sense of the critical difference between what Harel and Sowder\cite{harel1998students} call perceptual and transformational conceptualizations. +\section{Equivalence Class, Generic Particular, Abstraction in Proofs} +\section{Computer Science Education} +Thun\'e and Eckerdahl\cite{thune2009variation} have applied variation theory has been applied to the teaching of computer science. + + +\section{Educational Psychology} +\subsection{What do students need to construct?} + +Archavi et al.\cite[p. 13]{arcavi1998teaching} ``I'd like to have you doing some mathematics and I will do everything I can --- including using grading --- as a device for having you do that.'' + +\section{Phenomenography, Variation Theory} + +Marton and Booth\cite{marton1997learning} have written + +{\aa}kerlind \cite{aakerlind2012variation} has written on how + +Runesson\cite{runesson2005beyond} has applied variation theory to math + +\section{Constructivism} + + + +Vygotsky in Language and Thought said we do as individuals build up thoughts and then becoming socialized with shared language, some accommodation would need to be enforced onto the child. [p.17] the psychological problem is to become convinced that always, necessarily a given picture has to appear as one of a multiple of possible graphs of the same category (i.e. only as a representative of a class \ldots must be grasped not in a final fixed state but rather \textit{in construction} the point moving) + +Vygotsky\cite[p. 49]{vygotsky1978mind} noted that "one child selected a picture of an onion to recall the word 'dinner'. When asked why she chose the picture, she gave the perfectly satisfactory answer, 'Because i eat an onion'. However, she was unable to recall the word 'dinner' during the experiment. This example shows that the ability to form elementary associations is not sufficient to ensure that the associative relation will fulfill the \textit{instrumental} function necessary to produce recall." + +\subsection{What do students need to construct?} + +Archavi et al.\cite[p. 10]{arcavi1998teaching} ``To be successful, students must know both the appropriate heuristics and the mathematics required to solve the problem. +\subsection{Intuition} +Students have some knowledge constructed already, and it is not all conscious. + +overconfidence --- counter by ``search for reasons it might be wrong'' Koriat et al., 1980\\ +confidence --- doesn't correlate with correctness\\ +be as to conform Tweney Doherty Mynatt 1981\\ +``renouce several of his funcamental beliefs with regard to reality'' [p. 39]\\ +$$ Combine desire to gain points having taken the place of desire to learn, with propensity to learn how to take tests rather than how to believe, and obtain ``How we answer the tests, or, what we really think'' $$ +Piaget (Piaget-Beth 1966 [p. 195] ) ontogenetic construction of evidence a new domain integrates former domain as subdomain\\ +(n heast?) intuition tends to survive even when contradicted by systematic formal instruction [p. 47]\\ +Polanyi 1969 [p. 143-144]\\ +\ldots in the structure of tacit knowledge we have found a mechanism which can produce discoveries by steps we cannot specify this mechanism may then account for scientific intuition \ldots not the supreme immediate knowledge called intuition by Leibnitz or Spinoza or Husserl, but a work-a-day skill for scientific guessing with a chance of guessing right.\\ +Polanyi sees a deep analogy between integrative capacity +$$ there we have ``unsupervised'' specialization network formation during consolidation, perhaps $$ ``Where to turn for a logic by which such tacit powers can achieve and uphold true conclusions'' Polanyi 1967 [p. 137]\\ +$$I'm thinking about conscious / unconscious$$ +$$ When we do something consciously we can be checking, when we do something unconsciously, we might not be $$ +Fischbein [p.59] ``Inferential affirmatory intuition may have an inductive or deductive structures. After one has found that a certain number of elements (objects, substances, individual, mathematical entities, etc.) have certain properties in common one tends \textit{intuitively} to generalize and to affirm that the \textit{whole} category of elements possesses that property. This is not a mere logical operation. +The generalization appears more of less suddenly with a feeling of confidence. +This is a fundamental source of hypotheses in science. +According to Poincar\'e ``generalization by induction copied, so to speak, from the proedures of experimental sciences'' is one of the basic categories of intuition (Poincar\'e 1920 [p. 20]).\\ +\cite[p. 67]{fischbein1987intuition}(check) ``One morning walking on the bluff, the idea came to me with just the same characteristics of brevity, suddenness and immediate certainty that the arithmetical transformation of indeterminate ternary indefinite forms were identical to those of the non-Euclidean geometry Poincar\'e 1913 [p. 388]''\\ +David Tall mathematician and psychological analyst, moment of insight ``never felt that he made 'conjectures'; what he say were 'truths' evidenced by strong resonances in his mind. Even though they often later proved to be false, at the time he felt much emotion vested in their truth \ldots intense intuitive certainties. Yet at the same time his contact with them often seemed tenuous and trasient; initially he had to write them down, even though they might be imperfect, before they vanished like ghosts in the night (Tall 1980 [p. 33])\\ +$$ being unconscious seems to go with consolidation. Either unconscious because attending to something else like walking on a bluff, or asleep, being unconscious is relevant to these integration occurring and then moving into consciousness $$ + +$$''Really wants to know'' implies an openness to change the pre-determined ideas, and ``complyig with requriements'' does not imply readiness to revise $$\\ +\cite[p. 68]{Fischbein} citing Feller ``experienced player absorbs a complicated situation at a glance and is unable to account rationally for his intuition''\\ +$$ it (the part of the brain doing the reasoning) is functioning without conscious oversight (the neurons that did that when the capability was new have been deemed extraneous and removed)$$) +$$Brooks -- the first system is done carefully with all consciousness like the beginning chessplayer in Feller, the second system has some unwarranted conviction and the third system has mostly warranted $$\\ +\cite[p. 69]{Fischbein} Felix Kelin (1898) trained intuition\\ +Suppes 1966 train the intuition for finding an writing mathematical proofs''\\ +\cite[p. 72]{Fischbein} categorical syllogism type AAA seems easiest for which, EAE, AII 65\%, EIO +(These are categorical syllogism types. See www.philosophypages.com/lg/e07a.htm) +\cite[p. 77]{Fischbein} +AAA and modus ponens come earliest and are class inclusion, maybe $\bar{p} \rightarrow q$ never develop\\ +re \cite[p. 81]{Fischbein}, $$ people have data stored to supply their intuition and it our be wrong education involves opening it up to conscious inspection, fixing it, and restoring the rapid unconscious operation $$\\ +\cite[p. 106]{Fischbein} citing \cite[p. 228]{Wertheimer 1961} ``These thoughts did not come in any verbal formulation. I very rarely think in words at all. T thought comes, and I may try to express it in words afterwards''. Einstein\\ +\cite[p. 119]{Fischbein} A total fusion of the generality of a principle and a particular directly graspable (in this case figural) expression of it. It is this kind of fusion which is the essence of intuition.\\ +\cite[p. 120]{Fischbein} specific, directly convincing example and the general principle derived through similarity and proportionality from the particular case.\\ +\cite[p. 129]{Fischbein} Analogy frequently intervenes in mathematical reasoning, Polya writes about great analogies \\ +138 1985 software by David Tall\\ +\cite[p. 144]{Fischbein}ways in which people process concepts Smith Meliss 1981\\ +\cite[p. 147]{Fischbein} ``For many students the concepts of parallelogram, square and rectangle are not organized hierarchically. They represent classes of quadrilaterals of the same generality''. +$$ some programmers before unified method were guilty of writing code this way, with wasteful effect. Moreover, Liskov Substitutability Principle was enunciated to help people know how to populate hierarchies when they were learning to do so. Students who are not organizing their concepts hierarchically are similarly disadvantaged. As I believe this hierarchical situation is consolidated during sleep or relaxation, ti becomes a research question$$ +\cite[p. 159]{Fischbein} analogies already similar diagram post concept\\ +\cite[p. 165]{Fischbein} diagram relies on intervening structure (conceptual structure) else it does not communicate\\ +stability of intuition, Ajzen? 1983 epistemic freezing\\ +\cite[p. 214]{Fischbein} concept of intuitive loading --- have to know students first before knowing how to teach them + + +Intuition in Science and Mathematics An Educational Approach Efraim Fischbein 1987 Reidel Publish +Westcott combines theoretical analysis with experimental findings. +Andrea di Sessa building a theory of intuition +Bergson 1952 essence of lining changing phenomena +Kant intellectual (does not exist) and sensible intuition +1980 [p. 268] + +Poincar\'e useful +Hahn 1956 source of misconception +to use or to eliminate? +Berne professional quality work without awareness says Westcott 1968 [p.42-46] +immediacy --- is that because crosses into conscious unconscious\\ +consistency, brevity of expression $\rightarrow$ bearty +if the result of the brain's consolidation of knowledge, info, into uncouscious knowledge, usable, available for recall +going into intuition, what you taught us for the test, or what we really think +Fischbein p. 9 ``One may not be aware of the existence of such an explicit representation but it continues to act tacitly and to influence ways of reasoning. + +Seymour Papert 1980 apparently says something about intuition + +Brouwer, Weil, Kline 1980 [pp 306-327] + +The sum of the angles of a triangle is equal to two right angles'' +Connect bewteen intuitino and reasoning +\subsection{Social Constructivism} +Archavi et al.\cite[p. 6]{arcavi1998teaching} ``Students' mathematical activity takes place in an inherently social milieu.'' +\subsection{Tall: Set-befores and Met-befores} +McGowan and Tall \cite[p. 172]{ (2010 Jour. Math. Behav.)} ``If learning defaults to the goal of learning how, it can be successful. However, if it is accompanied by a lack of conceptual meaning so that mistakes occur, it can become fragile and more likely to fail in the longer term. At this stage the problems may proliferate as the student becomes confused as to which rule to use, where to use it, and how to interpret it. + +Tall and Mejia-Ramos \cite[p. 138]{2010, Explanation and Proof in Mathmatics, Springer} ``Here proof develops through generalized arithmetic and algebraic manipulation'', +different kinds of warrants for truth $$ so assess student by asking what kind of warrant$$ see Pinto and Tall (1999 and 2002) build on met-befores. +\begin{figure}[tbph] +\centering +\includegraphics[width=0.7\linewidth]{chp7p1} +\caption{How proof develops, Tall Mejia-Ramos} +\label{fig:chp7p1} +\end{figure} + +\subsection{Harel and Sowder} + + \cite[p. 237]{harel1998students}Rather than gradually refining students' conception of what constitutes evidence and justification in mathematics, we impose on them proof methods and implication rules that in many cases are utterly extraneous to what convinces them. +Editors Schoenfeld et al.\cite{kaput1998research} describe that Harel and Sowder\cite{harel1998students} ``characterize students' cognitive schemes of proof''. + +The subdivisions in the 1998 version of categories of conceptualizations \cite{harel1998students}, specifically intuitive -- axiomatic, structural and axiomatizing, +matter much in computer science, because intuitive -- axiomatic could be thought to be less used in computer science than in math, program's content could be less intuitive than Euclidean geometry, more subject to checking by assertion checking or debugger examination. + + \cite[p. 268]{harel1998students} contextual proof scheme: --students have learned to work in a context, e.g., $\mathbb{R}^n$, and so, interpret statements that have greater generality as restricted to be in the context they have learned ``he or shee has not yet abstracted the concept \ldots beyond this specific context''. Compare this with Pang's (is it Pong?) observation that for students who know only one language, ``speaking'' and ``speaking that language'' are concepts that are undifferentiated. + + \cite[p. 274]{harel1998students} ``An important distinction between the structured proof scheme and the intuitive proof scheme is the ability to separate the abstract statements of mathematics (e.g., $1+1=2$) from their corresponding quantitative observations (e.g., 1 apple + 1 apple = 2 apples) or the axiomatically -- based observations from their corresponding visual phenomena \ldots ``, ``axiomatic proof scheme is epistemologically an extension of transformational proof scheme. One might mistakenly think of the axiomatic proof scheme is the ability to reason formally \ldots ``. + + \subsection{Pirie and Kieren Model of Mathematical Understanding} + \cite{meel1998honors} + Verify these are due to Pirie and Kieren rather than to Meel. + + \paragraph{Primitive Knowing} + This is brought by the student, and is also known as intuitive knowledge, situated knowledge, prior knowledge and informal knowledge. + \paragraph{Image Making} + any mental image not necessarily pictorial + \paragraph{Image Having} + mental picture / objects, concept image, frame, knowledge representation structure, students' alternative frameworks + \paragraph{Property Noticing} + unselfconscious knowing, can notice distinctions combinations connections between mental objects + \paragraph{Formalizing} + abstract (this is a verb) common qualities from classes of images, classlike mental objects built from noticed properties, description of these class-like mental objects results in production of full mathematical definitions + \paragraph{Observing} + ability to consider one's own formal thinking, organize personal thought processes, recognize ramifications, + \paragraph{Structuring} + axiomatic system, conceive proofs of properties associated with a concept + \paragraph{Inventising} + create new questions, develop new concepts + \paragraph{folding back} + reorganizing lower level understanding to accommodate new information + + +\subsection{van Hiele Levels} +Abstraction is before deduction +\subsection{Performance Levels} +Baranchik and Cherkas\cite{baranchik1998supplementary} found three levels of understanding in a population taking algebra exams: +\begin{enumerate} +\item Early skills --- arithmetic and elementary algebra +\item Later Skills --- subsequent algebra and a variety of skills involving methematical abstraction, and +\item Formalism --- either devising a solution strategy or reformulating a problem into a standard form that permits a solution using early or later skills +\end{enumerate} +\subsection{Student Centered} +Carlson\cite{carlson1998cross} has concluded that ``\ldots an individual's view of the function concept evolves over a period of many years and requires an effort of 'sense making' to understand an orchestrate individual function components to work in concert.'' +\subsection{Use of Diagrams} +Gibson\cite{kaput1998research} states ``Diagrams aided students' thinking by corresponding more closely to the part of their understanding with which they were operating at the time and by reducing the burden that proving placed on their thinking.'' +\cite[p. 205]{kaput1998research} The nature of internal representations, however, is unclear because they are not observable.: +$$ nature of internal representations can be broad, and we can perhaps influence the nature of internal representations, which are ultimately neural nets, by how we teach, and nature of internal representations is such that some, e.g., perceptual, are nto as helpful as others, e.g., transformational. Get the superior colliculus involved, see the motion Ties in with variation theory. Also visualization parts of brain.(B17?) $$ + +Winn, B, (get the citation from Gibson article in Kaput RCME 1998)( Charts, graphs and diagrams in Ed. materials Psych Illus Basic Research Vol 1 Springer 1987 pp. 152-198) has a spectrum for internal representations from pictures to works, the word end is called abstract. + +Zimmerman, Visual thinking in Calculus Visualization Teaching Learning Math 1991 +Gibson\cite[p. 132]{kaput1998research} ``There is no doubt that diagrams play a heuristic role in motivating and understanding proofs'' +Tall 1991 Intuition and rigor, role of visualization in teaching learning mathematics +Gibson\cite[p. 288]{kaput1998research} ``When student used visual language I inferred that they were operating with the visual part of their understanding'' +Gibson\cite[p. 289]{kaput1998research} ``Students indicated that diagrams helped them understand information by appealing to their natural thinking. They said that diagrams seemed to coincide with the way their 'minds work' and that information represented visually seemed easier or clearer than verbal/symbolic representations.'' +more concrete than verbal/symbolic +Gibson\cite[p. 290]{kaput1998research} ``used it to help me see what would be happening'' +$$ executive parts of brain is engaging visual parts of brain$$ +easier than holding the mental image is look at the drawn image +Gibson\cite[p. 291]{kaput1998research}''When I read the definitions you can't think about the whole thing at once, but when you have a picture you can'' +Gibson\cite[p. 294]{kaput1998research}''Because students did not usually think of their criteria in terms of formal definitions, their ability to decide whether their criteria had been met was hindered when they worked with information represented in only verbal/symbolic form.'' +``They could obtain ideas more readily from diagrams than they could from verbal/symbolic representations'' +Gibson\cite[p. 297]{kaput1998research} Why always keep the picture in your mind when you can have it on the paper, allowing you to focus more on how to get to the end of the proof instead of always having to recall the picture in each individual step?'' +$$visual rather than mirror area is possible$$ +Gibson\cite[p. 298]{kaput1998research} ``students sometimes used diagrams to help them express their ideas'' symbolically +$$compare proofs without words$$ +Gibson\cite[p. 298]{kaput1998research} ``diagrams helped Laura write out her ideas by helping her connect her ideas to verbal/symbolic representations of these ideas'' +Gibson\cite[p. 299]{kaput1998research}''you need to down load that picture on here so that you can touch it and then allow your brain to think about the words you need to say'' +visualization does not always help, Gibson quoted some sources +Gibson\cite[p. 302]{kaput1998research} ``when attempting to solve unfamiliar problems, students can benefit from using diagrams'' +Moore\cite[p. 262]{moore1994making} ``The students' ability to use the definitions in the proofs depended on their knowledge of the formal definitions, which in turn depended on their informal concept images. The students often needed to develop their concept images through examples, diagrams, graphs and others means before they could understand the formal verbal or symbolic definitions'' %[p. 262], Moore R Making the transition to formal proof Ed Studeies in Math 1994. +Gibson\cite[p. 303]{kaput1998research}''That students would operate in this manner (with the visual part of their concept images) and that such behavior might be of benefit is reasonable when one considers the nature of the concepts in the proofs together with the students' experiences as visual beings and the physiology of their brains''. + + +\section{Cognitive Science} +Archavi et al.\cite[p. 6]{arcavi1998teaching} Mathematics requires abstraction, and problems should inspire generalization and specialization. + + + +\subsection{Intrinsic Reward} +Archavi et al.\cite[p. 9]{arcavi1998teaching} Good problems are ``\ldots non-routine and interesting mathematical tasks, which students want and like to solve, and for which they lack readily accessible means to achieve a solution''. +\subsection{What do students need to construct?} + +Archavi et al.\cite[p. 13]{arcavi1998teaching} `` There were occasions later in the course in which the whole-class discussion also dealt with issues of mathematical elegance and aesthetics.'' + + +Leslie Valiant\cite[p. 103]{valiant2000circuits} points out that representations, for models of cognition, are not all equally learnable. +(in polynomially many steps, p. 104) +Easily learnable representations (of concepts) ``include Boolean conjunctions (e.g., $x_1 \land x_5 \land \bar{x_y}$) and Boolean disjunctions (e.g., $x_1 \lor \bar{x_3} \lor x_8$) \ldots An important class that is not currently learnable is disjunctive normal form (or DNF for short)'', (e.g., $x_1\bar{x_2}x_3 \lor x_1x_2 \lor x_2x_4x_7$), describes a concept whose membership can be attained in one of three ways, in two of which $x_2$ must b true, but in the other of which $x_2$ may be false, so long as $x_1$ and $x_3$ are true. +He goes on to observe these may be learned in stages, stating ``more is required of the teacher or environment than in the simplest case of learning by example'' [p. 104] +He uses the idea later clarified by Marton and Pang\cite{} stating ``a teacher may have to teach the name of this subconcept and then identify positive and negative examples of it'' [p. 104]. +`` In this context, learning theory can be thought of as defining the granularity with which learning can proceed without intervention \ldots the largest chunks of information that can be learned feasibly without their having to be broken up into smaller chunks'' [p. 104] +(Combine this with the approximately 7 chunks in short term memory?) + +Generalization and analogy are directly addressed in mathematics teaching by assigning students to ``search for connections and extensions of problems''\cite{santos1998instructional}. +\subsection{Analogical Reasoning} +Gentner and Smith\cite{gentner2012analogical} define analogical reasoning as "the ability to perceive and use relational similarity between two situations and events", and have stated that analogical reasoning is fundamental to human cognition. +They state that \cite[p. 131]{gentner2012analogical} ``Analogy is often the most effective way for people to learn a new relational abstraction; this makes it highly valuable in education.'' +Because we wish to obtain the value inherent in reasoning by analogy, we note that it depends upon recognition of relationships, and abstraction, to compare relationships at a level divested of some specifics. +Abstraction, for students of computer science, has been observed to be difficult to learn\cite{or2004cognitive} in that context. +Nevertheless, application of proverbs, such as "Don't cry wolf.", is routinely expected in education of children\cite{lutzer1988comprehension}. +Or-Bach and Lavy show empirical data and provide insight into the difficulties of computer science students who have trouble extracting common features from a problem statement that emphasizes differences, and promoting those to a more general class, while maintaining the differences in the more specific classes. +The relationship from one class to a related class in an inheritance hierarchy, motivated as it has been by code reuse, is more stereotyped than the relationships in proverbs, which are not restricted to generalization/specialization. So, we should be careful about generalizing the difficulty students of computer science have with abstraction. +Gentner and Smith go on to say\cite[p. 131]{gentner2012analogical} that analogical reasoning is characterized by retrieval, in which a current topic in working memory may remind a person of a prior analogous situation in long term memory; mapping, which involves aligning the representations and projecting inferences from one analog to another; and evaluation, which judges the success of the alignment of the representations and inferences. +Thus we see that the relationships are key in analogical reasoning, compared with being stereotyped in establishing inheritance hierarchies. +Gentner and Smith\cite[p. 133]{gentner2012analogical} remark that "Another benefit of analogy is \textit{abstraction}: that is, we may derive a more general understanding based on abstracting the common relational pattern." and "analogies can also call attention to certain differences between the analogs." +Though we might wish to have people readily retrieve knowledge that would, by analogy, be helpful to solving a current problem, Gick and Holyoak\cite{gick1980analogical} showed that people do not always retrieve the knowledge they have, rendering it, at least temporarily and for this purpose, what Alfred North Whitehead called "inert knowledge"\cite{whitehead1959aims}. +Gentner and Toupin \cite{gentner1986systematicity} have observed, however that, older children (and not younger children) benefited from systematicity: a summary statement of the structure of the relationships. There is a shift that can occur from focussing on objects to focusing on relatiohships, called a "relational shift", which has been the subject of research\cite{gentner1988metaphor,rattermann1998more,bulloch2009makes}. +Dunbar\cite{dunbar2000scientists} outlines three important strategies that scientists use: attention to unexpected findings, analogic reasoning, and distributed reasoning. Dunbar states\cite[p. 54]{dunbar2000scientists} "our analyses suggest that analogy is a very powerful way of filling in gaps in current knowledge and suggesting experimental strategies that scientists should use" and "If scientific reasoning is viewed as a search in a problem space, then analogy allows the scientist to leap to different parts of the space rather than slowly searching through it until they find a solution". + +Day and Gentner\cite{day2007nonintentional} showed in +Day and Gentner\cite[p. 41]{day2007nonintentional} +"Gentner and Medina proposed that +schemas and other abstractions are often derived via a +process of repeated analogizing over instances (see also +Cheng \& Holyoak, 1985)." + +\textbf{This, schemas and abstractions, earned a double question mark. Maybe that means it should be reported in more length.} +Day and Gentner\cite[p. 41]{day2007nonintentional}"The goal of this research was to investigate an important open question: Can a single prior instance influence how a new episode is understood, and if so, does it do +so by using a structurally sensitive mapping process, as +in analogy?" +Day and Gentner\cite[p. 42]{day2007nonintentional}"The results are consistent with the claim that individuals +may use a single prior instance as a source for nonintentional inference based on structural commonalities. The +pattern of inferences is what would be expected if participants were structurally aligning the two representations +and drawing inferences about the target from relationally +similar aspects of the base. Participants' responses that the +inferred information had actually been stated in the target +story suggest that these inferences were not deliberately +considered and evaluated, but rather were spontaneously +incorporated into the target representations as they were +being created." + +\subsection{Generalization} +Generalization is thought to result when multiple instances of analogies, sharing the same structure of relationships, have been considered. *who was I reading before kowatari?" + +Ball states\cite[p. 38]{loewenberg2003mathematical} "Generalization involves searching for patterns, structures, and relationships in data or mathematical symbols. These patterns, structure, and relationships transcend the particulars of the data or symbols and point to more--general conclusions that can be made about all data or symbols in a particular class. Hypothesizing and testing generalizations about observations or data is a critical part of problem solving." + +She continues \cite[p. 38]{loewenberg2003mathematical} "In one of the simpler common exercises designed to develop young students' capabilities to generalize, students are presented with a series of numbers and are asked to predict what the next number in the series will be. \ldots Representational practice play an important role in generalizing. For example, being able to represent an odd number as $2k+1$ shows the general structure of an odd number. \ldots Representing the structure using symbolic notation premits a direct view of the general form." + +Lesh and Doerr\cite[p. 379]{lesh2000symbolizing} encourage students to construct models, that may include "a combination of spoken words, written symbols, pictures or diagrams, or references to other models or real-life experiences \ldots in any case, the representation tends to organize and simplify the situation so that additional information can be noticed, or so that attention can be direct toward underlying patterns and regularities, which may, in turn, drive changes in conceptions." + +Bowers\cite[p 390]{bower2000postscript} summarizes ideas on generalizing saying: "Bransford et al.\cite{bransford2000designs} describe several studies to support the claim that 'people's representations of problems and experiences have strong effects on the degree to which they will transfer their knowledge to new settings'. Similarly, Lesh and Doerr\cite{lesh2000symbolizing} argue that the models students produce when engaging in model-eliciting problems are not just solutions to the problem at hand, but instead stand as more generalized conceptual tools that can be 'shared and reused in other situations' ". + +Huth et al.\cite{huth2012continuous} + A continuous semantic space describes the representation + + % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % + + + $$ One reason we delve into this is that we want to know something about to what degree different factors that might possibly assist learning are significant. For example, how important is it to note the beauty of a proof, and what is the significance of the order in which a proof is presented (for example, lemmas first), and how quickly might we expect students to grasp a hierarchy of abstractions. We learn possibly surprising things, such as, if we provide a fragrance during learning a proof with ideas about spatial locations, and we provide that same fragrance during the early part of sleep, the memory will be consolidated more effectively, as demonstrated by subsequent recall during an awake state.$$ + + Cerf et al. \cite{cerf2014studying} opine "early neuron activity observed here could represent the early stages of the formation of a thought or recollection, the states at which we may not yet be fully aware of the content of the thought" + + \subsection{Cognitive Neuroscience} + Cognitive neuroscience provides evidence for believing that suspense, and concern for characters, is useful in helping students selectively attend to, and remember at an abstracted level, the material they are seeing. + For example, + Bezdek et al. \cite{Bezdek2015338} have measured brain responses corresponding to attention, and have shown that attention is modulated by the emotional flow of a narrative as it unfolds over time, and that suspense is associated with increased central processing (of the visual field) and decreased peripheral processing. Moreover they have reason to believe that this attention does produce downstream consequences, reflecting encoding of content at a level abstracted from visual features. They have brain metabolism imaging showing decreases in activity that has been associated with mind-wandering\cite{Christoff20098719}. + Kosslyn and K\"onig \cite[p. 56--57]{kosslyn1992wet} describe some layout of lower level functions in the brain, "The ventral (object-properties-encoding) system in the temporal lobes not only registers key properties of shapes, but also encodes color and texture; this information is matched to that of objects stored in visual memory. This temporal-lobe memory stores information in a visual code, which cannot be accessed by input from other sensory modalities. \ldots The outputs from the \ldots encoding systems come together at an \textit{associative memory} (which relies on tissue in various places in the brain) \ldots once the appropriate information is accessed, one knows the name of the object, categories to which it belongs, sounds it makes, and so forth. \ldots there is ample evidence that the frontal lobe plays a critical role in this process" + Kosslyn and K\"onig \cite[p. 78]{kosslyn1992wet} "For example, if a caterpillar is still against a twig, it may be very difficult to notice. But if it is moving, we may identify it immediately. Computing motion relations appears to be qualitatively distinct from computing the organization of portions of static images, and distinct regions of the brain apparently encode motion (particularly areas MT and MST). There appears to be a distinct \textit{motion relations subsystem}. The motion relations subsystem extracts key aspects of motion fields, and operations concurrently with the preprocessing subsystem." + $$ knowing whether two (simultaneously presented) parts of a proposed presentation will require attention from two systems able to operate concurrently, or will make conflicting demands upon one attention window is useful$$. + + + Kosslyn and K\"onig \cite[p. 102]{kosslyn1992wet} "the frontal lobe clearly has a role in setting up plans. More important, the frontal lobe clearly has a role in formulating and testing hypotheses."$$needed for creating proofs$$. + + Kosslyn and K\"onig \cite[p. 104]{kosslyn1992wet}"a subsystem that \textit{engages} attention. \ldots the thalamus. The thalamus is a kind of switching station, connecting many parts of the cortex." + + Kosslyn and K\"onig \cite[p. 112]{kosslyn1992wet} "Visual object agnosia is often divided into two types Patients who are diagnosed as having \textit{apperceptive} agnosia have difficulty putting together visual information to form an integrated perception on an object. Some such patients describe the world as fragmented or chaotic. These patients cannot determine whether two objects are the same or different, let alone identify an object they see. However, even these patients are not blind. Patients who are diagnosed as having \textit{associative agnosia}, in contrast, have difficulty associating the perceptual input with previously stored information. Patients who have 'pure' associative agnosia can discriminate between and properly compare shapes, even though they cannot identify the shapes \ldots could tell whether objects were the same or different, but not what they were. In short, a problem appreciating the shape of an object is apperceptive; a problem identifying the object while being able to distinguish its shape is associative." + $$ it would be nice to diagnose student difficulties to this degree, especially if an treatment could be associated with the diagnosis. Moreover, there could be a strong analogy here. Students could recognize two applications of rules of inference as being the same without being able to identify which rule of inference it is. This suggests a clicker question about rule applications, as to whether they are the same or not, or, choose the rule application that is different. $$. + + Kosslyn and K\"onig \cite[p. 114]{kosslyn1992wet} "Prosopagnosia is particularly puzzling because at least some of these patients apparently recognize faces unconsciously, and never become aware that they have done so; patients with prosopagnosia showed changes in the electrical properties of their skin (i.e., increased electrodemal skin conductance responses) when they were viewing familiar faces, compared to unfamiliar ones, even when they claimed to have no idea whom they were looking at. The stimulus must be matching a stored memory of a face in the pattern activiation subsystem (at least to some degree), but processing sops early -- and the persons is never aware that a match was made. + + Kosslyn and K\"onig \cite[p. 118]{kosslyn1992wet} There are processes that activate stored visual information to generate visual mental images. + + Posner for systems involved in attention + + Kosslyn and K\"onig \cite[p. 124]{kosslyn1992wet} "one of the most intriguing aspects of the neglect syndrome is that it appears to affect consciousness itself." + + Kosslyn and K\"onig \cite[p. 132]{kosslyn1992wet} "Patricia Goldman-Rakic and her colleagues showed that the frontal lobes contain a short-term spatial memory." + + Kosslyn and K\"onig \cite[p. 136]{kosslyn1992wet} "the (mental) image is not like a picture; it is facing almost as soon as it is generated."(Fink, Pinker, Farah, Chambers, Reisberg) + + Kosslyn and K\"onig \cite[p. 147]{kosslyn1992wet} "In either case, instructions somehow must be provided to the attention shifting subsystems; if the image is truly novel, the instructions cannot be previously stored. Thus, it is of interest that there are rich connections from the frontal lobes (which presumably direct the process), not only to parts of the parietal lobes known to be involved in attention, but also to the subcortical structures involved in shifting attention." Goldman-Rakic, posner, 1987, 1988, petersen 1990 + + $$Kosslyn et al. have inferred an organization of processing in the brain, and we as instructors could think about how we are deploying the learning task over this architecture. We could think about what path information has follow, to be learned to the extent that it can be called into future use. That is, while some instruction could produce inert knowledge, the goal is usable knowledge.$$ + + Kosslyn and K\"onig \cite[p. 152--153]{kosslyn1992wet} "By inferring that images occur in the same visual buffer that is used in perception, we expect that the properties of the buffer that affect perception also should affect imagery. If so, then the inability to maintain patterns in images for very long may be another consequence of this common mechanism. That is, in perception one does not want an image to linger; one wants to 'clear the buffer' every time the eyes move. Indeed, if images did not fade rapidly, they would smear and become overlaid. \ldots Thus, patterns in the visual buffer are transient in both perception and imagery." + + Kosslyn and K\"onig \cite[p. 154]{kosslyn1992wet} "A symbolic use of imagery involves the same subsystems used in other kinds of imagery reasoning; one must generate the image and retain it long enough to operate on it and 'see' the results. But there is one critical difference between this kind of symbolic imagery and the other sorts: One now must decide how to convert abstract material to particular patterns in the image. For example, one could visualize relative intelligence not only as dots on a line, but also as a set of circles" + + Kosslyn and K\"onig \cite[p. 163]{kosslyn1992wet} both hemispheres can generate images (posterior of brain), instructions come from frontal lobes. Left hemisphere naturally, right hemisphere with training. Kosslyn and K\"onig \cite[p. 164]{kosslyn1992wet} both hemispheres can generate images, but in different ways. + + Kosslyn and K\"onig \cite[p. 165]{kosslyn1992wet} image transformations require contributions from both hemispheres + + Kosslyn and K\"onig \cite[p. 168]{kosslyn1992wet} "Learning to read these (already familiar) words is learning to use an additional route into associative memory to access the information that is stored with words. The problem is, that we initially learned to access the relevant memories on the basis of hearing sounds, \ldots How are we able to use a pattern of lines to access these memories?" + + $$subvocalization is an inferior way to accomplish this$$ + + + + + + + Kosslyn and K\"onig \cite[p. 193]{kosslyn1992wet} "With repeated use, the entire pattern of lines becomes an entry in the pattern activation subsystem and the pattern recognized there is associated with a word in associative memory" + + $$repetition of relationships between items allows us to claim a pattern, a pattern suggests the thought of generalization, the symbolic representation of the pattern makes generalization more efficiently notated, maybe increases awareness of the possibility of generalization, and the ability to generalize is used in transfer of knowledge to further applications, counteracting inert knowledge$$ + + Kosslyn and K\"onig \cite[p. 204]{kosslyn1992wet} "the information that allows one to understand a word must be in associative memory, and possibly is stored only in the left hemisphere. (The left hemisphere is typically the site of the most language processing, and hence it would not be surprising if representations of word meaning were stored primarily in this hemisphere.) \ldots the left hemisphere associative memory \ldots presumably implements the word shape/sound associations." + + Kosslyn and K\"onig \cite[p. 207]{kosslyn1992wet} writing about a form of dyslexia that has provided insight, describe ideas of Coltheart\cite{schmalz2015getting} "He hypothesized that only words that name visible objects or properties are stored in the right hemisphere, \ldots " + + Kosslyn and K\"onig \cite[p. 214]{kosslyn1992wet} "Sounds are represented initially in a cortical structure in the inner part of the superior (upper) temporal lobes. Heschl's gyrus \ldots is the auditory analog to area V1; indeed, this structure is often called A1 (to indicate that it is the first cortical auditory area). As in vision, the raw auditory input is organized and represented in the auditory buffer prior to high-level processing. Ulrich Neisser posited that such a buffer serves as an 'echoic memory';" rather than being organized as visual input, where spatially adjacent cells have spatially adjacent receptive fields, instead it is organized so that spatially adjacent cells receive input at different sound pitches/frequencies. + + Kosslyn and K\"onig \cite[p. 215--216]{kosslyn1992wet} "Recall that in vision there are two subcortical pathways from the eyes to the brain, the geniculo-striate and the tecto-pulvinar; the tecto-pulvinar pathway (eye, superior colliculus, pulvinar, cortext) draws attention to potential regions of interest. The inferior colliculus projects to the deep and superior colliculus play a similar role in audition. \ldots the auditory receptive fields of neurons in the superior colliculus shift with changes in eye position, allowing the auditory and visual maps to remain aligned. (see, for example Maddox et al.,\cite{maddox2014directing}) Hence, one tupically pays attention to a single object, registering its appearance and sounds at the same time." + + Kosslyn and K\"onig \cite[p. 219--220]{kosslyn1992wet} "visual preprocessing subsystem becomes 'tuned' by experience to encode useful visual patterns \ldots the separate auditory areas are also connected with reciprocal connections \ldots the reasoning we used to infer than experience tunes the visual preprocessing subsystem also leads us to expect that stimulus properties that distinguish between words will be noted downstream, and feedback will reinforce the encoding of those properties in the auditory preprocessing subsystem" + + Kosslyn and K\"onig \cite[p. 220--221]{kosslyn1992wet} "\textit {categorical perception} \ldots lies at the heart of what is accomplished by the auditory preprocessing subsystem \ldots the same categories are extracted when a word is spoken by different people -- and these categories help one to understand the words spoken under different circumstances." + + $$ + Kandel talks about removing some extraneous (i.e., those that might be redundant, maybe less often on) connections, do I think this is having the effect of categorization/generalization? + Choosing consciously to de-emphasize consideration of details at a high level, and unconsciously removing less-often exercised distinctions could have a similar effect. + $$ + Kosslyn and K\"onig \cite[p. 225]{kosslyn1992wet} "We are led to infer than the representations of the sounds of individual words depend on temporal--parietal cortex" + + Kosslyn and K\"onig \cite[p. 227]{kosslyn1992wet} "If we assume that unimodal memories are stored in the subsystem that encodes them", which is done by Squire (1987)\cite{squire1987memory}(check this, there is a google book) + + + + David and Squire reviewed protein synthesis connected with memory formation\cite{davis1984protein} " Evidence from learning curves, examination of short-term retention, and posttraining drug injection indicate that initial acquisition is not dependent on such synthesis, but it appears that protein synthesis, during or shortly after training, is an essential step in the formation of long-term memory." + + Kosslyn and K\"onig \cite[p. 230]{kosslyn1992wet} "it is possible that semantic information is organized along these lines, with appearance-based meanings segregated from use-based meanings". + + Kosslyn and K\"onig \cite[p. 343]{kosslyn1992wet} "We inferred in Chapters 3 and 4 that unimodal visual information is probably stored in the inferior temporal lobe (in the object-properties-encoding subsystem), and we inferred in Chapter 6 that unimodal auditory information is probably stored in temporal-parietal cortex." + + Kosslyn and K\"onig \cite[p. 344]{kosslyn1992wet} "information in associative memory can be activated by input from any perceptual modality". Associations can be established between representations in perceptual memory and in associative memory. "memory formation subsystems rely on anatomical structures \ldots the principal members of this set being the \textit{hippocampus} (and related cortex), the \textit{limbic thalamus}, and the \textit{basal forebrain}." + + Kosslyn and K\"onig \cite[p. 345]{kosslyn1992wet} "the hippocampus receives input from a number of other structures (the septum and the hypothalamus, via the fornix; the anterior thalamic nucleus and the subcallosal area, via the cingulum; and the amygdala). \ldots The hypothalamus appears to be involved in motivation, and the amygdala appears to have a role in emotion; clearly both factors affect what we remember. \ldots The hippocampus plays a critical role in the storage of new perceptual representations. \ldots also plays a critical role in storing associations between representations. (Mishkin and Appenzeller 1987, Squire 1987)" Long term potentiation is a phenomenon of hippocampal cells in neural microcircuits involved in storing associations between representations. + + Kosslyn and K\"onig \cite[p. 346]{kosslyn1992wet} "many of the thalamic nuclei appear to be involved in attentional processes (Posner and others) \ldots The basal forebrain \ldots in turn issues a signal that new representations and/or associations should be stored.(Mishkin and Appenzeller 1987) This is a biochemical signal, consisting of the release of \textit{acetylcholine}." + + Kosslyn and K\"onig \cite[p. 347]{kosslyn1992wet} "we shall decompose the memory formation subsystem into two more precisely characterized subsystems, \ldots these subsystems are involved in initiating the learning sequence, and in changing selected connections strengths in particular neural networks, respectively." + + Kosslyn and K\"onig \cite[p. 347]{kosslyn1992wet} "the \textit{striatum} plays a critical role in skill acquisition \ldots the striatum receives information from cortical perceptual areas". + + Kosslyn and K\"onig \cite[p. 349--350]{kosslyn1992wet} "If novel stimuli are perceived, \ldots when a match is not obtained, this information is sent to the frontal lobes and provides input to the print-now sybsystem; recall that perceptual encoding subsystems have anatomical projects into the frontal lobe (Goldman-Rakic, 1987) Thus, the perceptual encoding subsystems send outputs to the associative memory \ldots Structural changes are initiated that will allow the systems later to reconstitute the pattern of activation evoked by the novel stimulus" + + Kosslyn and K\"onig \cite[p. 351]{kosslyn1992wet} "The key to storing new information in memory is the ability to change the 'strengths' of connections among neurons in just the right way. This \ldots has been documented in actual neural networks (Kandel and Schwartz, 1985, Shepherd 1988)" + + Kosslyn and K\"onig \cite[p. 353]{kosslyn1992wet} "These changes (in synaptic connections) apparently begin in the hippocampus \ldots this phenomenon is called long term potentiation. \ldots it can last hours, days, weeks, or even longer -- depending on \ldots as well as various properties of the stimulus." + + $$ It is the accumulation of surface area from vesicles delivering neurotransmitter at the presynaptic cell in the synaptic cleft that creates microenvironments that are more effective (NMDA activated/molecule of neurotransmitter delivered) in activating the postsynatic neuron, since the corresponding membrane that is used to restore the number of vesicles is taken from smooth area.$$ + + Kosslyn and K\"onig \cite[p. 358--359]{kosslyn1992wet} "The idea that associative memory hooks back into unimodal perceptual representations is consistent with a range of clinical findings. \ldots Squire suggests that the perceptual systems that encode information may actually store much of it.(Squire 87) \ldots We speculate that associative memory depends in part on the superior, posterior temporal lobe, if only because patients with lesions in this area often appear to have disrupted associations." + + Kosslyn and K\"onig \cite[p. 361]{kosslyn1992wet} "When an object is later perceived and input enters associative memory, the relevant associations are activated." + + Kosslyn and K\"onig \cite[p. 370--371]{kosslyn1992wet} "We often store new information even if it has no obvious relevance to any goal or problem at hand. This is called \textit{incidental} memory.\ldots if one pays attention to a stimulus, it is likely to be stored 'automatically,' with no decision to do so. \ldots one tries to memorize the information, and is more likely to remembers it than if the effort were not made. This sort of memory is called \textit{intentional} memory. \ldots the longer the information is attended to, the more likely it is that the memory formation subsystems eventually will store it in memory \ldots if the property lookup subsystems access more information about the to-be-remembered material, memory will be improved. \ldots called a \textit{depth of processing} effect" + + Kosslyn and K\"onig \cite[p. 372]{kosslyn1992wet} "one can store information relatively effectively by inventing distinctive \textit{retrieval cues}. \ldots Allan Paivio reviews a large amount of evidence that we remember information better when we use a 'dual code' (visual and verbal) than when we store it in only a single way (Paivio 1971)" + + Kosslyn and K\"onig \cite[p. 373]{kosslyn1992wet} "when one becomes an expert in any domain, one often cannot report how one performs the task. Much, if not most, of the information in memory cannot be directly accessed and communicated" + + $$ it is not in conscious memory anymore$$ + Lisman and Sternberg\cite{lisman2013habit} talk about Habit and nonhabit systems for unconscious and conscious behavior: Implications for multitasking. + + Peter Graf and Daniel Schacter implicit memory + + Kosslyn and K\"onig \cite[p. 375]{kosslyn1992wet} "There is considerable evidence that priming tasks and explicit memory tasks rely on distinct processing subsystems \ldots certain drugs impair both recall and recognition in explicit memory tasks, but do not affect the magnitude of priming." + + Kosslyn and K\"onig \cite[p. 376]{kosslyn1992wet} In priming, "The words apparently were processed to some level within the system, even when the patient was fully unconscious." + + Kosslyn and K\"onig \cite[p. 377]{kosslyn1992wet} "priming has two components, only one of which is perceptual" There is an advantage to having the priming and the recall be in the same sensory modality, only for the right hemisphere. " Graf and Schacter also showed that implicit memory for associations between words is modality-specific. + + $$ to make the most of the priming effect, the students should have practice writing responses to stimuli, like test questions. For memorization type test questions, that are answered by handwriting, note-taking in class, of the same item, should help.$$ + + Kosslyn and K\"onig \cite[p. 380]{kosslyn1992wet} "Mishkin suggests that a subcortical structure called the \textit{substantia nigra} is critically involved in the 'reinforcement' process that strengthens connections between perceptual states and responses." + Stimulus/response learning, striatum, dopamine. Is this what is used in flash cards? This kind of learning is restricted, [p. 383]"the response can only be evoked by the appropriate stimulus" + + Kosslyn and K\"onig \cite[p. 387]{kosslyn1992wet} "material in \textit{working memory} is used to aid reasoning processes (Baddeley 1986). Reasoning processes only can operate on information in short-term memory, but relatively little information can be stored in short-term memory." + $$ so lemmas are good$$ + + Kosslyn and K\"onig \cite[p. 387-388]{kosslyn1992wet} " We interpret the short-term memory structures Patricia Goldman-Rakic reports in the frontal lobe (Goldman-Rakic 1987, 1988) as extension of the perceptual encoding subsystems, which serve to make perceptual information immediately available to the decision processes; according to this view, it is debatable whether one want to conceive of the information as actually being stored -- as opposed to monitored -- in the frontal lobe. + Working memory, then, corresponds to the activated information in the long-term memories, the information in short-term memories, and the decision processes that manage which information is activated in the long-term memories and retained in the short-term memories(Kosslyn 1991)" + + Kosslyn and K\"onig \cite[p. 398]{kosslyn1992wet} "recall that Goldman-Rakic provides evidence that perceptual information ultimately projects to the frontal lobe(Goldman-Rakic 1987, 1988)" + + Kosslyn and K\"onig \cite[p. 398--399]{kosslyn1992wet}explicit memory formation uses acetylcholine, modifying the strengths of connections in the appropriate networks, implemented in part in the hippocampus and related cortex + + + + + + + + Squire and Dede\cite{} + Martin\cite{martin2015grapes}"some of the latest functional + neuroimaging findings on the organization of object concepts + in the human brain. I argue that these data provide strong + support for viewing concepts as the products of highly inter- + active neural circuits grounded in the action, perception, and + emotion systems. The nodes of these circuits are defined by + regions representing specific object properties (e.g., form, color, and motion) and thus are property-specific, rather than + strictly modality-specific. How these circuits are modified + by external and internal environmental demands, the distinction between representational content and format, and the + grounding of abstract social concepts are also discussed." + + $$we have modality specific long term, and we have association area for other than single modality, we have consolidation and reconsolidation, do we have spatial arrays of concepts$$ + \subsection{Brain Imaging} + Bachmann\cite{bachmann2015brain} discusses "brain-imaging markers of neural correlates of consciousness" + $$when we are learning a skill we can be conscious of exercising that skill, and later, we can become less conscious of specifically how we exercise that skill. Is the part of the brain that is conscious moving? Is the location of the memory moving? Could be both. Don't some motor skills move to the cerebellum, and aren't there problems with this for dyslexics? What's going on when something suddenly appears in consciousness? I think the metabolic activity brings the consciousness to the place where the memory is. But in these kind of answers popping into consciousness (Poincar\'e?) isn't it a new synthesis that pops into consciousness?$$ + + Luck and LeClerc \cite{luck2014potentiation} wrote about the Potentiation of Associative Memory by Emotions: An Event-Related FMRI Study, + + Brain imaging provides evidence for believing that creativity, (generating novel ideas, such as proofs) can be improved by training and takes time corresponding to reorganizing intercortical interactions\cite{kowatari2009neural}. (Look here kowatari more, there is something about predominance of right prefrontal over left.) + According to Huang et al.\cite{huang2015highest}, a large body of research suggests that an abstract cognitive processing style produces greater creativity. Empirically, decades of work have shown that both abstract thinking and creativity are consistently linked to right-hemispheric activation in the brain (e.g., Fink et al., 1996\cite{fink1996brain} and Mihov et al., 2010\cite{Mihov2010442}) + . + + Squire tells us \cite{squire2015conscious} "findings suggest that fMRI activity in the medial temporal lobe reflects processes related to the formation of long-term memory" + + Chambers et al.~\cite[p. 1045]{chambers2003developmental}, citing Gurden et al, Mulder et al., Robinson and Kolb, and Hyman and Melenka~\cite{gurden1999integrity,mulder1997short,hyman2001addiction} state "Dopamine transmission in nucleus accumbens and prefrontal cortex regions projecting to the nucleus accumbens has been implicated in mechanisms of learning and plasticity, including changes in long-term potentiation and morphology of neuronal dendritic trees." + + \subsection{Brain structure} + + \begin{quote} + Who can trace out the secret threads by which our concepts are united?\\ + Hermann von Helmholtz + \end{quote}\cite{kounios2015eureka} + + Takeuchi et al. \cite{takeuchi2010training} Training of working memory impacts structural connectivity + + Melby-Lerv{\aa}g \cite{melby2013working} "current findings cast doubt on both the clinical relevance of working memory training programs and their utility as methods of enhancing cognitive functioning in typically developing children and healthy adults. " + + Rutishauser et al.\cite[p. 104-105]{fried2014single} medial temporal lobe "lesion subjects showed reduced (but above chance) recognition memory performance but, strikingly, has a complete lack of performance improvement for items shown together with task-irrelevant novel attributes. In contrast, normal control subjects had a substantial gain in memory performance for these items (Kishiyama et al., 2004\cite{kishiyama2004restorff}). Together, this indicates that the 'von Restorff effect' (Wallace, 1965\cite{wallace1965review}; Kinsbourne \& George, 1974\cite{kinsbourne1974mechanism}; Hunt, 1995\cite{hunt1995subtlety}; Parker et al., 1998\cite{parker1998restorff}) is driven by neuronal mechanisms that reside in the MTL. \ldots Novelty responses are thus a sensitive measure to quantify learning and plasticity. \ldots We identified a subpopulation of single neurons in the hippocampus and the amygdala that showed striking differences in their spiking response. \ldots the same cell would indicate the novelty of a stimulus regardless of which category the stimulus was from. \ldots These cells \ldots also show changes in firing rate as a function of repeated presentation even when subjects are only passively viewing stimuli without an explicit memory task (Perdreira et al., 2010\cite{pedreira2010responses})" + + Rutishauser et al.\cite[p. 106]{rutishauser2014single} "it seems that there are at least two distinct classes of novelty-sensitive single neurons in the human MTL: abstract and visually tuned. The first class of untuned general novelty detectors could serve to signal the significance of stimuli during the acquisition of new memories (Lisman \& Otmakova, 2001\cite{lisman2001storage}). It has been suggested that such neurons trigger dopaminergic release through projections to the ventral tegmental area (Lisman \& Grace, 2005\cite{lisman2005hippocampal})" + + Mormann et al.\cite[p. 131--133]{mormann2014visual} reports "Another seminal study described neurons representing a specific semantic concept in the human MTL (Quian Quiroga et al., 2005). \ldots A problem in determining the precise turning curves of semantic neurons in the MTL is to find a suitable parameterization of the stimulus space \ldots compounded in human studies by the short duration of any one experiment in any one patient." + + Ouchi et al.\cite{ouchi2013reduced} "Adult neurogenesis is known to be important in hippocampus-dependent memory" + + Ojemann\cite[p. 262]{ojemann2014human} has demonstrated that lateral temporal neurons are involved in recent memory, making use of attention to increase the success of consolidation of memory of items competing with distractions. "The evolutionary changes in the brain orgainzation for recent memory involve an expansion of the cortical component with relative conservation of the medial temporal -- hippocampal component. \ldots A substantial literature has established the importance of the temporal lobe in learning. \ldots neurons with changes during \ldots memory were significantly more likely to have associative learning changes." + + Kowatari et al.\cite{kowatari2009neural} state "In the experts, creativity was quantitatively correlated with the degree of dominance of the right prefrontal cortex over that of the left, \ldots Our results supported the hypothesis that training increases creativity via reorganized intercortical interactions." + + Waisman et al.\cite{waisman2014brain} observe that "various studies demonstrate that when complexity of the (arithmetic) problems rises, more brain areas simultaneously support the solving process", citing Zamarian et al.\cite{zamarian2009neuroscience}. + + Waisman et al.\cite{waisman2014brain} investigated "cortical activity associated with solving problems that require translation between symbolic and graphical representations". + + Waisman et al.\cite[p.691]{waisman2014brain} stated that the "posterior parietal cortex is know to be activated when mental representations are manipulated (Zacks 2008)\cite{zacks2008neuroimaging}. + + Deng et al.\cite{deng2010new} report that "Neurons born in the subventricular zone(SVZ) differentiate and integrate into the local neural network as granule cells of the dentate gyrus." + + Knutson et al.\cite{knutson2001anticipation} state that a region in the nucleus accumbens codes for expected positive incentive value, and + \cite[p. 4]{knutson2001anticipation} it is "an apparently lateralized response of the right nucleus accumbens." + + Chambers et al.~\cite{chambers2003developmental} explained that "Adolescent neurodevelopment occurs in brain regions associated with motivation, impulsivity, and addiction. Adolescent impulsivity and/or novelty seeking as a transitional trait behavior can be explained in part by maturational changes in frontal cortical and subcortical monoaminergic systems. These developmental processes may advantagiously promote learning drives" + + $$ these systems that we are trying to use to shepherd information from the place it begins (sensory to MTL) to the right prefrontal cortex, from where it can be retrieved for creative application to problem solving, are developing$$ + + \begin{figure} + \centering + \includegraphics[width=0.7\linewidth]{./l73f1.jpeg} + \caption{In this figure reprinted from Chambers\cite{chambers2003developmental}, (I plan to do something, either redraw or ask permission) we see how sensory input, new information, is delivered to hippocampus. Other sources have shown us that new neurons are created in or near hippocampus that are a response to new information arriving and related to memory for the new information. We have seen how presence of dopamine, from striatum and VTA assist the consolidation of new memory into longer term memory. We have seen that for monomodal information, the long term memory is stored in cortex near where input is provided by that modality (visual cortex, auditory cortex) and for multimodal information, the long term memory is stored in association cortex. We have seen that information stored in association cortex is more readily retrieved, as any of the associated modalities can help retrieve it. We have seen that this consolidation requires protein and is facilitated by sleep, (I forget which of REM or slow wave sleep.) We have seen how reconsolidation can occur, assisted by nucleus accumbens, and can result in information accessible to the prefrontal cortex, on the right side. We have seen how anticipation of positive reward activates the nucleus accumbens on the right side. Could it be that anticipation of positive reward occurs in REM sleep, I wonder.} + \label{fig:poster23} + \end{figure} + + Anderson et al.~\cite[p. 53]{anderson2011cognitive} state "There is some reason to suspect that the angular gyrus (ANG) may also be engaged to serve the metacognitive activities of monitoring and reflecting." (on non-routine problem solving). They go on to say "Regions close to the right ANG have been found to play a variety of metacognitive functions, citing a review by Decety \& Lamm\cite{decety2007role}. + + Anderson et al.~\cite[p. 54]{anderson2011cognitive} state "Another region that is potentially involved in metacognition is Brodmann Area 10 or frontopolar cortex (FPC), particularly its lateral portion, (citing Fletcher and Henson\cite{fletcher2001frontal}). A number of converging lines of research suggest that this region of the brain may be critical in the ability to extend knowledge." + Anderson et al.~\cite[p. 58]{anderson2011cognitive} state "The left ANG is often distinguished from the right in many theories including the triple code, but the pattern of ANG effects in this experiment is basically the same in the two hemispheres." + Anderson et al.~\cite[p. 62]{anderson2011cognitive} state "In every case (brain areas related to metacognition), the patterns are roughly bilaterally symmetric." + + + \subsection{Brain function} + + + + R. Quian Quiroga\cite{quiroga2012concept} opines that concept cells are the building blocks of declarative memory functions. + + Suthana et al.\cite{suthana2012memory} report on Memory enhancement and deep-brain stimulation of the entorhinal area. + + Imamoglu (fix the accents) et al.\cite{imamoglu2012changes} discuss changes in functional connectivity support conscious object recognition. + + Murayama and Kitagami\cite{murayama2014consolidation}dopaminergic memory consolidation effect can result from extrinsic reward. + + $$ + so, give a little quiz at the end of lecture, covering the main points, and hand out tickets in exchange for handing in quizzes. Then, tickets can be handed in with homework to count for points. So they are paid at the time they are thinking about quiz contents, and that is expected to help with consolidation of the material on the quiz.$$ + + Born and Wilhelm\cite{born2012system} discuss System consolidation of memory during sleep + + Diekelmann et al.\cite{diekelmann2012offline} describe that Offline consolidation of memory varies with time in slow wave sleep and can be accelerated by cuing memory reactivations. + + Taylor et al.\cite{tayler2013reactivation} describe Reactivation of neural ensembles during the retrieval of recent and remote memory. + + Cowansage et al.\cite{cowansage2014direct} describe the Direct reactivation of a coherent neocortical memory of context + + Lustenberger et al.\cite{lustenberger2012triangular} discuss a triangular relationship between sleep spindle activity, general cognitive ability and the efficiency of declarative learning. + + Roux and Uhlhaas\cite{roux2014working} consider working memory and neural oscillations, questioning whether alpha--gamma versus theta--gamma codes for distinct WM information. + + Walker and Stickgold\cite{walker2014sleep} consider Sleep, memory and plasticity. + + Tonnoni and Cirelli\cite{tononi2014sleep} discuss Sleep and the price of plasticity: from synaptic and cellular homeostasis to memory consolidation and integration. + + Rutishauser et al.\cite[p. 107]{rutishauser2014single} "Neurons coded this information very reliably: The decoder (a function of a population of neurons) could tell (for correct trials) whether the stimulus was new or old on a single trial basis with an accuracy of 75\% \ldots (using) a single previously identified novelty/familiarity neuron. Performance increased to 93\% if six neurons were considered. \ldots for 75\% of error trials, the decoder predicted what the correct response would have been (but which was not given by the subject). \ldots a simple decoder outperform(s) the patient \ldots The neurons have better memory than the patient demonstrated behaviorally," + + Rutishauser et al.\cite[p. 111]{rutishauser2014single} "Many factors modulate the probability that a memory for a stimulus will be formed. Examples include attention, motivation, and saliency of the stimulus (Paller \& Wagner, 2002\cite{paller2002observing}). Structurally, the modification of synaptic circuits by plasticity mechanisms is thought to underlie memory formation (Martin et al., 2000\cite{martin2000synaptic}) + + Rutishauser et al.\cite[p. 112]{rutishauser2014single} "findings from animal literature indicate that theta oscillations and the timing of neuronal activity relative to the ongoing theta oscillations have a strong influence on plasticity as well as learning, suggesting the possibility that the two are functionally linked by theta oscillations. \ldots The power of theta oscillations measured on the scalp \ldots can be predictive of whether a memory is formed or not (Klimesch et al., 1996\cite{klimesch1996theta}; Sederberg et al., 2003\cite{sederberg2003theta}) \ldots what is relevant for whether a memory was formed or not is whether the preferred phase of a particular neuron was followed faithfully and not the absolute phase \ldots what was predictive (of whether a memory was formed or not) was whether the spikes that were fired were phase locked to ongoing theta or not." + + Rutishauser et al.\cite[p. 113]{rutishauser2014single} "the spike field coherence at the time of learning was already indicative of whether a memory was later strong or weak." + + Mormann et al.\cite[p. 140]{mormann2014visual} state "The process of encoding episodic memories consists of associating pieces of semantic information (what happened where with whom involved and so on) in a defined temporal order. Lesion studies in humans have shown that structures in the MTL are essential for the encoding of episodic memories (Squire et al., 2004; Squire et al., 2007; Squire, 2009; Milner et al. 1968, Scoville \& Millner 1957). Representations of semantic information at the single unit level are frequently found in these very structures and thus might provide a unique opportunity to investigate how our brain links pieces of semantic information together into episodic memories (Quiroga, 2012). \ldots Episodic memories are always context dependent whereas semantic memories are context invariant and can emerge via generalization of recurring context-dependent experiences (Buzsaki, 2005)." + + Mormann et al.\cite[p. 142--143]{mormann2014visual} state "Earlier notions that the amygdala might be specialized to elicit or mediate fear responses (LeDoux, 1996) have been supplemented by more abstract accounts whereby the amygdala processes ambiguity or unpredictability in the environment (Herry et al., 2007) and mediates an organism's vigilance and arousal (Davis and Whalen, 2001). \ldots In humans, neuroimaging studies of the amygdala argue for a broad role in processing stimuli that are strongly rewarding or punishing (Sander et al., 2003; Ohman et al., 2007) \ldots In human fMRI memory studies, increased blood flow in the amygdala during encoding was found to correlate with improved memory formation (e.g, Canli et al., 2000). In addition, the phase locking of human amygdala neurons to ongoing theta oscillations was found to predict memory formation (Rutishauser et al., 2011) \ldots there might be a distinction between the hippocampus and the amygdala in terms of the extent to which unconscious information reaches those areas. If indeed unconscious information can reach the amygdala but not the hippocampus and surrounding structures" + + Paz and Pare\cite{paz2013physiological}" in emotionally + arousing conditions, whether positively or negatively valenced, + the amygdala allows incoming information to be processed + more efficiently in distributed cerebral networks." + + $$ This is just corroborating that infectious enthusiasm for a subject, on the part of the instructor, helps students learn.$$ + + Schwabe et al. \cite{schwabe2014reconsolidation} there is reconsolidation + + Patel et al.\cite[p. 205]{patel2014human} "Functions such as reward processing, motivations, and learning have since been attribted to basal ganglia circuits." + + Patel et al.\cite[p. 207]{patel2014human} "Examined impact of emotional valance and cACC responsitveness to complext attnetion tasks \dots Examined reward properties of dopaminergic neurons using virtual financial reward" + + Patel et al.\cite[p. 208--209]{patel2014human} "when there is a differentce in expected and actual outcome -- a prediction error signal -- midbrain dopaminergic neurons rapidly fire at the onset of the unexpected reward. \ldots This feature is thought to drive reward-based learning and adaptive behavior. \ldots It is thought that phasic dopaminergic activity is the neural substrate of this type (classical conditioning) of learning (Schultz, 1998) \ldots dopamine release in the striatum" + + $$ So, deliver unexpected rewards. Maybe some clicker questions have greater value.$$ + + Patel et al.\cite[p. 210]{patel2014human} "the ventral striatum has become a focal point in studies of reinforcement learning, \ldots phasically active neurons, thought to be the medium spiny neurons that make up about 95\% of striatal neurons, are more relevant to strengthiening functional circuits diring instrumental conditioning and procedural learning (Graybiel 2008, Jog 1999)" anticipation of rewards, including secondary rewards (monetary) + + + Patel et al.\cite[p. 212]{patel2014human} "NAcc activity reliably encodes the anticipation of reward proportional to reward magnitude" + + Patel et al.\cite[p. 213--214]{patel2014human} "each subregion of the ACC is predomintely involved with distinct roles, such as motivation and cognition (Paus, 2001). Notably these areas are not homogeneous and demonstrate some overlap in function. \ldots The portion of the ACC implicated in reward processing and cognition is the dACC. \ldots The vmPFC is strongly connected to the limbic system, which is a group of brain structures that are implicated in several functions such as emotion, motivation, and memory." + + Ojemann\cite[p. 268]{ojemann2014human} states "there are also 'subconscious' or 'implicit' memory processes, processes of which the subject is unaware that change performance. One such process is repetition priming, a shortening of reaction time with repeated presentation of the same item. Changes in lateral temporal cortical single neuron activity related to this, \ldots that study included recordings throughout lateral temporal cortex, neurons with those significant implicit memory changes were all in superior temporal gyrus or superior portion of middle gyrus, significantly more superior and posterior than neurons with recent memory processes \ldots implicit memory then involves neural networks in posterior superior temporal cortex, largely separate from those for recent memory" + + Rutishauser et al.\cite[p. 348]{rutishauser2014next} state "Different timescales of memory formation have been described at the neuronal level." + + Rutishauser et al.\cite[p. 348]{rutishauser2014next} "It has remained very difficult to directly link the mechanisms of synaptic plasticity to memories" + %$$ Hebbian by deposition of membrane, increasing the folds per volume, because membrane harvesting for vesicle retrieval is from flat$$ + + Rutishauser et al.\cite[p. 348]{rutishauser2014next} "We are able to recall memories that were established years or even decades ago, but the processes by which such remote call(sic) works remain largely unknown." + %$$ it is one cell activating another, and glial cells bringing in blood to the region in response to activated neuron. Remember, it was Wilder Penfield stimulating neurons that brought back remote memories$$ + Rutishauser et al.\cite[p. 348]{rutishauser2014next} "emotions are likely to play a key role in memory formation (Cahill et al. 1995, Fanselow \& Gale 2004, Phelps, 2004) + + + Rutishauser et al.\cite[p. 351]{rutishauser2014next} "There is ample evidence that multiple tasks show sleep-dependent enhancement (Stickgold, 2005)" + + There is a journal Neurobiology of Learning and Memory.(single neuron book p 182) + + Patel et al.\cite[p. 211]{patel2014human} "Knutson and colleagues further demonstrated that NAcc activation increases proportional to the magnitude of the anticipated monetary reward (Knutson et al, 2001) + + According to Kowatari et al.\cite[p.1679]{kowatari2009neural}, Carlsson et al.\cite{carlsson2000neurobiology} "reported that both hemispheres were involved in highly creative subjects." Kowatari et al. \cite[p.1679]{kowatari2009neural} "found that professional training reorganized brain activation patterns, which was correlated with increased creativity." + Kowatari et al. \cite[p.1682]{kowatari2009neural} "In the expert group, a right and left hemispheric difference was obvious; only the right prefrontal cortex (PFC) and parietal cortex (PC) were activated in the expert group, whereas in the novice group, bilateral PFC and PC were activated." + Kowatari et al. \cite[p.1682]{kowatari2009neural} "This results indicated that the direct or indirect interaction between the right and left PFC might contribute to producing highly original designs in the expert group." + + Kowatari et al. \cite[p.1683]{kowatari2009neural} "Based on these observations, we postulated that \ldots brain regions that are involved in yielding high creativity indices shifted from the PC to the PFC." + + Deng et al.\cite[p. 341]{deng2010new} report that "the adult born dentate granule cells (DGCs) exhibit stronger synaptic plasticity than mature DGCs, as indicated by their lower threshold for the induction of long term potentiation (LTP) and their higher LTP amplitude. and, citing Toni et al.\cite{toni2008neurons}, say "structural modification of dendritic spines and axonal boutons continues to occur as the just-born DGCs become older" Toni et al.\cite{toni2008neurons} report that neurons born in the adult dentate gyrus form functional synapses with target cells. + + Deng et al.\cite[p. 343]{deng2010new} report, citing Kee et al. \cite{kee2007preferential} that "learning by a mouse when a set of adult-born DGC was at least 4-6 weeks of age led to preferential activation of these cells during memory retrieval" during the learned task "when the DGCs were 10 weeks old." + + Trouche et al. \cite{trouche2009recruitment} state that recruitment of adult-generated neurons into functional hippocampal networks contributes to updating and strengthening of spatial memory. + + Deng et al.\cite[p. 343]{deng2010new} surmise that finding suggest that, compared with their mature counterparts, adult-born DGCs may be specifically activated by an animal's experiences and thus can make unique contributions to learning and memory. + + Deng et al.\cite[p. 344]{deng2010new} opine that neurogenesis allows plasticity to be mostly localized to newborn immature DGCs, preserving the information that is represented by mature DGCs. %dentate granule cells + + Deng et al.\cite[p. 348]{deng2010new} report, citing Kitamura et al.\cite{kitamura2009adult} "a recent study in mice suggested that adult neurogenesis facilitated memory reorganization that led to a gradual reduction of the hippocampus-dependence of memories and the permanent storage of these memories in extra-hippocampal regions. + + Diekelmann and Born\cite{diekelmann2010memory} state that "Sleep has been identified as a state that optimizes the consolidation of newly acquired information in memory, depending on the specific conditions of learning and timing of sleep. Consolidation during sleep promotes both quantitative and qualitative changes of memory representations. Through specific patterns of neuromodulator activity and electric field potential oscillations, slow-wave sleep (SWS) and rapid eye movement (REM) sleep support system consolidation and synaptic consolidation, respectively. During SWS, slow oscillations, spindles and ripples -- at minimum cholinergic activity -- coordinated the re-activation and redistribution of hippocampus-dependent memories to neocortical sites, whereas during REM sleep, local increases in plasticity-related immediate-early gene activity -- at high cholinergic and theta activity -- might favour the subsequent synaptic consolidations of memories in the cortex." + + Diekelmann and Born\cite{diekelmann2010memory} state that, among its functions, "sleep's role in the establishment of memories seems to be particularly important". promoting primarily the consolidation of memory. + They tell us that "Consolidation refers to a process that transforms new and initially labile memories encoded in the awake state into more stable representations that become integrated into the network of pre-existing long-term memories.", involving active re-processing of 'fresh' memories within the neuronal networks that were used for encoding them. + + Diekelmann and Born\cite{diekelmann2010memory} , citing Stickgold et al., and Walker et al.\cite{stickgold2000visual.walker2003dissociable} state that "For optimal benefit on procedural memory consolidation, sleep does not need to occur immediately but should happen on the same day as initial training." + + Chambers et al.~\cite[p. 1044]{chambers2003developmental}, when discussing events activating loops within primary motivation circuity, state "These events may also facilitate mechanisms of neuroplasticity among nucleus accumbens neurons, and their afferents." + + Chambers et al.~\cite[p. 1045]{chambers2003developmental}, citing Masterman et al.~\cite{masterman1997frontal}, stated "dopamine release into the nucleus accumbens is associated with motivational stimuli, subjective reward, premotor cognition(thought), and learning of new behaviors" + + Chambers et al.~\cite[p. 1045]{chambers2003developmental}, citing Waelit et al.~\cite{waelti2001dopamine}, stated "Rewards delivered in intermittent, random, or unexpected fashions have greater capacity over repeated trials to maintain dopamine cell firing and reward-conditioned behavior." + + Chambers et al.~\cite[p. 1045]{chambers2003developmental} state that "well-learned motivated behaviors or habits performed under expected contingencies become less dependent on nucleus accumbens dopamine release." + + Chambers et al.~\cite[p. 1045]{chambers2003developmental}, citing Yates~\cite{yates1990theories} "In adolescence, the prefrontal cortex has not yet maximized a variety of cognitive functions \ldots Measures of prefrontal cortex function, including working memory, complex problem solving, abstract thinking, and sustained logical thinking, improve markedly during adolescence. + + Wittmann et al.~\cite{wittmann2005reward}show that activation of dopaminergic midbrain is associated with enhanced hippocampus-dependent long term memory formation. + + Wittmann et al.~\cite{wittmann2005reward} state that reward anticipation reliably elicits a dopaminergic response. They hypothesize that the known improved dopamine driven synaptic plasticity and long-term potentiation is associated with better memory consolidation in the hippocampus. Their findings are consistent with they hypothesis that activation of dopaminergic midbrain regions enhances hippocampus-dependent memory formation, possibly by enhancing consolidation. They have shown that activity of the ventral tegmental area and medial substantia nigra accompanied hippocampal activity related to memory formation, in that both structures were activated by novelty and in relation to subsequent free recall performance. The areas that respond to signals related to reward are the dopaminergic areas when the response is reward prediction and the mesial frontal cortex after learning the contingency between the predicting stimulus and the reward; there is a shift with learning.\cite{knutson2003region}. Their findings support their hypothesis that the hippocampus is a major site for the neuromodulatory influence of reward on long-term memory formation. + + Lee et al.~\cite{lee2007strategic} reported on neuroanatomical correlates of converting between symbolic algebra and a pictorial representation. They found that for conversion in either direction, the active areas include the both left frontal gyri, intraparietal sulci bilaterally, which are linked to working memory and quantitative processing. They also found that using the symbolic method activated the posterior superior parietal lobules and the precuneus, in contrast to the pictorial method. They conclude that the symbolic and pictorial strategies impose different attentional demands. + + Lee et al.~\cite{lee2007strategic}, citing Anderson et al.~\cite{anderson2003information}, observe that algebraic transformation is subserved by the left posterior parietal region and the left dorsal lateral prefrontal cortex. Lee et al.~\cite{lee2007strategic}, citing Sohn et al.~\cite{sohn2004behavioral}, who found that "anterior prefrontal activation + was greater in the story condition and posterior parietal + activation was greater in the equation condition". + + $$ then creativity is found in the same or nearby location to the storytelling way, rather than the symbolic representation way$$ + + Lee et al.~\cite[p. 167]{lee2007strategic} report the symbolic method was associated with activation in the left precuneus and bilateral posterior superior parietal lobules. This finding suggest the symbolic condition recruited attentional processes more extensively than did the model method. Also activated were various loci in the visual processing area and in the basal ganglia. The model condition did not activate any areas beyond those activated by the symbolic ". According to Lee et al.~\cite[p. 167]{lee2007strategic}, both the precuneus and the posterior superior parietal lobules are associated with attentional processes. + $$so, maybe it takes more attention to work with symbols. After all, they are more concise$$ + + Lee et al.~\cite[p. 168]{lee2007strategic} citing Owen et al.~\cite{owen2005n} say that dorsolateral prefrontal, overlapping middle frontal is involved in reorganizing material into pre-existing knowledge structures. + + $$ thinking about proof might use this $$ + + Lee et al.~\cite[p. 169]{lee2007strategic} found that "occipital areas were activated in the algebraic condition. This suggests participants spent more time viewing the questions in the algebraic conditions.\ldots activation in the posterior superior parietal lobules might be related" + + Wittmann et al.~\cite{wittmann2007anticipation} found that the "hippocampus differed from the response profile of SN/VTA in responding to expected and 'unexpected' novelty". Their results demonstrate parallels between the processing of novelty and reward in the SN/VTA. Their fMRI analysis revealed that cues predicting novel images elicited significantly higher SN/VTA activation than cues predicting familiar stimuli.[p. 198] + Wittmann et al.~\cite[p. 200]{wittmann2007anticipation} "Irrespective of whether the dopaminergic midbrain drives the hippocampus or vice versa, coactivation of the hippocampus and SN/VTA could be associated with increased dopaminergic input to the hippocampus during anticipation. This, in turn, could induce a state that enhances learning for upcoming novel stimuli" + + Keller and Menon~\cite{keller2009gender} studied brain activation during mathematical cognition, and compared men and women. They found that the same brain areas were used: right intr-parietal sulcus areas and angular gyrus regions, ventral stream of right lingual and parahippocampal gyri. "Females had greater regional density and greater regional volume where males showed greater fMRI activation. \ldots Our findings provide evidence for gender differences in the functional and structural organization of the right hemisphere brain areas involved in mathematical cognition. Together with the lack of behavioral differences, our results point to more efficient use of neural processing resources in females." + Keller and Menon~\cite[p. 348]{keller2009gender}stated "Gender differences were all localized to the right posterior regions of the brain." + + Diekelmann et al.~\cite[p. 116]{diekelmann2010memory} observes that "a great number of studies indicate that sleep supports consolidation of memory in all major memory systems \ldots There is growing evidence that explicit encoding, even in procedural tasks, involves a dialogue between the prefrontal cortex and the hippocampus, citing \cite{schendan2003fmri}, which also integrates intentional and motivational aspects \ldots Sleep changes memory representations quantitatively and qualitatively. \ldots a strengthening of associations \ldots qualitative changes in memory representations." + + $Isay$ In order to encourage the prefrontal cortex involvement, we should be explicit about motivation for choosing one inference rule over another as the demonstration / pedagogical proofs are exhibited.$$ + + Diekelmann et al.~\cite[p. 116]{diekelmann2010memory} observes that "subjects learned single relations between different objects which, unknown to the subject, relied upon an embedded hierarchy, citing \cite{ellenbogen2007human}. When learning was followed by sleep, subjects at a re-test were better at inferring the relationship between the most distant object, which had not been learned before. Likewise, after sleep subjects more easily solved a logical calculus problem that they were unable to solve before sleep or after corresponding intervals of wakefulness citing \cite{wagner2004sleep}. Of note, sleep facilitated the gain of insight into the problem only if adequate encoding of the task was ensured before sleep." + $$ need the definition of adequate encoding. Does this use of encoding refer to placing the representation into, say, hippocampus?$$ + Diekelmann et al.~\cite[p. 116]{diekelmann2010memory} "sleep can re-organize newly encoded memory representations, enabling generation of new assoications and the extraction of invariant features" + $$ here is generalization, abstraction, we want cs students to learn$$ + Diekelmann et al.~\cite[p. 116]{diekelmann2010memory} "from complex stimuli, and therby easing novel inferences and insights. Re-organization of memory representations during sleep also promotes the transformation of implicit into explicit knowledge \ldots procedural and declarative memory systems interact during sleep-dependent consolidation." + They also observe that once implicit memory has become explicit, subjects no longer showed improvement in implicit procedural skill. + $$I'm connecting explicit with conscious. Once the subject is conscious of (interiorized after internalized, using Harel and Sowder's 1998 scheme, thinking about what you are doing as you do it) the procedure, they carry it out consciously, which could be more slowly. Maybe eventually they will become unconscious of how they carry out this skill, maybe the slowing process of conscious execution will drop away$$ + + (what has Stickgold been learning about sleep, memory, lately?) + + Diekelmann et al.~\cite[p. 117]{diekelmann2010memory} "It is assumed that the re-activations during system consolidation stimulate the redistribution of hippocampal memories to neocortical storage sites" + + (probably there is more recent on redistribution) + Diekelmann et al.~\cite[p. 117]{diekelmann2010memory} "In addition to system consolidation, consolidation involves strengthening of memory representation at the synaptic level." + + $$Maybe this is so that when the memories are moved farther out from the hippocampus, there is a stronger trail to it$$ + + Diekelmann et al.~\cite[p. 121--122]{diekelmann2010memory} In the active system consolidation view, "It is assumed that in the waking brain events are initially encoded in parallel in neocortical networks and in the hippocampus. During subsequent periods of SWS the newly acquired memory traces are repeatedly re-activated and thereby become gradually redistributed such that connections within the neocortex are strengthened, forming more persistent memory representations. Re-activation of the new representations gradually adapt them to pre-existing neocortical 'knowledge networks', thereby promoting the extraction of invariant repeating features and qualitative changes in the memory representations." + + Diekelmann et al.~\cite[p. 122]{diekelmann2010memory}, citing Hasselmo et al.\cite{} suggest that "acetylcholine serves as a switch between modes of brain activity, from encoding during wakefulness to consolidation during SWS" + + Diekelmann et al.~\cite[p. 122]{diekelmann2010memory}, citing Wagner and Born \cite{wagner2008memory} observes that glucocorticoids (cortisol in humans) block the hippocampal information flow to the neocortex, and if the level of glucocorticoids is artificially increased during SWS, the consolidation of declarative memories is blocked. + + $Isay>$ so, we need to find out whether students who play video games on the way to retiring are decreasing or increasing their cortisol in the process.$$ + + + There is evidence that some games help reduce cortisol "The impact of playing computer games on cortisol concentration of saliva before and after the game showed that the amount of saliva plasma after playing the game has dropped significantly." \cite{aliyari2015effects}, casual video games decrease stress\cite{russoniello2009effectiveness} and there is evidence that Tetris in particular reduces stress\cite{mercer2015stress} and there is evidence that excessive use of violent video games by some young men \cite{eickhoff2015excessive} can negatively impact their cognitive effectiveness. Maass et al. conducted a study on 117 university students; Maass et al. state "The more time spent on media the poorer cognitive performance is. This association has mainly been found for general-audience, violent, and action-loaded contents but not for educational contents. \ldots A significant univariate difference was found for high- vs. low-arousing contents in general (independent of type of media), the high-arousing content leading to poorer ability to concentrate after media use. The expected mediating and moderating effects are not supported. The study yields evidence that short-term mechanisms might play a role in explaining the negative correlations between media use and cognitive performance." \cite{maass2015does} + + $Isay>$ might wish to advise students who play video games on the way to retiring to, close to retiring, use those that reduce stress$$ + + Diekelmann et al.~\cite[p. 122]{diekelmann2010memory} report "The concept of a redistribution of memories during sleep has been corroborated by human brain imaging studies (82,83,149,159,158) Interestingly, in these studies, hippocampus-dependent memories were particularly redistributed to medial prefrontal cortex regions (82,83,122)." + + $$ if it is the case that when students. trying to understand some math in computer science, but opting to memorize, given the time available (Is this what David Tall's bifurcation is about?) form memories that are not hippocampus-dependent (procedural memories are less hippocampus dependent)? If so, is it then also the case that they are less likely to be conveyed to the medial prefrontal cortex? Is the intervention then that instructors provide better explicit descriptions, give exercises to write something in code, which we hope is a bridge from internalized/implicit to interiorized/explicit, declarative, and avail ourselves of emotional support by evaluating the beauty, to aid in recruiting hippocampus-dependent memory formation? $$ + + Diekelmann et al.~\cite[p. 122]{diekelmann2010memory} state that "These regions not only have a key role in the recall and binding of these memories once the are stored for the long term, citing Frankland and Bontempi \cite{frankland2005organization}, \ldots prefrontal-hippocampal system might provide a selection mechanism that determines which memory enters sleep-dependent consolidation." + + Diekelmann et al.~\cite[p. 123]{diekelmann2010memory} state that the REM time interval upregulation of genes related to plasticity is dependent upon the learning experience in prior wakefulness, and is localized to the brain regions involved in prior learning, citing Ribeiro \cite{ribeiro2007novel,ribeiro2002induction}. + + \paragraph{Steps} + \begin{enumerate} + \item the input is a demonstration of a proof that explicitly describes the motivation for choosing each successive inference rule after another, and explicitly describes the motivation for choosing each lemma, i.e., why was that part of the proof handled separately, (e.g., we know we can, and we know it will be useful, but something about how we suspect it will be useful) Moreover, because we know emotional content is helpful in consolidation of memory during sleep, we remark upon such beauty as we may find in the proof. + Note that it is important to be, in the unfolding of time, orderly. + \item sensory input appears in sensory acquisition (accompany with pleasant scent\cite{born2012system}, and relevance to future plans\cite{born2012system}) + \item sensory input conveyed to medial temporal lobe, where it is sometimes seen with single neuron instrumentation + \item unexpected novelty stimulates dopamine, differently from expected novelty (as in exploration) + \item reward stimulates dopamine, or, if prediction of reward is learned, the stimulus role is transferred to the prediction + \item dopamine can come from SN/VTA + \item dopaminergic midbrain is associated with enhanced hippocampus-dependent long-term memory formation \cite{wittmann2005reward}, i.e., rewards help form long term memory, and it occurs in some sleep phase, maybe REM or slow wave + \item LTP is divided into early and late, and dopamine contributes to late~\cite{wittmann2005reward} + \item dopamine is also used for long term depression, which is also learning + \item new neurons are generated~\cite{deng2010new} related to new hippocampus memories + + \item explicit (vs. only implicit) learning favors access to sleep-dependent consolidation\cite[p. 115]{diekelmann2010memory} + \item motivational tagging of memories, might signal behavioral effort and relevance and mediate preferential consolidation\cite[p. 116]{diekelmann2010memory} + \item if video games are played just before retiring, they should probably not be violent. Conversely, if violent video games are played, something else, such as Tetris, should be used after the violent ones, prior to retiring. + \item sleep deprivation is expected to be detrimental to learning + \item sleep occurring 3 hours after learning was more effective than sleep delayed by more than 10 hours.\cite{gais2006sleep,talamini2008sleep,walker2003dissociable} + \item use same scent as in class, during SWS + \item slow oscillations typically seen in slow wave sleep (earlier part of sleep) have a causal role in the consolidation of hippocampus-dependent memories~\cite[p. 119]{diekelmann2010memory} + \item ripples typically seen in slow wave sleep (earlier part of sleep) have a causal role in the consolidation of memories~\cite[p. 119]{diekelmann2010memory} + \item it is not a particular sleep stage per se that mediate memory consolidation, but rather the neurophysiological mechanisms associated with those sleep stages~\cite[p. 116]{diekelmann2010memory} + \item re-activation of encoded memories occurs during slow wave sleep, in the order the remembered material was experienced Maquet\cite{maquet2000experience} cited in Diekelmann\cite[p. 117]{diekelmann2010memory} + \item theta oscillations, associated with REM sleep, have been found specifically over the right prefrontal cortex to be correlated with the consolidation of emotional memories\cite{nishida2009rem} + \item practice retrieval\cite{Bridge01082015}, better after sleep, can disrupt decoding if before sleep (check with Bridge 2015) + \item once attention to an item of knowledge has been rewarded, subsequent attention to that item is involuntary \cite{sali2014role} + \item consolidation occurs, differentiated by number of modalities, could be inert knowledge, as Whitehead~\cite{whitehead1959aims}, or retrievable, preferably. The number of related modalities, the more easily retrieved. + \item Retrieval allows for reconsolidation.\cite{sandrini2015modulating} (check it)\cite{schwabe2014reconsolidation}(check it) \cite{forcato2013role}(check it) + \cite{walker2003dissociable} (check it) + \item dopamine, nucleus accumbens helps reconsolidation occur with connection to right pre-frontal cortex \cite{knutson2001anticipation} and this is helped by anticipation of positive reward + \item right pre-frontal cortex is what experts use to be creative \cite{kowatari2009neural}, so we want to shepherd our taught material here, so that it is readily retrievable for inventing proofs + \end{enumerate} + + Chou et al.~\cite[p. 726]{chou2011sex} observes "the analytic brain for mathematical and logical cognition comprises the inferior frontal gyrus, parietal cortex and supramarginal gyrus", citing Dehaene et al, 1998k Goel et al., 1998, Zago et al., 2001. They measured, using fractional anisotropy (FA), microstructure of white matter that differed significantly in several areas, between men and women. + $$Of these, at least the bilateral precuneus has been identified as of interest in mental activity related to mathematical proofs.$$ + They state ~\cite[p. 731]{chou2011sex} "the interaction analysis of dispositional measures by sex demonstrated that FA of the WM \ldots underlying occipital gyrus and postcentral gyrus was negatively associated with systematic quotient (SQ)in females.\ldots males exhibited larger FA in the WM of hippocampus whereas females showed larger FA in WM of parahippocampal gyrus \ldots females typically hold an advantage in tasks related to declarative memory, in which the parahippocampal gyrus has been implicated, such as in the retrieval and recognition of longterm \ldots memories." + + Lisman et al.\cite{lisman2011neohebbian} report that "For novel information and motivational events such as rewards this signal at hippocampal CA1 synapses is mediated by the neuromodulator, dopamine." They summarize a consequence of the Hebb framework "if cell A represented object A and cell B represented object B, the co-occurrence of the two object would, by the Hebb rule, strengthen the synaptic linkage between these cells. This link would subsequently be evident when only object A was presented because it would lead to the firing of cell B, thus bringing object B to mind by association." + + Lisman et al.\cite[p. 537]{lisman2011neohebbian} state "Two types of experiments demonstrate that dopamine can strengthen the synaptic potentiation produced by learning itself." + %PPT pedunculopontine tegmentum + Lisman et al.\cite[p. 540]{lisman2011neohebbian} state "Thus, novelty, reward stimuli and aversive stimuli are all able to activate the dopamine system \ldots in humans. + + Lisman et al.\cite[p. 540]{lisman2011neohebbian} state "This reward-related memory enhancement was associated with a coactivation of SN/VTA, striatum, and hippocampus, as detected by fMRI (\cite{adcock2006reward,wittmann2005reward}) Memory enhancement after long retention intervals (e.g. 24h) has been consistently found (\cite{krebs2009personality,wittmann2011behavioral}). Moreover, the enhancement was greater at late timepoints than at early intervals (i.e. 3 wks vs 20 min)(47) + + $$ using novelty, reward (or punishment, i.e., if there are bad test grades, maybe they can be used), make these memories.$$ + + Lisman et al.\cite[p. 540]{lisman2011neohebbian}point out that it is necessary to be able to encode and consolidate after a single exposure. + + \begin{figure} + \centering + \includegraphics[width=0.7\linewidth]{./lismanNeoHebbian} + \caption{connectivity in medial temporal lobs and hippocampus ventral tegmentum loop processing info about object and spatial context. Allows perirhinal cortical info about novelity to general dopamine response} + \label{fig:lismanNeoHebbian} + \end{figure} + Lisman et al.\cite[p. 542]{lisman2011neohebbian} summarize a model, citing Frey and Morris~\cite{frey1997synaptic} "weak stimulation induces on ly early long term potentiation (LTP). By contrast, stronger stimulation produces the dompamine-dependent protein synthesis that allows late LTP.\ldots the memory for event sthat occur before or after the dopamine release would depend not only on their own properties, but also on whether they fell within the penumbra of a dopamine-releasing stimulus." + + $$Music that gives "chills" gives dopamine, so if we could play something like the prelude to the StarWars IV before class, students would be primed. Some national anthems might be chill inducing, such as the French, Marseillaise (lyrics?) How long is this penumbra? It could be different for each person. for rodents, 1/2 hour$$ + + Lisman et al.\cite[p. 542]{lisman2011neohebbian} "novel photographs of natural scenes ('strong events', such as those one would expect to see in the magazine National Geographic)" had a penumbra of at least 5 minutes. + + $$so, have some slides with nat geo pictures, at least every 5 minutes, new every time$$ + + Lisman et al.\cite[p. 542]{lisman2011neohebbian} "Cholinergic\cite{sarter2005unraveling} and noradrenergic\cite{frey2008synaptic} projections to Medial temporal lobes can also modulate Long term potentiation and long-term memory. \ldots Reward-related SN/VTA activation improves memory for the rewarded stimulus but not for the non-rewarded stimuli given in close temporal proximity \cite{wittmann2011behavioral} implying either a very short or a very stimulus-specific penumbra. This is at odds with the observation that novelty-related activation of the SN/VTA has a long (ca 30 min) penumbra that affect memory for unrelated information (e.g., exposure to novel scenes can affect memory for words)\cite{fenker2008novel} One possible resolution is that the duration or stimulus-specificity of the penumbra depends on the type of motivational event that triggers dopamine release." + + Lisman et al.\cite[p. 544]{lisman2011neohebbian} "the ability to recollect newly acquired information could be intrinsically rewarding. In fact, the study of human learning has revealed an interesting puzzle; long-term retention is not helped by simple re-exposure to recently learned material but is greatly helped by retesting even when subjects already know the answer \cite{karpicke2008critical}. One interesting possibility is that retesting provides an opportunity to generate intrinsic reward signals, thereby enhancing long-term persistence of newly learned material." + + Mains et al.~\cite{Mains01072015}"Embedding three (clicker) questions within a + 30 min lecture increased students' knowledge + immediately after the lecture and 2 weeks later. We + hypothesise that this increase was due to forced + information retrieval by students during the learning + process, a form of the testing effect." + + Bridge and Voss \cite{Bridge01082015} "Cueing with actively retrieved objects facilitated memory of associated objects, which was associated with unique patterns of viewing behavior during study and enhanced ERP correlates of retrieval during test, relative to other reminder cues that were not actively retrieved. Active short-term retrieval therefore enhanced binding of retrieved elements with others, thus creating powerful memory cues for entire episodes." + + $$students may make their own flash cards to practice retrieval, but it appears a more effective strategy would have multiple differing cues$$ + + Rottschy e al.\cite[p. 830]{rottschy2012modelling} define "working memory subsumes the capability to memorize, retrieve and utilize information for a limited period of time". + + Rottschy e al.\cite[p. 836]{rottschy2012modelling} "experiments using non-verbal material showed significantly higher convergence in the left (pre-)motor area and bilateral dorsal pre motor cortex." + + Rottschy e al.\cite[p. 843]{rottschy2012modelling} "selective attention system (Shulman et al.) \ldots is right-dominant and \ldots includes the temporo-parietal junction. \ldots apparent overlap between a distributed central executive for working memory, (and) the attention system" + + $$we can get their attention, for example by music that gives chills, and it will bring blood circulation to those areas, so, working memory will be supplied with circulation$$ + + + + Wittman et al.\cite{wittmann2011behavioral}"Recent functional imaging studies link reward-related activation of the midbrain substantia nigra –ventral tegmental area + (SN/VTA), the site of origin of ascending dopaminergic projections, with improved long-term episodic memory. Here, + we investigated in two behavioral experiments how (1) the contingency between item properties and reward, (2) the magnitude of reward, (3) the uncertainty of outcomes, and (4) the contextual availability of reward affect long-term memory. + We show that episodic memory is enhanced only when rewards are specifically predicted by the semantic identity of the + stimuli and changes nonlinearly with increasing reward magnitude. These effects are specific to reward and do not occur in + relation to outcome uncertainty alone. These behavioral specifications are relevant for the functional interpretation of how + reward-related activation of the SN/VTA, and more generally dopaminergic neuromodulation, contribute to long-term + memory." + + Abraham et al.\cite[p. 1906]{abraham2012creativity} investigates a specific aspect of creativity that they call conceptual expansion. They found that this activity selectively involved the anterior inferior frontal gyrus, the temporal poles and the lateral frontopolar cortex. These findings "go against \ldots dominance of the right hemisphere during creating thinking, and indicate \ldots anterior cingulate cortex \ldots (for) abstract facets of cognitive control." + + Abraham et al.\cite[p. 1907]{abraham2012creativity} explain that conceptual expansion "refers to the ability to widen the conceptual structures of acquired concepts, a process that is especially critical in the formulation of novel ideas", citing Ward~\cite{ward1994structured}. + Abraham et al.\cite[p. 1910 -- 1911]{abraham2012creativity} found that the regions involved were left anterior inferior frontal gyrus, lateral frontopolar cortex, temporal poles, posterior regions in the inferior frontal gyrus, the middle frontal gyrus, the anterior cingulate cortex, the dorsomedial prefrontal cortex and the inferior parietal lobule. The "activation pattern is strongly lateralized to the left hemisphere." For working memory, "the overall brain activation pattern as a function of working memory was stronger in the right hemisphere" + + \begin{figure} + \centering + \includegraphics[width=0.7\linewidth]{./frontopolar-cortex} + \caption{left frontal polar cortex playing a particularly relevant role in concept expansion, is thought to mediate cognitive control at the most abstract level of information processing. The left dorsolateral pre frontal cortex, and the dorsomedial prefrontal cortex (BA8/9 and 8) also showed stronger brain activity} + \label{fig:frontopolar-cortex} + \end{figure} + + \begin{figure} + \centering + \includegraphics[width=0.7\linewidth]{./inferiorfrontalgyrus} + \caption{Conceptual expension was associated with greater brain activity in left anterior inferior frontal gyrus (right is pictured)} + \label{fig:inferiorfrontalgyrus} + \end{figure} + + \begin{figure} + \centering + \includegraphics[width=0.9\linewidth]{./gr1} + \caption{The inferior frontal gyrus, temporal poles and frontopolar cortex are involved in coceptual expansion. The roles of the anterior cingulate cortex and the dorsolateralprefrontal cortex were found to be most responsive in conceptual expansion, and active in divergent thinking.} + \label{fig:gr1} + \end{figure} + + Abraham et al.\cite[p. 1912]{abraham2012creativity} In a measure intended to isolate cognitive control processes, results showed the dorsolateral prefrontal cortex and superior parietal lobule bilaterally, and "only the right dorsolateral prefrontal cortex and anterior cingulate were found to be involved." + + $$we are seeing that being creative calls upon left side structures, and when we get specifically to being creative with symbolic and diagrammatic representations, we may call upon also the right sides. $$ + + Abraham et al.\cite[p. 1913]{abraham2012creativity} "The roles of the anterior cingulate cortex and the dorsolateral pre-frontal cortex are particularly noteworthy given the patterns of activation in these regions in the current study. Not only were they found to be more activated during divergent thinking compared to working memory, more importantly, they were also found to be most responsive as a function of conceptual expansion. \ldots the posterior aspect of the dorsomedial pre frontal cortex was also activated as a function of conceptual expansion. \ldots As this region has been discussed with reference to concepts that are central to hypothetical reasoning, such as constructive processes in cognition~\cite{abraham2008thinking} which involve flexible recombination of representations from memory~\cite{schacter1998cognitive} and evaluative judgment~\cite{zysset2003functional}, the dorsomedial prefrontal cortex may prove to be highly relevant structure for select aspects of creative thinking. \ldots speaks against the ubiquitous idea the right brain is more 'creative' than the left. \ldots in the current study, we have explored the deliberate problem solving mode of creating thinking under time constraints. There is, however, another vast dimension of creative thinking where idea generation occurs spontaneously, effortlessly, and/or in a state of defocused attention~\cite{|} In fact, creating idea generation is far less likely to result from deliberate cogitation during real everyday problem solving, but instead, it occur spontaneously and unpredictably. This unconscious non-deliberate spectrum of creating thinking \ldots is less amenable to well-controlled investigation. " + + Born and Wilhelm~\cite[p. 192]{born2012system} "Experimental evidence for these three central implication is provided: It has been shown that reactivation of memories during slow-wave sleep(SWS) plays a causal role for consolidation, that sleep and specifically SWS consolidates preferentially memories with relevance for future plans, and that sleep produces qualitative change in memory representations such that extraction of explicit and conscious knowledge from implicitly learned materials is facilitated." + + $$In procedural memory we don't need to know why, there might not be any, (for example, remember the melody) but for some things they are accompanied by why. Is declarative everything other than procedural? What about implicit vs. explicit? These both are compatible with "why". There are times when we can use knowing why to save on what would otherwise need to be remembered. Does it have a name? Is it named in David Tall's article with bi-furcation in the name?$$ + + Born and Wilhelm~\cite[p. 195]{born2012system} "Via the olfactory system odour stimulation acquires immediate access to the hippocampus. \ldots we found that the odour when re-exposed during SWS after learning induced a distinct activation of the left hippocampus, i.e. the odour served as a cue that reactivated the new memories for the card locations encoded in the left hippocampus, thereby enhancing these memories \ldots hippocampal networks are particularly sensitive in SWS to inputs capable of reactivating memories." + + Born and Wilhelm~\cite[p. 197]{born2012system} "explicit encoding favours access to sleep-dependent memory consolidation (\cite{robertson2004awareness}). Involvement of the prefrontal-hippocampal system underlying explicit encoding has been proposed as prerequisite for consolidation to occur during sleep(\cite{marshall2007contribution}). \ldots emotionality of the encoded events can increase the memory benefit from sleep (\cite{kuriyama2004sleep,wagner2006brief})." + + Wagner et al. \cite{wagner2006brief} investigated memory after a four hour interval of sleep. "Sleep following learning compared with wakefulness enhanced memory for emotional texts after 4 years (p = .001). No such + enhancement was observed for neutral texts (p = .571)." + + Born and Wilhelm~\cite[p. 197]{born2012system} "Processing of anticipatory aspects of behaviour such as expaectancies and plans is particularly linked to executive functions of the prefrontal cortex that regulates activation of memory representations during anticipated retrieval and accommodates specifically the intentional and prospective aspects of a memory representation~\cite{polyn2008memory} \ldots prefrontal tagging of memories explicitly encoded under control of the prefrontal-hippocampal system could be decisive for the selectivity in off-line memory consolidation" + + Born and Wilhelm~\cite[p. 199]{born2012system} "there is convergent evidence \ldots that the system consolidation process during sleep supports the extraction of invariant and repeating features in newly encoded memories, and in this way, the conversion of implicit into an explicit and conscious form of memory \ldots more than twice as many subjects of the sleep group gained insight into the hidden structure as compared with the wake control group \ldots subjects who had slept after \ldots training were distinctly more able to deliberately generate the sequence underlying \ldots than the subjects who had stayed awake" + + Born and Wilhelm~\cite[p. 201]{born2012system} "sleep appears to prime the transformation of implicitly encoded information into explicit knowledge, i.e., something that is not conscious before sleep enters consciousness through sleep". + + Dudai~\cite[p. 229]{dudai2012restless} defines declarative memory as that which requires conscious awareness for retrieval (facts, events), and nondeclarative can be retrieved in absence of conscious awareness (habit, skill). + + Dudai~\cite[p. 231]{dudai2012restless} reminds "university students can improve their memory bye practicing self-testing, because retrieval practice is a powerful mnemonic enhancer", citing Karpicke and Roediger~\cite{karpicke2008critical}. + + Forcato et al.~\cite[p. 1]{forcato2013role} observe "the reconsolidation process alllows new information to be integrated into the background of the original memory; second it strengthens the original memory. \ldots at least one labilization-reconsolidation process strengthens a memory via evaluation 5 days after its re-stabilization. \ldots this effect is not triggered by retrieval only. \ldots repeated labilization-reconsolidation processes made the memory more resistant to interference during re-stabilzation." + + Forcato et al.~\cite[p. 1]{forcato2013role} "reconsolidation does not represent recapitulation of initiali consolidation, but rather, if refers to the functional role of this process: to stabilize memories." + + Forcato et al.~\cite[p. 2]{forcato2013role} "when the reminder only included contextual cues (context reminder), the memory was evoked but not labilized." + + A reminder has the effect of labilization which allows reconsolidation. "We found that just one labilization-reconsolidation process was enough to strengthen a memory that was evaluated 5 days following its re-stabilization. \ldots Memory persistence is increased by repeated triggering of labilization-reconsolidation." + + Wirebring et al. \cite{}"Repeated testing is known to produce superior long-term retention of the to-be-learned material compared with repeated encoding and + other learning techniques, much because it fosters repeated memory retrieval. This study demonstrates that repeated memory retrieval + might strengthen memory by inducing more differentiated or elaborated memory representations in the parietal cortex, and at the same + time reducing demands on prefrontal-cortex-mediated cognitive control processes during retrieval. The findings contrast with recent + demonstrations that repeated encoding induces less differentiated or elaborated memory representations. Together, this study suggests + a potential neurocognitive explanation of why repeated retrieval is more beneficial for long-term retention than repeated encoding, a + phenomenon known as the testing effect." + + \subsection{Educational Neuroscience} + + \section{Systems Biology} + be sure to get the Kandel, Dudai 2014 review + + \section{Physiologically Informed Constructivism} + Research has shown the progression of an idea, from a short term representation in the medial temporal lobe, to a consolidated memory in an area related to the single sensory modality with which that idea was received, to a representation in association cortex when multiple sensory modalities are involved, to a reconfiguration in right prefrontal cortex when an idea is used creatively. + How we as instructors shepherd these representations in the minds of our students can be suggested by cognitive neuroscience. One example is a deeper understanding of the utility, for memory and attention, to intrinsically rewarding learning activities. Another example is the application of multiple sensory modalities, such as representations in figures, text and mathematical symbols, and also pseudocode. Yet another is the deliberate construction of pseudo-monetary rewards, of unpredictable value, in the immediate aftermath of + instruction of a new idea. (This derives its effect from the dopamine response to the monetary reward, which is augmented by not knowing the amount.) + + We can expect information stored in the cortex near a single sensory modality to be retrieved when a similar situation recurs. When multiple modalities are involved there are more ways to arrive at increased metabolism reactivating this memory, bringing it into consciousness. + + Music is rewarding. The areas of the brain that are rewarded by memory are the same ones that help us remember. (Blood Zatorre salimpoor) We can reward the students / administer dopamine to help them remember. We have from Blood and Zatorre~\cite{blood2001intensely} that music so intensely pleasurable that it creates "chills" is correlated with activity in nucleus accumbens, which we also know, from (whom?) advances consolidation of memory representations in medial temporal cortex into longer term memory. %Here I am thinking how REM sleep is seen to be helpful to consolidation of memory, and I am thinking about the "subject matter" of REM sleep, and wondering whether a method more time-efficient than sleep could be developed + + $$ bagpipe music generates chills$$ + + We want the students to construct an understanding of the material, for example, the correctness or resource utilization of an algorithm, and to be able to creatively apply the information they have to construct a proof of it. + + The material first arrives in the student's awareness through some sensory modality, and we wish the information to be remembered, not only as inert knowledge, but as information that enters consciousness in response to a situation in which it is usefully employed in a proof. + + Lisman and Grace\cite{lisman2005hippocampal} describes the hippocampal--ventral-tegmental area loop, controlling the entry of information into long term memory, facilitated by dopamine(DA). One stimulus for production of DA is the arrival of an unexpected reward \cite{schultz2000neuronal}, another is novelty\cite{kafkas2015striatal} (check this). Instructors can provide unexpected rewards with, for example, increased points for clicker questions. Other ways to provoke DA release, in the part of the brain that produces memories, is with music.\cite{salimpoor2015predictions} (check that this is the right article). Lisman and Grace\cite{lisman2005hippocampal} describe reactions that they state "provide a basis for the dopaminergic modulation of early long term potentiation (LTP)", which is a change to the signaling behavior of neurons in the brain. They continue "It follows that novelty itself should enhance LTP." Moreover they claim\cite[p. 707]{lisman2005hippocampal} "There is reasonable evidence from animal experiments that DA enhances learning, as would be expected from its enhancement of LTP." and \cite[p. 708]{lisman2005hippocampal}"The spiny cells in the accumbens are a likely site for combining novelty signals and goal-dependent motivational signals." and \cite[p. 709]{lisman2005hippocampal} "reasonable working hypothesis that \ldots combines novelty signals with information about saliency and goals \ldots thereby enhance the entry of the information into memory." \cite[p. 709]{lisman2005hippocampal} "without dopamine, late LTP does not occur and early LTP decays within about an hour." + + Wittmann et al.\cite{wittmann2005reward}"Long=term potentiation in the hippocampus can be enhanced and prolonged by dopaminergic inputs from midbrain structures such as the substantia nigra. This improved synaptic plasticity is hypothesize to be associated with better memory consolidation in the hippocampus." They say that reward anticipation reliably elicits a dopaminergic response. They hypothesize that "activation of dopaminergic midbrain regions enhances hippocampus-dependent memory formation, possibly by enhancing consolidation." Wittmann et al.\cite[p. 464--465]{wittmann2005reward}"supports our hypothesis that the hippocampus is a major site for the neuromodulatory influence of reward on long-term memory formation \ldots supports the hypothesis that dopaminergic neuromodulation enhances hippocampus-dependent memory formation \ldots It is likely that a greater proportion of subsequently recallable items will undergo hippocampus-dependent consolidation than of subsequently recognizable items. \ldots these results provide evidence for a relationship between activation of dopaminergic areas and hippocampus-dependent long-term memory formation." + + Here I want to diagram a sequence of activities mapped onto the brain areas, showing the progression of memory from the medial temporal lobes with new dentate granular cells, followed by consolidation in one sensory cortext or association cortext followed by reconsolidation into prefrontal and parietal followed by prefrontal from where it can be used creatively in the construction of proofs. It would be good if I could cite references for each milestone, and associate with conceptualizations, such as those in Harel and Sowder and Tall. + + % % % % % % % % % % % % % % % % + + \subsection{Intuition} + help and obstacle\\ along with obstacles arising from intuition there exist epistemological obstacles Bachelard 1938 Brosseau 1983 preventing acquisiton of new knowledge. There exist didactic obstacles. + epistemological obstacles + \subsection{Met-befores} + Tall\\ + McGowan Tall\cite{Metaphor or met-before 2010 Jour Math Behavior}, The idea met-before (formerly met afore) emphasize that a metaphor relates new knowledge (the 'target') in terms of existing knowledge (the 'source') developed from previous experience, so that new ideas can be related to familiar knowledge already in the grasp of the learner. + + Small example of a met-before: Integrated circuits are available at many levels of integhration, from singletons of transistors to tens of millions. + One simple circuit function ins the counter. The number of cluck edges since the last reset, modulo the number given by $2^n$, where $n$ is the number of bits in the counter, is represented digitally, that is, by a voltage level whose domain has been partitioned into that representing 1 and that representing 0. A counter may reset immediately (so to speak) upon the reset signal. This kind is called asynchronous reset. The synchronous reset kind resets after a clock edge occurs during a reset input. + Having, for the purposes of this example, set the context this way, now consider this problem, from Santos-Trigueros\cite{[p. 74]{}} + ``Nine counters with digits from 1 through 9 are placed on a table.'' + Had we not just been discussing counters in another setting, this sentence might have been understood as intended, more quickly. That is, the context can serve as a distraction, to be overcome. It can help the instructor to realize that students arrive in class, not only lacking wished-for preparation, but also bringing unhelpful contexts. + \subsection{Harel and Sowder} + \subsection{van Hiele Levels} + \subsection{Student Centered} + something about students' perspectives are not always well-matched to their needs + + Students might have in mind material they would like to learn, and there may be also a lack of appreciation for material in required courses + Students may have a rate at which they would like to learn --- points at which they would like to pause and integrate new material with things they already know. + \subsection{Attention} + Lindquist \cite{lindquist2013mind} describes Mind wandering during lectures: Observations of the prevalence and correlates of attentional lapses, and their relationships with task characteristics and memory. + + Brosch et al.\cite{brosch2013impact} examined the impact of emotion on perception, attention, memory, and decision-making. + + Sali et al.\cite{sali2014role} show that "Previously rewarded stimuli involuntarily capture attention." + + \subsection{Memory} + Ojemann\cite[p. 257]{ojemann2014human} " \textit{episodic, explicit} or \textit{declarative} memory, memory for the specific event (specific name, word) that occurs at encoding " + + Ojemann\cite[p. 257]{ojemann2014human} "Our memory paradigm requires storage over a short period and thus a measure of \textit{recent} or \textit{short-term} memory. In the functional neuroimaging literature memory for this duration is often referred to as \textit{working} memory. + + Ojemann\cite[p. 257--258]{ojemann2014human} "Lateral temporal cortex \ldots models of brain stubstrate for recent verbal memory \ldots during the encoding phase." + + \subsection{Social Constructivism} + Attempts at communication, as in conversations about material, are regarded + as helpful to learning. + + McGowan Tall 2013 Jour Math Behav + ``One student wrote that she knew her answer was correct (it was actually incorrect) because the other members of her group agreed with her. These students consistently evaluated both the numerical expression 'minus a number squared' and a quadratic function with a negative-valued input incorrectly throughout the remaining twelve weeks of the semester. This cautions us to realize that cooperative leaning amongst students who are failing to make sense of the mathematics may reinforce their problematic conceptions rather than reconstruct them'' [p. 533] + + Though it is easy to assume that communication between practitioners is carried out verbally, there are examples of proofs without words \cite{nelson1993proofs}. + + D\"orfler\cite[p. 129]{dorfler2000means} proposes that by discussion, abstraction can be promoted "the abstract terms might serve the purpose of talking about a variety of concrete, even physical experiences such as describing observations \ldots This abstracted manner of talking then acquires some independence from the experiences and experiential phenomena referred to, so that the abstract objects gain discursive existence. First, the abstract description lends 'meaning' to the experiences. Later, the abstract objects derive their 'meaning' from their taken-to-be representations or applications." + + D\"orfler\cite[p. 129]{dorfler2000means} summarizes "our experiences with material objects are schematized in an image schema that is projected metaphorically to terms such as \textit{abstract object}. \ldots At times, these image schemas might even be exteriorized by symbolic expressions that, in turn, can serve as generic terms. \ldots A cognitively oriented explanation for the failure to be inducted into the mathematical discourse is, therefore, the lack of image schemata on which to base the discursive extension \ldots this lack might results from an absence of pertinent experiences \ldots there can be differences in the abilities of each individual to schematize his or her experiences and to make metaphoric use of words." + + + + \subsection{Beliefs about Diagrams} %after social constructivism, because people attempt to communicate with diagrams + helpful, hindering, post-conceptual (i.e., they refer to existing concepts, maybe do not convey new ones), + Hilbert's ``who does not'' with a, b, c\\ + perceptual to transformative\\ + animation + \subsection{Semiotics} + Because sybmolization supports generalization\cite{loewenberg2003mathematical} and operations in mathematics\cite{schoenfeld1998reflections}, and because symbols are used also in efficient communication with others, symbolization is a skill our students need. + + Van Oers\cite[p. 133--134]{van2000appropriation} emphasizes the role of the adoption of symbol use, as students learn mathematics, stating "Mathematics as a discipline is now generally conceived of as an activity in which constructive representation, with the help of symbols, plays a decisive role" citing Bishop, Freudenthal and Kaput. + + Van Oers\cite[p. 133--134]{van2000appropriation} summarizes: "According to Freudentahl, mathematics is basically an activity of mathematizing: that is, organizing a (concrete empirical or abstract mental) domain, representing it with the hop of symbols \ldots experimenting with symbolic means". Freudenthal\cite[p. 10]{freudenthal1973mathematics} describes some of the history of symbol use: "Another algebraic idea is symbolism, the use of signs which do not belong to everyday language, to indicate variables. 'Think of a number' is how the problems are introduced in old narrative algebra. In Diaphantus' work the word 'number' becomes more and more a computation symbol. This continues in Indian and Arabian mathematics. The \textit{cossists} of the late Middle Ages has a whole system of symbols for the unknown and its powers \ldots " + + Van Oers\cite[p. 135]{van2000appropriation} states that van Hiele "emphasized the importance of symbols in mathematics for grasping the mathematical meaning. \ldots appropriating the meaning of symbols is primarily a communicative process" requiring a mutual understanding of the meaning. + + Van Oers\cite[p. 136]{van2000appropriation} states: "Symbols are indispensable as means for coding the results of thinking. More importantly, however, symbols also function as ways of organizing in the course of thinking". He goes on to report " According to empirical investigations of the development of mathematical thinking in pupils, the failure of meaningful appropriation of mathematical symbols has turned out to be one of the main problems in mathematics learning.", citing Hughes, 1986 and Walkerdine, 1988, also Miles and Miles 1992. + + Van Oers\cite[p. 170]{van2000appropriation} states: "The analysis of symbolizing in a mathematical context has led us to the realization that symbol use is intrinsically related to meaning, negotiation of meaning, and communication." + + Nemirovsky and Monk described symbolizing\cite[p. 177--178]{nemirovskymonk} "Conceiving of symbolizing as the creation of a space in which the absent is made present and ready at hand elicits at least two major issues: (a) the nature of such a space, and (b) the ways in which the absent is made present and ready at hand". + They observe \cite[p. 178]{nemirovskymonk} "our play as children is a crucial activity through which each one of us has practiced and learned to symbolize", citing Piaget 1962, Slade and Wolf 1994 and Winnicott, 1971/1992. + Nemirovsky and Monk give an example \cite[p. 204]{nemirovskymonk} "Lin shifted her attention from being immersed in creating something \ldots to reflecting on it as a particular manner of doing things". They go on to say, "Symbolizing is making possible the sudden and unanticipated encounter with past experience that can radically transform the 'here and now' of the symbolizer. + + $$ sudden, unanticipated, coming into consciousness is a relevant idea $$. + + Nemirovsky and Monk state\cite[p. 212]{nemirovskymonk} "Insights that 'come' or 'happen' to us is a way of saying that we often experience what we become as surprising and unexpected." + + Bransford et al.\cite{bransford2000designs} mention "a common problem of expertise, namely, that things become so intuitively obvious that one forgets the difficulties that novices have in grasping new ideas". + + $$ consider that procedures to which we have become so accustomed that we do not need conscious attention (e.g., shifting gears for a car or bike) could be represented more efficiently in neurons, see Kandel and Squire on Memory and Attention\cite{squire2000memory} and Squire \cite{squire2015conscious}$$ \ No newline at end of file diff --git a/ch9.tex b/ch9.tex index 906427a..f734585 100644 --- a/ch9.tex +++ b/ch9.tex @@ -1,242 +1,124 @@ -\chapter{Future Work} -Anecdotal evidence suggests that pleasantness and fun, which are intrinsic rewards, help students pay attention and remember. Make use of the natural experiment --- some students will have enjoyed the material. Are these the ones who remember better? and does the nature of enjoyment make any difference in understanding productivity aesthetics? -\section{Helping Students Discern Derivation for Proof of Correctness} -Recall that variation theory holds that students cannot discern a thing unless contrast is provided. -Pang has pointed out that, for persons aware of only one language, ``speaking'' and ``speaking their language'' are conflated. -Only when the existence of a second language is known, does the idea of speaking become separated from the ide of speaking a specific language. -We may wish to alert students to the ability to derive code from requirements mathematically. -We could show them, for example, an even function. -We could show them how to construct a function that is even. -We could show them how to construct a function that is odd. -This might hep them see the difference between writing code and testing it afterwards, and deriving the code to be correct by construction. - - -Variation theory and cognitive science suggest the teacher should make an effort to show positive and negative examples and to point out what causes one to be positive and the other negative to help discern the relevant features. Have you noticed this practice? Did you find it helpful?\\ - -Engagement with the material\\ -Please rank order the factors that might help you e engaged wit theh material\\ -job related\\ -prereque\\ -have been curious about\\ -fun\\ -competition\\ -beautiful\\ -(Does this happen often)\\ -(Does this matter to you?)\\ -What, if anything, makes you engage with the material?\\ - Neurophysiology suggest that animation in overheads makes it easier to be alert to material, that figures draw attention more easily than text. - Plesae comment on what if any things make animation helpful to you, or whether you find some animation helpful. - If you find figures helpful, can you comment on how they help? Do you find it easier to pay attention to figures than to text? - -\section{Use of Conceptualizations} - -Suppose we wish to incorporate into our curriculum on software engineering, material on both verification by formal methods, and also code derivation from specifications. -We might wish to have our students able to recognize a specification that can be satisfied -by a transition system, so that they can ascertain that a representation of that transition system that can be converted automatically into code implementation is an available approach. -We would need it to be, that the mathematical arguments for the transformations of requirements -to Floyd-Hoare-triples, for example, convince the students. -Knowing the range of the state of preparedness of the incoming students allows us to plan the amount of time to devote to background material, -and which background material should be provided. -Suppose we wished to make use of higher order logic, so that we could employ relations as arguments -to functions. -We might, for example, wish to pass a -graph constraining how a computation should be carried out. -We would like the students to appreciate the mapping between a recursive -implementation and proof by mathematical induction. -We would like the students to discern when a recursion is a tail recursion. -We would like the students to appreciate the mapping between a tail recursion and iteration. -On the one hand we have an idea which mathematical tools we wish to use, and on -the other hand we have an idea -of the extent of the students' conceptualization. -With these we can begin to design an instructional approach. -\subsection{Recognize a specification that can be satisfied by a transition system} -For this, they should know what a transition system is. -There is a state, so we wish to represent a store, and the states it may take on. -There is a way to change the state, so we wish to represent transitions, and the idea of a path from -state to state. -They should understand how a transition system can be represented in code, and that this can be carried out automatically. -Path algebra furnishes a rigorous description of these processes. -Path algebra out to be accessible to the students. -We could introduce (typed) lambda calculus to the students with path algebra as its first use. -Now that we have Java 8 which provides anonymous functions, we can also offer a coded version for -exploration. -Having the use of typed lambda calculus for path algebra, we can then use it to describe the effect of instruction in the context of Floyd-Hoare triples. -Program derivation can be illustrated. -Because we always prefer the students to apply what they comprehend, -it is important to have exercises. -Students should perform derivation, so that they come to appreciate that it can be automated. - - -\subsection{Composition} -Because derived components can be expected to be composed, either in sequence or in parallel, it -is important that the students understand this at a rigorous level. -As we can expect to be passing as arguments, procedures to be composed, the need for -higher order logic is evident. -\subsection{Invariants} -The role of invariants in assuring the correctness of composition provides motivation for consideration of invariants. -\subsection{Provability} -With transition systems as a first example of what can be correct by construction, and invariant -as a tool for provability, students can see that a way to design code with the prospect of fewer bugs is possible, at least in some circumstances. -\subsection{User of Bridging Material} -The material on mathematical tools we wish to use should be introduced at a level that addresses the -conceptualizations the students bring to the course. -Because the students bring different levels of preparedness to the class, some -self-assessment might be included, to guide the student to the starting poing corresponding to his or her current ideas. -The self-assessment can make use of the concepts used in the course. -For example, during the course, the idea of invariants is used. -Self-assessment can include questions posed to determine a student's understanding -of the nature and role of an invariant, and direct the student to explanations that may be -helpful, according to the results of the self-assessment. -From this vantage point, other techniques of proving, including contradiction and contrapositive, are motivated. -\subsection{Summary} -The less-developed end of the range in the students' conceptualizations identifies the starting point of explanatory background material that should be provided. -The needs of the course determine the ending point of the explanatory background -material. -The major themes in the students' conceptualizations make evident the -critical aspects, that distinguish on conceptualization from another, and therefore -suggest the self-assessment questions that should be furnished, to help students know where to begin improving. -An alternative to having analyzed the conceptualizations among the students is to use on a -student case-by-case basis, a sort of failure modes and effects analysis, whereby the effect of lacking a certain understanding can be predicted to cause a certain pattern of confusion on the part of the -student, and that confusion is used to discover what way is best to help that particular individual. -Different parts of the background material become motivated at different times within the course. -It could help the students to provide a key from possible trouble spots to relevant background material. -\subsection{Example} -We use the model of students' conceptualizations shown in () to provide background material for a course in software engineering organized around the concept of provability-driven development. -Finding the calculational approach of Gries\cite{}, which provides relatively succinct proofs, matching well with the phenomenology of Rota, who found that short proofs were more beautiful and more memorable, we prefer this style. -From the range of conceptualizations, we choose to include symbolization, and formation of valid arguments stressing that rules of inference satisfy guards upon the transformations we can make as we transition from one statement to another. -Which of the following describes a while loop executing?\\ -boolean done = false;\\ -while(!done)\{\\ -statements which eventually change done to true\} - +\chapter{Conclusion} +\begin{quote}Harel and Sowder \cite[p. 277?] {harel1998students}by their natures, teaching experiments and interview studies do not give definitive conclusions. They can, however, offer indications of the state of affairs and a framework in which to interpret other work.\end{quote} + + + +Are CS students' conceptualizations more like Harel and Sowder, or more like Tall? +Are the several schemes (Pirie Kieren, etc. complementary? reconcilable? Is one more likely than another based on cognitive neuroscience of language? (proofs are in a language after all)) + + + +This research suggests that suitable question for a larger study +\section{ Recognizing an Endpoint} +A qualitative study is thought to be finished when an internally consistent +narrative, compatible with the data, both situating the data and explaining +them, has been produced. +For our research questions, a model, accompanied by a narrative combining +the information obtained from inquiry about these topics will complete the +work. Data from our extended student body, that provide a persuasive model +containing categories of conceptualizations, and that are closely enough related +that some insight about concepts differentiating adjacent categories can +be inferred, are thought sufficient to generate this narrative. The proposed +differentiating concepts are thought to have the potential to become material +for a larger survey, thereby providing a starting point for new work. +I expect to find a model similar to that of Harel and Sowder 1998[?], but +modified because of the different emphases on material in computer science +compared to mathematics. First, students of computer science should be very +familiar with the idea of consciously constructing, examining and evaluating a +process, from their study of algorithms. Because of this, the category internalization +might be subsumed by the category interiorization. +From empirical data, we know that there are students of computer science +who think that proofs might be irrelevant to their career; it would be hard to +imagine a mathematics student who thought so. CS students who do not think +proof is part of their career might be relatively content with conceptualizations +corresponding to outside sources of conviction. We found computer science +students whose conception of proof includes that a single example is sufficient +for proving a universally quantified statement. We found computer science +students whose conception of proof is that definitions are barely interesting, +and who find demonstrations based on definitions unconvincing. Because our +findings were not quantitative, we could not compare the population of categories. +Nevertheless, the relationships between categories, and the resulting +critical factors, might be different, especially in the area of Harel and Sowder's +internalization and interiorization. +Because the scope is broader, involving proof for deciding whether or not an +algorithm is suitable for a problem, I expect we will find more categories, +related to algorithms and their applicability. +The product of a phenomenographical investigation is categories of conceptualization +and critical aspects that distinguish one category from the previous. +One hopes that by identifying critical aspects, suggestions about what to emphasize +when teaching, and what to seek in assessments are also clarified. This +investigation is intended to develop insight into students understandings of +proofs, that are the meanings they have fashioned for themselves, based on how +they have interpreted what they have heard or read. By examining some of +these understandings, we might find directions in which to improve our teaching. +Moreover, observations about the conceptualizations of students early in +the curriculum can forewarn instructors, helping them recognize the preparation +of incoming students. Perhaps we could use this to prepare remediation +materials. +For example, we can use UML diagrams and ``trie'' data structures to emphasize +definitions for families of concepts. We can choose groups of examples, +and non-examples of proofs whose correctness turns on the qualification that +distinguishes a subclass from its immediate superclass. +Beyond this, one may hope that qualitative research suggests worthwhile questions +for larger scale investigations. + +Application of findings about students +of mathematics to students of computer science is fraught by differences +in the preparation and interests related to algorithms. One likely difference is +motivation: students of mathematics know that proof is the principal means +of discourse in their community, but students of computer science might not +be aware of the importance of proof to their work. Not all differences favor +students of mathematics. In particular, the categories internalization and +interiorization of Harel and Sowder’s 1998 model\cite{harel1998students} are apt to be, in students +interested in algorithms, more closely related, than in students of mathematics. +There may be a difference regarding abstraction. Both mathematics and +computer science deal in abstraction, and students in both disciplines struggle +with it. \cite{mason1989mathematical,hazzan2003students}. In mathematics, following Vi\`ete, \cite{viete2006analytic}, single letter variable names +are used. These are thought to support the learning of abstraction, for example, Gray and Tall \cite[p. 121]{gray1994duality} observe ``we want to encompass the growing compressibility of knowledge characteristic of successful mathematicians. Here, not only is a single symbol viewed in a flexible way '' + and +in computer science abstraction, one way to exhibit abstraction is UML diagrams. Because the +``trie'' structure and International Standards Organization ISO standard 11179 +are computer science approaches to management of definitions, it could be that +computer science students would be more accessible to noticing the desirability +of concept definitions over concept images (see R\"osken and Rolka, \cite{rosken2007integrating} and +Rasslan and Tall \cite{rasslan2002definitions}). It would be interesting to know whether any of several +approaches reported by Weber [?] could be used, perhaps in modified form, for +instruction of students of computer science. The Action Process Object Schema +approach of Dubinsky \cite{dubinsky2002apos} sounds compatible with computer science students' +interests. An approach due to Leron and Dubinsky uses computer programming +\cite{leron1983structuring}, another \cite{leron1995abstract} is directed more to learning group theory than to learning proof +construction. Also specific to students concerned with algorithms, we may +wish to extend the notion of social constructivism from that of Piaget \cite{}, [?] and +of Vygotsky, [?] where it was necessarily a person with whom the learner was +communicating, and therefore with whom it was necessary to share a basis for +communication, to include a compiler and runtime execution environment, as +students of computing disciplines must also comply with rules (e.g., syntax) +used in these systems. Recalling the work of Papert and Harel\cite{harel1991constructionism}, we might +call this constructivism with constructionism. Constructionism is an approach +to learning in which the person learns through design and programming. +A cluster of related problems exists, which includes what students conceptualizations +are, about some elements of proof they should understand: \begin{itemize} -\item ($\neg$ done)(statements)(done) -\item ($\neg$)(statements)(d) -\item($\neg$p)(statements)(p) -\item(p)(statements)($\neg$p) -\item all of the above -\item none of the above +\item what internal representations do students use? +\item Is there a gamut of internal representations, and does that help with abstraction? +\item mathematization, which is the ability to represent problems in mathematical +notation +\item interiorization, which is the ability to examine and discuss the process of +creating proof +\item comprehension of simple proofs, which is the ability to see that, and why, +an argument is convincing +\item proof analysis, which includes the ability to analyze simple proofs to +recognize structure +\item problem recognition, which is the ability to see that a problem is one that +matches a known solution technique +\item transformational approach, which is considering the consequences of +varying features of the problem +\item axiomatic approach, which is the exploration of the consequences of +definitions +\item construction of valid arguments, which is to synthesize deductions with +component parts, including warrants \end{itemize} -Which of the following describes two statements executing in sequence (where, for any $i$, $s_i$ denotes a state of the variables being used by the program)? Please note that reading the value of variable does not change the state. - -\begin{itemize} -\item (in state $s_0$)(print ``Hello, world'')(print x) (in state $s_0$) -\item (in state $s_0$)(x=x+1)(print x) (in state $s_0$) -\item (in state $s_0$)(x=x+1)(print x) (in state $s_1$) -\item 1 and 3 but not 2 -\end{itemize} - -Please note that the questions have been designed to help you notice the content being explored. If you find that the phrasing of the questions is helping you arrive at the correct answer, you should choose to study the material. Variables, which can be individual letters, or names, either in computer programs or mathematical statements, are used to represent ideas. In programs, meaningful -variable names have been shown to increase the speed and correctness of comprehension \cite{} In mathematical statements, single letter variable names have been shown to improve the skill of abstraction. -Thus we choose to employ meaningful variable names for now, but will tend to replace them with single letters later. -You have probably seen variables in high school algebra, for example in the equation for a line, typically written as $y=mx+b$. -Here, $y$, the vertical position on a graph, is calculated from $m$, the slope of the line, and $b$, the vertical offset when $x=0$. A family of parallel lines can be generated by keeping one value for $m$, and setting different values for $b$. -Thus we see that a variable can refer to a single value, and a variable can also refer to different single values at a time. -For any one point along a line, when the value of $x$ refers to a single value, the single values of $x,m$ and $b$ can be used to calculate a single value for $y$. -It is, however, also possible to think of the values of horizontal axis all together, and refer to that idea by a single variable $x$. -If you are familiar with a programming language, such as MATLAB\textregistered, that supports vector variables, you will have seen single variable that can refer to many individual values at once. -Likewise, you may have worked with arrays of one ore more dimensions; reflecting on this, the array name (without specified indices) can be though of as referring to the entire array. -Keeping that idea in mind, we can contemplate the mathematical expression $0 \leq x \leq 1$. -When $x$ is an integer, we have that the integers 0 and 1 satisfy this description. -Depending upon context, $x$ may represent the range including these. -changing the universe of discourse from integers to reals, the expression $x<3$ represents, depending upon context, not necessarily only a single, yet-to-be-determined real number that also has the property of being less than 3, but also it can be that $x$ represents all real numbers less than 3. - - -\subsection{Logical Operators} -We wish students to understand strengthening and weakening of conditions. -We wish student to understand the idea of a precondition as a disclaimer, and also as a guard. -We wish students to be able to negate statements with multiple quantifiers. -We have seen from investigating the conceptualizations that students can have trouble even negating statements with single quantifiers. -\subsection{Relations} -We want students to understand relations, because we want them to see the pairing of an input state and an output state, as a result of a transformation by a computation, as a relation. -\subsection{Rules of Inference} -Calculational proofs\cite{are these Gries?} seem to suggest strongly the way to proceed. -Stepping through examples of these might help students comprehend -\subsection{Internalization and Interiorization} -Recall that internalization has been achieved when a student can perform a process correctly, not necessarily recognizing the circumstances in which that process is appropriate. -What distinguishes interiorization from internalization is that the student can examine and discuss the process, beyond being able to carry it out. -It is important for the software engineering course that students can examine and discuss the process of program derivation from requirements. -Recognition of the circumstances in which automatic program derivation can be performed is an important part of understanding requirements. -We combine internalization and interiorization because our interviews have not addressed topics that the participants cannot discuss. -We have however found student testimony that process that can be carried out confidently are nevertheless not used due to lack of a means to determine whether the process is appropriate to circumstances. -Which, if any, of the requirements described below can be satisfied by a finite state machine? - -\begin{itemize} -\item Recognize whether or not a string could have been generated by a regular expression -\item Recognize whether or not a string could have been generated by at least one of a set of regular expressions -\item Recognize whether or not a string could have been generated by juxtaposition of strings, each of which could have been generated by at least one of a set of regular expressions -\item all of the above -\item none of the above -\end{itemize} -Is this a mapping from a recursive algorithm to a proof by mathematical induction? -The base case or cases of the recursive definition map to the base case or cases of the proof, and the recursive call or calls are always invoking with a problem of a smaller size, and the inductive step is always working with a premise that is assumed to be true. -In both recursive algorithms and inductive steps, the next problem is solved making use of a previous solution. -Is this a mapping from a concrete example of a proof to a proof in a more abstract circumstance?\\ -Concrete\\ -There is a commercially available car, a deLorean.\\ -There is an unreachable goal, time-travel.\\ -Spielberg has shown us that if only we had a ``flux-capacitor'' to attach to the deLorean, we could have time-travel.\\ -Therefore we conclude that the flux-capacitor must also be unreachable, by the following logic:\\ -If we had the flux-capacitor and the deLorean, we would get time travel.\\ -We cannot get time-travel.\\ -So, we cannot have the flux-capacitor and the ddLorean.\\ -We can get the deLorean.\\ -So, it must be that we cannot get the flux-capacitor.\\ -Relatively Abstract\\ -There is a simple algorithm, $M$, by which we can determine whether two finite sets have any elements in common.\\ -There is an unreachable goal, $A_{TM}$.\\ -Sipser has shown us that, if only we had $E_TM$ and $M$, we could have $A_{TM}$.\\ -We cannot get $A_{TM}$.\\ -So, we cannot have $E_{TM}$ together with $M$.\\ -We can have $M$.\\ -So, it must be that we cannot get $E_{TM}$.\\ -(I'm thinking pictures would help.) -In symbols:\\ -$\neg A_{TM}$\\ -$M \land E_{TM} \rightarrow A_{TM} $\\ -$\neg A_{TM} \rightarrow \neg(M \land E_{TM})$\\ -$\neg(M \land E_{TM})$\\ -$\neg M \lor \neg E_{TM}$\\ -$M$\\ -$\neg E_{TM}$ - -\subsection{Perceptual} -\subsection{Transformational} -\subsection{Axiomatic} -To detect -\subsection{What Do You Say After You Say HelloWorld?} -This is a software engineering course, organized around the idea of provability driven development. -We want to impart the desire to approach software engineering problems with methods of formalizing calculation checking. -Mathematics provides what is necessary for proofs, by means that include precise definitions. -We would write definitions of program function in a style admitting proof. -We will use finite automata. -So, a section on proof, followed by a section on finite automata, formal definitions of things we use, -clients, servers, files, streams, What sort of things are we aiming to prove? -liveness, safety, accuracy, i/o relations for component based systems resource utilization including time utility functions. -The construction technique needs to be proven. -BL, statecharts, Wise computing This goes with composition of systems. -Unlink (in a book on formal methods)Wing, we do not pick one, rather, we relate (multiple possibilities to one another?) them, like ISO standard communication stack and those defined things live in the space of abstractions, i.e., many different operational sequences covered by one invariant. -What impact does provability have on these? -structure in types data structures meets provability (recursive data structure, proof by induction) -can we represent data structures as finite automata? -Is there a reason why not to do this? e.g., would it not be sufficiently general? -representing datatypes with algebras Hoare 1985 Milner 1980 models, \cite{milner1978theory} with math, might have more properties that we want to use, vs. -theories, with which we create only our desired properties. -Z, VDM, emphasize models over theories, -Z VDM Larch algebraic specification language would we want to use Tempo instead of OBJ? -A formal method has an assertion language, for example, first order predicate logic. -A formal method has a specification language, wich in turn has a syntax and a semantics. -The syntax contains rules for formulating syntactically correct (legal) terms. - -Mitchell \cite[p. 35] (Foundation of Programming Languages) says Hilbert-style proofs system consists of axioms and proof rules. -axiom provable be definition\\ -a proof is a structured object build from formulae, according to constraints established by a set of axioms and rules of inference. - -Arcavi et al.\cite[p. 56]{arcavi1998teaching}''Work in cognitive science\cite{mcleod1992research} has shown that students' beliefs about the nature of a subject may have profound effects on their learning of it % (McLeod 1992) Research of affect in mathmatics education: A reconceptualization In Doublas A. Grouws (Ed.) Handbook of research on mathematics teaching and learning [pp. 575-596 NY Macmillan] - -One approach to teaching students to generalize is used by Schoenfeld (page 60 and earlier), who teaches his students heuristics, including one from Polya, about problems with fewer requirements +\section{Application of Findings} -Researchers have noticed that students, having learned procedures, have not necessarily learned to apply them. -Cipra 1995 UME Trends The bumpy road to reform -Culotta 1992 The calculus of education reform science 255 \ No newline at end of file +\section{ Perspective on Future Directions} \ No newline at end of file diff --git a/thesis2.pdf b/thesis2.pdf index 073fc50..c189222 100644 Binary files a/thesis2.pdf and b/thesis2.pdf differ diff --git a/thesis2.tex b/thesis2.tex index b9f46c8..1d656e4 100644 --- a/thesis2.tex +++ b/thesis2.tex @@ -39,14 +39,14 @@ Connecticut, USA, 2014} %%%---- Input content ----%%% \input{ch1.tex} %intro \input{ch2.tex} %research perspective and epistemological framework -\input{ch3.tex} %methodology including analysis -\input{ch4.tex} %results -\input{ch5.tex} %interpretation/discussion -\input{ch6.tex} %validity and reliability -\input{ch7.tex} %related work -\input{ch8.tex} %conclusions -\input{ch9.tex} %future work -%\input{ch10.tex} % +\input{ch3.tex} %methodology +\input{ch4.tex} % analysis +\input{ch5.tex} %results +\input{ch6.tex} %interpretation/discussion +\input{ch7.tex} %validity and reliability +\input{ch8.tex} %related work +\input{ch9.tex} %conclusions +\input{ch10.tex} %future work %\input{ch11.tex} % %\input{TODOs.tex} %... etc.