From 0aa1a6533d18cf2ebdbb2b62ebceb72cf35ec50d Mon Sep 17 00:00:00 2001 From: theresesmith Date: Sat, 14 Nov 2015 15:50:22 -0500 Subject: [PATCH] analysis --- ch3.tex | 37 ++++++++- ch4.tex | 206 +++++++++++++++++++++++++++++++++++++++++++++---- ch5.tex | 4 + literature.bib | 9 +++ 4 files changed, 239 insertions(+), 17 deletions(-) diff --git a/ch3.tex b/ch3.tex index 9cfe886..00667cc 100644 --- a/ch3.tex +++ b/ch3.tex @@ -363,7 +363,8 @@ The audio portion of all interviews was collected by electronic recorder and sub \hline \hline \endlastfoot - % \begin{tabular}{|p{7cm}|p{8.5cm}|}\hline + % \begin{tabular}{|p{7cm}|p{8.5cm}|} + \hline Category & Representative\\\hline\hline @@ -531,9 +532,25 @@ The audio portion of all interviews was collected by electronic recorder and sub \subsection{Analysis of Help Session and Tutoring} + + Help sessions were scheduled weekly; attendance was optional. Typically six to twelve students would participate. + Originally these were called help sessions, but the demographics of the attendees did not represent the enrolled students. + Subsequently the name was changed to consultation sessions. + This change had the desired effect, that the population attending better reflected the enrolled students. + At these sessions, students would raise topics about which they had questions. + Frequently the student would be requested to work at the white board, and leading questions were asked, and problems of very small size were posed, to urge the student along the right path of development of a solution. + Occasionally these suggested paths were met with resistance from the students, which is to say, misunderstandings were encountered and discussed. + Ideas mentioned in these discussions that were relevant to manuscripts in process at the time were noted, anonymized, into the manuscripts. + + The idea that a variable used in the pumping lemma could take on only one value, as if it were a single root of linear equation, rather than representing the domain of values possible for strings in a language, was proposed by a student in a help session. The context was a student proposition that examples were sufficient for proofs of universal statements, because the the variable in the pumping lemma could take on only a single value. + some students, who do know that any statement must and can, be either true or false, thought implications must be true. + Due to attention being focused on interacting with students in the normal course of teaching, these field notes are incomplete. + + One use of such data is that they can give evidence that categories of conceptualization of proof already created in the mathematics literature can be found also in computer science students. This is similar to a deductive rather than inductive process, in that we are aware of the categories created by Harel and Sowder\cite{harel1998students} and by Tall\cite{Tall?} and student utterances that seem well matched to those categories draw our attention to those categories, validating them for students of computer science. + % The study proceeded using prior interview experiences to suggest further investigation. @@ -572,14 +589,26 @@ The audio portion of all interviews was collected by electronic recorder and sub 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 Addressing Validation} + Triangulation is a technique for increasing the confidence that the results of analysis are reliable. + + In this study we applied triangulation in several ways. + We interviewed faculty teaching the courses involving proofs. We interviewed TAs assisting in the courses involving proofs. The students in these courses are from our same population. To get an idea of the background preparation of our students, we substitute taught geometry and algebra II classes in a high school. The high school population was quite similar to our university population, but differed by consisting almost entirely of domestic students, studying in their first language, and by having a larger percentage of women students, and of declared transgender students. Though the community served by this high school is diverse over socio-economic status, this component of diversity is probably greater in our university population. + + Consistency with the work of other researchers is a check on the validity of an analysis. + In this study we compared our results with those achieved by some other researchers in computer science education and also by some researchers in the mathematics education community. + Checking possible interpretations is a technique that may aid in increasing confidence in validity. + + We prepared a list of questions that was addressed by several faculty and several students, that began an examination of the role of specific representation styles (mathematical notation, figures and pseudocode) for proof related problem statements. \section{Method of Presentation of Results} - The product of analysis in a phenomenographic study, is a set of categories, and relationships \textit{among} them. + The product of analysis in a phenomenographic study is a set of categories, and relationships \textit{among} them. These categories and relationships are often depicted in a graph. This product may be accompanied by a "thick and rich" narrative description of the categories and relationships. This narrative must be consistent with the individual text fragments, excerpts from transcriptions, field notes or documents obtained for the study. + + 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." - 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 + This tells us that the narrative should describe the categories of composition hierarchies found in the students' understandings. The faces or facets of the learning object have their importance and relationships as envisioned by the teacher. The students' conceptualizations may be less complete, contain superfluous items, and differ as to the relationships of the parts, especially by lacking profundity in understanding of relationships. \ No newline at end of file diff --git a/ch4.tex b/ch4.tex index d1e49b6..8c0605f 100644 --- a/ch4.tex +++ b/ch4.tex @@ -9,7 +9,7 @@ \subsection{Application of Phenomenographic Analysis in this Study} - We applied phenomenographic analysis, + We applied phenomenographic analysis to transcripts, field notes and documents. We addressed several research questions. The analyses are organized herein by the question addressed. 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. @@ -19,30 +19,170 @@ \subsubsection{Phenomenographic Analysis of What Students Think Proofs Are} + Carnap writes eloquently on proof, a subset of logical deduction: \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} + The essential character of logical deduction, i.e. concluding from a sentence $\mathfrak{S}_i$ a sentence $\mathfrak{S}_j$ that is L-implied by it, consists in the fact that the content of $\mathfrak{S}_j$ is contained in the content of $\mathfrak{S}_i$ (because the range of $\mathfrak{S}_i$ is contained in that of $\mathfrak{S}_j$). We see thereby that logical deduction can never provide us with new knowledge about the world. In every deduction the range either enlarges or remains the same, which is to say the content either diminishes or remains the case \textit{Content can never be increased by a purely logical procedure.} + To gain factual knowledge, therefore, a non-logical procedure is always necessary. \ldots Though logic cannot lead us to anything new in the logical sense, it may well lead to something new in the psychological sense. Because of limitations on man's psychological abilities, the discovery of a sentence that is L-true or of a relation of L-implication is often an important cognition. - Probably I don't want to keep this, but it's fun at the moment. + But this cognition is not a factual one, and is not an insight into the state of the world; rather it is a clarification of logical relations subsisting between concepts., i.e. a clarification of relations between meanings. Suppose someone knows $\mathfrak{S}_i$ to begin with; and suppose that thereafter, by a laborious logical procedure, he finds that $\mathfrak{S}_j$ is L-implied by $\mathfrak{S}_i$. Our subject may now regard $\mathfrak{S}_j$ as known, but he may not count it as logically new: for the content of $\mathfrak{S}_j$, even though initially concealed, was from the beginning part of the content of $\mathfrak{S}_i$. Thus logical procedure, by disclosing $\mathfrak{S}_j$ and making it known, enables practical activities to be based on $\mathfrak{S}_j$. Again, two L-equivalent sentences have the same range and hence the same content; consequently they are different formulations of this common logical content. However, the psychological content (the totality of associations)of one of these sentences may be entirely different from that of the other. Carnap, \cite[p. 21-22]{carnap1958introduction} \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. + Moving to 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. + 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. Those we found 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". + The source of such sequences served as a dimension of variation among the concepts we found in our students. Some students stressed the role of a procedure in synthesizing proofs. Proof by mathematic induction was considered preferable; a synthesis by procedure property was assigned to it. By contrast, proofs involving sequences of statements warranted by rules of inference, but otherwise unconstrained as to form, were considered less desirable. + + 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". + + Some students recognized patterns in sequences of statements. Contrapositive, contradiction, categorization into cases, proof by mathematic induction have been seen as patterns, consisting of steps that can be followed. These are contrasted with what were called "logic proofs". + + It could be difficult to distinguish between a correct succession of logical steps from the premise(s) to the desired consequence(s) that "reaches a psychologically useful revised formulation" from "carries out a pattern". Indeed, the objectives of the course teaching proof may be met, while the preparation for the course using proof to explain the nature of, say, complexity classes, might not. + + + 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. + \paragraph{Codes} - + \begin{longtable}{|p{7cm}|p{8.5cm}|}\hline + \caption{Phenomenographic Analysis of What Students Think Proofs Are}\label{exemplar} + \endfirsthead + + \multicolumn{2}{c}% + {{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\ + \hline \multicolumn{1}{|c|}{\textbf{Code}} & + \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 + Code & Representative\\\hline\hline + + Abstraction, Logical Abstraction &\\ + Comprehending and Applying & \\ + Connecting Recursion and Induction & \\ + Construct Using Patterns & \\ + Context for Use &\\ + Definition &\\ + Evaluating Proofs &\\ + Generalization from instances &\\ + Learning proof by induction &\\ + Logic &\\ + Logical progression, warrant &\\ + Mathematical formulation &\\ + Proof and programming &\\ + Proof is logical steps &\\ + Proof is magical incantation &\\ + Proof is validation &\\ + Proof relies on definitions &\\ + Quantifiers &\\ + Representations &\\ + Structure &\\ + Two too fast, relation or confusion &\\ + + + + \end{longtable} + \paragraph{Preliminary Categories} + \begin{longtable}{|p{7cm}|p{8.5cm}|}\hline + \caption{Phenomenographic Analysis of What Students Think Proofs Are}\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 + \end{longtable} + \paragraph{Main theme and its relationships to minor themes} + \begin{longtable}{|p{7cm}|p{8.5cm}|}\hline + \caption{Phenomenographic Analysis of What Students Think Proofs Are}\label{exemplar} + \endfirsthead + + \multicolumn{2}{c}% + {{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\ + \hline \multicolumn{1}{|c|}{\textbf{Theme}} & + \multicolumn{1}{c|}{\textbf{Relation to Main Theme}} \\ \hline + \endhead + + \hline \multicolumn{2}{|r|}{{Continued on next page}} \\ \hline + \endfoot + + \hline \hline + \endlastfoot + + % \begin{tabular}{|p{7cm}|p{8.5cm}|} + \hline + Theme & Relation to Main Theme\\\hline\hline + \end{longtable} + \paragraph{Categories and Relationships} + \begin{longtable}{|p{7cm}|p{8.5cm}|}\hline + \caption{Phenomenographic Analysis of What Students Think Proofs Are}\label{exemplar} + \endfirsthead + + \multicolumn{2}{c}% + {{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\ + \hline \multicolumn{1}{|c|}{\textbf{Theme}} & + \multicolumn{1}{c|}{\textbf{Relation to Main Theme}} \\ \hline + \endhead + + \hline \multicolumn{2}{|r|}{{Continued on next page}} \\ \hline + \endfoot + + \hline \hline + \endlastfoot + + % \begin{tabular}{|p{7cm}|p{8.5cm}|} + \hline + Theme & Relation to Main Theme\\\hline\hline + \end{longtable} + \paragraph{Dimension of Variation and Critical Factors} +\begin{longtable}{|p{7cm}|p{8.5cm}|}\hline + \caption{Phenomenographic Analysis of What Students Think Proofs Are}\label{exemplar} + \endfirsthead + + \multicolumn{2}{c}% + {{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\ + \hline \multicolumn{1}{|c|}{\textbf{Dimension of Variation}} & + \multicolumn{1}{c|}{\textbf{Critical Factor}} \\ \hline + \endhead + + \hline \multicolumn{2}{|r|}{{Continued on next page}} \\ \hline + \endfoot + + \hline \hline + \endlastfoot + + % \begin{tabular}{|p{7cm}|p{8.5cm}|} + \hline + Dimension of Variation & Critical Factor\\\hline\hline + \end{longtable} % % %comprehension @@ -83,6 +223,12 @@ % % % structural relevance + \paragraph{Codes} + \paragraph{Preliminary Categories} + \paragraph{Main theme and its relationships to minor themes} + \paragraph{Categories and Relationships} + \paragraph{Dimension of Variation and Critical Factors} + \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. @@ -139,7 +285,11 @@ - + \paragraph{Codes} + \paragraph{Preliminary Categories} + \paragraph{Main theme and its relationships to minor themes} + \paragraph{Categories and Relationships} + \paragraph{Dimension of Variation and Critical Factors} % % %application @@ -149,23 +299,37 @@ Some students claimed they never constructed proofs when not assigned. - + \paragraph{Codes} + \paragraph{Preliminary Categories} + \paragraph{Main theme and its relationships to minor themes} + \paragraph{Categories and Relationships} + \paragraph{Dimension of Variation and Critical Factors} \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 - + \paragraph{Codes} + \paragraph{Preliminary Categories} + \paragraph{Main theme and its relationships to minor themes} + \paragraph{Categories and Relationships} + \paragraph{Dimension of Variation and Critical Factors} \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." + Students claimed to prefer proofs by mathematical induction on the basis that they were formulaic; supposedly, a procedure could be used to synthesize them. + 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. - + \paragraph{Codes} + \paragraph{Preliminary Categories} + \paragraph{Main theme and its relationships to minor themes} + \paragraph{Categories and Relationships} + \paragraph{Dimension of Variation and Critical Factors} \subsubsection{Phenomenographic Analysis of Whether students notice structural elements in proofs} @@ -175,6 +339,12 @@ Some students claimed to know how to write recursive algorithms but said they ne Some students describe proofs as a sequence of statements, not commenting on any structure. + \paragraph{Codes} + \paragraph{Preliminary Categories} + \paragraph{Main theme and its relationships to minor themes} + \paragraph{Categories and Relationships} + \paragraph{Dimension of Variation and Critical Factors} + \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. @@ -184,6 +354,11 @@ Some students claimed to know how to write recursive algorithms but said they ne Some students do not notice that proof by contradiction introduces (for purposes of contradiction) a premise. % % %synthesis + \paragraph{Codes} + \paragraph{Preliminary Categories} + \paragraph{Main theme and its relationships to minor themes} + \paragraph{Categories and Relationships} + \paragraph{Dimension of Variation and Critical Factors} \subsubsection{Phenomenographic Analysis of Whether students incorporate structural elements in proofs} @@ -206,8 +381,13 @@ Students have asked whether, when using categorization into cases, they must app 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. + \paragraph{Codes} + \paragraph{Preliminary Categories} + \paragraph{Main theme and its relationships to minor themes} + \paragraph{Categories and Relationships} + \paragraph{Dimension of Variation and Critical Factors} - \subsubsection{Themes / Categories} + \subsubsection{Combined 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. @@ -228,7 +408,7 @@ Students have asked whether, when using categorization into cases, they must app \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} + \subsubsection{Combined Relationships} \subsection{Analysis of Homework and Tests} \subsubsection{Proofs} diff --git a/ch5.tex b/ch5.tex index 3ee5c92..d1b70c9 100644 --- a/ch5.tex +++ b/ch5.tex @@ -6,6 +6,8 @@ Booth states\cite{booth2001learning} "The results are communicated as descriptions of the essential aspects of each category, illustrated by pertinent extracts from the data". +\section{Criteria for Results} + 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. @@ -199,6 +201,8 @@ Some students, and some instructors, do not emphasize that a single presentation There are a couple of ways students work with exercises in proof, that are incomplete. +One of the groups of conceptualizations, which is consistent with that observed by Harel and Sowder\cite{harel1998students} and others (OK, get some more citations), it that a proof-attempt can be the result of filling in a template, and it can earn full points, without providing assurance to the person who enacted the procedure and filled in the template. Some students receive full credit, and some receive partial credit without explanation. Some students do not further pursue either situation (full credit without internal conviction, or partial credit) because they say they do not find the time; those who do not know why proof is taught in the curriculum might assign a low priority to following up these situations. + 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. diff --git a/literature.bib b/literature.bib index d7271ed..40cec54 100644 --- a/literature.bib +++ b/literature.bib @@ -1414,6 +1414,15 @@ year={1987} publisher={LWW} } % % % % % % % % % % % % % % % % % % % % % % % % % +% 1958 +% % % % % % % % % % % % % % % % % % % % % % % % +@book{carnap1958introduction, + title={Introduction to symbolic logic and its applications}, + author={Carnap, Rudolf}, + year={1958}, + publisher={Dover Books} +} +% % % % % % % % % % % % % % % % % % % % % % % % % % 1952 % % % % % % % % % % % % % % % % % % % % % % % % @book{piaget1952origins,