From 3fbcc4426a3d403a4a0eab71facfd2198b5c6cbc Mon Sep 17 00:00:00 2001 From: theresesmith Date: Sun, 18 Oct 2015 14:19:31 -0400 Subject: [PATCH] in analysis on what students think proofs are --- ch3.tex | 510 ++++++++++++++++++++++++++++++++++--------------- literature.bib | 7 + 2 files changed, 361 insertions(+), 156 deletions(-) diff --git a/ch3.tex b/ch3.tex index 6b60ac7..41b8015 100644 --- a/ch3.tex +++ b/ch3.tex @@ -9,133 +9,180 @@ be used, if in an anonymous, aggregate form. Both deductive and inductive analysis provide qualitative data. + + \section{Design of the Study} + + + Information learned in tutoring and lecturing undergraduates inspired the research questions. + + 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. + %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. + + 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. + + We used early interviews 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 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 + with different (specific) proof techniques: induction, construction, contradiction, + and what students think it takes to make an argument valid. + We used yet later interviews to discover whether students used proof techniques on their own, and how students ascertained whether circumstances were appropriate for the application of algorithms they knew, and how students ascertained certain properties of algorithms. + + %\chapter{Design of the Study} - This work is a qualitative study, the underlying philosophy is constructivist, - the research perspective is phenomenography, as extended to variation theory, - and the epistemological framework is a layered collection of intellectual disciplines. - At the highest level of integration, computer science and mathematics - reside, supported by studies in memory and attention, including computational - complexity applied to cognitive neuroscience, and neurophysiology. - 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. + %This work is a qualitative study, the underlying philosophy is constructivist, + %the research perspective is phenomenography, as extended to variation theory and structural relevance, + %and the epistemological framework is + %social constructivism, in particular, that mental preparation is influential, readying students to learn from instructors and peer interaction, some material better than other material. + + %a layered collection of intellectual disciplines, + %including +% complexity applied to cognitive neuroscience, and neurophysiology. + %At the highest level of integration, computer science and mathematics +% reside, supported by studies in memory and attention, including computational + %complexity applied to cognitive neuroscience, and neurophysiology. + % the focus is on determining what questions %would be posed, in the process of continuous curriculum adaptation and improvement %the meaning students are making of their specific educational experiences. + \section{Parts of the Study} + + The parts of the study reflect the several research questions. + The first part of the study was about what undergraduates think proof is, and how they go about understanding them. + 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. + The second part was about what undergraduates think proof is for. + The third part was about what students do when a situation might be well-addressed by proof. + 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. + + These parts are summarized in Table \ref{parts}. + + \begin{table} + \caption{Parts of the Study} + \begin{tabular}{|p{4cm}|p{8cm}|}\hline + Part & Purposes \\\hline \hline + first & what proof is\\ + & how to understand proofs\\ + & use of structure\\ + & how is validity attained\\\hline + second & purpose of proof\\\hline + third & student use of proof\\ + & comfort level with proof\\ + & consequences of not applying proof\\ \hline + \end{tabular} + \label{parts} + \end{table} + + + \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. - \section{Chronology of the Design} - The design of this study began while teaching Introduction to the Theory of - Computing. While helping students learn the pumping lemma for regular - languages, and trying to understand from where the several difficulties arose, - I became curious about the bases of these difficulties. One example was that - a student felt strongly that a variable, a letter, denoting repetitions in a mathematical - formulation, could only stand for a single numeric value, rather than a - domain. Subsequently I have learned that symbolization is a category identified - by Harel and Sowder \cite{harel1998students}, for students of mathematics learning proofs. Our - student is a vignette of our computer science student population harboring - some of the same conceptualizations. As a consequence of this opinion, the - student felt that showing that a mathematical formulation had a true value - was equivalent to demonstrating a true value for a single example, rather than - demonstrating a true value for a domain. Here we see evidence for the category - Harel and Sowder \cite{harel1998students} call (is it inductive, perceptual?) where an example - is thought to provide proof of a universal statement. Later, while helping students - study the relationship between context free grammars and pushdown - automata, I learned from the students that many of them did not find inductive - proofs convincing. Subsequently I have learned that Harel and Sowder \cite{harel1998students} created - a category called axiomatic reasoning. In axiomatic reasoning, students - begin with accepted information, such as axioms and premises, and apply rules - of inference to deduce the desired goal. This category had not always been - reached by their students, similarly to ours. As will be seen, later interview data showed, - some of our students learn to produce the artifact of a proof by mathematical - induction by procedure. They learn the parts, and they supply the parts - when asked, but are not themselves convinced. (McGowan and Tall report a similar situation.) This matches with two other - categories created by Harel and Sowder \cite{harel1998students}, internalization and interiorization. - Still later, when leading a course on ethical reasoning for issues related to computer - science, I found that most of the students did not notice that methods - of valid deductive argumentation were tools that they might apply to defend - their opinions. - Thus the idea of exploring the nature of the students' degrees of preparation - for understanding and creating proofs appeared. - First, interviews about proofs in general were conducted, with a broad interview - script. + For the benefit of readers wondering to what extent the results might be transferable, demographics of some commonly seen properties of populations are provided: + + Number of interviews transcribed:\\ + Number of interview subjects:\\ + Percentage by race:\\ + Percentage by sex:\\ + Percentage by age:\\ + Percentage by language:\\ + Percentage by domestic/international:\\ + +% \section{Chronology of the Design} +% The design of this study began while teaching Introduction to the Theory of +% Computing. While helping students learn the pumping lemma for regular +% languages, and trying to understand from where the several difficulties arose, +% I became curious about the bases of these difficulties. One example was that +% a student felt strongly that a variable, a letter, denoting repetitions in a mathematical +% formulation, could only stand for a single numeric value, rather than a +% domain. Subsequently I have learned that symbolization is a category identified +% by Harel and Sowder \cite{harel1998students}, for students of mathematics learning proofs. Our +% student is a vignette of our computer science student population harboring +% some of the same conceptualizations. As a consequence of this opinion, the +% student felt that showing that a mathematical formulation had a true value +% was equivalent to demonstrating a true value for a single example, rather than +% demonstrating a true value for a domain. Here we see evidence for the category +% Harel and Sowder \cite{harel1998students} call (is it inductive, perceptual?) where an example +% is thought to provide proof of a universal statement. Later, while helping students +% study the relationship between context free grammars and pushdown +% automata, I learned from the students that many of them did not find inductive +% proofs convincing. Subsequently I have learned that Harel and Sowder \cite{harel1998students} created +% a category called axiomatic reasoning. In axiomatic reasoning, students +% begin with accepted information, such as axioms and premises, and apply rules +% of inference to deduce the desired goal. This category had not always been +% reached by their students, similarly to ours. As will be seen, later interview data showed, +% some of our students learn to produce the artifact of a proof by mathematical +% induction by procedure. They learn the parts, and they supply the parts +% when asked, but are not themselves convinced. (McGowan and Tall report a similar situation.) This matches with two other +% categories created by Harel and Sowder \cite{harel1998students}, internalization and interiorization. +% Still later, when leading a course on ethical reasoning for issues related to computer +% science, I found that most of the students did not notice that methods +% of valid deductive argumentation were tools that they might apply to defend +% their opinions. +% Thus the idea of exploring the nature of the students' degrees of preparation +% for understanding and creating proofs appeared. +% First, interviews about proofs in general were conducted, with a broad interview +% script. %The students almost all selected proofs by mathematical induction. - During analysis of these data, a more elaborate interview script was developed, - aiming at the ideas of domain, range, relation, mapping, function, the ideas of - variable, as in programs and mathematical formulations, and abstraction. - Some students emphasized that mathematical definitions are analogous to - definitions in natural languages, and that mathematical discourse is carried - out in the mathematical language created by these definitions. - The capabilities for expression and care bestowed by these definitions invest - mathematical reasoning with its persuasive power. - Thus both the reasoning processes, using concepts and the clearly defined - mathematical concepts together provide the ability of mathematical argumentation - to be convincing. Students who appreciated this found it invigorating. - Other students had different reactions to definitions. Thus, the role of definitions - and language became another area of exploration. - The difference between a domain and a single point in a domain can be seen as - a level of abstraction. If something is true for a single point in a domain, but is - also true for every single point in the domain, then the point can be seen as a - generic particular point, representative of the domain. This concept of ability - to represent is related to the idea of abstraction. - We saw data in this study that affirmed the observations of others, that students - do not always easily recognize the possibility of abstraction. - - \section{Parts of the Study} - 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. - - - - \section{Design of the Study} - We - conducted over 30 interviews. - Our interview participants included undergraduate and graduate students of (\textbf{how do we want to say this}). 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. - - Information learned in tutoring and lecturing inspired the research questions. - We used exams to study errors in application of the pumping lemma for regular - languages. We used early interviews to explore proof, adapting to the - student preference for proof by mathematical induction and incorporating the use of recursive - algorithms. We used homework - to observe student attempts at proofs. - We used later interviews to investigate the remaining questions mentioned earlier. - \textbf{Need to make this true: We used homework to observe student familiarity/facility - with different (specific) proof techniques: induction, construction, contradiction, - and what students think it takes to make an argument valid.} - +% During analysis of these data, a more elaborate interview script was developed, +% aiming at the ideas of domain, range, relation, mapping, function, the ideas of +% variable, as in programs and mathematical formulations, and abstraction. +% Some students emphasized that mathematical definitions are analogous to +% definitions in natural languages, and that mathematical discourse is carried +% out in the mathematical language created by these definitions. +% The capabilities for expression and care bestowed by these definitions invest +% mathematical reasoning with its persuasive power. +% Thus both the reasoning processes, using concepts and the clearly defined +% mathematical concepts together provide the ability of mathematical argumentation +% to be convincing. Students who appreciated this found it invigorating. +% Other students had different reactions to definitions. Thus, the role of definitions +% and language became another area of exploration. +% The difference between a domain and a single point in a domain can be seen as +% a level of abstraction. If something is true for a single point in a domain, but is +% also true for every single point in the domain, then the point can be seen as a +% generic particular point, representative of the domain. This concept of ability +% to represent is related to the idea of abstraction. +% We saw data in this study that affirmed the observations of others, that students +% do not always easily recognize the possibility of abstraction. + + + \section{Sample Selection} - \textbf{Need even more detail than what's here. It could alternatively be put in subsections.} - Students from the University of Connecticut who have taken or are taking the - relevant courses were offered the opportunity to be interviewed. The students - who volunteered were mostly male, mostly traditionally aged undergraduates, - though some graduate students also volunteered. Some students were - domestic, and some international. Some students were African-American, - some Asian, some Caucasian, some Latino/a, some with learning disabilities - such as being diagnosed as on the autistic spectrum. - %\subsection{Proofs Using the Pumping Lemma for Regular Languages} + 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. + + 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. + +% 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 + who volunteered were mostly male, mostly traditionally aged students. Some students were domestic, and some international. + Some students were African-American, + some Asian, some Caucasian, some Latino/a, some with learning disabilities such as being diagnosed as on the autistic spectrum. + + \subsection{Proofs Using the Pumping Lemma for Regular Languages} + This study is in the first part, about what proof is. + The participants for the study of proofs using the pumping lemma for regular languages were - %In a recent course offering to forty-two students, of whom thirty-four were men and eight women, forty-one traditional aged, %one former Marine somewhat older, one collegiate athlete (a @@ -148,7 +195,10 @@ and 4 women. Three quarters of those (15/42) selecting the pumping lemma got it wrong. These students, who chose the pumping lemma problem and subsequently erred on it, form the population of our study. - %\subsection{Proofs by Mathematic Induction} + + \subsection{Proofs by Mathematic Induction} + This study is in the first part, about what proof is. + The participants for the study of proof by mathematic induction %We studied students who were taking, or @@ -162,38 +212,46 @@ included 2 women and 9 men. Two were international students, a third was a recent immigrant. - %\subsection{Domain, Range, Mapping, Relation, Function, Equivalence in Proofs} + + + \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 - %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 second part, about why proof appears in the curriculum. + + 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{Student Use of Proof for Applicability of Algorithms} + This study is in 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 - and real tests were analyzed for proof attempts. Data from individual tutoring - sessions and group help sessions was also informative. Aggregate, anonymous - data was used. + observations from individual tutoring sessions, and group help sessions. %Interview transcripts were analyzed with thematic analysis. + Homework, 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. + Aggregations of anonymous data were used. \subsection{Interviews} - \subsection{Documents} - \subsubsection{Proofs Using the Pumping Lemma for Regular Languages} - The study was carried out on the exam documents. The interpretation was informed - by remembering events that occurred in the natural conduct of lectures, - help sessions and tutoring. - One method of assessing whether students understood the ease of application - of the pumping lemma to a language to be proved not regular was offering a - choice between using the Myhill-Nerode theorem with a strong hint or using - the pumping lemma. The pumping lemma problem, which could very easily - have been solved by application of the Myhill-Nerode theorem, especially with - the supplied hint, was designed, when tackled with the pumping lemma, to - require, for each possible segmentation, a different value of $i$ (the number of - repetitions) that would create a string outside of the language. The intent was - to separate students who understood the meaning of the equation's symbols, - and the equation itself, from those students engaged in a manipulation with at - most superficial understanding. + An application to the Institutional Review Board was approved, for the conduct of the interviews. The protocol numbers include H13-065 and H14-112. + + %Consistent with phenomenographic studies, we wished to sample widely, so we sampled not only current students of courses involving proof, but also teaching assistants, faculty and former students associated with these courses. Some students are strictly CS/E majors, others are dual majors or minors in CS/E and math. Some students are not CS/E majors. Some former students are professionally employed in development, and others have left the major. + +% All interview participants were volunteers. + +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} Interviews were solicited in class by general announcement, and by email. Interviews were conducted in person, using a voice recorder. No further @@ -219,45 +277,148 @@ Almost every student introduced and described proof by mathematic induction as experienced in their current or recent class. - \section{Expanded semi-structured interview protocol for domain, range, language, equivalence class in Proofs} - \section{Expanded semi-structured interview protocol for definitions, language, reasoning in Proofs} - \section{Analysis} + \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 + 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 + of the pumping lemma to a language to be proved not regular was offering a + choice between using the Myhill-Nerode theorem with a strong hint or using + the pumping lemma. The pumping lemma problem, which could very easily + have been solved by application of the Myhill-Nerode theorem, especially with + the supplied hint, was designed, when tackled with the pumping lemma, to + require, for each possible segmentation, a different value of $i$ (the number of + repetitions) that would create a string outside of the language. The intent was + to separate students who understood the meaning of the equation's symbols, + 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? - 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: - Apply the principle of focusing on one aspect of the object and seeking its dimension of variation while holding other aspects frozen. %partial derivative - Remember to apply both perspectives, that pertaining to the individual and that pertaining to the collective. - Establish a perspective with boundaries, within which looking for variation. + \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 seartch for extracts from data, that might pertain to perspective + \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} - Keep going, clarity will emerge. + 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}. - Result: 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. - assume that what people say is logical from their point of view\cite[p. 134]{marton1997learning}, citing Smedlund\cite{smedslund1970circular} + 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. + + \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} + \end{quote}Probably I don't want to keep this, but it's fun at the moment. + + 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". + + 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". + + 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{harel} offered extrinsic and intrinsic conviction, and their most advanced definitional/transformational class as broader categories, and (how many?) useful subcategories of these, yielding (how many?) critical aspects that suggest what teachers could usefully vary, to help learners discern items that would advance their knowledge. + + + + \subsubsection{Phenomenographic Analysis of How Students Attempt to Understand Proofs} + + \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{table} + \begin{tabular}{|p{6cm}|p{9cm}|}\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.\\\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 +\end{tabular} + \end{table} + + + 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 @@ -326,7 +487,19 @@ [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") - \section{Analysis of Interviews} +\subsubsection{Phenomenographic Analysis of How Students Attempt to Apply Proofs (When not assigned)} + + +\subsubsection{Phenomenographic Analysis of Whether students exhibit consequences of inability (such as avoiding recursion)} + +\subsubsection{Phenomenographic Analysis of Whether students notice structural elements in proofs} + + + • How familiar and/or comfortable are students with different (specific) proof + techniques: induction, construction, contradiction? + • What do students think it takes to make an argument valid? + + \subsection{Analysis of Interviews} 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 @@ -360,7 +533,7 @@ capture the meaning. The products of the analysis were the narrative and the diagram. - \section{Interview} +% \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. @@ -372,7 +545,7 @@ In interviews, the students almost all chose to discuss proofs by mathematical induction. - \subsection{Themes / Categories} + \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. @@ -392,16 +565,16 @@ \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 students recognized they were undertaking proofs. \end{itemize} - \subsection{Relationships} + \subsubsection{Relationships} - \section{Analysis of Homework and Tests} - \subsection{Proofs} + \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. - \subsection{Pumping Lemmas} + \subsubsection{Pumping Lemmas} We wrote descriptions for each error. Some example descriptions are in Table II. @@ -443,13 +616,38 @@ - \section{Help Session and Tutoring} + \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 diff --git a/literature.bib b/literature.bib index db7f2b5..d7271ed 100644 --- a/literature.bib +++ b/literature.bib @@ -1,3 +1,10 @@ +@book{ellenberg2014not, + title={How not to be wrong: The power of mathematical thinking}, + author={Ellenberg, Jordan}, + year={2014}, + publisher={Penguin} +} + @book{merriam2009qualitative, title={Qualitative research: A guide to design and implementation}, author={Merriam, Sharan B},