diff --git a/ch3.tex b/ch3.tex index 48ba390..80904af 100644 --- a/ch3.tex +++ b/ch3.tex @@ -64,26 +64,29 @@ \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. + The parts of the study are organized taking inspiration from Bloom's Taxonomy of Cognitive Domain.\cite{bloom1956taxonomy} + The first part of the study was about recognition: what undergraduates think proof is, and comprehension: how they go about understanding them, and about structural relevance (part of phenomenography\cite{marton1997learning}: why students think proof is taught. + 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. - 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. +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 + \begin{tabular}{|p{4cm}|p{10cm}|}\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 + recognition, & what proof is\\ + comprehension and& how students approach understanding proofs\\ + structural relevance & purpose of teaching proof\\\hline + application & how students apply proofs they have been taught\\ \hline + analysis, & use of structure\\ + & how is validity attained\\ + synthesis and & comfort level with proof\\ + & student use of proof\\ + & consequences of not applying proof\\ + evaluation &\\ \hline \end{tabular} \label{parts} \end{table} @@ -179,25 +182,9 @@ 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 - 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 - %woman), - there were three students having Latin-heritage surnames, 1/4 of the - students had Asian heritage, 2 had African heritage, and 8 were international - students. Each student individually took the final exam. A choice among - five questions was part of the final exam; one required applying the pumping - lemma. Half the students (21/42) selected this problem. These were 17 men - 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} - This study is in the first part, about what proof is. + 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. The participants for the study of proof by mathematic induction %We studied students who @@ -223,10 +210,27 @@ to be interviewed. \subsection{Purpose of Proof} - This study is in the second part, about why proof appears in the curriculum. + This study is in the first 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{Proofs Using the Pumping Lemma for Regular Languages} + This study is in the second part, 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, + forty-one traditional aged, + %one former Marine somewhat older, one collegiate athlete (a + %woman), + there were three students having Latin-heritage surnames, 1/4 of the + students had Asian heritage, 2 had African heritage, and 8 were international + students. Each student individually took the final exam. A choice among + five questions was part of the final exam; one required applying the pumping + lemma. Half the students (21/42) selected this problem. These were 17 men + 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{Student Use of Proof for Applicability of Algorithms} This study is in the third part, about student use of proof. @@ -359,13 +363,19 @@ The audio portion of all interviews was collected by electronic recorder and sub 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} - \end{quote}Probably I don't want to keep this, but it's fun at the moment. - 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. + 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. @@ -374,13 +384,18 @@ The audio portion of all interviews was collected by electronic recorder and sub 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". + 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{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. + 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. @@ -404,7 +419,7 @@ The audio portion of all interviews was collected by electronic recorder and sub 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. + 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. @@ -412,8 +427,12 @@ The audio portion of all interviews was collected by electronic recorder and sub 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. @@ -536,21 +555,65 @@ The audio portion of all interviews was collected by electronic recorder and sub 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") + [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} + + 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 notice structural elements in proofs} +\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. - • 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? + + +% %evaluation - \subsection{Analysis of Interviews} +\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 @@ -613,7 +676,8 @@ The audio portion of all interviews was collected by electronic recorder and sub 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 students recognized they were undertaking proofs. + \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} diff --git a/chapter3StandAlone.pdf b/chapter3StandAlone.pdf index 02a8fa5..e259899 100644 Binary files a/chapter3StandAlone.pdf and b/chapter3StandAlone.pdf differ diff --git a/thesis2.pdf b/thesis2.pdf index 1a7bca9..073fc50 100644 Binary files a/thesis2.pdf and b/thesis2.pdf differ