From 22076a292995d3b7c4924413daa751ed06d0dda8 Mon Sep 17 00:00:00 2001 From: theresesmith Date: Sat, 14 Nov 2015 21:13:56 -0500 Subject: [PATCH] more analysis, is coding --- ch4.tex | 940 +++++++++++++++++++++++++++++++++++++++++++++++++++++--- ch6.tex | 74 +++++ 2 files changed, 975 insertions(+), 39 deletions(-) diff --git a/ch4.tex b/ch4.tex index 8c0605f..5c41c16 100644 --- a/ch4.tex +++ b/ch4.tex @@ -11,7 +11,7 @@ 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. + 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} @@ -24,13 +24,16 @@ 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. - 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} + 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} + We take this understanding of proof as complete, for the purpose of comparison with student conceptualizations, which we expect, from Marton\cite{marton1976aqualitative}(is this a suitable ref?), to be partial rather than complete, and superficial, rather than deeply appreciative of the relations among the parts. + 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 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. + "leaving it at the formal definition is kind of aaahh, I kind of work backwards with those, like I get an example, then ok this relates to this step that's what this means" 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. @@ -80,6 +83,7 @@ Construct Using Patterns & \\ Context for Use &\\ Definition &\\ + Difficulty with Mathematical Formulation & they are using many letters for base cases, k, k+1, let's say, and then they are using different letters, t for t and then for k+1 then t and k+1, so, it shows you they don't understand\\ Evaluating Proofs &\\ Generalization from instances &\\ Learning proof by induction &\\ @@ -188,6 +192,16 @@ % % %comprehension \subsubsection{Phenomenographic Analysis of How Students Attempt to Understand Proofs} + + It could be that some students are not attempting to understand proofs. + "part of that is that there are kids in computer science who don't really want to be in CS, do the bare minimum or whatever, so i think that's part of the problem, not going to get kids who want to do proofs in cs, if they don't really want to do cs" + + "i will pay attention where most kids will zone out" + + Students can experience anxiety about mathematical notation. +" the second I see a summation, I'm like oh god this is some really long thing, or my professor's going to ask me to put in you know ask me to find the equation for this summation and it's just 'cause we did a I think it was calc2 um they had us do they gave us sequences and series and summations and they're like write the eqn for this and they were awful, just and summations are just weird because you're writing out this really long thing, used them in uh, some of the stuff we used in like 2500 for one of them we had to write out the actual summation i don't know but when you look at that little squiggly (o god)$*6$. + + Some students are attempting to understand proofs while not recognizing that they are studying a proof. "Were we studying proofs today?" "No" "Were we discussing certain contexts, and why certain ideas will always be true in those contexts?" "Yes" "Doesn't that seem like proof, then?" "Yes"\\ "So, you're taking introduction to the theory of computing this semester. Do you seen any use of proofs in that course?" "No"\\ @@ -195,9 +209,12 @@ Some students read proofs. + "you just have to go through a variety of proofs a variety of contexts" + Some students look up the definitions of terms used in the proofs and some do not. Some students think (or hope) they can solve problems involving producing proofs, without knowing the definitions of the terms used to pose the problem. - Some students are aware that definitions are given, but "zone out" until examples are given. When examples are given, the students attempt to infer definitions themselves. Some students will compare the definitions they infer with the mathematical community's definitions. Some students do not. + Some students are aware that definitions are given, but "zone out" until examples are given. + "she'll read through the formal definition, half the class will kind of zone out, which is fine, everybody has to understand there's a formal definition but going over it personally I would use a lot of examples, i love examples" When examples are given, the students attempt to infer definitions themselves. Some students will compare the definitions they infer with the mathematical community's definitions. Some students do not. Some students think that the reading of proofs is normally conducted at the same speed as other reading, such as informal sources of information. @@ -209,13 +226,18 @@ 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. + "i think the thing a lot of people hadn't really had to deal with before was just the level of abstraction that comes with proof writing, which is inherent with computer science, but a lot of time when we talked about problems it's always through analogies, i mean the traveling salesman problem is about cities and moving but that's not really what it's about, it's about graphs and paths" + Some students who achieve with difficulty the ability to recognize the application of rules of inference in one argument about abstract entities, become quicker at recognizing arguments of similar form about other abstract entities, and some continue to achieve with difficulty, as if learning the first argument did not facilitate learning the second argument of the same form. Students would attend to diagrammatic representation of proofs, such as a block digram depicting machine descriptions packaged as input for yet other machines to process, but were not observed to employ such diagrams. Students have been seen to employ decision tree diagrams. - Students would attend to algorithm representation of proofs, such as a recursive process that determines a prime factorization, but were not observed to employ such algorithmic descriptions. + Students 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." I wonder whether if proofs like mathematical formulations, could be rewritten as algorithms would the computer science students find them more readily understood." + "absolutely" + + "that makes a little more sense than some of the assertions, the equalities, an algorithm you can trace through, you can write it out, things like that, it's very beneficial" 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". @@ -224,10 +246,110 @@ % % % structural relevance \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{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} \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{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} \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{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} \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{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} \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} \subsubsection{Phenomenographic Analysis of Reasons for Teaching Proof} @@ -258,10 +380,14 @@ & we didn't see ok why do i really have to know the proof of the theorem to do that right? We didn't see the point, because no one taught us the point, so, that's a very important part that was missing.\\\hline + + &just learning math and not learning where to apply, you don't really appreciate 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 see how proofs are applied to algorithms & we're going over graphs from a mathematical and you know theoretical i guess perspective in 2500 and then in 2100 we're going over them in a practical like usage in terms of like solving a maze is what we're going to do with them, so it was really cool when we started doing them in 2100 seemed like ``I know these, I already learned how to do this''\\\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 @@ -279,17 +405,121 @@ & I know that recursion has the same structure as proof by mathematical induction. \ldots If I had an algorithm with a recursive data structure like a tree, and I had to prove something like termination about it, I'm not sure what approach I would use, it would depend.\\\hline - Some students see that they could employ proof to explore whether an algorithm can be expected to solve a problem in a given context that includes bounds upon resources that are available for consumption. & mostly design the algorithm first, we had some expectation of what that complexity results would be and then we try to find an approach to prove. + Some students see that they could employ proof to explore whether an algorithm can be expected to solve a problem in a given context that includes bounds upon resources that are available for consumption. & mostly design the algorithm first, we had some expectation of what that complexity results would be and then we try to find an approach to prove.\\\hline + & i think i mean proofs are about um you know building the next layer of truth kind of showing what you could do or compute um so in computer science for example uh by proving something we're just showing that its possible um we're kind of setting bounds on what we can do and can't do\\\hline + & it might not be the highest lower bound, but it's the highest lower bound we know of, that we can prove, yet the best algorithm we have been able to find takes much more resources, time, memory, messages, then those proofs show us a window\\\hline %\end{tabular} + +Some students find that creating proofs related to algorithms can be more difficult than problems they practiced on when learning & i have to prove that uh if there is an edge um connecting two points and there is a path connects two points with edges with all weights less than this edge have to show that that edge wouldn't be in any minimum spanning tree, a lot of things going on there, not like a theorem with a couple simple assumptions and you have to show result, you know you have to show there are multiple minimum spanning trees possibly things like that it's not as uh i mean the way proofs come up isn't as straightforward i find, makes a little bit confusing sometimes,\\\hline \end{longtable} - \paragraph{Codes} + \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{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} \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{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} \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{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} \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{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} \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} % % %application @@ -299,54 +529,486 @@ 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} + \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{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} + \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{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} + \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{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} + \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{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} + \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} \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} + \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{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} + \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{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} + \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{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} + \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{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} + \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} \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." + "I'm not particularly fond of them \ldots there are different ways of strong and weak induction a whole procedure and try and so yeah there's a lot of details that go into it" + + "i'm probably not going to go home and do mathematical induction practice for fun" + + "i think like mathematical induction um uh because they verify themselves, like you use them to check your answer, and you know if you've arrived at the expected answer most likely did it right. I generally like to do any kind of proof that has either like a given like set of steps like mathematical induction has a set of steps where you have your eventual condition, you verify your um you verify and then you do the inductive step and then you can conclude" + + "I'm a fan of like having like a set of steps to do something with, rather than so i know like what to do next" + + "because i think the thing that draws people to coding is problem solving, the feeling of achievement when something works, i mean, and so why couldn't the same thing be a part of proofs, I guess, I can see it in that way" + + "it's always good too because if you're even it's really reassuring when you're expecting to find something and then you find that and find out why in the process, so you have like you know what you're looking for so it's almost like working backwards, i know what i'm looking for and i know where i'm starting but if i can work both ways i can find the path pretty easily" + + When asked about specific proof techniques, some students mentioned proof by mathematical induction "when I start to create a proof, most commonly mathematical induction because this method of proof seems most straightforward to me, and most of these assignments we did mathematical induction, so that's what goes through my mind first." Students claimed to prefer proofs by mathematical induction on the basis that they were formulaic; supposedly, a procedure could be used to synthesize them. + "the thing that that induction is there are steps to it, you prove for this case, you prove for that case, plus one, I can go through those steps and by going through the steps I'm sure it's correct, because it's the right steps but in my mind it's a little shaky" + + + "this feeling of well, I did it, but I'm not necessarily convinced" + "yes, that's precisely the feeling I've had" 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. +"laws of logic proofs, they're a little more difficult it's almost like a puzzle" + +"even if it's almost like in straight programming, there's a few ways to choose from, you know how to solve a certain problem, if you can choose between like one or two or three you know different steps that's fine to but when it's kind of like solve this problem here's the formal definition go for it i'm like whoa" + 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} + \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{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} + \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{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} + \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{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} + \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{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} + \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} \subsubsection{Phenomenographic Analysis of Whether students notice structural elements in proofs} + Maybe for an ideal, get something from Leslie Lamport's description of using structure. + 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. - \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} + Some students appreciate structure: "what i had to start doing with my physics problems was breaking them down into i have this chunk, i have this chunk, i have this chunk, i'm going to label and use this chunk, i'm going to label and use this chunk and then i'm going to see how they all fit together" + + \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{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} + \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{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} + \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{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} + \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{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} + \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} \subsubsection{Phenomenographic Analysis of What do students think it takes to make an argument valid?} + Some students are not sure how to construct an argument. + "when we hit 2100 and it was no longer like write this method, write this statement, you would then have to do this, do this, it was just a paragraph, write a stock trader that will handle this input and output this output, i panicked, i had no idea, i didn't even know, we learned how to code, but we didn't learn, we learned how to write code but we didn't learn how to code, the same we learned proofs, but we didn't learn how to write proofs, the only place we saw that was 2500 it helped close a gap for me that i, i'm still not perfect at it, it definitely helped to bring me along which was good" + + Some students do not understand that statements should be warranted. Without an appreciation of definitions, understanding warrants can be expected to be fraught with difficulty. "that's where you'll find most of your problems understanding proofs, that's the first really the first class where you get it, where it's not just you know, this is the proof, take it on faith, that's what they tell you to do, but that's where you should really see why this works, let's see how this is proven, understand this as a whole, once you get it, that's where the gap is where everyone kind of loses it". + + Some students do not recognize a good argument when they are looking at one. "you can't expect it to be totally rigorous in decidability" + 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. @@ -354,11 +1016,111 @@ 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} + \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{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} + \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{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} + \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{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} + \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{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} + \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} \subsubsection{Phenomenographic Analysis of Whether students incorporate structural elements in proofs} @@ -366,11 +1128,11 @@ Students have asked whether, when using categorization into cases, they must app Probably needs additional interviews. - +and they you split, what do i have to do to get to that point, so you have to actually find what are the required pieces for you to solve the problem so and every single piece, they you have to prove by itself, so that can i get to the second step, can i get to the third step, because if i lose the first proof, i will never get to the second, because i already established that my second piece depends upon my first piece, so i cannot move forward so i have to divide into small pieces and try to prove them % %evaluation - % \section{Interview} + \section{Does this go anywhere? 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. @@ -381,11 +1143,111 @@ 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} + \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{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} + \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{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} + \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{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} + \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{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} + \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} \subsubsection{Combined Themes / Categories} \begin{itemize} diff --git a/ch6.tex b/ch6.tex index 39fcc33..4a29882 100644 --- a/ch6.tex +++ b/ch6.tex @@ -1,6 +1,80 @@ \chapter{Interpretation/Discussion} \section{Interpretation} + +\subsubsection{Interpretation of What Students Think Proofs Are} +Students know that proofs are about truth, and are about demonstrating that something is true. Students know that such demonstrations involve logic, and make use of mathematical notation. Students know that there are forms or patterns that can be used, such as proof by mathematical induction. + +However, the lack of importance that students assign to definitions, and the lack of facility with representation in mathematical formulation hinders students, especially in the construction of proofs. + +Students know that proofs contain sequences of logical statements. +There is a diversity of concepts for what a logical statement is. +Some students know that proofs use logical statements when warranted. + + + + +\subsubsection{Interpretation of How Students Attempt to Understand Proofs} + +Students do not always attempt to understand proofs they are shown. +"people have trouble with they see a proof, they see it, that's a theorem, that's a proof, that's true, i believe it, they don't look to see how is it a proof, everyone understands when you're staring at the screen, my recursion should work, my mathematical reduction works, but it's the steps in between that no one has an idea about, it's like a bridge, you start at a, you get to c, but b is the journey, and everyone skips that, they understand a, c but they don't" + + "while knowing what they can and can't do through proofs is of course important i just keep saying it get's a bit confusing in this class, nebulous sometimes" + + "the biggest thing that changed in my proof writing in the math side i didn't really have a good understanding of logical statements, like an if and only if" + +When students attempt to understand proofs, they sometimes get stuck. +They reported preferring a representation in code, which they could exercise in a development system. They did not know of a similar system that would help them tinker with or otherwise examine a proof. + +If construction by students attempted without an appreciation of the role of definitions (students zone out during definitions, students prefer examples), is formulaic, i.e., by learned steps but not also problem solving (as can happen when proof by mathematic induction is selected as a technique because it is a known technique, rather than some sense of its appropriateness is signaled by the problem), that would be consistent with comprehension not involving warrants. + + + + +\subsubsection{Interpretation of Reasons for Teaching Proof} + +Some students do not know why proofs appear in the curriculum. +Though we might expect this to be remediated upon the student seeing proofs used in subsequent classes, sometimes students don't recognize proofs when they see them. + +Some students know that proofs are used for justifying claims about the correctness and resource consumption of algorithms. No student was found who considered provability to be a guide for algorithm construction. + +Some students' conceptions of what proofs are for are at a lower level of detail than others. For example, a student reported that proof by counterexample is for proving something is not true. + +Other students connect learning proof by induction with explaining why algorithms work: "yes, of course, the first thing i thought of when i saw induction, was recursion" + +\subsubsection{Interpretation of How Students Attempt to Apply Proofs (When not assigned)} + +We can also look at how students attempt to apply proofs when assigned. +What are conceptualizations of application of proofs? +They need to use proofs, like pumping lemma, to solve problems posed to them, such as "Can we say this language is not regular?". +Rather than understanding what the proof says about features a regular language must possess, (Scouts must be thrifty, brave, clean and reverent.) and inverting the statement to find that a language not having these features cannot be regular (If a person is not thrifty or not brave or not clean or not reverent, that person cannot be a scout.), students seem to prefer to learn a procedure. Such a procedure would include, test for all possible segmentations, show for any such segmentation, it is not the case that segment $y$, when replicated any natural number of times, contributes to a string that is present in the language. Some students stumble in the application of the procedure, especially if the procedure is described using a mathematical formulation. Some students try to memorize and reproduce the mathematical formulation without understanding it. + +Quantifiers introduce additional complexity and challenge for students. + +There is insufficient evidence of students applying proofs when not assigned. + +\subsubsection{Interpretation of Whether students exhibit consequences of inability (such as avoiding recursion)} + +Students claim to be unsure of when certain algorithmic approaches, such as recursion, are appropriate, and do not think of proof as helpful in a determination. + +\subsubsection{Interpretation of How familiar and/or comfortable are students with different (specific) proof techniques: induction, construction, contradiction?} + +Students are familiar, in the sense of name recognition, with proof by mathematical induction, contrapositive, contradiction and cases. +Students see these as patterns for proof construction, i.e., steps that can be carried out. +Students are aware that, but are not always possessed of intrinsic conviction that the patterns achieve the desired result. +Thus the students do not always achieve confidence in what Carnap\cite[p. 21-22]{carnap1958introduction} calls "the psychological content (the totality of associations)" of the consequence of the logical deductions. +When some proof is subsequently viewed in a context where it is intended to be explanatory, its power over the students' conceptions is correspondingly diminished. + +\subsubsection{Interpretation of Whether students notice structural elements in proofs} + + + +\subsubsection{Interpretation of What do students think it takes to make an argument valid?} +\subsubsection{Interpretation of Whether students incorporate structural elements in proofs} + +When students consider whether a rule of inference is applicable to, and helpful for the transformation of logical representation they are attempting to accomplish, we might ask whether there are two levels of abstraction. One for the rule of inference in isolation, and another for that same rule of inference when in use in the specific proof. + + Here I want to put the ideas about definitions and abstraction. Without abstraction definitions are more cumbersome to remember and operate with. This discourages use of the axiomatic proof conceptions, because they are based on