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\chapter{Phenomenographic Analysis}
\section{Application of Phenomenographic Analysis in this Study}
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 questions are ordered guided by the 1956 version of Bloom's Taxonomy, namely, recognition, comprehension, application, analysis, synthesis, evaluation.\cite{bloom1956taxonomy}
% % %recognition
\subsection{Phenomenographic Analysis of What Students Think Proofs Are}
The categories developed in the orthodox phenomenographic analysis are:
\begin{table}
\begin{tabular}{|p{6cm}|p{6cm}|}\hline
Category & Description\\\hline\hline
Element of Domain of Mental Constructs & \\\hline
Contain Certain Syntactic Elements & \\\hline
Composed of Mathematical Statements & \\\hline
Combinations of Standard Argument Forms & \\\hline
Arguments in support of an idea or claim & \\\hline
Make claims obviously correct & \\\hline
\end{tabular}
\end{table}
Ideas that would have been welcome but did not appear:
\begin{table}
\begin{tabular}{|p{6cm}|p{6cm}|}\hline
Idea & Description\\\hline\hline
Consequence of Definitions & \\\hline
Relationship to Examples & \\\hline
Relative Value vs. Experiment & we do not expect students to say that proofs wouldn't entirely replace experimentation, but would back up experiment\\\hline
\end{tabular}
\end{table}
The arrangement of the categories follows that of the table, and is shown in Figure \ref{fig:WhatProof}.
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{./WhatProof}
\caption{Categories from what is proof}
\label{fig:WhatProof}
\end{figure}
Carnap writes eloquently on proof, a subset of logical deduction:
\begin{quote}
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}
\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.
Our participants seemed to regard axioms as strings of symbols that do mean something, though that meaning the participant ascribed might not be correct (especially as participants did not always know definitions of the entities being related), or the participant might feel unable to ascribe any meaning. We did find participants who appreciated the significance of definitions. Those we found were dual majors in math.
The source of such sequences served as a dimension of variation among the concepts we found in our students. Some students stressed the role of a procedure in synthesizing proofs. Proof by mathematic induction was considered preferable; a synthesis by procedure property was assigned to it. By contrast, proofs involving sequences of statements warranted by rules of inference, but otherwise unconstrained as to form, were considered less desirable.
Some students do not see the sequence of statements as carrying out a transformation process on a representation: " i get that a lot in straight programming, a lot of people have this mathematical explanations, and then in code it all of a sudden makes sense I think part of it has to do with the uh it has to do with the procedural nature of programming we're in state a, we move to state b, state c, state d and in the end we get an answer but equation is like an absolute constant total truth".
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".
Absence of attention to warrants has been reported by Alcock and Weber\cite{alcock}. Some of our students have noted this difficulty "Q: what made it difficult? A: probably not sufficiently understanding how the logic worked i guess, for certain techniques of proofs".
Another example of not finding the "connective tissue" between statements in a proof, and not noticing how a pair of statements warrants a conclusion, and not being convinced by the sequence of statements: " i'm not too fond of induction, for whatever reason, i don't know why i think that one made the least sense when i was learning you could just say there's a base case i increment once and i guess abstracting from that, and it's true for everything it seems i don't know, it seems kind of weird, sometimes when you think about it"
By contrast, some students clearly appreciate warrants: "like my (debating) points need to be clear and concise and they need to be connected one to the next. it is very much related to proofs"
Some students recognized patterns in sequences of statements. Contrapositive, contradiction, categorization into cases, proof by mathematic induction have been seen as patterns, consisting of steps that can be followed. These are contrasted with what were called "logic proofs".
It could be difficult to distinguish between a correct succession of logical steps from the premise(s) to the desired consequence(s) that "reaches a psychologically useful revised formulation" from "carries out a pattern". Indeed, the objectives of the course teaching proof may be met, while the preparation for the course using proof to explain the nature of, say, complexity classes, might not.
A yet more complete concept is "finite sequence of statements, starting with the premise and ending with what we want to prove, and justified in each step." A more profound conceptualization was found "finite sequence of statements, starting with axioms and premises, proceeding by logical deduction using (valid) rules of inference to what we wanted to prove, that shows us a consequence of the definitions with which we began, an exploration in which we discover the truth value of what we wanted to show, serving after its creation as an explanation of why the theorem is true".
Some students have a concept of the exploration purpose of proof: "like everything, uh, like everything that there is to prove, it already exists. So proof is just like a way to get there."
A few categories, such as those above, serve to identify a dimension of variation. When our purposes include discovering which points we may want to emphasize, we can examine the categories seeking to identify how they are related and how they differ.
It can certainly be that having more categories provides more critical aspects. For example, Harel and Sowder\cite{harel1998students} offered extrinsic vs. intrinsic conviction, empirical proof schemes and their most advanced deductive proof schemes as broader categories, and seven useful subcategories of these, yielding six critical aspects that suggest what teachers could usefully vary, to help learners discern items that would advance their knowledge.
\paragraph{Codes}
\begin{longtable}{|p{7cm}|p{8.5cm}|}\hline
\caption{Phenomenographic Analysis of What Students Think Proofs Are}\label{exemplar}
\endfirsthead
\multicolumn{2}{c}%
{{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\
\hline \multicolumn{1}{|c|}{\textbf{Code}} &
\multicolumn{1}{c|}{\textbf{Representative}} \\ \hline
\endhead
\hline \multicolumn{2}{|r|}{{Continued on next page}} \\ \hline
\endfoot
\hline \hline
\endlastfoot
% \begin{tabular}{|p{7cm}|p{8.5cm}|}
\hline
Code & Representative\\\hline\hline
Abstraction, Logical Abstraction &\\
Comprehending and Applying & \\
Connecting Recursion and Induction & \\
Construct Using Patterns & \\
Context for Use &\\
Definition &\\
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 &\\
Logic &\\
Logical progression, warrant &\\
Mathematical formulation &\\
Proof and programming &\\
Proof is logical steps &\\
Proof is magical incantation &\\
Proof is validation &\\
Proof relies on definitions &\\
Quantifiers &\\
Representations & visual proofs were just always easier, even to this day, I find that things that I can visualize I tend to do a lot better with, so I you know I had very little trouble for example with graph algorithms, because graphs for me personally were very, very easy to visualize, but heaps for example don't have like heaps are not a distinguished by their visual element\\
Structure &\\
Two too fast, relation or confusion &\\
\end{longtable}
\paragraph{Preliminary Categories}
\begin{longtable}{|p{7cm}|p{8.5cm}|}\hline
\caption{Phenomenographic Analysis of What Students Think Proofs Are}\label{exemplar}
\endfirsthead
\multicolumn{2}{c}%
{{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\
\hline \multicolumn{1}{|c|}{\textbf{Category}} &
\multicolumn{1}{c|}{\textbf{Representative}} \\ \hline
\endhead
\hline \multicolumn{2}{|r|}{{Continued on next page}} \\ \hline
\endfoot
\hline \hline
\endlastfoot
% \begin{tabular}{|p{7cm}|p{8.5cm}|}
\hline
Category & Representative\\\hline\hline
\end{longtable}
\paragraph{Main theme and its relationships to minor themes}
\begin{longtable}{|p{7cm}|p{8.5cm}|}\hline
\caption{Phenomenographic Analysis of What Students Think Proofs Are}\label{exemplar}
\endfirsthead
\multicolumn{2}{c}%
{{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\
\hline \multicolumn{1}{|c|}{\textbf{Theme}} &
\multicolumn{1}{c|}{\textbf{Relation to Main Theme}} \\ \hline
\endhead
\hline \multicolumn{2}{|r|}{{Continued on next page}} \\ \hline
\endfoot
\hline \hline
\endlastfoot
% \begin{tabular}{|p{7cm}|p{8.5cm}|}
\hline
Theme & Relation to Main Theme\\\hline\hline
\end{longtable}
\paragraph{Categories and Relationships}
\begin{longtable}{|p{7cm}|p{8.5cm}|}\hline
\caption{Phenomenographic Analysis of What Students Think Proofs Are}\label{exemplar}
\endfirsthead
\multicolumn{2}{c}%
{{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\
\hline \multicolumn{1}{|c|}{\textbf{Theme}} &
\multicolumn{1}{c|}{\textbf{Relation to Main Theme}} \\ \hline
\endhead
\hline \multicolumn{2}{|r|}{{Continued on next page}} \\ \hline
\endfoot
\hline \hline
\endlastfoot
% \begin{tabular}{|p{7cm}|p{8.5cm}|}
\hline
Theme & Relation to Main Theme\\\hline\hline
\end{longtable}
\paragraph{Dimension of Variation and Critical Factors}
\begin{longtable}{|p{7cm}|p{8.5cm}|}\hline
\caption{Phenomenographic Analysis of What Students Think Proofs Are}\label{exemplar}
\endfirsthead
\multicolumn{2}{c}%
{{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\
\hline \multicolumn{1}{|c|}{\textbf{Dimension of Variation}} &
\multicolumn{1}{c|}{\textbf{Critical Factor}} \\ \hline
\endhead
\hline \multicolumn{2}{|r|}{{Continued on next page}} \\ \hline
\endfoot
\hline \hline
\endlastfoot
% \begin{tabular}{|p{7cm}|p{8.5cm}|}
\hline
Dimension of Variation & Critical Factor\\\hline\hline
\end{longtable}
% % %comprehension
\subsection{Phenomenographic Analysis of How Students Attempt to Understand Proofs}
The categories developed in the orthodox phenomenographic analysis are:
\begin{table}
\begin{tabular}{|p{6cm}|p{6cm}|}\hline
Category & Description\\\hline\hline
Just Like the Examples from Class & \\\hline
Apply the Proof Pattern from Class & seen mathematic induction most often, so try that\\\hline
Go over all the logical elements from Class, related axioms and theorems & \\\hline
Use a diagram, visualization & \\\hline
Look up the definitions and use them (Math major) & \\\hline
\end{tabular}
\end{table}
Ideas that would have been welcome but did not appear:
\begin{table}
\begin{tabular}{|p{6cm}|p{6cm}|}\hline
Idea & Description\\\hline\hline
Notice the premises & \\\hline
Notice the desired outcome & \\\hline
Consider what might be deduced from the premises that might be closer to the desired outcome & \\\hline
\end{tabular}
\end{table}
The arrangement of the categories follows that of the table, and is shown in Figure \ref{fig:HowApproach}.
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{./HowApproach}
\caption{Categories from how do students approach comprehending proof}
\label{fig:HowApproach}
\end{figure}
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"\\
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.
"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.
Of students who read proofs attentively, some try to determine what rule of inference was used in moving from one statement to the next, and some do not.
Some students experience transient understanding of proof techniques: "aha moment have always been proofs written for induction, despite the fact that I've done them multiple times, they go over my head and I have to relearn proofs by induction".
Some students notice that lemmas can be proved and then used as building blocks in a larger argument, and some do not.
Some students can identify the forms of proof learned in discrete systems, when they see them employed in proofs, such as the combination of arguing by contrapositive and modus ponens. Some cannot.
Some students can identify these forms in an argument if the argument is made about concrete objects, such as cars or specific people. Some of these students have difficulty transferring this ability with concrete objects to application to abstract entities such as sets, algorithms or symbols.
"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." 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".
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
\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}
\subsection{Phenomenographic Analysis of Reasons for Teaching Proof}
What is proof for? What subset of what proof is for gives us reason for teaching it?
The categories developed in the orthodox phenomenographic analysis are:
\begin{table}
\begin{tabular}{|p{6cm}|p{6cm}|}\hline
Category & Description\\\hline\hline
Nothing of relevance & \\\hline
Nothing desirable & \\\hline
Do not know why & ``we do not accomplish anything''\\\hline
Increase confidence in experimental results & \\\hline
Find out whether hypothesis is false & \\\hline
Obtain more knowledge & \\\hline
Demonstrate claims (conclusively) & \\\hline
Distinguish the possible from the impossible & \\\hline
Understanding Algorithms and Their Properties & \\\hline
Ensuring we know why an algorithm works & \\\hline
Show that an algorithm meets requirements & \\\hline
Establish bounds on resource utilization & \\\hline
Tailor an algorithm so that its properties can be proven & \\\hline
Derive algorithms for efficiency & \\\hline
Derive mathematical formulation of intuitive ideas & \\\hline
Understand the consequences of definition & \\\hline
Effective Communication of Mathematical Thoughts & \\\hline
\end{tabular}
\end{table}
Ideas that would have been welcome but did not appear:
\begin{table}
\begin{tabular}{|p{6cm}|p{6cm}|}\hline
Idea & Description\\\hline\hline
Reasoning carefully about algorithms & \\\hline
\end{tabular}
\end{table}
The arrangement of the categories follows that of the table, and is shown in Figure \ref{fig:ForWhat}.
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{./ForWhat}
\caption{Categories from What do students think a proof is for}
\label{fig:ForWhat}
\end{figure}
Excerpts of student transcripts were selected on the basis of being related to this question. A dimension of variation emerged from the data, such that the excerpts seemed readily organized along this dimension.
\begin{longtable}{|p{7cm}|p{8.5cm}|}\hline
\endfirsthead
\multicolumn{2}{c}%
{{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\
\hline \multicolumn{1}{|c|}{\textbf{Category}} &
\multicolumn{1}{c|}{\textbf{Representative}} \\ \hline
\endhead
\hline \multicolumn{2}{|r|}{{Continued on next page}} \\ \hline
\endfoot
\hline \hline
\endlastfoot
% \begin{tabular}{|p{7cm}|p{8.5cm}|}\hline
Category & Representative\\\hline\hline
some students do not see any point to proof&
They teach it to us because they were mathematicians and they like it.\\
& we didn't see ok why do i really have to know the proof of the theorem to do that right? We didn't see the point, because no one taught us the point, so, that's a very important part that was missing.\\\hline
& Just stuff that I had to learn. \\\hline
Some students are not sure whether the material in 2500 is the more significant, or whether the proof techniques are the more significant: " i think a proof is um is just steps, little steps, induction, you start with your assumptions from there you build whatever it is whatever it is you want to prove, before i didn't have a clear concept of what a proof was, i had an idea, but and even now i don't feel like i have a very solid "This is a proof". I have the idea, i know how to go through the motion and how to prove a little bit, um, but, i wish there was more um probably like even if it's possible, have like a separate class about to do proofs"
&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 do not generalize the domain of applicability for proof techniques: " i suspect that part of the reason that they didn't make connection between inductive proofs and recursive programs is that normally in a workplace setting and probably in a lot of courses, too, you know the programs you write that use recursion probably aren't simply for like evaluating mathematical expressions or um they're not very theoretical so, you probably would never have to so there are a lot more complicated than an inductive proof like most of the problems you need us to prove for induction, so it's not immediately obvious that they're connected, and also people probably rarely you know he's been out of school for a couple of years probably and people in the work force are not probably asked to prove their programs like by induction or anything"
"Q: you could think of a tree as being recursively defined, right?\\
A: yes, to an extent i do when i think about the first kind of way we implemented trees i see them as graphs too in java, was binary tree you would have a node and that node would be connected to the you know child nodes,
and that i can't say that it's a rec(ursive), well, it's sort of a recursive in a way
Interviewer: a tree is defined to be a node that can have subtrees.\\
Participant: yeah. That's kind of a weird way of defining an interesting way of defining it, i guess"
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
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.\\
Some students appreciate proofs can be used for ascertaining correctness:& "if something holds potential of being extremely useful in a lot of situations, we want to know that our solution is correct, so that is why we write proofs in computer science, also historically probably the first computer scientists some of the first ones i think Lady Ada Lovelace, Turing, were mathematicians, so like probably started off you know a history of writing a lot of proofs"\\
& " proofs are used when you want to when you have concept and you want to prove it, like you want to make sure that that it's true"\\
& 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.\\\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
Some students agreed with the idea that proofs could guide algorithm creation: & "Q: Do you think it changes the way you invent algorithms?
A: I haven't thought about that actually, but, it does. It does."\\\hline
& "that way you know that you're code, what you're how it's going to be."\\\hline
\end{longtable}
\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
\subsection{Phenomenographic Analysis of How Students Attempt to Apply Proofs (When not assigned)}
There is only one category for student responses to this question. They do not attempt proofs when not assigned.
Some students claimed they never constructed proofs when not assigned.
Code-instead "do you ever decide on your own that you want to do a proof?
p no, I just tend I tend to just write code"
Some students did exercises related to proofs, without being assigned: "Q: Do you ever find yourself doing proofs? associated with computer science? that haven't been assigned?\\
A: That have not been assigned?\\
Q: Right, for fun, or because you want to know something?\\
A: um, he-he, well, i did find myself doing proofs, they were silly proofs, just like things about like things stuff up, yeah since i didn't have very solve it base, it was just like statements, not really just proof, just where you want to get to, so like the end result that you want to get to"
\paragraph{Codes}
\begin{longtable}{|p{7cm}|p{8.5cm}|}\hline
\caption{Phenomenographic Analysis of What Students Think Proofs Are}\label{exemplar4}
\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}
\subsection{Phenomenographic Analysis of Whether students exhibit consequences of inability (such as avoiding recursion)}
There is only one category from the orthodox phenomenographic method: Students do not know when they can apply recursion. They felt they were asked to produce recursive algorithms in situations in which it applied, and that they could. They felt that such situations did not occur subsequently. Some asked their employed friends who echoed this opinion.
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}
\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}
\subsection{Phenomenographic Analysis of How familiar and/or comfortable are students with different (specific) proof techniques: induction, construction, contradiction?}
This question was not pursued with the orthodox phenomenographic method.
"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}
\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}
\subsection{Phenomenographic Analysis of Which structural elements students notice in proofs}
The categories developed in the orthodox phenomenographic analysis are:
\begin{table}
\begin{tabular}{|p{6cm}|p{6cm}|}\hline
Category & Description\\\hline\hline
Components & \\\hline
Puzzle & \\\hline
Pattern(s) & \\\hline
Process Steps, State Transitions & \\\hline
Like Programs & \\\hline
\end{tabular}
\end{table}
Ideas that would have been welcome but did not appear:
\begin{table}
\begin{tabular}{|p{6cm}|p{6cm}|}\hline
Idea & Description\\\hline\hline
Good sentence structure & \\\hline
Scoping Like Lexical Scoping & \\\hline
\end{tabular}
\end{table}
The arrangement of the categories follows that of the table, and is shown in Figure \ref{fig:WhatStructure}.
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{./WhatStructure}
\caption{Categories from What structure do students notice in proofs?}
\label{fig:WhatStructure}
\end{figure}
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.
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"
"very much the same logical sense, um, like with programming there's no ambiguity, everything is very structured, like proofs are structured in much the same way i enjoy programming more than regular proofs, particularly why, maybe because it's more fun to see results, when you program something"
\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}
\subsection{Phenomenographic Analysis of What do students think it takes to make an argument valid?}
The categories developed in the orthodox phenomenographic analysis are:
\begin{table}
\begin{tabular}{|p{6cm}|p{6cm}|}\hline
Category & Description\\\hline\hline
Know what's true and why & all those theorems\\\hline
Re-use proof patterns & \\\hline
Stick to valid rules of inference & \\\hline
\end{tabular}
\end{table}
Ideas that would have been welcome but did not appear:
\begin{table}
\begin{tabular}{|p{6cm}|p{6cm}|}\hline
Idea & Description\\\hline\hline
Take note of the difference between the idea in the hypothesis, and the consequence, and consider what warranted transformations might bring the representation of the hypothesis closer to that of the consequence & \\\hline
\end{tabular}
\end{table}
The arrangement of the categories follows that of the table, and is shown in Figure \ref{fig:Valid}.
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{./Valid}
\caption{Categories from what do students think it takes ot make an argument valid?}
\label{fig:Valid}
\end{figure}
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.
Some students do not notice that proof by contradiction introduces (for purposes of contradiction) a premise.
% % %synthesis
\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}
\subsection{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.
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
\subsection{Phenomenographic Analysis of Combined Data}
definitions vs examples, examples are easier, value of definitions not necessarily appreciated.
Use of examples implies hope that generalization will occur.
Recognition that generalization is difficult.
\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.
Some students remembered having to furnish proofs of geometrical facts, also facts about prime numbers and sets.
Some students knew that CSE2500 treated proofs because they would be used in later courses. Students did not know why proofs would be used later, and were generally happy to hear some example uses.
Though students were asked whether they made use of proofs spontaneously, none of those interviewed gave an example.
Some students preferred to articulate with code, and some (who were dual computer science / math) sometimes preferred mathematical symbols, depending upon the context.
Some students do wish to convince themselves of things, such as tractable execution times, and correctness. Though students were asked whether they made use of proofs for this purpose, none of those interviewed claimed to do so, rather they mentioned going carefully over their algorithm construction, and considering cases.
In interviews, the students almost all chose to discuss proofs by mathematical induction.
\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}
\item Definitions\\
Students divided into (1)those who found definitions boring, difficult to pay attention to, and undesirable compared to examples, from which they preferred to induce their own definitions, and (2) those who had caught on to the idea that definitions were the carefully crafted building blocks of reasoning.
\item Procedures
Students sometimes learned what was desired in a proof, but learned to produce it by procedure, and were not themselves convinced.
\item Context
Students asked whether the topics for examples and exercises, such as prime numbers, had relevance to programming, with which they had experience, but not unrelated to the topics.
Students did not know the context in which the proofs, or procedure version of proof, was applicable, so, for example, did not apply proof by mathematic induction to recursive algorithms, and did not know how to tell whether recursive algorithms would be applicable.
\item Concrete vs. Abstract
Some students felt quite comfortable with the application of rules of inference to concrete items, but had difficulty transferring application of those rules to mathematical symbols.
\item Symbolization
consistent with Harel and Sowder's 1998 categorization of concepts, we found students who would attempt to write in symbols, but not understand what was denoted, and consequently were uncertain about appropriate operations. Some of these students were glad to see a progression from pseudocode with long variable names to pseudocode with short variable names to mathematical symbolization (formula translation (FORTRAN) in reverse).
\item Applicability of single examples
Some students believed that a few examples constituted a proof. These examples were not generic particular, nor were they transformational, in the sense of Harel and Sowder's 1998 model.
\item Substructure
Students familiar with methods, in the sense of object-oriented programming, and with construction of programs involving multiple method calls, did not always recognize that proofs could be built from multiple lemmas, although they did understand that axioms could be applied.
\item Proofs are used, in computer science, to show resource consumption (complexity class), properties of models of computation, and computability/decidability. No occasion was identified, other than assignment, when undergraduate students recognized they were undertaking proofs.
\item Among graduate students, proofs were undertaken in the context of preparing manuscripts for publication. These were scheduled to be approached after algorithm design, though retroactive adjustment of algorithms did occur for simplifying the proof.
\end{itemize}
\subsubsection{Combined Relationships}
\subsection{Analysis of Homework and Tests}
\subsubsection{Proofs}
Proofs submitted on homework and tests were analyzed in several respects.
The overall approach should be valid. For example, students who undertook to prove that the converse was true did not use a valid approach.
The individual statements should each be warranted.
Use of structure, such as lemmas, and care that cases form a partition of the relevant set are gladly noticed.
Proof attempts that lose track of the goal, and proof attempts that assert with insufficient justification, the goal are noted.
\subsubsection{Pumping Lemmas}
We wrote descriptions for each error. Some example descriptions
are in Table II.
Table : Some example errors
Let x be empty
$|xy| \leq p, so xy = 0^p$\\
$|xy| \leq p; let \; x = 0^{p+r}, y = 0^{p+r}, 0 < r < p$\\
Let’s choose $|xy| = p$\\
$0^{p+1}0^b1^p \neq 0^{p+1}1^p \therefore xy^2z \not\in \mathcal{L}$
where $\mathcal{L} = \{0^i1^j, i \neq j\}$\\
we choose $s = 0^{p+1}1^p$ within $|xy|$\\
thus $\neq 0^p1^{p+1}$\\
Let $x = 0^a, y = 0^b1^a$\\
$x = 0^{p-h}, y = 0^h$\\
$x = 0^i, y = 0^i, z = 0^i1^j$
A handful of students did exhibit their reasoning that for
all segmentations there would exist at least one value of $i$ that
would generate a string outside the language.
We categorized the errors as misunderstandings of one or
more of:
\cite[get some page reference]{sipser2012introduction}
1) ∣𝑥𝑦∣ ≤ 𝑝 permits ∣𝑥𝑦∣ < 𝑝\\
2) 𝑥 is the part of the string prior to the cycle\\
3) 𝑦 is the part of the string which returns the state of
the automaton to a previously visited state\\
4) 𝑧 is the part of the string after the (last) cycle up to
acceptance\\
5) 𝑝 − 1 characters is the maximum size of a string
that need not contain a cycle, (strings of length 𝑝
or greater must reuse a state)\\
6) 𝑖 is the number of executions of 𝑦\\
7) There must be no segmentation for which pumping
is possible, if pumping cannot occur.\\
8) A language is a set of strings.\\
9) A language class is a set of languages.\\
Categories are shown in the chapter on results (labelled table iii).\\