diff --git a/ch4.tex b/ch4.tex index 28eb0ad..89d67a9 100644 --- a/ch4.tex +++ b/ch4.tex @@ -1,13 +1,6 @@ - \chapter{Phenomenographic Analysis} +\chapter{Phenomenographic Analysis} - - - - - - - - \section{Application of Phenomenographic Analysis in this Study} +%\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. @@ -17,11 +10,12 @@ % % %recognition - \subsection{Phenomenographic Analysis of What Students Think Proofs Are} +\section{Phenomenographic Analysis of What Students Think Proofs Are} The categories developed in the traditional phenomenographic analysis are: \begin{table} + \caption{Categories for Student Conceptualizations of What Proofs Are} \begin{tabular}{|p{6cm}|p{6cm}|}\hline Category & Description\\\hline\hline Make claims obviously correct & \\\hline @@ -36,6 +30,7 @@ Ideas that would have been welcome but did not appear: \begin{table} + \caption{Relationships for Student Conceptualizations of What Proofs Are} \begin{tabular}{|p{6cm}|p{6cm}|}\hline Idea & Description\\\hline\hline Consequence of Definitions & \\\hline @@ -44,9 +39,9 @@ \end{tabular} \end{table} - The arrangement of the categories follows that of the table, and is shown in Figure \ref{fig:WhatProof}. +The arrangement of the categories follows that of the table, and is shown in Figure \ref{fig:WhatProof}. - \begin{figure} +\begin{figure} \centering \includegraphics[width=0.7\linewidth]{./WhatProof} \caption{Categories from what is proof} @@ -123,151 +118,17 @@ Considering the categories "Argument in Support of Idea or Claim" and "Makes a C 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 + \newpage - \subsection{Phenomenographic Analysis of How Students Attempt to Understand Proofs} + \section{Phenomenographic Analysis of How Students Attempt to Understand Proofs} The categories developed in the traditional phenomenographic analysis are: \begin{table} + \caption{Categories for Student Conceptualizations of How to Understand Proofs} \begin{tabular}{|p{6cm}|p{6cm}|}\hline Category & Description\\\hline\hline Look up the definitions and use them (Math major) & \\\hline @@ -286,6 +147,7 @@ Considering the categories "Argument in Support of Idea or Claim" and "Makes a C Ideas that would have been welcome but did not appear: \begin{table} + \caption{Relationships for Student Conceptualizations of How to Understand Proofs} \begin{tabular}{|p{6cm}|p{6cm}|}\hline Idea & Description\\\hline\hline Notice the premises & \\\hline @@ -373,136 +235,33 @@ Considering the categories "Argument in Support of Idea or Claim" and "Makes a C % % % 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} + \newpage - \subsection{Phenomenographic Analysis of Reasons for Teaching Proof} + \section{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 traditional phenomenographic analysis are: \begin{table} + \caption{Categories for Student Conceptualizations of Reasons for Teaching Proof} \begin{tabular}{|p{6cm}|p{6cm}|}\hline Category & Description\\\hline\hline Effective Communication of Mathematical Thoughts & \\\hline Understand the consequences of definition & \\\hline - Derive mathematical formulation of intuitive ideas & \\\hline - Derive algorithms for efficiency & \\\hline - Tailor an algorithm so that its properties can be proven & \\\hline - Establish bounds on resource utilization & \\\hline - Show that an algorithm meets requirements & \\\hline - Ensuring we know why an algorithm works & \\\hline - Understanding Algorithms and Their Properties & \\\hline + Derive mathematical formulation of intuitive ideas & \\\hline \hline + Derive algorithms for efficiency & \\\hline + Tailor an algorithm so that its properties can be proven & \\\hline \hline + Show that an algorithm meets requirements & \\\hline + Establish bounds on resource utilization & \\\hline + Understanding Algorithms and Their Properties & \\\hline + Ensuring we know why an algorithm works & \\\hline \hline + Demonstrate claims (conclusively) & \\\hline Distinguish the possible from the impossible & \\\hline - Demonstrate claims (conclusively) & \\\hline - Obtain more knowledge & \\ \hline + Obtain more knowledge & \\ \hline \hline Find out whether hypothesis is false & \\\hline - Increase confidence in experimental results & \\\hline + Increase confidence in experimental results & \\\hline Do not know why & ``we do not accomplish anything''\\\hline Nothing desirable & \\\hline Nothing of relevance & \\\hline @@ -512,10 +271,12 @@ Considering the categories "Argument in Support of Idea or Claim" and "Makes a C Ideas that would have been welcome but did not appear: \begin{table} + \caption{Relationships for Student Conceptualizations of Reasons for Teaching Proof} \begin{tabular}{|p{6cm}|p{6cm}|}\hline Idea & Description\\\hline\hline Reasoning carefully about algorithms & \\\hline - + These reasons include being certain about algorithm properties & \\\hline + There are good and relevant reasons for teaching proof in the computer science curriculum & \\\hline \end{tabular} \end{table} @@ -533,9 +294,9 @@ Considering the categories "Argument in Support of Idea or Claim" and "Makes a C \label{tab:forWhat} \begin{tabular}{|p{8cm}|}\hline Critical Aspect\\ \hline\hline - Explain purpose of proof, including starting from given, including irrefutable, demonstrate the truth value of what is to be shown, with examples from CSE \\ \hline + Discuss in detail occasions in which constructing proofs serves the creation of algorithms, such as pre-conditions, post-conditions, Gries-like construction \\\hline Sketch in more detail the domain of CSE in which students can expect to encounter proofs, also opportunities they might experience, to construct proofs \\ \hline - Discuss in detail occasions in which constructing proofs serves the creation of algorithms, such as pre-conditions, post-conditions, Gries-like construction \\\hline + Explain purpose of proof, including starting from given, including irrefutable, demonstrate the truth value of what is to be shown, with examples from CSE \\ \hline \end{tabular} \end{table} @@ -547,201 +308,11 @@ Considering the categories "Argument in Support of Idea or Claim" and "Makes a C 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 + \newpage -\subsection{Phenomenographic Analysis of How Students Attempt to Apply Proofs (When not assigned)} +\section{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. @@ -756,225 +327,19 @@ Some students did exercises related to proofs, without being assigned: "Q: Do yo 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)} +\newpage + +\section{Phenomenographic Analysis of Whether students exhibit consequences of inability (such as avoiding recursion)} There is only one category from the traditional 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?} + + \newpage + + \section{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 traditional phenomenographic method. @@ -1008,117 +373,16 @@ Some students claimed to know how to write recursive algorithms but said they ne 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} + + + \newpage - \subsection{Phenomenographic Analysis of Which structural elements students notice in proofs} + \section{Phenomenographic Analysis of Which structural elements students notice in proofs} The categories developed in the traditional phenomenographic analysis are: \begin{table} + \caption{Categories for Student Conceptualizations of What Structural Elements are Found in Proofs} \begin{tabular}{|p{6cm}|p{6cm}|}\hline Category & Description\\\hline\hline Components & \\\hline @@ -1133,6 +397,7 @@ Some students claimed to know how to write recursive algorithms but said they ne Ideas that would have been welcome but did not appear: \begin{table} + \caption{Relationships for Student Conceptualizations of What Structural Elements are Found in Proof} \begin{tabular}{|p{6cm}|p{6cm}|}\hline Idea & Description\\\hline\hline Good sentence structure & \\\hline @@ -1176,117 +441,15 @@ Some students claimed to know how to write recursive algorithms but said they ne "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} + + \newpage - \subsection{Phenomenographic Analysis of What do students think it takes to make an argument valid?} + \section{Phenomenographic Analysis of What do students think it takes to make an argument valid?} The categories developed in the traditional phenomenographic analysis are: \begin{table} + \caption{Categories for Student Conceptualizations of Valid Argumentation} \begin{tabular}{|p{6cm}|p{6cm}|}\hline Category & Description\\\hline\hline Know what's true and why & all those theorems\\\hline @@ -1300,6 +463,7 @@ Some students claimed to know how to write recursive algorithms but said they ne Ideas that would have been welcome but did not appear: \begin{table} + \caption{Relationships for Student Conceptualizations of Valid Argumentation} \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 @@ -1343,113 +507,9 @@ 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} - \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} +\section{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. @@ -1458,8 +518,9 @@ Students have asked whether, when using categorization into cases, they must app 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 +\newpage -\subsection{Phenomenographic Analysis of Combined Data} +\section{Phenomenographic Analysis of Combined Data} This work includes two methods of analysis of combined data, the first being influenced by the traditional phenomenographic approach, so, we take each research question in turn, we take text fragments relevant to the individual research question, we obtain categories, and arrange them and infer critical aspects. Then with the arrangements and critical aspects we look for insights spanning the multiple questions. In the second method, the text fragments are not segregated by research question. Categories emerge from the whole collection of text fragments, and relationships between categories are examined using axial coding, as found in grounded theory \cite{which one uses axial?}. @@ -1475,193 +536,3 @@ 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) ∣π‘₯π‘¦βˆ£ ≀ 𝑝 permits ∣π‘₯π‘¦βˆ£ < 𝑝\\ - 2) π‘₯ 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 βˆ’ 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).\\ - - - \ No newline at end of file diff --git a/thesis2.pdf b/thesis2.pdf index 784e9ae..9d2a1f2 100644 Binary files a/thesis2.pdf and b/thesis2.pdf differ diff --git a/thesis2.tex b/thesis2.tex index 1d656e4..9a2fff9 100644 --- a/thesis2.tex +++ b/thesis2.tex @@ -27,6 +27,8 @@ Connecticut, USA, 2014} \input{frontmatter.tex} \singlespacing \tableofcontents +\listoffigures %tms20151102 +\listoftables%tms20151102 \thispagestyle{plain} \mainmatter \doublespacing @@ -65,7 +67,9 @@ Connecticut, USA, 2014} %\newpage \appendix{Incoming Assessment for Discrete Math} \input{QlistCSE2500.tex} -%\newpage +\newpage +\appendix{Details of Analysis: Codes and Representatives} +\input{DetailsAnalysis.tex} %\appendix{Incoming Assessment for Algorithms} %\input{incomingAlgos.tex} %\newpage