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the orthodox method initial description in chapter 3 and initial
analysis in chapter 4
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8 changes: 8 additions & 0 deletions ch3.tex
Expand Up @@ -496,6 +496,14 @@ The audio portion of all interviews was collected by electronic recorder and sub
\subsection{Analysis of Interviews}

Items excerpted from interviews for analysis should be analyzed in the context of the specific interview and also in the context of the ensemble.\cite{marton1997learning}.

Data were analyzed multiple ways. Both an orthodox phenomenographic analysis, and a modified thematic analysis were carried out.

\subsubsection{Orthodox Phenomenographic Analysis}

In the orthodox phenomenographic analysis of interviews, the transcriptions are printed, and text fragments corresponding to units of meaning are cut out (as, with scissors). These pieces are then grouped (making copies if necessary) according to a sense of similarity. During a stage in the process, categories are learned, as researchers sense of features that distinguish categories evolves. During this stage, text fragments are moved from one category to another. After this category development phase, researchers, look into each category, to recognize and describe each category. Subsequently the perspective is shifted so that relations between categories are sought. Thus the categories are arranged relative to one another, and pairwise relations, where they exist, are identified and described. This produces a graph. From the graph, critical features of the learning objective are inferred.

\subsubsection{Modified Thematic Analysis}

Data were analyzed using a modified version of thematic analysis, which is
in turn a form of basic inductive analysis.\cite{Merriam2002,Merriam2009,braun2006using,fereday2008demonstrating,boyatzis1998transforming} Using thematic analysis, we
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214 changes: 203 additions & 11 deletions ch4.tex
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\subsection{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.

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% % %recognition

\subsubsection{Phenomenographic Analysis of What Students Think Proofs Are}
\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}
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% % %comprehension

\subsubsection{Phenomenographic Analysis of How Students Attempt to Understand Proofs}
\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"
Expand Down Expand Up @@ -367,7 +437,53 @@
Dimension of Variation & Critical Factor\\\hline\hline
\end{longtable}

\subsubsection{Phenomenographic Analysis of Reasons for Teaching Proof}
\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.

Expand Down Expand Up @@ -565,7 +681,11 @@ A: I haven't thought about that actually, but, it does. It does."\\\hline

% % %application

\subsubsection{Phenomenographic Analysis of How Students Attempt to Apply Proofs (When not assigned)}
\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.

Expand Down Expand Up @@ -682,7 +802,9 @@ A: um, he-he, well, i did find myself doing proofs, they were silly proofs, just
\hline
Dimension of Variation & Critical Factor\\\hline\hline
\end{longtable}
\subsubsection{Phenomenographic Analysis of Whether students exhibit consequences of inability (such as avoiding recursion)}
\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.

Expand Down Expand Up @@ -792,7 +914,9 @@ Some students claimed to know how to write recursive algorithms but said they ne
\hline
Dimension of Variation & Critical Factor\\\hline\hline
\end{longtable}
\subsubsection{Phenomenographic Analysis of How familiar and/or comfortable are students with different (specific) proof techniques: induction, construction, contradiction?}
\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"

Expand Down Expand Up @@ -930,7 +1054,42 @@ Some students claimed to know how to write recursive algorithms but said they ne
Dimension of Variation & Critical Factor\\\hline\hline
\end{longtable}

\subsubsection{Phenomenographic Analysis of Whether students notice structural elements in proofs}
\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.

Expand Down Expand Up @@ -1050,7 +1209,40 @@ Some students claimed to know how to write recursive algorithms but said they ne
Dimension of Variation & Critical Factor\\\hline\hline
\end{longtable}

\subsubsection{Phenomenographic Analysis of What do students think it takes to make an argument valid?}
\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"
Expand Down Expand Up @@ -1172,7 +1364,7 @@ Some students claimed to know how to write recursive algorithms but said they ne
Dimension of Variation & Critical Factor\\\hline\hline
\end{longtable}

\subsubsection{Phenomenographic Analysis of Whether students incorporate structural elements in proofs}
\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.

Expand All @@ -1182,7 +1374,7 @@ and they you split, what do i have to do to get to that point, so you have to ac

% %evaluation

\subsubsection{Phenomenographic Analysis of Combined Data}
\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.
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