diff --git a/ch3.tex b/ch3.tex
index 00667cc..a2c5439 100644
--- a/ch3.tex
+++ b/ch3.tex
@@ -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
diff --git a/ch4.tex b/ch4.tex
index 110f26c..ce2c23b 100644
--- a/ch4.tex
+++ b/ch4.tex
@@ -7,7 +7,7 @@
  
  
  
- \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.
  
@@ -17,7 +17,43 @@
  
  % % %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}
@@ -205,7 +241,41 @@
   
   % % %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"
@@ -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.
  
@@ -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.
 
@@ -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.
 
@@ -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"
   
@@ -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.
   
@@ -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"
@@ -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.
  
@@ -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.
diff --git a/thesis2.pdf b/thesis2.pdf
index b38c283..784e9ae 100644
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