diff --git a/ch4.tex b/ch4.tex
index 5c41c16..110f26c 100644
--- a/ch4.tex
+++ b/ch4.tex
@@ -41,8 +41,16 @@
   
   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.
@@ -50,6 +58,8 @@
   
   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.
@@ -96,13 +106,17 @@
      Proof is validation &\\
      Proof relies on definitions &\\
      Quantifiers &\\
-     Representations &\\
+     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}
@@ -220,6 +234,8 @@
  
  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.
@@ -381,11 +397,23 @@
  
  & 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&
@@ -402,6 +430,10 @@
   
   & 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
  
@@ -411,11 +443,21 @@
 %\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
@@ -527,11 +569,17 @@ Some students find that creating proofs related to algorithms can be more diffic
 
 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{exemplar}
+               \caption{Phenomenographic Analysis of What Students Think Proofs Are}\label{exemplar4}
                \endfirsthead
                
                \multicolumn{2}{c}%
@@ -894,6 +942,8 @@ Some students claimed to know how to write recursive algorithms but said they ne
   
   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}
@@ -1131,6 +1181,15 @@ 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
+
+\subsubsection{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.
diff --git a/ch6.tex b/ch6.tex
index 4bad6f3..889658e 100644
--- a/ch6.tex
+++ b/ch6.tex
@@ -3,6 +3,11 @@
 In this chapter we consider the results summarized in the previous chapter. We consider alternate dimensions of variation, comparing them with the chosen. 
 We consider alternate critical factors, both along alternate dimensions of variation, but also along the proposed dimensions of variation.
 
+Critical factors are ideas that are necessary for a better conceptualization\cite{marton and pang?}. 
+We conjecture that not all points along a dimension of variation are equally valuable.
+Some ideas encounter more resistance from students than others. (Is a cite needed for this?)
+Those ideas that are less readily adopted by students might be preferable as critical factors, in that these enable the helping role of teachers.
+
 \section{Interpretation}
 
 \subsubsection{Interpretation of What Students Think Proofs Are}
@@ -365,6 +370,13 @@ What can we learn from students of computer science who excelled in reasoning to
 %\chapter{Discussion}
 
 \section{Discussion}
+
+The idea of representation appears to be extremely important.
+       Depending upon how subject matter is represented, students may or may not achieve the mental connections instructors intend.
+ Students might not recognize that a heap is a graph.
+ Students might not recognize that a tree, being a recursive data structure, enables the use of recursive algorithms, that are in turn provable using mathematical induction.
+ 
+ 
  \subsection{Importance}
 Importance goes here, rather than in analysis\\
 
@@ -1110,6 +1122,22 @@ Some students, and some instructors, do not emphasize that a single presentation
 
 \section{Combined Description}
 
+\subsection{Generalization meets  preference for examples over definitions}
+Harel and Sowder\cite{harel}, and also Polya\cite{polya} have described a category of conceptualization in which students work with examples,   concrete examples, in preference for definitions. An axiomatic, Hilbert-style approach is not appreciated in this conceptualization.
+When dealing with this conceptualization, 
+
+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.
+
+When deferring the axiomatic approach until generalization is easier, we should do this consciously, and plan to support generalization.
+Though we hope generalization will occur spontaneously, some students have expressed, while taking introduction to the theory of computing, CSE3502, that they do not understand logical reasoning unless it is  carried out with concrete instances. 
+This poses difficulties in CSE2502. 
+
+We can imagine a trajectory which begins in concrete examples, generalizes, accompanied by developing an appreciation of careful definitions. Also we can imagine building in structural relevance, so that a relationship between the generalized knowledge and the intended use of it, in computer science and engineering, is facilitated. During assessment we wish for the existence of a path from otherwise inert knowledge to application. Some problem-solving activity begins with the problem statement, which might well be concrete, and the generalization ability gets exercised, to recognize and example of a principle that should be known, and the knowledge, that should not be inert, should be activated and brought to bear on the problem. Because practicing taking quizzes exercises these latter capabilities, whose exercise is probably useful for test taking, we find a compatibility with existing knowledge on the utility of test preparation by taking quizzes.\cite{marcell2008effectiveness}
+
 There are a couple of ways students work with exercises in proof, that are incomplete.
 
 Some students reason well with concrete entities, yet are confused with abstractions. These students are not appreciating the value of careful definitions, because they do not use them as tools, or the basis for reasoning. They are more comfortable with examples, because they are operating in a concrete world.
@@ -1118,6 +1146,30 @@ Some students do not connect the world of concrete objects with the abstract, sy
 
 Some students do understand application of facts, axioms and rules of inference, and are at home with careful definitions and symbolic concision. Some of these students also study math.
 
+\subsection{Recognize problem solutions, comprehend problem solutions, create problem solutions}
+
+Some students know they have seem demonstrations of working code. 
+Some students can mimic using supplied code, and by producing variations on it, and can comprehend how supplied code works.
+While for some, a list of statements lacks properties that emerge from the collection, for others the relations and combinations are noticed.
+Synthesis of problem solutions can be a difficulty for some students, including for some of those who appreciate these interactions.
+
+This difficulty is reported  in synthesis both of logical arguments and  of solutions to problems posed in data structures and in algorithms courses.
+
+Hunter/gatherer vs. farmer, for solutions to problems. What about the Freudian sublimation process of creation of solutions to problems? 
+
+Schoenfeld\cite{schoenfeld1998reflections} problem solving class. Is it that, only those motivated to solve problems sign up?
+
+Some students have motivation to solve problems that existed when they wanted to solve networking problems and configuation editing problems to support playing multiplayer games.
+Some students have found gratification in problem solving in general.
+Some students have a concern for earning the grades in the class.
+Some of those who feel rewarded by and/or have become successful at problem solving will choose to solve problems themselves, even if looking up solutions is easier than creating them.
+Others have not developed this preference for problem solving might be tempted to look up solutions.
+
+Some of synthesis is retrieval of a learned technique in response to a problem setting that might be novel.
+Some synthesis requires combination of multiple learned techniques in response to a problem setting.
+Some synthesis requires analysis of the problem setting, so that divide and conquer can be applied.
+Maybe we need more exercise of retrieval and problem analysis, and of synthesis.
+
 
 \newpage
 \section{Diagram of Conceptualizations}
diff --git a/literature.bib b/literature.bib
index 40cec54..ace137b 100644
--- a/literature.bib
+++ b/literature.bib
@@ -519,6 +519,15 @@ publisher={Elsevier}
  address = {New York, NY, USA},
  keywords = {concept inventory, digital logic, discrete math, formal logic, misconceptions},
 } 
+@article{marcell2008effectiveness,
+  title={Effectiveness of regular online quizzing in increasing class participation and preparation},
+  author={Marcell, Michael},
+  journal={International Journal for the Scholarship of Teaching and Learning},
+  volume={2},
+  number={1},
+  pages={7},
+  year={2008}
+}
 @article{selden2008overcoming,
   title={Overcoming students’ difficulties in learning to understand and construct proofs},
   author={Selden, Annie and Selden, John},
diff --git a/thesis2.pdf b/thesis2.pdf
index c189222..6a10b2a 100644
Binary files a/thesis2.pdf and b/thesis2.pdf differ
diff --git a/thesis4Through6.pdf b/thesis4Through6.pdf
new file mode 100644
index 0000000..76cdafc
Binary files /dev/null and b/thesis4Through6.pdf differ