scaffolding with programming, uneven attention

ch4.tex

@@ -19,18 +19,17 @@
 
   \subsection{Phenomenographic Analysis of What Students Think Proofs Are}
 
-  The categories developed in the orthodox phenomenographic analysis are:
+  The categories developed in the traditional phenomenographic analysis are:
 
   \begin{table}
   \begin{tabular}{|p{6cm}|p{6cm}|}\hline
   Category & Description\\\hline\hline
+   Make claims obviously correct & \\\hline
+    Arguments in support of an idea or claim & \\\hline
+     Combinations of Standard Argument Forms & \\\hline
+      Composed of Mathematical Statements & \\\hline
+       Contain Certain Syntactic Elements & \\\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}
 
@@ -41,7 +40,7 @@
     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
+    Relative Value vs. Experiment & we do not expect students to say that proofs wouldn't entirely replace experimentation, but would back up experiment. We prefer students would know that proof is sufficient on its own.\\\hline
     \end{tabular}
     \end{table}
 
@@ -54,6 +53,29 @@
 \label{fig:WhatProof}
 \end{figure}
 
+
+Following the traditional phenomenographic method, we exam pairs of categories. We choose categories that appear to be adjacent in the space of features with which categories are distinguished from one another. This calls attention to features whose values differ. These differing values are candidates for critical aspects. We can consider whether a particular difference in feature value is important in distinguishing one category of conceptualization from another. The confidence with which we hold that difference in feature value to be important is the confidence with which we feel that difference is a critical aspect.
+
+For example, there is a conceptualization found in the cohort of students that in a spoken proof attempt, will produce the phrase "You know what I mean." The aspect of argumentation that certain forms, such as mathematical formulation, can be suitable for proof, and other forms, such as "You know what I mean." are not, in general, appropriate seems very important. We propose that "express ideas with logical statements, including mathematical formulation" is a critical aspect differentiating the category "Domain of Mental Constructs" from "Composed of Statements".
+Critical aspects are listed in Table \ref{tab:critWhatIs}.
+
+\begin{table}[ht]
+\caption{Critical factors for What is proof?}
+\label{tab:critWhatIs}
+\begin{tabular}{|p{8cm}|}\hline
+Critical Aspect\\ \hline\hline
+Express ideas with logical statements, including mathematical formulation.\\ \hline
+Later statements must be justified by what has gone before.\\ \hline
+A guiding principle in the development of the argument is to reach the state in which the desired claim has been made obviously true, or false.\\ \hline
+One criterion for evaluating an argument is that it renders the claim obviously true or false.\\ \hline
+\end{tabular}
+\end{table}
+
+Considering the categories "Composed of Statements" and "Combination of Standard Argument Forms", the aspect of relationships between statements, specifically that later statements must be justified by what has gone before, seems critical.
+
+Considering the categories "Combination of Standard Argument Forms" and "Argument in Support of an Idea or Claim", the proposed critical aspect is that guidance about the development of the argument comes from the goal claim. There is evidence of students setting out to create an argument, but getting lost, such as forgetting that a scope has been set in which a statement is temporarily held to be false, and forgetting to exit that scope.
+
+Considering the categories "Argument in Support of Idea or Claim" and "Makes a Claim Obviously Correct", we propose that the rendering obvious, of the claim, by the argument, is the most important distinguishing feature.
 
   Carnap writes eloquently on proof, a subset of logical deduction:
   \begin{quote}
@@ -243,16 +265,20 @@
 
  \subsection{Phenomenographic Analysis of How Students Attempt to Understand Proofs}
 
-  The categories developed in the orthodox phenomenographic analysis are:
+  The categories developed in the traditional phenomenographic analysis are:
 
   \begin{table}
    \begin{tabular}{|p{6cm}|p{6cm}|}\hline
    Category & Description\\\hline\hline
+    Look up the definitions and use them (Math major) & \\\hline
+     Use a diagram, visualization & \\\hline
+    Go over all the logical elements from Class, related axioms and theorems & \\\hline
+        Apply the Proof Pattern from Class & seen mathematic induction most often, so try that\\\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}
@@ -277,6 +303,22 @@
  \label{fig:HowApproach}
  \end{figure}
 
+ \begin{table}[ht]
+ \caption{Critical factors for How do students approach comprehending proof?}
+ \label{tab:critApproach}
+ \begin{tabular}{|p{8cm}|}\hline
+ Critical Aspect\\ \hline\hline
+ Generalization from instance to pattern \\ \hline
+ Attempt at visualization \\ \hline
+ \end{tabular}
+ \end{table}
+
+ The first category of approach to comprehending a proof is to check whether it is one they have already examined in class.
+ The next category is to check whether the proof follows a pattern that has been treated in class. The most important difference seems to be that generalization from an instance to a pattern occurs.
+
+ The  next category after a pattern that has been discussed in class seems to be to engage the visual domain. It is not clear that students view the argument as a process by which one representation of a truth gets transformed into another representation, that renders the claim obvious. Thus, it is not clear what the students are attempting to visualize.
+
+
  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"
 
@@ -441,28 +483,29 @@
 
  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:
+   The categories developed in the traditional 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
+
+     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
-    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
+     Ensuring we know why an algorithm works &  \\\hline
+      Understanding Algorithms and Their Properties & \\\hline
+   Distinguish the possible from the impossible & \\\hline   
+  Demonstrate claims (conclusively) & \\\hline
+     Obtain more knowledge & \\ \hline
+      Find out whether hypothesis is false & \\\hline
+     Increase confidence in experimental results & \\\hline
+    Do not know why & ``we do not accomplish anything''\\\hline
+        Nothing desirable & \\\hline        
+    Nothing of relevance & \\\hline 
     \end{tabular}
     \end{table}
 
@@ -484,6 +527,23 @@
   \caption{Categories from What do students think a proof is for}
   \label{fig:ForWhat}
   \end{figure}
+
+   \begin{table}[ht]
+   \caption{Critical factors for What do students think proof is for?}
+   \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
+   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
+   \end{tabular}
+   \end{table}
+
+   Because there is an unusually large number of categories in response to this research question, the categories have been grouped.
+   The least sophisticated group of categories includes misunderstandings, not just incomplete, but containing misinformation. The critical aspect differentiating this group of categories from the next group is the idea that proof establishes, unequivocally, the truth value of the claim. The context of assumptions in which the claim is made is included in the understanding of the claim, in so far as that context has supplied warrants for the argument.
+
+   The next group of categories are true conceptualizations that are not very detailed, such as, a proof demonstrates that a claim is true.
+   The group of categories beyond that 
 
  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.
 
@@ -507,7 +567,7 @@
  Category & Representative\\\hline\hline
 
 
-  some students do not see any point to proof&
+  some students do not see any point to proof &
  They teach it to us because they were mathematicians and they like it.\\
 
 
@@ -804,7 +864,7 @@ A: um, he-he, well, i did find myself doing proofs, they were silly proofs, just
                 \end{longtable}
 \subsection{Phenomenographic Analysis of Whether students exhibit consequences of inability (such as avoiding recursion)}
 
-There is only one category from the orthodox phenomenographic method: Students do not know when they can apply recursion. They felt they were asked to produce recursive algorithms in situations in which it applied, and that they could. They felt that such situations did not occur subsequently. Some asked their employed friends who echoed this opinion.
+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.
 
@@ -916,7 +976,7 @@ Some students claimed to know how to write recursive algorithms but said they ne
               \end{longtable}
   \subsection{Phenomenographic Analysis of How familiar and/or comfortable are students with different (specific) proof techniques: induction, construction, contradiction?}
 
-  This question was not pursued with the orthodox phenomenographic method.
+  This question was not pursued with the traditional 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"
 
@@ -1056,7 +1116,7 @@ Some students claimed to know how to write recursive algorithms but said they ne
 
   \subsection{Phenomenographic Analysis of Which  structural elements students notice in proofs}
 
-  The categories developed in the orthodox phenomenographic analysis are:
+  The categories developed in the traditional phenomenographic analysis are:
 
      \begin{table}
       \begin{tabular}{|p{6cm}|p{6cm}|}\hline
@@ -1089,6 +1149,19 @@ Some students claimed to know how to write recursive algorithms but said they ne
     \caption{Categories from What structure do students notice in proofs?}
     \label{fig:WhatStructure}
     \end{figure}
+
+
+    Critical aspects are listed in Table \ref{tab:critWhatStruct}.
+
+    \begin{table}[ht]
+    \caption{Critical factors for What structure do students notice in proofs?}
+    \label{tab:critWhatStruct}
+    \begin{tabular}{|p{8cm}|}\hline
+    Critical Aspect\\ \hline\hline
+     The parts of proofs include statements, warrants, and lemmas. The idea of scope, as in proof by contradiction in which we make the temporary assumption that the consequence of an implication to be proved is false, is worth mentioning.\\ \hline
+     Statements can be put in sequence. Some sequences are better than others. In particular, only sequences in which the later statements are justified by the former statements, or axioms or premises, are acceptable.\\ \hline
+    \end{tabular}
+    \end{table}
 
 
   Maybe for an ideal, get something from Leslie Lamport's description of using structure.
@@ -1211,7 +1284,7 @@ Some students claimed to know how to write recursive algorithms but said they ne
 
   \subsection{Phenomenographic Analysis of What do students think it takes to make an argument valid?}
 
-  The categories developed in the orthodox phenomenographic analysis are:
+  The categories developed in the traditional phenomenographic analysis are:
 
     \begin{table}
     \begin{tabular}{|p{6cm}|p{6cm}|}\hline
@@ -1239,10 +1312,22 @@ Some students claimed to know how to write recursive algorithms but said they ne
       \begin{figure}
   \centering
   \includegraphics[width=0.7\linewidth]{./Valid}
-  \caption{Categories from what do students think it takes ot make an argument valid?}
+  \caption{Categories from what do students think it takes to make an argument valid?}
   \label{fig:Valid}
   \end{figure}
 
+      Critical aspects are listed in Table \ref{tab:critValid}.
+
+      \begin{table}[ht]
+      \caption{Critical factors for What structure do students think it takes to make an argument valid?}
+      \label{tab:critValid}
+      \begin{tabular}{|p{8cm}|}\hline
+      Critical Aspect\\ \hline\hline
+      Instructors do tell students about valid forms of arguments. There are some students who can recite the names of valid forms, but cannot produce arguments using them. More practice distinguishing valid arguments from invalid ones, and more practice writing arguments in valid forms might help.\\ \hline
+      Students do reiterate valid proofs from class. If the assigned example matches the proof from lecture sufficiently, false positive results for understanding can occur. Students need to pay attention to how the context of definitions and the item to be proved relate to the progression of statements that demonstrates what is to be shown. Practice explicitly providing warrants might help.  \\ \hline
+      \end{tabular}
+      \end{table}
+
 
   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"
@@ -1375,6 +1460,13 @@ and they you split, what do i have to do to get to that point, so you have to ac
 % %evaluation
 
 \subsection{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?}. 
+
+In the traditional phenomenographic analysis, in three research questions the idea of patterns appears. The ideas of process steps and of visualization also feature prominently. Some students seem to be attempting to understand the structure and validity of proofs by using their knowledge of patterns, process steps and visualization.
+This suggests that explicit use of patterns, process steps and visualization, comparing and contrasting these existing ideas with ideas from proof will help students learn the new ideas.
+
+
 definitions vs examples, examples are easier, value of definitions not necessarily appreciated.
 
 Use of examples implies hope that generalization will occur.