analysis

ch3.tex

@@ -363,7 +363,8 @@ The audio portion of all interviews was collected by electronic recorder and sub
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    \endlastfoot
 
-  % \begin{tabular}{|p{7cm}|p{8.5cm}|}\hline
+  % \begin{tabular}{|p{7cm}|p{8.5cm}|}
+  \hline
    Category & Representative\\\hline\hline
 
 
@@ -531,9 +532,25 @@ The audio portion of all interviews was collected by electronic recorder and sub
 
 
        \subsection{Analysis of Help Session and Tutoring}
+
+       Help sessions were scheduled weekly; attendance was optional. Typically six to twelve students would participate.
+       Originally these were called help sessions, but the demographics of the attendees did not represent the enrolled students.
+       Subsequently the name was changed to consultation sessions.
+       This change had the desired effect, that the population attending better reflected the enrolled students.
+       At these sessions, students would raise topics about which they had questions.
+       Frequently the student would be requested to work at the white board, and leading questions were asked, and problems of very small size were posed, to urge the student along the right path of development of a solution.
+       Occasionally these suggested paths were met with resistance from the students, which is to say, misunderstandings were encountered and discussed.
+       Ideas mentioned in these discussions that were relevant to manuscripts in process at the time were noted, anonymized,  into the manuscripts.
+
+       The idea that a variable used in the pumping lemma could take on only one value, as if it were a single root of linear equation, rather than representing the domain of values possible for strings in a language, was proposed by a student in a help session. The context was a student proposition that examples were sufficient for proofs of universal statements, because the the variable in the pumping lemma could take on only a single value.
+
        some students, who do know that any statement must and can, be
        either true or false, thought implications must be true.
 
+       Due to attention being focused on interacting with students in the normal course of teaching, these field notes are incomplete. 
+
+       One use of such data is that they can give evidence that categories of conceptualization of proof already created in the mathematics literature can be found also in computer science students. This is similar to a deductive rather than inductive process, in that we are aware of the categories created by Harel and Sowder\cite{harel1998students} and by Tall\cite{Tall?} and student utterances that seem well matched to those categories draw our attention to those categories, validating them for students of computer science.
+
 
 
       %  The study proceeded using prior interview experiences to suggest further investigation.
@@ -572,14 +589,26 @@ The audio portion of all interviews was collected by electronic recorder and sub
       We applied phenomenographic analysis by focusing on the aspect of relevance of proof for learning computer science and practicing as a software developer. In this case we had already identified the dimension of variation to be the depth of understanding of why we teach proof. Thus we could select fragments of student utterances and rank them according to depth of understanding. We then presented them in a sequence by rank.
 
 
-
+  \section{Method of Addressing Validation}
+  Triangulation is a technique for increasing the confidence that the results of analysis are reliable.
+
+  In this study we applied triangulation in several ways.
+  We interviewed faculty teaching the courses involving proofs. We interviewed TAs assisting in the courses involving proofs. The students in these courses are from our same population. To get an idea of the background preparation of our students, we substitute taught geometry and algebra II classes in a high school. The high school population was quite similar to our university population, but differed by consisting almost entirely of domestic students, studying in their first language, and by having a larger percentage of women students, and of declared transgender students. Though the community served by this high school is diverse over socio-economic status, this component of diversity is probably greater in our university population.
+
+  Consistency with the work of other researchers is a check on the validity of an analysis.
 
+  In this study we compared our results with those achieved by some other researchers in computer science education and also by some researchers in the mathematics education community.
 
+  Checking possible interpretations is a technique that may aid in increasing confidence in validity. 
+
+  We prepared a list of questions that was addressed by several faculty and several students, that began an examination of the role of specific representation styles (mathematical notation, figures and pseudocode) for proof related problem statements.
 
 
   \section{Method of Presentation of Results}
 
 
-        The product of analysis in a phenomenographic study, is a set of categories, and relationships \textit{among} them.
+        The product of analysis in a phenomenographic study is a set of categories, and relationships \textit{among} them. These categories and relationships are often depicted in a graph. This product may be accompanied by a "thick and rich" narrative description of the categories and relationships. This narrative must be consistent with the individual text fragments, excerpts from transcriptions, field notes or documents obtained for the study. 
+
+        Marton and Booth\cite[p. 135]{marton1997learning} state "in the late stages of analysis, our researcher [has] a sharply structured object of research, with clearly related faces, rich in meaning. She is able to bring into focus now one aspect, now another; she is able to see how they fit together like pieces of a multidimensional jigsaw puzzle; she is able to turn it around and see it against the background of the different situations that it now transcends."
 
-        Marton and Booth\cite[p. 135]{marton1997learning} state "in the late stages of analysis, our researcher [has] a sharply structured object of research, with clearly related faces, rich in meaning. She is able to bring into focus now one aspect, now another; she is able to see how they fit together like pieces of a multidimensional jigsaw puzzle; she is able to turn it around and see it against the background of the different situations that it now transcends."
+        This tells us that the narrative should describe the categories of composition hierarchies found in the students' understandings. The faces or facets of the learning object have their importance and relationships as envisioned by the teacher. The students' conceptualizations may be less complete, contain superfluous items, and differ as to the relationships of the parts, especially by lacking profundity in understanding of relationships.