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\chapter{Methodology} | |
Knowledge about how students conceptualize has a qualitative nature. For | |
qualitative research, methodology varies, but has standard parts: design of the | |
study, sources and their selection, data, the process of analysis, the interpretation, | |
and the approach to validation. Sample selection is recorded and reported | |
so that others may judge transferability to their own context. Interviews are | |
the principle technique used by phenomenographical research. Documents | |
can also be used. Normal conduct of teaching can also provide data that can | |
be used, if in an anonymous, aggregate form. Both deductive and inductive | |
analysis provide qualitative data. | |
\section{Design of the Studies} | |
We | |
conducted over 30 interviews. | |
Our interview participants included undergraduate and graduate students of (\textbf{how do we want to say this}). Most of the graduate student interviews were among teaching assistants in courses that taught and/or used proofs. We also included faculty of courses that involved proofs. | |
Information learned in tutoring and lecturing inspired the research questions. | |
We used exams to study errors in application of the pumping lemma for regular | |
languages. We used early interviews to explore proof, adapting to the | |
student preference for proof by mathematical induction and incorporating the use of recursive | |
algorithms. We used homework | |
to observe student attempts at proofs. | |
We used later interviews to investigate the remaining questions mentioned earlier. | |
\textbf{Need to make this true: We used homework to observe student familiarity/facility | |
with different (specific) proof techniques: induction, construction, contradiction, | |
and what students think it takes to make an argument valid.} | |
\section{Sample Selection} | |
\textbf{Need even more detail than what's here. It could alternatively be put in subsections.} | |
Students from the University of Connecticut who have taken or are taking the | |
relevant courses were offered the opportunity to be interviewed. The students | |
who volunteered were mostly male, mostly traditionally aged undergraduates, | |
though some graduate students also volunteered. Some students were | |
domestic, and some international. Some students were African-American, | |
some Asian, some Caucasian, some Latino/a, some with learning disabilities | |
such as being diagnosed as on the autistic spectrum. | |
%\subsection{Proofs Using the Pumping Lemma for Regular Languages} | |
The participants for the study of proofs using the pumping lemma for regular languages were | |
%In a recent course offering to | |
forty-two students, of whom thirty-four were men and eight women, | |
forty-one traditional aged, | |
%one former Marine somewhat older, one collegiate athlete (a | |
%woman), | |
there were three students having Latin-heritage surnames, 1/4 of the | |
students had Asian heritage, 2 had African heritage, and 8 were international | |
students. Each student individually took the final exam. A choice among | |
five questions was part of the final exam; one required applying the pumping | |
lemma. Half the students (21/42) selected this problem. These were 17 men | |
and 4 women. Three quarters of those (15/42) selecting the pumping lemma | |
got it wrong. These students, who chose the pumping lemma problem and | |
subsequently erred on it, form the population of our study. | |
%\subsection{Proofs by Mathematic Induction} | |
The participants for the study of proof by mathematic induction | |
%We studied students who | |
were taking, or | |
% who | |
had recently taken, a course | |
on Discrete Systems required of all computer science, and computer science and | |
engineering students. | |
Volunteers were solicited from all students attending the Discrete Systems | |
courses. | |
Interviews of eleven students were transcribed for this study. Participants | |
included 2 women and 9 men. Two were international students, a third was a | |
recent immigrant. | |
%\subsection{Domain, Range, Mapping, Relation, Function, Equivalence in Proofs} | |
For the study about domain, range, mapping, relation, function and equivalence in proofs, students | |
%Students | |
taking, or having taken, discrete systems, especially students who | |
had sought help while taking introductory object oriented programming volunteered | |
to be interviewed. | |
\section{Data Collection} | |
Our corpus included interview transcripts, homework, practice and real tests, | |
observations from individual tutoring sessions, and group help sessions. Interview | |
transcripts were analyzed with thematic analysis. Homework, practice | |
and real tests were analyzed for proof attempts. Data from individual tutoring | |
sessions and group help sessions was also informative. Aggregate, anonymous | |
data was used. | |
\subsection{Interviews} | |
\subsection{Documents} | |
\subsubsection{Proofs Using the Pumping Lemma for Regular Languages} | |
The study was carried out on the exam documents. The interpretation was informed | |
by remembering events that occurred in the natural conduct of lectures, | |
help sessions and tutoring. | |
One method of assessing whether students understood the ease of application | |
of the pumping lemma to a language to be proved not regular was offering a | |
choice between using the Myhill-Nerode theorem with a strong hint or using | |
the pumping lemma. The pumping lemma problem, which could very easily | |
have been solved by application of the Myhill-Nerode theorem, especially with | |
the supplied hint, was designed, when tackled with the pumping lemma, to | |
require, for each possible segmentation, a different value of $i$ (the number of | |
repetitions) that would create a string outside of the language. The intent was | |
to separate students who understood the meaning of the equation's symbols, | |
and the equation itself, from those students engaged in a manipulation with at | |
most superficial understanding. | |
\subsubsection{Proofs by Mathematic Induction} | |
Interviews were solicited in class by general announcement, and by email. | |
Interviews were conducted in person, using a voice recorder. No further | |
interview script, beyond these following few questions, was used. The interviews | |
began with a general invitation to discuss students' experience with and | |
thoughts on proofs from any time, such as high school, generally starting with | |
\begin{itemize} | |
\item ``Tell me anything that comes to your mind on the subject of using proofs, | |
creating proofs, things like that.'' | |
\end{itemize} | |
and then following up with appropriate questions to get the students to elaborate | |
on their answers. | |
Additional questions from the script that were used when appropriate included | |
\begin{itemize} | |
\item ``Why do you think proofs are included in the computer science curriculum?'', | |
\item ``Do you like creating proofs?'' | |
\end{itemize} | |
and, after proof by induction was discussed, | |
\begin{itemize} | |
\item “Do you see any relation between proof by induction and recursive algorithms?”. | |
\end{itemize} | |
Almost every student introduced and described proof by mathematic induction as experienced | |
in their current or recent class. | |
\section{Expanded semi-structured interview protocol for domain, range, language, equivalence class in Proofs} | |
\section{Expanded semi-structured interview protocol for definitions, language, reasoning in Proofs} | |
\section{Data Analysis} | |
Describing how analysis was done in detail is really important. | |
How do you do phenomenography? | |
Is this the way everything was analyzed? | |
Marton and Booth\cite[p. 133]{marton1997learning} describe a desirable analysis technique: | |
Apply the principle of focusing on one aspect of the object and seeking its dimension of variation while holding other aspects frozen. %partial derivative | |
Remember to apply both perspectives, that pertaining to the individual and that pertaining to the collective. | |
Establish a perspective with boundaries, within which looking for variation. | |
\begin{enumerate} | |
\item seartch for extracts from data, that might pertain to perspective | |
\item inspect them in context of own interview | |
\item inspect them in context of other extracts all interviews on the same theme | |
\end{enumerate} | |
\begin{enumerate} | |
\item select one aspect of the phenomenon and inspect across all subjects | |
\item select another aspect | |
\item whole interview -- to see where these two aspects lie relative to other aspects, and to background | |
\end{enumerate} | |
\begin{enumerate} | |
\item all of research problems, one problem at a time, whole transcripts that have particularly interesting ways of handling problem | |
\end{enumerate} | |
Keep going, clarity will emerge. | |
Result: identify a number of qualitatively different ways in which phenomenon has been experienced (not forgetting different methods of expression)\cite[p. 133]{marton1997learning}. | |
Overlap of the material at the collective level is expected. | |
assume that what people say is logical from their point of view\cite[p. 134]{marton1997learning}, citing Smedlund\cite{smedslund1970circular} | |
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 | |
read texts, including transcripts, looked for “units of meaning”, and extracted | |
these phrases. Deductive categorization began with defined categories, and | |
sorted data into them. Inductive categorization “learned” the categories, in | |
the sense of machine learning, which is to say, the categories were determined | |
from the data, as features and relationships found among the data suggested | |
more and less closely related elements of the data. A check on the development | |
of categories compared the categories with the collection of units of meaning. | |
Each category was named by either an actual unit of meaning (obtained during | |
open coding) or a synonym (developed to capture the essence of the category). | |
A memo was written to capture the summary meaning of the category. | |
Next a process called axial coding, found in the literature on grounded theory, | |
\cite{strauss1990basics,kendall1999axial,glaser2008conceptualization} was applied. This process considered each category in turn as a central | |
hub; attention focused on pairwise relations between that central category | |
with each of the others. The strength and character of the posited relationship | |
between each pair of categories was assessed. On the basis of the relationships | |
characterized in this exercise, the categories with the strongest interesting relationships | |
were promoted to main themes. A diagram showing the main | |
themes and their relationships, qualified by the other, subsidiary themes and | |
the relationships between the subsidiary and main themes was prepared to | |
present the findings. Using the process of constant comparison, the structure | |
of these relationships was reviewed in the light of the meanings of the categories. | |
A memo was written about each relationship in the diagram, referring | |
to the meaning of the categories and declaring the meaning of the relationship. | |
A narrative was written to capture the content of the diagram. Using the | |
process of constant comparison, the narrative was reviewed to see whether it | |
captured the sense of the diagram. Units of meaning were compared with the | |
narrative and their original context, to see whether the narrative seemed to | |
capture the meaning. The products of the analysis were the narrative and the | |
diagram. |