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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} | |

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. We used homework | |

to observe student attempts at proofs. We used later interviews to explore the | |

understanding of proof by mathematical induction and the use of recursive | |

algorithms. We plan to use homework to observe student familiarity/facility | |

with dferent (specific) proof techniques: induction, construction, contradiction, | |

and what students think it takes to make an argument valid. We plan to | |

conduct interviews to investigate the remaining questions mentioned earlier. | |

\section{Sample Selection} | |

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. | |

\section{Proofs Using the Pumping Lemma for Regular Languages} | |

In a recent course offering to forty-two students, of whom thirty-four were men and eight woment, | |

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. | |

\section{Proofs 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. | |

\section{Domain, Range, Mapping, Relation, Function, Equivalence | |

in Proofs} | |

Students taking, or having taken, discrete systems, especially students who | |

had sought help while taking introductory object oriented programming volunteered | |

to be interviewed. | |

\section{Definition, Language, Reasoning} | |

\section{Equivalence Classes, Generic Particular, Abstraction in Proofs} | |

\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. | |

\section{Interviews} | |

\section{Documents} | |

\subsection{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. | |

\section{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} | |

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 focussed 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. | |

\section{Validity and Reliability} | |

We checked for internal consistency and reinforcement, and for external compatibility | |

of our findings with existing educational literature in computer science | |

and in mathematics. We noted the phenomenological work of Gian-Carlo | |

Rota \cite{rota1997phenomenology} who has reported that memory for mathematical proof and its elements | |

is noticeably improved when a proof is deemed to be beautiful. We were encouraged | |

by the overlap in description among interview participants. In the | |

literature of mathematics education, we found researchers [?] reporting quite | |

similar conceptions of proof by mathematical induction in students of mathematics. | |

In the literature of computer science education we found research \cite{booth1997phenomenography} | |

on a different topic, but with similar results. Booth reported categories of | |

conceptions of recursion similar to our categories of conception of proof by | |

mathematical induction. | |

\section{Researcher Bias and Assumptions} | |

Researcher Bias and Assumptions | |

strategies for trustworthy, valid, reliable | |

what about generalizability? (e.g., to people with ASD) | |

\subsection{Proofs Using the Pumping Lemma for Regular Languages} | |

The author believes diagrams aid the abstraction process. The researchers | |

tend to believe that students want to learn, and will try to comprehend and to | |

become able to apply the material, and that the limitations temporarily present | |

in the student can be overcome by explanation and practice. | |

\subsection{Proofs by Induction} | |

\subsection{Domain, Range, Mapping, Relation, Function, Equivalence Relation | |

in Proofs} | |

\subsection{Definitions, Language, Reasoning in Proofs} | |

\subsection{Equivalence Class, Generic Particular, Abstraction in Proofs} |