\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}