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\chapter{Design of the Study} | |

This work is a qualitative study, the underlying philosophy is constructivist, | |

the research perspective is phenomenography, as extended to variation theory, | |

and the epistemological framework is a layered collection of intellectual disciplines. | |

At the highest level of integration, computer science and mathematics | |

reside, supported by studies in memory and attention, including computational | |

complexity applied to cognitive neuroscience, and neurophysiology. | |

This study is qualitative because the focus is on determining what questions | |

would be posed, in the process of continuous curriculum adaptation and improvement | |

the meaning students are making of their specific educational experiences. | |

\section{Chronology of the Design} | |

The design of this study began while teaching Introduction to the Theory of | |

Computing. While helping students learn the pumping lemma for regular | |

languages, and trying to understand from where the several difficulties arose, | |

I became curious about the bases of these difficulties. One example was that | |

a student felt strongly that a variable, a letter, denoting repetitions in a mathematical | |

formulation, could only stand for a single numeric value, rather than a | |

domain. Subsequently I have learned that symbolization is a category identified | |

by Harel and Sowder \cite{harel1998students}, for students of mathematics learning proofs. Our | |

student is a vignette of our computer science student population harboring | |

some of the same conceptualizations. As a consequence of this opinion, the | |

student felt that showing that a mathematical formulation had a true value | |

was equivalent to demonstrating a true value for a single example, rather than | |

demonstrating a true value for a domain. Here we see evidence for the category | |

Harel and Sowder \cite{harel1998students} call (is it inductive, perceptual?) where an example | |

is thought to provide proof of a universal statement. Later, while helping students | |

study the relationship between context free grammars and pushdown | |

automata, I learned from the students that many of them did not find inductive | |

proofs convincing. Subsequently I have learned that Harel and Sowder \cite{harel1998students} created | |

a category called axiomatic reasoning. In axiomatic reasoning, students | |

begin with accepted information, such as axioms and premises, and apply rules | |

of inference to deduce the desired goal. This category had not always been | |

reached by their students, similarly to ours. As will be seen, later interview data showed, | |

some of our students learn to produce the artifact of a proof by mathematical | |

induction by procedure. They learn the parts, and they supply the parts | |

when asked, but are not themselves convinced. (McGowan and Tall report a similar situation.) This matches with two other | |

categories created by Harel and Sowder \cite{harel1998students}, internalization and interiorization. | |

Still later, when leading a course on ethical reasoning for issues related to computer | |

science, I found that most of the students did not notice that methods | |

of valid deductive argumentation were tools that they might apply to defend | |

their opinions. | |

Thus the idea of exploring the nature of the students' degrees of preparation | |

for understanding and creating proofs appeared. | |

First, interviews about proofs in general were conducted, with a broad interview | |

script. The students almost all selected proofs by mathematical induction. | |

During analysis of these data, a more elaborate interview script was developed, | |

aiming at the ideas of domain, range, relation, mapping, function, the ideas of | |

variable, as in programs and mathematical formulations, and abstraction. | |

Some students emphasized that mathematical definitions are analogous to | |

definitions in natural languages, and that mathematical discourse is carried | |

out in the mathematical language created by these definitions. | |

The capabilities for expression and care bestowed by these definitions invest | |

mathematical reasoning with its persuasive power. | |

Thus both the reasoning processes, using concepts and the clearly defined | |

mathematical concepts together provide the ability of mathematical argumentation | |

to be convincing. Students who appreciated this found it invigorating. | |

Other student had different reactions to definitions. Thus, the role of definitions | |

and language became another area of exploration. | |

The difference between a domain and a single point in a domain can be seen as | |

a level of abstraction. If something is true for a single point in a domain, but is | |

also true for every single point in the domain, then the point can be seen as a | |

generic particular point, representative of the domain. This concept of ability | |

to represent is related to the idea of abstraction. | |

We saw data in this study that affirmed the observations of others, that students | |

do not always easily recognize the possibility of abstraction. | |

\subsection{Parts of the Study} | |

Part of the study was devoted to proofs. | |

Part of the study was aimed at the idea of domain, directed at the concept that | |

though a variable could identify a scalar, it might also represent a set. | |

Part of the study was aimed at the activity of abstraction, because some students | |

exhibited the ability to operate at one level of abstraction, not necessarily a | |

concrete level, yet the ability to traverse between that level of abstraction and | |

a concrete level seemed to be absent. Other students claimed to be able to | |

understand concrete examples with ease, but to encounter difficulty when | |

short variable names were used within the same logical argument. | |

\subsection{Order of Exploration} | |

The order of exploration was data driven, thus the material was sought sometimes | |

in reverse order of the curriculum, almost as if seeking bedrock by starting | |

at a surface, and working downwards. | |

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

We taught the introduction to the theory of computing course from Sipser's | |

third edition\cite{sipser2012introduction}, using chapters 1 through 5 and 7. The pumping lemmas were | |

given emphasis in class, help sessions and tutoring, in homework, exams, and | |

in review. We treated the pumping lemma in the context of logic, emphasizing | |

the inversion of quantifiers. We discovered that some students seem to | |

tire of attending to statements with more than one quantifier, consistent with | |

Devlin \cite{devlin2012mathematical}. We also treated the pumping lemma with diagrams of machines | |

from Sipser's book\cite{sipser2012introduction}. We encouraged student collaboration on all learning | |

activities, including homework. | |

For grading, we used only work (exams) known to be individual. | |

To encourage active learning \cite{prince2004does} beyond using the classroom response system, | |

we assigned participation in a discussion of a specified question, weekly. We | |

discovered that students preferred to have their contributions to these discussions | |

be anonymous to other students. | |

\section{Proofs by Induction} | |

We carried out a qualitative study, inspired by the ideas of Marton et al.\cite{marton1981phenomenography,svensson1997theoretical,marton1997learning,marton2005unit}] on phenomenography and variation theory. We are using qualitative | |

techniques because we seek to be able to describe the nature of the various | |

understandings achieved by the students, rather than the relative frequency | |

with which any particular understanding is obtained. Phenomenography and | |

its extension, variation theory, are applicable to this study, because the variety | |

of outcomes in student understanding can be used to guide future offerings of | |

the course. | |

Marton and other researchers, e.g., Bussey, using phenomenography and variation | |

theory \cite{bussey2013variation} direct attention to the information intended by the instructor | |

for delivery to the student and the information received by the student. These | |

are not necessarily the same: the student may take in material with a different | |

emphasis than intended by the lecturer, such as specifics of an example that are | |

unimportant for illustration of the intended point. Items that seem obvious to | |

the instructor might not be to the student. Bussey et al. observe that “variation | |

theory is a useful framework for guiding qualitative educational research studies | |

that attempt to identify gaps between teaching and learning.” \cite{bussey2013variation}We attempt | |

to address the third goal identified by Bussey et al., ``describe the variation in | |

student understanding of a given object of learning after the learning event | |

has taken place''. We want to identify what Suhonen et al. \cite{suhonen2007applications} call ``critical | |

aspects'', in the area of proof by mathematic induction, those that seem to be conceptually | |

difficult, or are seen not to have been grasped by the students. | |

We used semi-structured interviews with students to learn their conceptualizations | |

of proof. | |

\section{Conceptions of Domain, range, mapping, relation, function, | |

equivalence in Proofs} | |

\section{Conceptions of Definitions, Language, Reasoning in Proofs} | |

\section{Conceptions of Equivalence, Abstraction in Proofs} | |

\section{Instructive Problems} | |

``Instructive problem'':begin with a problem situation that embodies key aspects of the topic, and mathematical techniues are developed as reasonable responses'' This note was made the weekend I was reading research in collegiate mathematics education 3, and I left no reference. | |

\section{Instrument for Initial (Before course) Assessment for Discrete Structures} | |

Discrete structures is the course in the curriculum that (re)acquaints students with proof, which will be used in other courses, including Algorithms and Introduction to the Theory of Computing, and possibly Data Structures, depending upon how it is taught, and possible Software Engineering, depending upon how it is taught. | |

The purpose of the instrument is to determine what level of skill with proof exists in the students, as they arrive. | |

The problems include problem situations posed in a variety of representations: words, and/or symbolically, and/or by figures and/or as pseudocode. | |

The students are asked to, in some cases, match the problems that are expressed in multiple ways, and in some cases provide the missing representation form. | |

This instrument was developed using inspiration from Gibson\cite{gibson1998students}, as well as Nelson\cite{nelson1993proofs}. | |