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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}.
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