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\chapter{Introduction}
\begin{quote}
Black and Williams 1998 stated ``When instructors understand what students know and how they think --- and the use that knowledge to make more effective instructional decisions --- significant increases in student learning occur'' \cite{black1998inside}%Black, Paul and Dylan William, Inside the Black box: Raising standards through classroom assessment Granada Learning 1998
\end{quote}
Selden and Selden\cite{kaput1998research} include, in their questions regarding teaching and learning mathematics, that instructors aim for their students to ``achieve the kind of organizing and integrated use of language'' used in the mathematics community.
$<IsThisSo>$While there are many aspects of students' conceptualizations of proofs that are interesting, we concentrate our attention onto proofs that seem to be useful in showing the correctness, progress, termination, safety and resource utilization of algorithms.$</IsThisSo>$
It is important for students of computer science, and of computer science and
engineering (called, in the following, computer science) to comprehend and
apply simple proofs, and to be able to synthesize simple proofs. These skills
are needed because proofs are used to demonstrate the resource needs and
performance effects of algorithms, as well as for safety, liveness, and correctness
/accuracy. Some students, having learned an algorithm, are not certain of
the problem environment in which this kind of algorithm is effective, and as a
result are reluctant to apply the algorithm. It is desirable for students to be able,
correctly, to develop internal conviction, and to ascertain that an algorithm is a
good match for a problem, otherwise their knowledge of the algorithm is less
useful.
It is important for instructors to impart, efficiently and effectively, knowledge
about proof to the students. We will be using phenomenography.
We wish to point out a distinction between
phenomenography and phenomenology.
Phenomenology might be more familiar: it has been used by mathematician
Gian-Carlo Rota to describe the beauty in mathematics, particularly in proofs.
Phenomenology has also been invoked by mathematician Alan Schoenfeld in modeling teachin behavior.
His lesson segments are chosen for phenomological integrity.\cite[p. 91]{kaput1998research}.
He states \cite[p. 91]{kaput1998research} ``develop knowledge and skills, pursue connection, extensions, generalizations to know how to make good conjectures and know how to prove them, have a sense of what it means to understand mathematics and good judgement about when they do. Have the tools that will enable them to do so. That means having a rich knowledge base, a wide range of problem solving strategies and good metacognitive behavior''
He had, earlier on the same page, described metacognitive behavior as reflecting and acting on what you know.
By contrast, phenomenography and its outgrowth, variation
theory, \cite{marton1981phenomenography,svensson1997theoretical,marton1997learning,marton2005unit} provide insight into ways to help students discern specific
points. The points, whose emphasis is conjectured to be most beneficial, are
identified by a qualitative research process. Application of findings about students
of mathematics to students of computer science is fraught by differences
in the preparation and interests related to algorithms. One likely difference is
motivation: students of mathematics know that proof is the principal means
of discourse in their community, but students of computer science might not
be aware of the importance of proof to their work. Not all differences favor
students of mathematics. In particular, the categories internalization and
interiorization of Harel and Sowder’s 1998 model\cite{harel1998students} are apt to be, in students
interested in algorithms, more closely related, than in students of mathematics.
There may be a difference regarding abstraction. Both mathematics and
computer science deal in abstraction, and students in both disciplines struggle
with it. \cite{mason1989mathematical,hazzan2003students}. In mathematics, following Vi\`ete, \cite{viete2006analytic}, single letter variable names
are used. These are thought to support the learning of abstraction, for example, Gray and Tall \cite[p. 121]{gray1994duality} observe ``we want to encompass the growing compressibility of knowledge characteristic of successful mathematicians. Here, not only is a single symbol viewed in a flexible way ''
and
in computer science abstraction, one way to exhibit abstraction is UML diagrams. Because the
``trie'' structure and International Standards Organization ISO standard 11179
are computer science approaches to management of definitions, it could be that
computer science students would be more accessible to noticing the desirability
of concept definitions over concept images (see R\"osken and Rolka, \cite{rosken2007integrating} and
Rasslan and Tall \cite{rasslan2002definitions}). It would be interesting to know whether any of several
approaches reported by Weber [?] could be used, perhaps in modified form, for
instruction of students of computer science. The Action Process Object Schema
approach of Dubinsky \cite{dubinsky2002apos} sounds compatible with computer science students'
interests. An approach due to Leron and Dubinsky uses computer programming
\cite{leron1983structuring}, another \cite{leron1995abstract} is directed more to learning group theory than to learning proof
construction. Also specific to students concerned with algorithms, we may
wish to extend the notion of social constructivism from that of Piaget \cite{}, [?] and
of Vygotsky, [?] where it was necessarily a person with whom the learner was
communicating, and therefore with whom it was necessary to share a basis for
communication, to include a compiler and runtime execution environment, as
students of computing disciplines must also comply with rules (e.g., syntax)
used in these systems. Recalling the work of Papert and Harel\cite{harel1991constructionism}, we might
call this constructivism with constructionism. Constructionism is an approach
to learning in which the person learns through design and programming.
A cluster of related problems exists, which includes what students conceptualizations
are, about some elements of proof they should understand:
\begin{itemize}
\item what internal representations do students use?
\item Is there a gamut of internal representations, and does that help with abstraction?
\item mathematization, which is the ability to represent problems in mathematical
notation
\item interiorization, which is the ability to examine and discuss the process of
creating proof
\item comprehension of simple proofs, which is the ability to see that, and why,
an argument is convincing
\item proof analysis, which includes the ability to analyze simple proofs to
recognize structure
\item problem recognition, which is the ability to see that a problem is one that
matches a known solution technique
\item transformational approach, which is considering the consequences of
varying features of the problem
\item axiomatic approach, which is the exploration of the consequences of
definitions
\item construction of valid arguments, which is to synthesize deductions with
component parts, including warrants
\end{itemize}
We propose to research these questions:
\begin{itemize}
\item What do students think a proof is? (We want to know, do they understand deduction, abstraction, where are they on van Hiele levels )
\item How do students attempt to understand proofs?
\item What do students think a proof is for?
\item What do students use proof for (if anything), in particular in circumstances
other than when assigned?
\item Do students exhibit any consequence of inability in proof, such as, avoiding
using recursion?
\item What kind of structure do students notice, do student make use of, in
proof?
\item How familiar and/or comfortable are students with different (specific)
proof techniques: induction, construction, contradiction?
\item What do students think it takes to make an argument valid?
\end{itemize}
According to Gray and Tall \cite[p. 117]{gray1994duality}, Hiebert and Lefevre observed ``a connected web \ldots a network in which the linking relationships are as prominent as the discrete pieces of information \ldots a unit of conceptual knowledge cannot be an isolated piece of information;
by definition is it part of conceptual knowledge only if the holder recognizes its relationship to other pieces of information ``\cite[p. 3-4]{hiebert2013conceptual}
conceptual knowledge is harder to assess than other kinds of knowledge.
These questions are interesting because with the curriculum we are trying to
build capabilities into the students, that will enable them to tackle various
problems they may encounter. Moreover, we wish the students to develop the
ability to have conviction with an internal source, and to be correct in their
convictions. As new situations emerge, and as students who have graduated
find the occasion to modify an algorithm to a new situation, we want these
individuals to be able to know that their modified algorithms are appropriate.
Thus it is important to know to what extent the students are absorbing the
knowledge about proof we are trying to impart. It is important that they
understand this algorithm-applicability purpose of proof, so that they can
judge applicability for themselves, and it is important to know what hindrances
they are experiencing, so that we can help the students overcome them. It is
important that they recognize that there is structure in proofs, and that they
can architect their own proofs, because we cannot foresee every situation our
students may experience.
Because we are greatly concerned that students should apply their knowledge
of proof to algorithm related contexts they may subsequently encounter, the
split between what is performed for assessment, and what students prefer for
their own use is significant to us.
Phenomenographic research yields critical factors, which are ideas whose emphasis
is thought to be particularly helpful in deepening student understanding.
Thus the relevance of this research to the curriculum is that the work will
generate suggestions about points to emphasize.
Chapter 2 discusses the design of the research study. Chapter 3 discusses the
phenomenographic research perspective, and the epistemological framework.
Chapter 4 discusses the methodologies applied in the several studies, including
sections on sample selection, data collection, techniques of data analysis and
approaches to validity and reliability, including reflection on researcher bias
and assumptions. Chapter 5 describes the unprocessed results of each study.
Chapter 6 discusses data analysis of each study, and the interpretation. Chapter
7 discusses validation and reliability. Chapter 8 discusses related work.
Chapter 9 concludes the description of completed work.
Chapter 10 offers a perspective on future directions.
An appendix contains an assessment instrument for incoming to discrete math.
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