ch1.tex

\chapter{Introduction}

This is a qualitative study.\\
Its analytic lens is phenomenography.\\
The research question is what are the conceptions of proof we find in the population of students of computer science (and engineering).

\section{Qualitative Research}

\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}

A student's approach to learning has been seen, empirically\cite[around 35]{marton1997learning} to be predictive of their learning outcome.
In looking at, and developing categories for, students' ways of experiencing their learning, we obtain insight into their approach, and can hope to improve their outcomes.

Marton\cite[p. 36]{marton1997learning} has defined that one conception (of a thing, $x$) differs from another, for the purposes of phenomenography, by the existence of a distinct manner in which participants were found to voice the way they thought about $x$. The categories of conceptions (also, conceptualizations) include two overriding categories,\cite[p. 35]{marton1997learning} the first being "a learning task, some facts to memorize", and the second having as objective "a way to change oneself, to see things in a new light, to relate to earlier learning, and to relate to a (changed) world. At the next level of drawing distinctions, S{\"a}lj{\"o}\cite{1979} has found five qualitatively distinct conceptualizations, and Marton\cite{1993} has found six distinct conceptualizations falling into the two overriding, task and objective.

\begin{enumerate}
\item learn as increase knowledge
\item learn as increase and be able to reproduce knowledge
\item be able to apply new knowledge
\item acquiring new meaning, multiple ways of thinking about things, changed perspective, improved understanding, thinking more logically 
\item modified perspective, multiple perspectives, dynamic perspective
\item changing the person
\end{enumerate}


Marton and Booth\cite[p. 78]{marton1997learning} observe that successive understandings increase in completeness as they move toward a theoretical understanding.


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.

D\"orfler\cite[p. 122]{dorfler2000means} complements the idea of concept, saying " ' What is the concept $xy$?' should be substituted, or at least complemented, by such questions as 'Which actions can be recorded and/or guided by the concept $xy$?' \ldots Learners must indulge in the discourse \ldots mathematical objects \ldots are discursive objects. This means they come into existence exclusively by and within the discourse, even if this discourse ascribes to them existence an properties of an objective and independent character. "

Wittgenstein said \cite[p. 19--20]{wittgenstein1989wittgenstein} "To understand a phrase, we might say, is to understand its use. \ldots Similarly, you only understand an expression when you know how to use it". 


So, we are inquiring into student conceptualizations, as shown by the students' use of their concepts, and by the students' reflections (in interviews) upon their concepts.

 $<IsThisSo>$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,
apply, and synthesize 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. 
We claim herein that 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, which can sometimes be proved, 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.
 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. 
We propose to research these questions:
\begin{itemize}
\item What do students think a proof is? 
\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}

\section{{Research Questions}}

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, in the terminology of Harel and Sowder\cite{harel1998students}, 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 construct % 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.

\section{Phenomenography for these Research Questions}

\section{Overview}

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.