ch1.tex

\chapter{Introduction}


The purpose of the research is to discover the understandings of proof we find in the population of students of computer science (and engineering), so as to provide information that might be helpful for teaching.
It is a qualitative study, because we seek the variety in the nature of the various understandings. 
We do not attempt to establish the relative frequencies with which these conceptualizations occur.\\
The analytic lens of the study is phenomenography, including variation theory.\\

In this chapter we
list our research questions addressing conceptions of proof in students of computer science. 
We briefly summarize what distinguishes qualitative research from other research. 
We provide a short description of phenomenography as extended to variation theory. 
We explain how phenomenography / variation theory are suited to investigate the questions we have pursued.

\section{Research Questions}

In order to address the students effectively, it can help to know their preparation and their approach to gaining new knowledge. 
This preparation may include useful ideas, and also may include unhelpful conceptions.
Their approach to learning might not yet include the degree of attentiveness to precision and thoroughness that is appropriate for deductive logic.
Besides their current knowledge and learning approach, their opinion of the structure of relevancy is interest.
We have identified facets of student conceptions surrounding the idea of proof, which led to specific questions.

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}

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.
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.
Thus it may be helpful to supplement what assessments tell us,
about the extent to which the students are absorbing the
knowledge about proof we are trying to impart, 
with information from interviews.
Interviews impose a burden of analysis which is usually too extreme for a lecture class.
Progression across multiple courses is beyond the scope of a lecture class.

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{Qualitative Research}

\begin{quote}
Black and Williams 1998 stated ``When instructors understand what students know and how they think --- and then 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}

How students think, for example, how they approach the study of proof, starting from a preparation in which an appreciation of definitions as foundational for deduction from axioms and premises is not yet present, is a qualitative question. 

Before the existence of categories of description for this preparation, we cannot have measures.
The qualitative approach aims to create categories of description.
These can help phrase questions for quantitative studies.

Qualitative analysis has types; some of these were created to address specific domains of research.
These types include basic qualitative research, phenomenology, ethnography, grounded theory, and narrative analysis,  each of which is interpretive.~\cite{merriam2009qualitative}
Critical research, while qualitative, intends to reform the object of its attention.
One type of research, which delimits its scope to the description of ways of experiencing, by a student, the communication from a source of instruction, is phenomenography. 
Svensson~\cite{svensson1997theoretical} reports that phenomenography was extended to include variation theory.
Variation is key to the phenomenography/variation theory qualitative research approach.
Variation of the communicated information is deemed necessary for discernment by the student of what is being mentioned.
Variation among the students, in their approach to receiving information is considered predictive of their success in learning.

\section{Phenomenography with Variation Theory}

Marton and S\"alj\"o~\cite{marton1976qualitative} performed an experiment which showed that a students' approaches to learning have been  predictive of their learning outcome, reiterated in Marton and Booth.~\cite[p. 22]{marton1997learning}
In looking at, and developing categories for, students' ways of experiencing their learning, we may 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{saljo1979learning} has found five qualitatively distinct conceptualizations, and Marton~\cite{marton1997learning} has found six distinct conceptualizations falling into the two overriding, task and objective.

\begin{table}[h]
\caption{Distinct Ways of Experiencing Learning, from Marton and Booth~\cite{marton1997learning}.}
\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}
\end{table}

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

\subsection{Conceptualizations}

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. 




\section{Phenomenography / Variation Theory for these Research Questions}

Conceptualizations, as we have seen above, are important for our research questions, and are a central object of attention for phenomenography.

Phenomenography and variation theory (henceforth "phenomenography") address mental concepts without reference to any sensory modality through which they may have been acquired.~\cite[p. 160]{marton1997learning}
As a deductive, logical argument, a proof is a mental concept that can exist without reference to any sensory modality.

Phenomenography concerns itself with students' approaches to learning. This allows us to hope to effect a change in approach, and thereby effect an improvement in students' outcomes.



\section{Overview}

Chapter 2   discusses the
phenomenographic research perspective, and the epistemological framework.
Chapter 3 discusses the methodology applied in the study, including
sections on sample selection, data collection, and analysis.
Chapter 4 describes the results of the analysis.
Chapter 5 provides interpretation and discussion. 
Chapter 6 discusses validation and reliability. 
Chapter 7 discusses some related work.
Chapter 8 concludes the description of completed work.
Chapter 9 describes some possible future work.