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updated chapter 1 after meeting 20150922
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140 changes: 108 additions & 32 deletions ch1.tex
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\chapter{Introduction}
% %the point is to motivate the rest of the work, pique interest in proofs in CS

Proofs are taught in the computer science curriculum, with the intent that students will be able to use these to ascertain whether the algorithms they have learned are applicable to the problems they wish to solve. Moreover, it is desirable that students will be able to show that the resource consumption of their implementation is suitable for the circumstances of their implementations. Certainly there may be additional reasons.

It can be the case that not every student gains the whole of the capabilities that we might wish. Insight into the partial understandings students do achieve might enlighten us as to specific aspects of the material, increased emphasis on which might improve student learning.

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.
Expand All @@ -10,7 +14,32 @@ 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.
We explain how phenomenography and variation theory are suited to investigate the questions we have pursued.

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{Research Questions}

Expand Down Expand Up @@ -54,8 +83,7 @@ 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.
split between what students perform for assessment, and what students perform for their own purposes 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,
Expand All @@ -82,21 +110,39 @@ 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.
Critical research, while qualitative, goes beyond interpretation, as it intends to reform the object of its attention.
Each of the above is a type of qualitative research associated with a qualitative research methodology.
One type of research, which differs from those above in that it delimits its scope to the description of ways of experiencing, by a student, the communication from a source of instruction, is phenomenography.
While the formerly mentioned types of qualitative research are associated with a qualitative research methodolgy, phenomenography is characterized by an aim, namely to describe people's conceptions.~\cite{svensson1997theoretical}.
It is necessary that the descriptions be subject to comparison and to systematization. As for the methods associated with phenomenography, Svensson states~\cite[p. 161--162]{svensson1997theoretical} ""development of methods was included as a main aim of the research and the development of methods was an integrated part of the tradition. However, methods have been dealt with in a problem-solving attitude rather than by giving prescriptions. The methods have been considered to have been derived from the basis of the general aim, the general character of the phenomena of conceptions and also from the basis of the specifics of the phenomena and the situation under investigation. Therefore phenomenographic research represents a research approach in a double sense. It mens an emphasis on approaching the research objects in the sense of creating methods adapted to the objects. \ldots The approach to describing conceptions is closely related to the view of the research objects and is not a system of generally defined methods. The most significant characteristics of the approach are the aiming at categories of description, the open explorative form of data collection and the interpretive character of the analysis of data."

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}
Phenomenography is an empirical research approach.~\cite{svensson1997theoretical}
Svensson states ~\cite[p. 169--171]{svensson1997theoretical} that phenomenography's "methodological assumptions also tend to have generality. The most central characteristics are the explorative character of the data collection and the contextual analytic character of the treatment of data. \ldots The analysis, then must not only mean an aggregation of specific data within generally given interpretations, but a delimitation of specific data related to each other as referring to parts of the same phenomena.\ldots The content is, then, not primarily considered in terms of meaning of linguistic units, but from the point of view of expressing a relation to parts of the world. \ldots scientific knowledge about conceptions is based on differentiation, abstraction, reduction and comparison of meaning".

Booth~\cite{booth2001learning} states "When I say `methodology' I mean that phenomenographic studies are
based on certain principles but that the actual methods used vary according to
the specific question being addressed; it is not prescriptive. Phenomenography
is not an experimental methodology -- phenomenographers do not set up
controlled trials and attempt to measure the results of change; but it is
relatively naturalistic, researchers engage with the learners themselves, in
close proximity (both spatial and conceptual) to the learning situations they
find themselves in. Nor is it a quantitative methodology -- the results are
descriptive and lie at a collective level, in the sense that individuals are seen as
contributing fragments of data that together constitute a whole and collective
experience, which can be subjected to research analysis."

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.
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. (See Table \ref{WaysLearning}.)

\begin{table}[h]
\begin{table}[h]\label{WaysLearning}
\caption{Distinct Ways of Experiencing Learning, from Marton and Booth~\cite{marton1997learning}.}
\begin{enumerate}
\item learn as increase knowledge
Expand All @@ -122,37 +168,67 @@ Wittgenstein said \cite[p. 19--20]{wittgenstein1989wittgenstein} "To understand
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.

\subsection{Relevance Structure}
%define it, add flesh, define/mention part of phenomenography
Relevance structure is a part of phenomenography: Marton and Booth state~\cite[p. 143]{marton1997learning} "Each situation, whether we consider it a learning situation or a situation in which one is applying something learned, has a certain \textit{relevance structure:} the person's experience of what the situation calls for, what it demands. It is a sense of aim, of direction, in relation to which different aspects of the situation appear more or less relevant. It is the way the learner experiences the situation as a whole, \ldots that renders the perspective in its component parts."

Knowledge may alter the perspective with which students view new material. For example, students who know they are about to work on projects, in which they are expected to apply new material they are studying, are enabled to receive and organize, mentally, that new material, assisted more or less by its relation to their anticipated use of it.

Marton and Booth suggest that relevance structure may be "the driving force of learning"\cite[p. 145]{marton1997learning} and assert that "its chief mechanism is variation". Changes in the curiosity of the learner, as may be brought about by the presentation of problems, can change the student's capability of experiencing something, and thereby change the student's way of experiencing it, which is learning. Sudden insight is an example of a change in a person's way of experiencing something.

Booth has written on relevance structure in learning computer science.~\cite{booth2001learning}
She writes "In pedagogical terms, the course aims to provide the heterogeneous group
of new students with a relevance structure \ldots a
whole sense of the programme, what it is aimed at, what it demands and where
it will lead -- not only by bringing about genuine needs to use the practical
tools they are learning, but also by raising questions that relate to theoretical
studies to come and to social issues"~\cite[p. 176]{booth2001learning}

There is a connection between the ideas relevance structure and motivation, as illustrated in a remark of Booth's "to create a relevance structure, or motivation if preferred, for the coming
education by offering the students from the start a whole picture -- however
vague and incomplete at the outset -- that encompasses the overall goals of
the programme"~\cite[p. 178]{booth2001learning}

Thota\cite{thota2014programming} refers to the relevance structure when considering "pedagogic principles \ldots that consider how to tie students' experiences to the course goals (relevance structure)" \ldots The two principles of teaching that
are espoused in phenomenographic pedagogy are (a) the
relevance structure -- making the learner personally experience
the relevance of the learning situation, and (b) variation theory -
emphasizing the critical aspects that lead to a change in the learner's understanding of a learning concept."
Thota claims that "The learning that occurs is dependent on how the learner
personally experiences the relevance of the learning situation".\cite[p. 127]{thota2014programming}

Thota, citing Bowden and Marton\cite{bowden2004university} observes "exposure to the critical
aspects of professional situations, likely to be encountered in the
future, enables students to develop holistic capabilities that link
disciplinary knowledge and professional skills."
Thota relates that a relevance structure can be built, in part, by relating and contextualizing learning.\cite[p. 128]{thota2014programming}

Thota reports\cite[p. 131]{thota2014programming}, of students "they become more
engaged in learning when they see the relevance of
programming to their chosen major and future career
needs."

Thota summarizes\cite[p. 131]{thota2014programming} "The building of the structure of relevance can be achieved by:
(a) awareness of pedagogical content knowledge as it should be
understood by learners; (b) ensuring learners reveal their
experience of ways of learning (including the what and how of
learning); (c) relating learning to the learner's own personal
interests and motivations and contextualizing the set assignments
to create authentic situations."

Relevance structure plays a role in motivating student learning.
Relevance structure can be present in a course at different levels of detail.
Present not only at the level of "this test driven development practice might aid your employability", relevance can be shown at the level of an individual algorithm.
For example, the purpose of learning to prove whether a problem has optimal substructure can be to determine whether a dynamic programming approach is suitable for a solution.
Students who do not know that optimal substructure is important for the use of a dynamic programming approach might not make mental connections between these ideas.
Without this connection, it might not occur to a student to
make use of a dynamic programming approach when that is warranted.
It is claimed herein that relevance structure provides another avenue for retrieval of knowledge, lessening the risk of knowledge being inert.

Without the relevancy of optimal substructure for dynamic programming, it might not occur to a student to
make use of a dynamic programming approach in a situation when that is warranted.

I



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6 changes: 5 additions & 1 deletion ch4.tex
@@ -1,6 +1,10 @@
\chapter{Results}

The results of a phenomenographic study comprise a set of categories of description of ways of experiencing (or capability for experiencing [p. 126]) a phenomenon, and relations among those categories.
The results of a phenomenographic study comprise a set of categories of description of ways of experiencing (or capability for experiencing \cite[p. 126]{marton1997learning}) a phenomenon, and relations among those categories.

Booth states\cite{booth2001learning} "The results are communicated as
descriptions of the essential aspects of each category, illustrated by pertinent
extracts from the data".

Marton and Booth\cite[p. 125]{marton1997learning} give criteria for the quality of a set of descriptive categories.
The collective experience, over all participants in the study should be included.
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