working on chapter 1

TODOs.tex

@@ -1,6 +1,29 @@
 \newpage
 
 TODOs
+
+read @article{marton1976qualitative,
+  title={On Qualitative Differences in Learning: I—Outcome and process*},
+  author={Marton, Ference and S{\"a}lj{\"o}, Roger},
+  journal={British journal of educational psychology},
+  volume={46},
+  number={1},
+  pages={4--11},
+  year={1976},
+  publisher={Wiley Online Library}
+}
+@article{marton1976aqualitative,
+  title={ON QUALITATIVE DIFFERENCES IN LEARNING -- {II} {O}UTCOME AS A FUNCTION OF THE LEARNER'S CONCEPTION OF THE TASK},
+  author={Marton, Ference and S{\"a}lj{\"o}, R},
+  journal={British Journal of Educational Psychology},
+  volume={46},
+  number={2},
+  pages={115--127},
+  year={1976},
+  publisher={Wiley Online Library}
+}
+figure out which is being quoted in chapter 1
+
 read more on van Hiele levels and their theory
 
 Professor McC asks should this include communication? Is he talking about proofs being communicative, or is he talking about proving that communication occurs among algorithms?

ch1.tex

@@ -1,20 +1,103 @@
 \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).
+
+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 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 
+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}
 
-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.
+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.
 
-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.
+\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
@@ -23,10 +106,11 @@ Marton\cite[p. 36]{marton1997learning} has defined that one conception (of a thi
 \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.
 
@@ -37,7 +121,7 @@ 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.$</IsThisSo>$
+% $<IsThisSo>$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)
@@ -46,8 +130,7 @@ 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. 
+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. 
@@ -63,65 +146,34 @@ about proof to the students. We will be using phenomenography.
 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{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 design of the research study. Chapter 3 discusses the
+Chapter 2   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.
+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.
 
+
 
 

literature.bib

@@ -1,3 +1,9 @@
+@book{merriam2009qualitative,
+  title={Qualitative research: A guide to design and implementation},
+  author={Merriam, Sharan B},
+  year={2009},
+  publisher={John Wiley \& Sons}
+}
 @book{marton2013classroom,
   title={Classroom discourse and the space of learning},
   author={Marton, Ference and Tsui, Amy BM and Chik, Pakey PM and Ko, Po Yuk and Lo, Mun Ling},
@@ -947,6 +953,12 @@ publisher={Elsevier}
   year={1993},
   publisher={Springer}
 }
+@misc{marton1994phenomenography,
+  title={Phenomenography. I T. Hus{\'e}n \& NT Postlethwaite (Red.) International encyclopedia of education, 8 vols (s. 4424-4429)},
+  author={Marton, Ference},
+  year={1994},
+  publisher={Oxford: Pergamon Press}
+}
 @article{ramsden1993theories,
   title={Theories of learning and teaching and the practice of excellence in higher education},
   author={Ramsden, Paul},
@@ -1231,6 +1243,16 @@ year={1987}
 % 1976
 % % % % % % % % % % % % % % % % % % % % % % % %
 @article{marton1976qualitative,
+  title={On Qualitative Differences in Learning: I—Outcome and process*},
+  author={Marton, Ference and S{\"a}lj{\"o}, Roger},
+  journal={British journal of educational psychology},
+  volume={46},
+  number={1},
+  pages={4--11},
+  year={1976},
+  publisher={Wiley Online Library}
+}
+@article{marton1976aqualitative,
   title={ON QUALITATIVE DIFFERENCES IN LEARNING -- {II} {O}UTCOME AS A FUNCTION OF THE LEARNER'S CONCEPTION OF THE TASK},
   author={Marton, Ference and S{\"a}lj{\"o}, R},
   journal={British Journal of Educational Psychology},