ch3.tex

\chapter{Research Perspective and Epistemological Framework}

The work of others has contributed to the research perspective and the epistemological framework\footnote{In the psychological sense, rather than that of Brouwer}.

Marton developed the phenomenographic research perspective, which is broadly applicable to education.

Many researchers have contributed to the literature on educating students in the use of proof.

Significant work has been done on teaching computer science students about mathematical proof.







\section{Phenomenography and Variation Theory}

%These (now combined) are important.

%First there was phenomenography, which focused on the interface between the
%student and the material.
 
%Then there was variation theory.
Svensson\cite{svensson1997theoretical} has written on the theoretical foundations of phenomenography. From Svensson we learn that
phenomenography is a research orientation.
This orientation has as its purpose, to describe conceptions. The context in which these conceptions are of interest is mainly, but not entirely, education.
This orientation is associated with an approach; this pair forms a research specialization.
Svensson also tells use that the name phenomenography was coined in 1981 by Ference Marton.
The approach in phenomenography is about the way of arriving at the descriptions of conceptions.
 It is a kind of contextual analysis.
Phenomenographic descriptions are amenable to comparison. These descriptions make use of categories. An open, explorative form of data collection is used. During analysis, the data are interpreted. The nature of the conceptualizations which are collectively the object of the description is the meaning that something has to the individual, i.e., the individual's understanding. Thus it is empirical and subjective. Moreover, the conceptualizations are regarded as occurring in a social context, and are qualitative in nature. Phenomenography does not include a position on the nature of reality.\cite[p. 165]{svensson1997theoretical} Nevertheless, phenomenography is concerned with the relation between dependence of conceptions / knowledge upon external reality. Because our conceptions are about mathematical proof, we may evade the distinctions between ideal and real external world, as we are dealing with mental constructions. However, the social nature of creating convincing arguments ensures that these mental constructions must be shared.

The approach will necessarily involve descriptions of meaning, and of similarities and differences in meaning.
By empirically determining similarities and differences of meaning, we might detect the absence of general agreement on meaning.
Categories of description are developed. These cluster more similar conceptualizations, so that distinctions among categories emerge. Because the category represents its members, abstraction occurs, reducing the number of individual ideas being considered, and summarizing the data.

Dahlin\cite[p. 328]{dahlin2007enriching} recounts that variation theory was a development upon phenomenography that brought in dynamic elements to the description of conceptualizations. "The concepts of discernment, variation and simultaneity are the core of variation theory. In order for learning to take place, the learner has to discern a critical aspect or dimension of variation in the phenomenon; she has to see how this aspect can vary; and she has to become simultaneously aware of the possible 'values' along this dimension of variation in order to compare them."

Variation is seen as occurring among conceptualizations, and as occurring during the learning. That is, the teacher may emphasize variation of an aspect of the material being taught, and may emphasize that values taken on along this dimension of variation are significant for the material being taught. This emphasis serves to help students discern not only the dimension of variation, but the factor that is changing; change of the factor calls attention to the factor. Were that factor constant, it might not be noticed. Variation among the categories of description extends the outcome space, such that more distinct conceptualizations are found.\cite[p. 124--125]{marton1997learning}. The outcome space is a complex of categories of description, capturing different ways of experiencing, comprising distinct groupings of aspects of the phenomenon and relationships among them. Often, but not always, in the form of set inclusion, these relationships can capture conceptualizations that are more includsive, or complex, or built including more layers of experience.


Variation theory\cite{marton2013meanings}  suggests that critical aspects, which are particular ideas, are necessary\cite{marton2006some} for meaning making (understanding) to progress from one level of conceptualization to a more advanced level. 

The research approach associated with variation theory has the goal of identifying these specific ideas, which, on the basis of an identification of the conceptualizations present in a student population, empirically  are seen to differentiate one level of conceptualization from another.

Variation theory uses these so-called critical factors. They are emphasized in teaching, specifically by varying them, and considering the consequences. For example, we may highlight the significance of climate zones by considering the  variation in annual rainfall from one zone to another. We may make salient the distinction between the ideas of language as contrasted with speech, the difference between speaking (in general), and speaking in a specific language (in particular) by acquainting children with the existence of a second language. (In the context of only one language, the distinction still exists, but might not be so readily described or learned.)
 
Marton and Pang\cite{marton2006some} "identify some necessary conditions of learning. To learn
something, the learner must discern what is to be learned (the object of learning). Discerning
the object of learning amounts to discerning its critical aspects. To discern an
aspect, the learner must experience potential alternatives, that is, variation in a dimension
corresponding to that aspect, against the background of invariance in other aspects
of the same object of learning. (One could not discern the color of things, for instance,
if there was only one color.) The study results illustrate that what students learn in a sequence
of lessons is indeed a function of the pattern of variation and invariance constituted
in that sequence. All teachers make use of variation and invariance in their teaching,
but this study shows that teachers informed by a systematic framework do it more
systematically, with striking effects on their students' learning."

\subsection{An example of applying variation theory}

A commonly used example of a proof utilizing one application of modus ponens is:

All men are motal.\\
Socrates is a man.\\
Socrates is mortal.\\

Variation theory tells us we must vary critical factors, for students to discern them. Some examples of variation are:\\

Some men are mortal.\\
Socrates is a man.\\
Socrates is mortal, maybe, but not necessarily. \\

The quantifier "All" matters. We don't get the desired result when we use some.  When we have removed insignificant items from our proof, what's left matters. It's easy to find elements to vary that will affect the outcome.

All men are mortal.\\
John Doe is a man.\\
Socrates is mortal.\\

What's different here is, we have lost the warrant for Socrates being mortal. Without that, we cannot know for sure that Socrates is mortal.

All men are mortal.\\
Socrates is a person.\\
Socrates is mortal.\\

This time we've kept our attention on Socrates, but we have lost the warrant. To have a warrant, we must remain within the domain granted by the axiom.

All men are mortal.\\
Socrates is a man.\\
Socrates is an orator.\\

The final statement, though true, is not justified by any warrant.

All men are mortal.\\
Socrates is a man.\\
Socrates is mortal.

 
\subsection{Variation Theory and Conjunctions}

Marton and Booth\cite{marton1997learning} have observed that increased differentiation, i.e., specialization, and also integration in the ways in which we experience the world are the results of learning.

The mind quickly learns certain specializations.
We know this from single neuron experiments\cite{fried2014single}.
Consistent with this, from cognitive neuroscience we know that specializations that are conjunctions of positive literals of existing concepts are easy to learn and that conjunctions containing literals that are not existing concepts, but are negations of existing concepts may not be\cite{valiant2000circuits}.
By examining the conceptualizations present in the population of learners, we can hope to find clusters from which we can learn features whose values differentiate the clusters. It is these features, called in variation theory "critical factors", which instructors should emphasize, showing in their positive and negative form. Showing this variable in its positive and negative literals, and the effect of this variation on the conjunction being studied, is expected to be very helpful to the students\cite{marton1997learning}.




\section{Constructivism}

%Describe what 
Constructivism entails
the idea that students learn by
aggregating new information onto their present conceptions.

\begin{quote}
Whilst part of what we perceive comes through our senses from the object before us, another part (and it may be the larger part) always comes out of our own head\\--William James
\end{quote}\cite{van2011slow}

\subsection{Constructivism of Piaget}
Piaget, an eminent evolutionary biologist\cite{dixon200320}, intensively studied his children, making daily notes for over 3000 days, studying the development of their intelligence, the ability through which they adapted to their environment. Piaget focused on developmental breakthroughs demonstrated by his children.[p.28]. He saw their intellectual functioning as adaptation through natural selection, just as in other aspects of biology. Reflexes are identified as the starting point from which knowledge structures that underlie all thinking are built. Similar to an axiom in an axiomatized context, reflexes do not have to be proved/developed prior to use. Reflexes adapt to the environment. The infant learns to associate additional stimuli beyond those present at birth, to retrieve and enact the behavior of feeding. They learn to distinguish which person holds them in a way that, with one person but not the other, corresponds to feeding.

It is the nature of adaptation that implies constructivism. Dixon\cite[p. 30]{dixon200320} reminds us of Piaget's view, that an organism begins with what is present in its brain, and adapts that as a foundation upon which can be built a collection of knowledge.

Dixon\cite[p. 30]{dixon200320} calls our attention to Piaget's observation that "they would exercise their schemas apparently just for the enjoyment of the exercise." Enjoyment, in light of more recent knowledge of the role of dopamine in reinforcing memory, is a key observation.

Dixon\cite[p. 31]{dixon200320} reports Piaget's surmise that "the \ldots is not longer a reflexive island, passively responding to environmental stimulation, but is being coordinated with other activities \ldots there was a certain amount of circularity involved". Those of an engineering bent of mind may observe that positive feedback in a loop in which pleasure reinforces memory will encourage the construction, by adaptation, of preferred behaviors.

Dixon\cite[p. 31]{dixon200320} describes Piaget's categorization of adaptation of reflexes involving only the child, vs. incorporating external objects. The orienting reflex forms the basis from which the child adapts to choosing to pay attention. Children would try to re-create an interesting visual event. Piaget mentions the joy associated with this secondary circular reaction. This adaptation occurs with auditory as well as visual input.  Dixon\cite[p. 32]{dixon200320} reports "Lucienne at once moves her whole body, and especially her feet, to make the noise last. She has a demented expression of mingled fear and pleasure, but she continues."  

In light of today's understanding of the role of cholinergic as well as dopaminergic involvement in memory and attention\cite{}, the observations of mingled fear and pleasure seem prescient.

Subsequently, Dixon\cite[p. 32]{dixon200320} reports "babies start showing that they can do things on purpose \ldots we start seeing the integration of some schemas into the service of other schemas". Piaget arranged experiments whereby babies exhibited means-ends action.  Dixon\cite[p. 32]{dixon200320} reports "What Piaget is describing here is that Jacqueline can use one schema, \ldots to help her enact another schema \ldots. This is intellectual adaptation of the best kind -- getting what you want!"  The emphasis on reward might be due to Dixon, or to Piaget, but the idea of reward is present in both. Constructions, at least some of the time, are associated with reward, in a feedback loop. Moreover, we can see utility in having rewarding combinations be remembered. We can also see the utility of frightening combinations being remembered.

According to Dixon\cite[p. 34]{dixon200320} Piaget believed that babies explore, learning the expand the domain over which their schemas have been seen to operate. In the next, i.e. sixth, substage children exhibit use of their schemas through imagination. The example given, in which Lucienne wishes to open a box, can be seen as evidence of making an analogy. Dixon\cite[p. 34]{dixon200320} reports "She looks at the slit with great attention: then several times in succession, she opens and shuts her mouth, at first slightly, then wider and wider: Apparently Lucienne understands the existence of a  cavity subjacent to the slit and wishes to enlarge the cavity." Dixon\cite[p. 35]{dixon200320} describes "represented the problem in a different way -- using her imagination. Once she removed the problems from its physical form and represented it mentally, she was able to invent a solution that wasn't previously possible. She pretended her mouth was the slit of the matchbox. By bringing this mental image into play, Lucienne was able to manipulate the image in a new way. Specifically, she was able to pretend she was opening and closing the matchbox by opening and closing her mouth. And once she was able to do this, she made the connection that to get the chain out of the matchbox all she had to do was open the matchbox wider than it already was".

 The use of analogy implies the existence of a process that develops, stores and retrieves analogies. One may wonder how analogies are retrieved. 

%Piaget, Vygotsky and Bruner worked with the idea that students learn by
%aggregating new information onto their present conceptions.

%didactical obstacle see McGowan Tall 2010 Jour Math Behav
McGowen and Tall\cite{mcgowen2010metaphor} suggest that "it is even more important to take into account the particular mental structures available to the individual that have been built from experience that the individual has 'met-before'." They say [p. 170] "New experiences that build on prior experiences are much better remembered and what does not fit into prior experience is either not learned or learned temporarily and easily forgotten."

Is the easier remembering related to analogy making? Exactly like what we learned before, in analogy to what we learned before vs. disruptive of what we learned before 

%McGowen and Tall\cite{mcgowen2013flexible}, citing 
Thompson \cite{thompson1994students} states ``\ldots an instructor who fails to understand how his/her students are thinking about a situation will probably speak past their difficulties. Any symbolic talk that assumes students have an image like that of the instructor will not communicate. Students need a different kind of remediation, a remediation that orients them to construct the situation in a mathematically more appropriate way % Thompson 1994 p. 32.
%Thompson P N 1994 Students, functions and the undergraduate curricular in Dubinsky, Schoenfeld and Kaput Research in collegiate mathematics education I, CBMS issue in math education vol 4 pp 21-44.

\subsection{Social Constructivism}
%Need to define social constructivism
Lev Vygotsky founded the idea of social constructivism, which can be summarized as learning is facilitated by interactions in a group.

According to Cole and Scribner\cite[p. 1]{vygotsky1978mind}, Vygotsky "and his colleagues sought to develop a Marxist theory of human intellectual functioning". They say[p. 5--6] that "What Vygotsky sought was a comprehensive approach that would make possible description \textit{and} explanation of higher psychological functions in terms acceptable to natural science. To Vygotsky, explanation meant a great deal. It included identification of the brain mechanisms underlying a particular function; it included a detailed explication of their developmental history to establish the relation between simple and complex forms of what appeared to be the same behavior; and, importantly, it included specification of the societal context in which the behavior developed."

According to Cole and Scribner\cite[p. 6]{vygotsky1978mind}, "In stressing the social origins of language and thinking, Vygotsky was following the lead of influential French sociologists, but to our knowledge he was the first modern psychologist to suggest the mechanisms by which culture becomes a part of each person's nature. Insisting that psychological functions are a product of the brain's activity, he became an early advocate of combining experimental cognitive psychology with neurology and physiology. Finally, by claiming that all of these should be understood in terms of a Marxist theory of the history of human society, he laid the foundation for a unified behavioral science."

According to Cole and Scribner\cite[p. 7]{vygotsky1978mind}, " Vygotsky believed that the internalization of culturally produced sign systems brings about behavioral transformations and forms the bridge between early and later forms of individual development. Thus for Vygotsky, in the tradition of Marx and Engels, the mechanism of individual developmental change is rooted in society and culture."

According to Cole and Scribner\cite[p. 22]{vygotsky1978mind}, Vygotsky, in Tool and  in Child Development, took note of the work of Shapiro and Gerke. "In their view, social experience exerts its effect through imitation; when the child imitates the way adults use tools and objects, she masters the very principle involved in a particular activity. They suggest that repeated actions pile up, one upon another, as in a multi-exposure photograph; the common traits becomes clear and the differences become blurred. \ldots they do not take into account the changes occurring in the internal structure of the child's intellectual operations. "

According to Cole and Scribner\cite[p. 24]{vygotsky1978mind},Vygotsky wrote that "Although practical intelligence and sign use can operate independently of each other in young  children, the dialectical unity of these systems in the human adult is the very essence of complex human behavior. Our analysis accords symbolic activity a specific \textit{organizing} function that penetrates the process of tool use and produces fundamentally new forms of behavior."

According to Cole and Scribner\cite[p. 25--26]{vygotsky1978mind},Vygotsky wrote that "A child's speech is as important as the role of action in attaining the goal. Children not only speak about what they are doing; their speech and action are part of \textit{one and the same complex psychological function}, directed toward the solution of the problem at hand. \ldots Using words \ldots the child achieves a much broader range of activity \ldots  planning future action."

According to Cole and Scribner\cite[p. 36--37]{vygotsky1978mind}, Vygotsky believed that "The possibility of combining elements of the past and presetn visual fields (for instance, tool and goal) in one field of attention leads in turn to a basic reconstruction of another vital function, \textit{memory}. Through verbal formulations of past situations and activities, the child frees himself from the limitations of direct recall; he succeeds in synthesizing the past and present to suit his purposes. The changes that occur in memory are similar to those that occur in the child's perceptual field \ldots The child's memory not only makes fragment of the past more available, but also results in a \textit{new method of uniting the elements of past experience with the present}. Created with the help of speech, the time field for action extends both forward and backward. \ldots This emerging psychological system in the child now encompasses two new functions: \textit{intentions and symbolic representation of purposeful action}".

According to Cole and Scribner\cite[p. 37]{vygotsky1978mind}, Vygotsky noted that "Lewin\cite{lewin1938will} \textbf{right person wrong book} gives a clear-cut definition of voluntary activity as a product of the historical-cultural development of behavior".

Vygotsky believed that speech reorganized perception, and created new relations among psychological functions.\cite[p. 38]{vygotsky1978mind}. He believed  that people, though capable of memory unrelated to words, \textit{eidetic} memory, proceeded to a new memory utilizing signs, and that "these sign operations are the product of specific conditions of \textit{social} development".\cite[p. 39]{vygotsky1978mind}

Vygotsky\cite[p. 49]{vygotsky1978mind} wrote with a change indeveloopmental level there occurs a change not so much in the structre of a single function (whic, for example we may call memory) as in the character of those functions with the aid of which remembering takes place; what changes is the \textit{interfunctional} relations that connect memory with other functions.

An example of how the meaning of signs evolves in a social context is given by  Vygotsky\cite[p. 56]{vygotsky1978mind}: A child reaches for an object, and the gesture is only a reach, but a person responds to the reach with assistance in obtaining the desired object, and the reach acquires the meaning of pointing.
In this way, the meaning is socially constructed.



\section{Heritage from Mathematics Education}
Part of the research perspective is formed by the goals for what students learning proof should know: according to Ball et al.\cite[p, 32 -- 34]{loewenberg2003mathematical} "These activities -- mathematical representation, attentive use of mathematical language and definitions, articulated and reasoned claims, rationally negotiated disagreement, generalizing ideas, and recognizing patterns -- are examples of what we mean by \textit{mathematical practices}. \dots These practices and others are essential for anyone learning and doing mathematics proficiently. \ldots investing in understanding these 'process' dimensions of mathematics could have a high payoff for improving the ability of the nations' schools to help all students develop mathematical proficiency".
Ball goes on to say\cite[p. 37]{loewenberg2003mathematical} "Another critical practice -- the fluent use of symbolic notations -- is included in the domain of representational practice. Mathematics employs a unique and highly developer symbolic language upon which many forms of mathematical work and thinking depend. Symbolic notation allows for precision in expression. It is also efficient -- it compresses complex ideas into a form that makes them easier to comprehend and manipulate. Mathematics learning and use is critically dependent upon one's being able to fluently and flexibly encode ideas and relationships. Equally important is the ability to accurately decode what others have written."

Even more tightly focused on proofs, Ball continues \cite[p. 37--38]{loewenberg2003mathematical} "A second core mathematical practice for which we recommend research and development is the work of justifying claims, solutions, and methods. \textit{Justification} centers on how mathematical knowledge is certified and established as 'knowledge'. Understanding a mathematical idea means both knowing it and also knowing why it is true. For example, knowing that rolling a 7 with two dice is more likely than rolling a 12 is different from being able to explain why this is so. Although 'understanding' is part of contemporary education reform rhetoric, the reasoning of justification, upon which understanding critically depends, is largely missing in much mathematics teaching and learning. Many students, even those at university level, lack not only the capacity to construct proofs -- the mathematician's form of justification -- but even lack an appreciation of what a mathematical proof is."
\subsection{Phenomenology Applied to Mathematical Proofs}

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.
Rota\cite{rota1997phenomenology} points out that proofs that are perceived by mathematicians as beautiful are easier to remember.
\subsection{Phenomenology Applied to  Problem Solving}
Phenomenology has also been invoked by mathematician Alan Schoenfeld in modeling teaching behavior.
His lesson segments are chosen for phenomenological 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 judgment 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 meta-cognitive behavior''
He had, earlier on the same page, described meta-cognitive behavior as reflecting and acting on what you know.

\section{Cognitive Science}
Cognitive science includes modeling plausible functioning of the brain. It helps us notice constraints that computational complexity illuminates, on how learning, including memory and retrieval of information, might be carried out.

\section{Neuroscience / Neurophysiology}

Though Marton and Booth\cite[p. 114]{marton1997learning} state that description of experience is autonomous from the student's nervous system, nevertheless we believe that supportive evidence from natural science can be of interest.

In education with assessment, instructors transmit information towards the student and attempt to elicit signs of learning from the student.
Neuroscience enables us to investigate stages of progress in learning that may occur starting from the moments at or just after the transmission through the event of assessment.
For example, single neuron measurements show that conjuncts of variables having positive literals, as described by Valiant\cite{valiant2000circuits} in cognitive science, are learned in one showing, into medial temporal lobe tissue.
Functional magnetic resonance imaging (fMRI) combined with electro-encephalography (EEG) show that slow wave sleep is a process that supports memory formation in other locations in the brain. 
Moreover, these technologies show us that exposing students to educational stimuli of multiple modalities rather than single modalities (e.g., sight and sound vs. only sight) changes the area within the brain in which the information is stored. These show us that transmitting with multiple modalities increases the ease of recall of information. These show us that if we wish information to be subject to recall when relevant, rather than what Whitehead\cite{whitehead1959aims} calls inert knowledge, we should concern ourselves about how the information is transmitted. These provide guidance about how we can plan for better assessment results by facilitating recall, in turn by shaping our transmission strategies.