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\chapter{Research Perspective and Epistemological Framework}
This work is a phenomenographic study, adopting the epistemological framework of social constructivism.
A qualitative study ought, for transparency, to contain a description of the author's viewpoint, or at least, the viewpoint that was exercised in the study. Ernest\cite[p. x]{ernest1994constructing}, crediting von Glasersfeld\cite[p. 41]{von1987learning}, notes:
To introduce epistemological considerations into a discussion of education has always been dynamite. Socrates did it, and he was promptly given hemlock. Giambattista Vico did it in the 18th century, and the philosophical establishment could not bury him fast enough.
The work of others has contributed to both the research perspective and the epistemological framework\footnote{In the psychological sense, rather than that of Brouwer}.
\item Marton developed the phenomenographic research perspective, which is broadly applicable to education.
\item Many researchers have contributed to the literature on educating students in the use of proof.
\item Significant work has been done on teaching computer science students about mathematical proof.
Vygotsky pointed out the ``zone of proximal development'', contrasting what students could achieve on their own with that which students could achieve with the assistance of others. He wrote:\cite[p. 85]{vygotsky1978mind}
In studies of children's mental development it is generally assumed that only those things that children can do on their own are indicative of mental abilities. We give children a battery of tests or a variety of tasks of varying degrees of difficulty, and we judge the extent of their mental development on the basis of how they solve them and at what level of difficulty. On the other hand, if we offer leading questions or show how the problem is to be solved and the child then solves it, or if the teacher initiates the solution and the child completes it or solves it in collaboration with other children -- in short, if the child barely misses an independent solution of the problem -- the solution is not regarded as indicative of his mental development. This 'truth' was familiar and reinforced by common sense. Over a decade even the profoundest thinkers never questioned the assumption; they never entertained the notion that what children can do with the assistance of others might be in some sense even more indicative of their mental development than what they can do alone.
He wrote\cite[p. 86]{vygotsky1978mind} \begin{quote}
When it was it was first shown that the capability of children with equal levels of mental development to learn under a teacher's guidance varied to a high degree, it became apparent that those children were not mentally the same age and that the subsequent course of their learning would obviously be different. This difference \ldots is what we call the zone of proximal development. It is the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers.
By choosing a version of social constructivism that directs attention to finding that zone in which a teacher, sometimes employing collaboration with peers, may most effectively advance the student's development,
we inherit the situation described by Ben-Ari\cite{ben1998constructivism}:
``The task of the teacher is significantly more difficult than in the classical paradigm, because the guidance must be based on the understanding of each student's currently existing cognitive structures.''
This choice about social constructivism affects our perspective by establishing the scope to be addressing specifically how a teacher may assist a student to learn, and how a teacher may make use of collaboration among peers to assist a student to learn. Using Vygotsky's definition of the zone of proximal development affects our perspective by focusing our efforts on matching teaching interventions with students' readiness to benefit from those interventions. By choosing as a basis, this social constructivism, the perspective is directed towards curriculum design, so that the students' zones of proximal development may be encouraged to advance along a trajectory supported by sequenced teaching activities.
\section{Phenomenography and Variation Theory}
% a paragraph explaining what phenomenography is
Phenomenography, as described by Svensson and others\cite{svensson1997theoretical,marton1997learning}, is a research orientation, which has a subject of investigation, that is, conceptions present in a group of students.
These conceptions are of the meaning of some thing, in particular of some thing which is being taught, called, the learning objective.
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.
The learning objective is considered to have parts, called aspects.
The number of aspects might be so large, that not all of them are in the forefront of consideration at all times.
Moreover, a conceptualization of a learning object may be incomplete.
Phenomenography has an intermediate goal, which is to develop categories of these conceptualizations.
Using these categories of conceptualizations, phenomenography obtains a latter goal, critical aspects, also called critical factors.
Phenomenography has an approach, which is aimed at identifying these categories of description and critical aspects.
The approach uses an open, explorative form of data collection.
Data are student verbal productions.
During analysis, the data are interpreted and categorized.
With these verbal productions, human judgment supplies the measure of distance from one verbal production to another.
In the process of categorization features are chosen.
In machine learning, we would say features are learned.
Subsequently human judgment infers relationships between features.
Together, the categories and relationships are called the outcome space.
The goal of the approach is to determine the aspects that are noticed by the students. Different categories of conceptions include more or fewer of these aspects. Different categories of conceptions appreciate the relationships among the aspects with varying degrees of superficiality or profundity.
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 inclusive, or complex, or built including more layers of experience. %did i write this, or is it a quote?
From these categories and relations, researchers infer dimensions of variation.
Critical aspects can be used to inform teaching; in particular a dimension of variation is determined by considering a critical aspect, and considering the consequences of differing values that may be assigned to the critical aspect.
% %Now, what do I want to say about phenomenography
Phenomenography addresses the ideas in the student population.
It draws attention to the variation in how the students experience lessons.
In particular it observes that students may miss aspects of the material in the lessons, and that some students may receive the material in the lessons in a more superficial manner than others.
Phenomenography and its extension, variation theory provide a framework (namely, dimensions of variation) by which researchers can organize the information garnered from student verbal productions.
This organization of the data yields clues, namely critical factors, that can be used to inform teaching practice.
%Now that i've thought about what i want to say, here is a rewrite.
Phenomenography, a research orientation created and developed by Ference Marton and colleagues\cite{svensson1997theoretical}, has as its purpose the discovery and elucidation of categories of conceptions, mainly in the context of education.
For raw data, researchers work with interview transcripts and other verbal productions by learners, extracting text fragments conveying meaning, and exploring possible categorizations of these excerpts.
The categories are about the meaning an object of learning has to an individual.
(Incidentally, as the categories are not of people, but of meanings, a person's multiple meanings attached to a given learning object, may occupy multiple categories.)
It is important that the descriptions of these categories make the categories comparable.
Once the categories are established, comparison is used to determine relationships between pairs of them. Usually, between three to eight categories are created(\cite{Lister}?),
though there are certainly occasions when a categorization with greater cardinality is very useful (see for example the work of Harel and Sowder\cite{harel1998students}).
Because categories are often related such that one contains additional aspects compared to another, subset inclusion is a common relationship.
Because categories are often related such that one reflects deeper understanding than another, greater and lesser depth of understanding is a common relationship.
Phenomenography, and its extension variation theory, are appropriate mental constructs for comprehending student conceptualizations about proof in computer science, because the subject is the understandings, including those obtained after reflection, of proof present in the student population. These understandings have been seen to vary in completeness and in depth. Comparison of categorizations has suggested items to clarify for students, and students have responded positively to these suggested ideas.
Marton and Booth\cite{marton1997learning} extended phenomenography and variation theory to include the idea of structural relevance. Using this idea, we can hope to help students discern an aspect of a learning objective, by pointing out a connection between this relatively new aspect and a pre-existing notion that provides, for example, motivation to take note of this aspect. With proofs we may bring to students' attention that these will be used in future course work to understand algorithm's properties, such as computational complexity.
By attending to the construction of structural relevance as well as to the construction of the curriculum's learning objectives, we can provide help to students, in the form of motivation.
%Social constructivism is used because of the nature of proof as a communication. Interaction between people composing and understanding a proof helps people learn how to compose and how to understand proofs.
%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.
%Svensson tells us that the name phenomenography was coined in 1981 by Ference Marton.
%From Svensson we learn that
%phenomenography is a research orientation.
%This orientation has as its purpose, to describe conceptions,
%or categories of conceptions.
%Moreover, phenomenographic descriptions must be amenable to comparison.
%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.
%The approach in phenomenography is about the way of arriving at the descriptions of conceptions.
%It is a kind of contextual analysis.
%The approach in phenomenography begins with data collection.
%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, or knowledge. upon external reality.
%Because the conceptions in this study 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 thereon.
%Categories of description are developed.
%These cluster together relatively 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}.
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[sic] 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 mortal.\\
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 possibly 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.
This was predicted by Valiant\cite{valiant2000circuits}.
This was verified experimentally by Fried, using single neuron experiments\cite{fried2014single}.
Cognitive neuroscience predicts 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}.
%Describe what
Constructivism, an idea related to social constructivism, entails
the idea that students learn by
aggregating new information onto their present conceptions.
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 (Quoted in \cite{van2011slow})
Brooks and Brooks\cite[p. 4]{brooks1999search} observe: ``Each of us makes sense of our world by synthesizing new experiences into what we have previously come to understand.''
There are many specializations of constructivism.\cite{ernest1994constructivism}
These range over the radical constructivism of von Glasersfeld\cite{von2013radical} and social constructivism.\cite{vygotsky1978mind}
Some debate has gone on, in the mathematics education literature as to which type of constructivism is most suitable for mathematics education.\cite{lerman2012articulating,ben1998constructivism}
\subsection{Piagetian Constructivism}
Piaget\cite{piaget1952origins} studied learning, and proposed the idea that learning was a form of adaptation to the environment. He suggested that learning took place as development, by additions to and modifications of what was already present. This has been called constructivism. Others have been influenced by constructivist ideas.
%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.''
%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.
It appears that within constructivism there are degrees to which the teachers regard their role as facilitative.
Wenger\cite[p. 279--280]{wenger1999communities} notes ``Constructivist theories focus on the processes by which learners build their own mental structures when interacting with an environment. Their pedagogical focus is task-oriented. They favor hands-on, self-directed activities oriented toward design and discovery. They are useful for structuring learning environments, such as simulated worlds, so as to afford the construction of certain conceptual structures through engagement in self-directed tasks.''
Brooks and Brooks\cite{brooks1999search} enumerate, in their description of constructivism applied in the classroom, five overarching principles:
Teachers seek and value their students' points of view.
\item Classroom activities challenge students' suppositions.\\
\item Teachers pose problems of emerging relevance.\\
\item Teachers build lessons around primary concepts and ``big'' ideas.\\
\item Teachers assess student learning in the context of daily teaching.\\
They\cite[p. x]{brooks1999search} go on to say ``Engagement in meaningful work, initiated and mediated by skillful teachers, is the only high road to real thinking and learning.'' They illustrate the student's viewpoint of such interactions: ``teachers \ldots made difficult concepts accessible by seeking to understand what [the student] knew at the time \ldots these remarkable teachers mattered so much because they were less concerned about covering material than they were about helping students connect their current ideas with new ones.''
Ben-Ari\cite{ben1998constructivism} articulated a slightly different version of constructivism:
``Passive learning will likely fail, because each student
brings a different knowledge framework to the
classroom, and will construct new knowledge in a
different manner. Learning must be active: the student
‘must construct knowledge assisted by guidance
from’ the teacher and feedback from other students.''
The value of social interaction is shown clearly by an experiment by Bausell et al.\cite{bausell1972factorial}, carefully comparing tutoring with classroom instruction, in which tutoring produced significantly greater achievement.
Marton contrasts what he calls individual constructivism with his idea of social constructivism.~\cite{marton1997learning}
\subsection{Social Constructivism}
%Need to define social constructivism
There are multiple perspectives on social constructivism.
Wenger\cite[p. 4]{wenger1999communities} states: ``My assumptions as to what matters about learning and as to the nature of knowledge, knowing, and knowers can be succinctly summarized as \ldots
\item We are social beings. Far from being trivially true, this fact is a central aspect of learning.
\item Knowledge is a matter of competence with respect to valued enterprises -- such as singing in tune, discovering scientific facts, fixing machines, writing poetry, being convivial, growing up as a boy or a girl, and so forth
\item Knowing is a matter of participating in the pursuit of such enterprises, that is, of active engagement in the world.
\item Meaning -- our ability to experience the world and our engagement with it as meaningful -- is ultimately what learning is to produce.
As a reflection of these assumptions, the primary focus of this theory is on learning as social participation. \ldots being active participants in the \textit{practices} of social communities and constructing \textit{identities} in relation to these communities.''
Ernest\cite[p. 65]{ernest1994constructing} reports ``social constructivism is used to refer to widely divergent positions. What they share is the notion that the social domain impacts on the developing individual in some formative way, and that the individual constructs (or appropriates) his or her meanings in response to his or her experiences in social contexts.''
He\cite[p. 66]{ernest1994constructing} says, ``In simplified terms, the key distinction among social constructivist theories of learning mathematics is that between individualistic and cognitively based theories (e.g., Piagetian or radical constructivist theories), on the one hand, and socially based theories (e.g., Vygotskian theories of learning mathematics), on the other.''
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.''
Vygotsky defined the ``zone of proximal development. The \textit{zone of proximal development} [extends from] the level as determined by independent problem solving [to] the level of potential development as determined through problem solving under \ldots guidance or in collaboration with more capable peers.~\cite[p. 85--86]{vygotsky1978mind}.
The zone of proximal development characterizes mental development prospectively.''~\cite[p. 87]{vygotsky1978mind}.
%Wishing to nurture the growth of students' understanding via guidance by or collaboration with more capable peers,
%and to facilitate the construction of new knowledge in the context of the knowledge framework brought to the classroom by the students,
% % %From somewhere get support for people have to talk to each other.
% % %Because it is the adopted theoretical perspective/epistemological framework
Marton and Booth\cite[p. 11]{marton1997learning} ``prefer to use 'social constructivism' as an umbrella term for a rather diverse set of research orientations that have in common an emphasis on what surrounds the individual, focusing on relations between individuals, groups, communities, situations, practices, language, culture and society.'' They give further examples\cite[p. 201]{marton1997learning} ``an emphasis on cultural, linguistic, social, historical situations''. Of social constructivism, they\cite[p. 12]{marton1997learning} say ``Individual constructivism is a form of cognitivism in the sense that it regards the outer (act, behavior) as being in need of explanation and the inner (mental acts) as explanatory, whereas, as we have pointed out, the reverse is true of \textit{social constructivism}.'' This understanding must be taken in context, because Marton and Booth go on to say\cite[p. 12]{marton1997learning} that ``in this book the dividing line between 'the outer' and 'the inner' disappears. \ldots The world \ldots is \textit{constituted} as an internal relation between them.''
It can also be observed that a person's inner life can be influenced by the external world, and that the inner life can in turn motivate an individual's behavior, and that the social surround of a person may react to that behavior, with the reaction impacting the individual. Thus, a cyclic feedback situation may result. The idea of a person embedded in a feedback situation seems to encompass more possibilities than either of the ideas Marton places in contrast, ``individual constructivism'' or ``social constructivism''. Moreover, feedback loops, which seem a feasible model for human interaction with society, enjoy more complex dynamics than open loop systems, which may endow them with greater explanatory power.
From this range of inquiry about and understanding of social constructivism, we extract a domain over which we can imagine the style of thinking our students are using.
Some students may wish to think individually, developing an opinion in their own way, exploring relevant materials and activities on their own, before engaging socially in discussion on a topic.
Other students might find exploration of possibilities in a social context helpful, as they are in the process of deciding how they are experiencing and integrating new concepts with what they already know.
In this study we employ a social constructivism into our theoretical and epistemological framework: We are aware the students can vary as to their ways of acquiring, consolidating, relabilizing and reconsolidating meaning.
\section{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 developed 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{Radical Constructivism Applied to Mathematical Education}
%Lerman\cite{lerman2012articulating} tells us
\subsection{Phenomenology Applied to Mathematical Proofs}
Two common relationships among phenomenographic categories are set inclusion and depth of understanding.
When aspects of a phenomenon are omitted, they have not been remembered.
In this section we see that mathematicians have contemplated the memorability of proofs.
In the next section we see that mathematicians have contemplated conveying depth of understanding by good meta-cognitive behavior and a rich knowledge base.
We wish to point out a distinction between
phenomenography and phenomenology.
Phenomenology 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.
%Brief description/definition phenomenology
Phenomenology is about experience, but the experience is examined by the person undergoing it, i.e., described in the first person.
Examples suited to the purpose of this study would be a person reporting that a proof is beautiful, or an algorithm is sleek.
\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.
You can’t perform that action at this time.