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\chapter{Research Perspective and Epistemological Framework}\footnote{In the psychological sense, rather than that of Brouwer}
Part of the research perspective is formed by the goals for what students 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."
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."
Marton developed the phenomenographic research perspective
First there was phenomenography, which focused on the interface between the
student and the material.
\section{Variation Theory}
Then there was variation theory.
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
From McGowan, Tall 2013 Jour Math Behav ``\ldots an instrucotr 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 mathmatically more appropriate way Thompson 1994 p. 32.
Thomposn P N 1994 Students, functions and the undergraduate curriculur in Dubinsky, Schoenfeld and Kaput Research in collegiate mathematics education I, CBMS issue in math education vol 4 pp 21-44.
help and obstacle\\ along with obstacles arising from intuition there exist epistemological obstacles Bachelard 1938 Brosseau 1983 preventing acquisiton of new knowledge. There exist didactic obstacles.
epistemological obstacles
McGowan Tall\cite{Metaphor or met-before 2010 Jour Math Behavior}, The idea met-before (formerly met afore) emphasize that a metaphor relates new knowledge (the 'target') in terms of existing knowledge (the 'source') developed from previous experience, so that new ideas can be related to familiar knowledge already in the grasp of the learner.
Small example of a met-before: Integrated circuits are available at many levels of integhration, from singletons of transistors to tens of millions.
One simple circuit function ins the counter. The number of cluck edges since the last reset, modulo the number given by $2^n$, where $n$ is the number of bits in the counter, is represented digitally, that is, by a voltage level whose domain has been partitioned into that representing 1 and that representing 0. A counter may reset immediately (so to speak) upon the reset signal. This kind is called asynchronous reset. The synchronous reset kind resets after a clock edge occurs during a reset input.
Having, for the purposes of this example, set the context this way, now consider this problem, from Santos-Trigueros\cite{[p. 74]{}}
``Nine counters with digits from 1 through 9 are placed on a table.''
Had we not just been discussing counters in another setting, this sentence might have been understood as intended, more quickly. That is, the context can serve as a distraction, to be overcome. It can help the instructor to realize that students arrive in class, not only lacking wished-for preparation, but also bringing unhelpful contexts.
\subsection{Harel and Sowder}
\subsection{van Hiele Levels}
\subsection{Student Centered}
something about students' perspectives are not always well-matched to their needs
Students might have in mind material they would like to learn, and there may be also a lack of appreciation for material in required courses
Students may have a rate at which they would like to learn --- points at which they would like to pause and integrate new material with things they already know.
\subsection{Social Constructivism}
Attempts at communication, as in conversations about material, are regarded
as helpful to learning.
McGowan Tall 2013 Jour Math Behav
``One student wrote that she knew her answer was correct (it was actually incorrect) because the other members of her group agreed with her. These students consistently evaluated both the numerical expression 'minus a number squared' and a quadratic function with a negative-valued input incorrectly throughout the remaining twelve weeks of the semester. This cautions us to realize that cooperative leaning amongst students who are failing to make sense of the mathematics may reinforce their problematic conceptions rather than reconstruct them'' [p. 533]
Thoug it is easy to assume that communication between practitioners is carried out verbally, there are examples of proofs without words \cite{nelson1993proofs}.
\subsection{Beliefs about Diagrams} %after social constructivism, because people attempt to communicate with diagrams
helpful, hindering, post-conceptual (i.e., they refer to existing concepts, maybe do not convey new ones),
Hilbert's ``who does not'' with a, b, c\\
perceptual to transformative\\
Because sybmolization supports generalization\cite{loewenberg2003mathematical} and operations in mathematics\cite{schoenfeld1998reflections}, and because symbols are used also in efficient communication with others, symbolization is a skill our students need.
\section{Cognitive Science}
Leslie Valiant, in Circuits of the Mind, Oxford University Press [p. 103] points out that representations, for models of cognition, are not all equally learnable.
(in polynomially many steps, p. 104)
Easily learnable representations (of concepts) ``include Boolean conjunctions (e.g., $x_1 \land x_5 \land \bar{x_y}$) and Boolean disjunctions (e.g., $x_1 \lor \bar{x_3} \lor x_8$) \ldots An important class that is not currently learnable is disjunctive normal form (or DNF for short)'', (e.g., $x_1\bar{x_2}x_3 \lor x_1x_2 \lor x_2x_4x_7$), describes a concept whose membership can be attained in one of three ways, in two of which $x_2$ must b true, but in the other of which $x_2$ may be false, so long as $x_1$ and $x_3$ are true.
He goes on to observe these may be learned in stages, stating ``more is required of the teacher or environment than in the simplest case of learning by example'' [p. 104]
He uses the idea later clarified by Marton and Pang\cite{} stating ``a teacher may have to teach the name of this subconcept and then identify positive and negative examples of it'' [p. 104].
`` In this context, learning theory can be thought of as defining the granularity with which learning can proceed without intervention \ldots the largest chunks of information that can be learned feasibly without their having to be broken up into smaller chunks'' [p. 104]
(Combine this with the approximately 7 chunks in short term memory?)
Generalization and analogy are directly addressed in mathematics teaching by assigning students to ``search for connections and extensions of problems''\cite{santos1998instructional}.
\subsection{Analogical Reasoning}
Gentner and Smith\cite{gentner2012analogical} define analogical reasoning as "the ability to perceive and use relational similarity between two situations and events", and have stated that analogical reasoning is fundamental to human cognition.
They state that \cite[p. 131]{gentner2012analogical} ``Analogy is often the most effective way for people to learn a new relational abstraction; this makes it highly valuable in education.''
Because we wish to obtain the value inherent in reasoning by analogy, we note that it depends upon recognition of relationships, and abstraction, to compare relationships at a level divested of some specifics.
Abstraction, for students of computer science, has been observed to be difficult to learn\cite{or2004cognitive} in that context.
Nevertheless, application of proverbs, such as "Don't cry wolf.", is routinely expected in education of children\cite{lutzer1988comprehension}.
Or-Bach and Lavy show empirical data and provide insight into the difficulties of computer science students who have trouble extracting common features from a problem statement that emphasizes differences, and promoting those to a more general class, while maintaining the differences in the more specific classes.
The relationship from one class to a related class in an inheritance hierarchy, motivated as it has been by code reuse, is more stereotyped than the relationships in proverbs, which are not restricted to generalization/specialization. So, we should be careful about generalizing the difficulty students of computer science have with abstraction.
Gentner and Smith go on to say\cite[p. 131]{gentner2012analogical} that analogical reasoning is characterized by retrieval, in which a current topic in working memory may remind a person of a prior analogous situation in long term memory; mapping, which involves aligning the representations and projecting inferences from one analog to another; and evaluation, which judges the success of the alignment of the representations and inferences.
Thus we see that the relationships are key in analogical reasoning, compared with being stereotyped in establishing inheritance hierarchies.
Gentner and Smith\cite[p. 133]{gentner2012analogical} remark that "Another benefit of analogy is \textit{abstraction}: that is, we may derive a more general understanding based on abstracting the common relational pattern." and "analogies can also call attention to certain differences between the analogs."
Though we might wish to have people readily retrieve knowledge that would, by analogy, be helpful to solving a current problem, Gick and Holyoak\cite{gick1980analogical} showed that people do not always retrieve the knowledge they have, rendering it, at least temporarily and for this purpose, what Alfred North Whitehead called "inert knowledge"\cite{whitehead1959aims}.
Gentner and Toupin \cite{gentner1986systematicity} have observed, however that, older children (and not younger children) benefited from systematicity: a summary statement of the structure of the relationships. There is a shift that can occur from focussing on objects to focusing on relatiohships, called a "relational shift", which has been the subject of research\cite{gentner1988metaphor,rattermann1998more,bulloch2009makes}.
Dunbar\cite{dunbar2000scientists} outlines three important strategies that scientists use: attention to unexpected findings, analogic reasoning, and distributed reasoning. Dunbar states\cite[p. 54]{dunbar2000scientists} "our analyses suggest that analogy is a very powerful way of filling in gaps in current knowledge and suggesting experimental strategies that scientists should use" and "If scientific reasoning is viewed as a search in a problem space, then analogy allows the scientist to leap to different parts of the space rather than slowly searching through it until they find a solution".
Day and Gentner\cite{day2007nonintentional} showed in
Day and Gentner\cite[p. 41]{day2007nonintentional}
"Gentner and Medina proposed that
schemas and other abstractions are often derived via a
process of repeated analogizing over instances (see also
Cheng \& Holyoak, 1985)."
Day and Gentner\cite[p. 41]{day2007nonintentional}"The goal of this research was to investigate an important open question: Can a single prior instance influence how a new episode is understood, and if so, does it do
so by using a structurally sensitive mapping process, as
in analogy?"
Day and Gentner\cite[p. 42]{day2007nonintentional}"The results are consistent with the claim that individuals
may use a single prior instance as a source for nonintentional inference based on structural commonalities. The
pattern of inferences is what would be expected if participants were structurally aligning the two representations
and drawing inferences about the target from relationally
similar aspects of the base. Participants' responses that the
inferred information had actually been stated in the target
story suggest that these inferences were not deliberately
considered and evaluated, but rather were spontaneously
incorporated into the target representations as they were
being created."
Generalization is thought to result when multiple instances of analogies, sharing the same structure of relationships, have been considered. *who was I reading before kowatari?"
Ball states\cite[p. 38]{loewenberg2003mathematical} "Generalization involves searching for patterns, structures, and relationships in data or mathematical symbols. These patterns, structure, and relationships transcend the particulars of the data or symbols and point to more--general conclusions that can be made about all data or symbols in a particular class. Hypothesizing and testing generalizations about observations or data is a critical part of problem solving."
She continues \cite[p. 38]{loewenberg2003mathematical} "In one of the simpler common exercises designed to develop young students' capabilities to generalize, students are presented with a series of numbers and are asked to predict what the next number in the series will be. \ldots Representational practice play an important role in generalizing. For example, being able to represent an odd number as $2k+1$ shows the general structure of an odd number. \ldots Representing the structure using symbolic notation premits a direct view of the general form."
\subsection{Cognitive Neuroscience}
Cognitive neuroscience provides evidence for believing that suspense, and concern for characters, is useful in helping students selectively attend to, and remember at an abstracted level, the material they are seeing.
For example,
Bezdek et al. \cite{Bezdek2015338} have measured brain responses corresponding to attention, and have shown that attention is modulated by the emotional flow of a narrative as it unfolds over time, and that suspense is associated with increased central processing (of the visual field) and decreased peripheral processing. Moreover they have reason to believe that this attention does produce downstream consequences, reflecting encoding of content at a level abstracted from visual features. They have brain metabolism imaging showing decreases in activity that has been associated with mind-wandering\cite{Christoff20098719}.
\subsection{Brain Imaging}
Brain imaging provides evidence for believing that creativity, (generating novel ideas, such as proofs) can be improved by training and takes time corresponding to reorganizing intercortical interactions\cite{kowatari2009neural}. (Look here kowatari more, there is something about predominance of right prefrontal over left.)
According to Huang et al.\cite{huang2015highest}, a large body of research suggests that an abstract cognitive processing style produces greater creativity. Empirically, decades of work have shown that both abstract thinking and creativity are consistently linked to right-hemispheric activation in the brain (e.g., Fink et al., 1996\cite{fink1996brain} and Mihov et al., 2010\cite{Mihov2010442}).
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