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\begin{abstract}
Using an analogy, I claim, is saying there is a set of relations among things $a_i$ that we agree upon, furthermore, I might wish to teach that there is a corresponding set of relations among things $b_i$. I might wish to say, use the relation we agree upon for municipalities provide addresses for homes that can be used for surface mail, and I might wish to teach that there is a corresponding provision of addresses for items a computer programmer might wish to use for storage and recall. We can note that addresses make use of a hierarchy of place names, countries, states, cities, streets, street numbers, apartment numbers. We can note that structured data types can correspondingly make use of instances and fields and indices that can be arranged in a tree.

In the absence of abstraction, the surface mail address hierarchy might not pose much more difficulty, but the data structure might, because the fields therein are more subject to change than municipalities. In the absence of abstraction, the comparison between one hierarchical arrangement with another would be more difficult, because it is the structure of the abstraction itself, namely, the choices of features regarded as significant throughout the tree, that is to be recalled and used as a scaffold for the new information.

"An alternative pathway towards abstraction involves recognizing an analogy between two structures in different domains, which then focuses one's attention on the abstract structure they share. This new abstraction then becomes a 'concrete' concept that one can study"~\cite[p449]{hofstadter}.
\end{abstract}

\section{Vertical Integration and Explanation}
It is accepted that, in discrete math, it is helpful to work problems.
We may inquire, what is it about working problems that helps?
We can, at the neurophysiology level, expect that long term potentiation of synapses, for the synapses collocated with the long term memory for the concepts in the problem, is being carried out, as the thinking about the problem occurs.
We can recall that a sense of reward, as might be gained from success at a problem, or pleasantness in a problem statement, helps consolidate the memory for the ideas that have been gained. 
We can recall that depression resulting from avoidance of sadness at failure to solve a problem reduces the ability of the hippocampus to support the formation of long term memory.
We can, at the cognitive neuroscience level, expect that the opportunity for like structures to be recognized occurs, and analogies, are made, and the abstraction hierarchy of concepts related to the problem is remodeled, extended, to more closely mirror the mathematical definitions being used.
We can, at the phenomenography level, suppose that fine distinctions between concepts will be more likely to be noted, because the mental structures that support these are forming.
We can, at the computer science education level, consider how to bring activity into lecture, that poses the analogies and distinctions we wish the student to gain, by varying the examples of the concepts such that a representative example is contrasted with a non-example, in an ambience that fosters curiosity and rewards progress.

\section{Validation}
We draw a connection between epistemology and validation.
Epistemology is why we believe what we believe.
Thus, it makes sense to apply our epistemological framework to explain why we believe what we believe about our results and interpretation.

Our epistemology is informed by the work of others over a wide range of disciplines:
Computer science education, mathematics education, and education generally, especially phenomenography.
Cognitive aspects, including memory and attention, are shared by and form a bridge between phenomenography and cognitive neuroscience.
Neuroscience  provides interesting relevant information.

Ira Black, MD, in ~\cite[p40]{Gazzaniga,Cognitive Neuroscience} "A satisfactory mechanistic description of any well-framed cognitive process requires that we simultaneously explain it at multiple levels of analysis. Different levels provide complementary insights to characterization and causality that are unobtainable from any snigle line of analysis."

\subsection{Validation at the Level of Computer Science Education}
Interviews with computer science educators, both instructors and teaching assistants have provided a diversity of viewpoints, and generally support the interpretations we have given. For example, one instructor, when asked what students thought proofs were, said "some kind of magic incantation", and teaching assistants have said "students really struggle with this".

\subsection{Validation at the Level of Mathematics Education}
The literature of mathematics education includes work on students' learning about proof. Our work with computer science students has had the benefit of some students who are dual majors of math and computer science, which has allowed us to trace the similarities and differences of these cohorts of students. The significance of definitions, the necessity and utility of proof, the role played by interest in forming procedures and functions, the difference between functional and procedural programming have differed in these three cohorts, in so far as we have been able to examine. We did not explore the interest in developing procedures/functions or prcedural or functional programming in mathematics majors who were not also computer science majors.

\subsection{Validation at the Level of Phenomenography/Variation Theory}
Variation theory supports our observation that comparing and contrasting fine distinctions in material being taught aids the process of learning.
We used the difference between assignment and equality testing, manifest in the java expression of "==" vs. "=". 
We compared a software procedure representation with a mathematical formulation (the latter using only "="), for comprehensibility by students. 
This helped us to see that barriers to student understanding exist, for some students of computer science, at the level of formulation.
It also helped us see that the barrier between the internalization and interiorization of Harel and Sowder might be less of a barrier in students of computer science who are routinely conscious of the need to analyze procedures. 


\subsection{Validation at the Level of Cognitive Neuroscience}
Consider these, which are from the bibliography of Hofstadter:\\
Gentner, Dedre, Keith Holyoak, et al. eds. The Analogical Mind, Perspectives from Cognitive Science, MIT Press, 2001.\\
Helman, David, ed., Analogical Reasoning: Perspectives of Artificial Intelligence, Cog Scie, Philo, Kluwer 1988\\
Holyoak, Thagard, Mental Leaps: MIT Press, 1995\\
Murphy 2002 Big Book of Concepts MIT Press\\
Vosniadou, Ortony eds 1989, Similarity and Analogical Reasoning, Cambridge U Press\\
Bartha, 2010, By Parallel Reasoning, Oxford U Press\\
Goldstone, 1994, Similarity, interactive activation, mapping, Jour Exp Psychol Learning Memory, Cognition pp3-28//
Cambridge Handbook of Thinking and Reasoning, Cambridge U Press\\
Huth, 2012 A continuous semantic space describes the representation of thousands of objects and action categories across the human brain, Neuronpp1210-1224\\
Barsalou, 1985 Ideas, central tendency, frequency of instantiation, graded structures in categories, Jour Exp Psychology, Learning, Memory and Cognition\\
Carey 2009 Origin of Concepts, Oxford U Press\\
Gentner, eds, 2003, Language in Mind: Advances in the study of language and thought, MIT Press\\
Gentner, eds, 2001, The Analogical Mind: Perspectives Cog Sci, MIT Press\\
Keil, 1979, Semantic and Conceptual Development, Harvard U Press\\
Keil, 1989, Concepts, Kinds and Cog Devel, Harvard U Press\\
Lakoff, 1987, Women, Fire, Dangerous Things, What Categories reveal about the mind, U Chicago Press\\
Lamberts 1997 Knowledge, Concepts, Categories MIT Press\\
Mandler 2004, Foundations of Mind: Origin of Conceptual Thought, Oxford U Press\\
Nosofsky,1986, Attention, similarity, identification categorization relationship, Jour Exp Psychology\\
Pothos, 2011, Formal Approaches in Categorization, Cambridge U Press\\
Prinz, 2002, Furnishing the Mind: Concepts and Perceptual Basis, MIT Press\\
Festinger, 1957, Cognitive Dissonance, Stanford U Press\\
Shank, 1982, Dynamic Memory A Theory of Reminding and Learning in Computers and People, Cambridge U Press\\
Shank, 1999, Dynamic Memory Revisited, Cambridge U Press\\
Foundalis, 2013, Unification of Clustering, concept formation, categorization and analogymaking, TR, Indiana University\\
French, 1995, subtlety of sameness, theory and computer model of analogy-making, MIT Press\\
Gentner, 2009, Reviving inert knowledge, Analogical abstraction supports relationsal retrieval pf past events, Cognitive Science, \\
Kanerva, Sparse Dstributed Memory, MIT Press\\
Mitchell 1993 Analogy making as Perception, MIT Press\\
Thagard, 1990, Analog retrieval by constraint satisfaction, AI\\
Laurence and Margolis, 2012, Abstraction and the origin of general ideas, Philosopher's Imprint \\
Margolis, Laurence 2007, Creations of the Mind, Theories of Artifacts and their Represe, Oxford U Press\\
Sander 2005, Analogyy and transfer, encoding the problem at the right level of abstraction, Proc. 27ty annual conf cog scienc society Stresa Italy pp1925-1930\\
Feeney, 2007, Inductive Reasoning: Experimental, Developmental and Computational approaches, Cambridge U Press\\
Glatzeder, 2010, On Thinking Vol 2, Towards a Theory of Thinking, Springer\\
Harnad, 1990, Symbol grounding problem, Physica D\\
Holland, 1986, Induction, Processes of Inference, Learning, Discovery, MIT Press\\
Holyoak, 2010, Analogical and category-based inference theory Bayesian causal models, Jour Exp Psychol pp702-727\\
Holyoak, 2012, Oxford handbook of Thinking and Reasoning, Oxford U Press\\
Fauconnier, Gilles 1985, Mental spaces, aspects of meaning construction in natural language MIT Press\\
Fauconnier, 1997, Mappngs in thought and language, Cambridge U Press\\
Coulson, 2001, Semantic leaps: Frame-shifting and Conceptual Blending in Meaning Construction, Cambridge U Press\\
Aubusson 2005 Metaphor and Analogy in Science Ed., Springer\\
kieran, 1981, concepts associated with the equality symbol, Educational studies in math\\
schoenfeld, 1982, problem perception and knowledge structure expert novice math problem solvers, Jour Exp PsychLearning Memory Cognition\\
Silver, 1981, Solving Related Problems, Jour. Research Math ED., \\
Spalding, 1996, Effects ackground Knowl Problem Solving, Jour Exp Psych, Learning, Mem, Cogn\\
McAllister, 1996, Beauty and Revolution in Science, Cornell U Press\\
 Timmermanns, 2012, Histoire philosophique de l'algebre moderne Les origines de le pensee abstraite, Classiques Garnier\\
 Weiner 2008, Analogies in Physics and life, a scientific autobiography, World Scientific\\
 Barsalou, 1982, Context-independent and context-dependent information in concepts, Memory and Cognition\\
 Dietrich, 2010, Analogical insight: Toward uifying categorization and analogy , Cognitive Processing\\
 Gentner 1988, evidence for relational selectivity in the interpretation of analogy and metaphor,  in Bower, ed., Psychology of Learning and Motivation Advances in Research and Theory, Academic Press
 Gick, 1983, Schema Induction and analogical transfer, Cog Psy\\
 Murphy, 1994, Prediction from uncertain categorizations, Cog Psy\\
 Oppenheimer, 1956, Analogy in Science, American Psy\\
 Sander, 2003, Analogie et categorisation, revue d'intelligence artificelle,\\
 Spalding, 1996, Effects of background knowledge on category construction, jour exp psy, learing, memory, cogn\\
 Turner, 1988 Categories and Analogies\\
 Wisniewski, 1995 Prior knowledge and functionally relevant features in concept learning, Jour Exp Psy Learn, Memory Cog\\
 
 
\subsection{Validation at the Level of Neuroscience}
Neuroscience~\cite{} % Kandel? Mandel? and Squires
teaches us that the body remodels neural connections into those that more efficiently carry out the task for which they are serving.

Computer science shows us that  storage of a set of related definitions is more efficiently obtained when there is a hierarchical arrangement of the defined terms \cite{}%()trie, Professor Fredkin).
%get a citation out of the abstraction/essences Hofstadter

Neuroscience
teaches us that people remember better material about which they are curious, especially if it is intrinsically pleasing, and that this facilitation is related to the release of dopamine ~\cite{}%Gruber

Neuroscience 
teaches us that foreground changes against a fixed background, %find something about snakes
such as the difference between an even function and an odd function as seen in the domain of real numbers,
can be expected to be more noticeable and therefore memorable,
than background changes supporting a fixed foreground,
such as even functions seen in integers, real numbers, and complex numbers.

Gazzaniga p.56, the limbic system participates in emotional processing, learning and memory.
Gazzaniga p. 57-58, the basal ganglia, subthalamic nucleus, substantia nigra mediate aspects of cognitive functions such as the short-term memory processes
Gazzaniga p. 59 the hippocampus has been implicated in emotional processing (because of its interconnections with cingulate and mamillary bodies) and memory.
Gazzaniga p. 60, David Amaral, PhD "We now know that virtually all of the sensory input that it receives arises from higher-order, multimodal cortical regions. This would indicate that whatever processing is done by tthe hippocampuls in the service of forming long-term memories is accomplished with fairly abstract, gestalt-like representation of experience.  It is also now clear that in addition to its subcortical connections, the hippocampal formation has massive return connections to the neocortex. This fits well with the emerging view that the hippocampal formation is not the final repository of long-term memories. \ldots effect of hippocampal lesions on memory suggest the hippocampus ought to be a neural machine designed for forming associations. \ldots by wa of the extensive associational connections, hippocampal neurons can form essentially limitless networks to create representations of perceived experiences. We have learned that one of the cardinal features of the intrinsic anatomy of the hippocampal formation is the high degree of divergence and convergence of its stepwise connections \ldots connections originating in one portion of the hippocampal formation project to as much as half of the entire region of the next processing step. /ldots the neuroanatomical fact of high levels of associational connections predicts that individual neurosn can be addressed by myria inputs; that is, their response properties are not hard wired as a neuron in V1 might be.\ldots two regions of neoortex, the perirhinal and parahippocampal formation \ldots subserve some forms of memory function on their own \ldots one of the highest brain densities of N-methyl-D-aspartate (NMDA) receptors is found in the CA1 filed of the hippocampus has heightened interest in the role of this glutamate receptor in memory function.
Gazzaniga p. 62 Recent evidence implicated the pulvinar as a critical structure in attentional processing because of its heavy interconnectivity with regions of the cortex known to be involved with attentional control (posterior area of the parietal lobe) and the areas of visual cortex where feature analysis and object recognition are accomplished (ventral projection pathway)
Gazzaniga p. 64 brainstem nuclei control \ldots states of consciousness

Gazzaniga p. 207 "Everyone knows what attention is. It is the taking possession of the mind, in clear and vivid form, of one our of what seem several simultaneously possible objects or trains of thought. Focalization, concentration of consciousness are of its essence. It implies withdrawal from some things in order to deal effectively with others, and is a condition which has a real opposite in the confused, dazed, scatterbrain state \ldots quoting William James from 1890.

$<$Isay$>$   Notice that there is a similarity to abstraction. In abstraction we selectively attend to some features and ignore others. 
Moreover, the consequence of not being able to attend is described. It suggests that our students who cannot abstract well, therefore cannot attend well, can be expected to be confused. If we want our students to be able to attend well, we can try to help by drawing their attention, through variation theory, to that subset of what we are saying, that is critical. 

Why do we have to select? Why can't we just take it all? There must be some limit, some processing limit, that is less than the combined bandwidth of all of our sensors, when we include as sensors, the results of integration, and of conclusions, or at least speculations associated with the true external inputs. (Don't we wish we had delayed evaluation.)

As we must select, how do we select? Here variation theory is saying something. We want to see this vertically through our epistemology.
	$</$Isay$>$
	
Gazzaniga p 214-215 discusses the research of Treisman on the nature of feature variance. When one feature (such as being red) differentiates the thing to be discerned from the distractors, the time to discover this one instance was a constant over the number of distractors, but when the feature was a conjunction, and some of the conjuncts were shared by some of the distractors, the time to discover the one instance was proportional to the number of distractors.

Gazzaniga p 219 event related potentials for specific stimuli change when people are directed to attend to those stimuli.

Gazzaniga p 223, cuing attention does increase event related potentials, but only if the cue is within 300 milliseconds of the event. Bigger gaps can be deleterious.

Gazzaniga p 223, incoming sesory signals may be altered within sensory-specific cortex when stimuli having relevant physical feature are encountered. The earliest form of this selection can be defined by location in the auditory and visual systems. The implication of these data is that descending projections from attentional control systems affect the excitability of neurons coding the feature of the to-be-attended or ignored stimuli. These data inform us about the site where attention effects are manifest during perceptual processing.

	$<$Isay$>$ It is very useful to have this physical layer measurement, so that we have a hint to start out with, when applying at the computer science education level, with variation theory	$</$Isay$>$

Gazzaniga p 225, 

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