\chapter{Related Work} \section{Goal Definition} Schoenfeld gives this expression of a goal for instructors teaching proof: He states that mathematics literature includes knowledge and perspectives of world-class mathematicians vs. more or less ordinary PhDs''\cite[p. 74]{schoenfeld1998reflections}. Do we have such characterization for computer scientists? If so, what are these characteristics that are relevant for computer science students, and what projection do these have into students' conceptualizations? \section{Methods} Archavi et al.\cite[p. 4]{arcavi1998teaching} report using microgenetic analysis, which they describe as having roots in both cognitive science and ethnography. Schoenfeld, Smith and Arcavi\cite{schoenfeld1993learning} describe it as striving 'for explanations that are both locally and globally consistent, accounting for as much observed detail as possible and not contradicting any other related explanations'''. Schoenfeld is, among other categorizations, a cognitive scientist, and uses this knowledge to inform his teaching.Archavi et al.\cite[p. 4]{arcavi1998teaching} . Research on teaching and learning about proof in mathematics education has produced an extensive literature. Only a small sampling is mentioned below. Mathematics educators, including Keith Weber[?], Harel and Sowder in 1998[?], and David Tall[?] have studied students' learning of proof in the mathematics curriculum. Leron, in 1983, [?] has described the structural method for proof construction, attributing it to recent ideas from computer science. Lamport, in 1995, [?] in work on proof construction, has given one approach that computer science students might find compatible with their background. Velleman, in 2006, has written software and a textbook [?] about proving with a structured approach. Weber has reported the success of several approaches to pedagogy [?]. Barnard [?] has commented upon students negating statements with quantifiers. Edwards and Ward [?, p. 223] have discussed the role of definitions for undergraduate mathematics courses, stating the enculturation of college mathematics students into the field of mathematics includes their acceptance and understanding of the role of mathematical definitions''. Bills and Tall [?] have distinguished student understanding of definitions that is sufficient that the student can use them in proofs. Harel and Sowder [?] and Harel and Brown [?] have conducted qualitative research on mathematics students' conceptualization of proofs. They have developed three main categories, each with several subcategories. Evidence from our studies is consistent with the presence of these categories of conceptualizations in the population of CS(E) students. Tall[?, ?] has also categorized mathematics students' understanding of proof. He has studied the development of cognitive abilities used in proof, starting, as did Piaget,[?] with abilities believed present at birth. Yang and Lin have modeled reading comprehension.[?] Leron[?] has written about encouraging students to attend to proof structure by teaching with generic proofs (proofs that use a generic particular). Mejia-Ramos et al.[?] have built a model for proof comprehension. They have observed that students who are assessed on appreciation of structural and other appropriate features of a proof, rather than on rote reproduction, are more likely to develop a deeper understanding of proof. Knipping and Reid[41] have examined proof in mathematics education. Weber[?, ?, ?, ?, ?, ?, ?, ?, ?] has investigated students' approaches to and difficulties with proof. When studying student proof attempts in group theory, Weber has found that some typical students' inabilities to construct proofs arise despite having adequate factual and procedural knowledge, the ability to apply that knowledge in a productive manner was lacking. [?] More specifically applying the knowledge was seen to include selecting among facts, guided by knowledge of which were important, for those most likely to be useful. [?] Alcock and Weber,[?] have studied students' understanding of warrants, the support for the use of a particular inference. Weber has published a framework for describing the processes that undergraduate students use to construct proofs. [?] Almstrum[?] has investigated the understanding of undergraduate computer science students of problems related to logic, compared to problems only weakly related to logic, and has shown that some students have trouble with the notion of truth or falsity. Healy and Hoyles[?] have reported on algebra students' preferences for the content of convincing arguments, and their distinction between preferences for ascertaining vs. preferences about what was likely to be well-received on assessments. {\.I}mamo{\u g}lu[?, ?] has studied the conceptualizations of proof of students who were preparing to become mathematics and science teachers, in their freshman and senior years. Knuth has applied qualitative research to the conceptualizations of proof by high school mathematics teachers [?, ?]. Because our work with proof also has explored the consequences for the student in terms of algorithm choice, including recursive algorithms with proof by mathematical induction, the work of Booth[?], who has used phenomenography to develop a model of students' understanding of recursive algorithms is related. Zhang and Wildemuth[?] have described qualitative analysis of content. \section{ Proofs Using the Pumping Lemma for Regular Languages} Mattuck[36] states analysis replaces the equalities of calculus with inequalities: certainty with uncertainty. This represents for students a step up in maturity.''[page xiii] and these are things which I find that many of my students don't seem to know, or don't know explicitly. They subtract inequalities \ldots . \subsection{Quantifiers} In 2010 Pillay [44] asserted that there has been no research into the actual learning difficulties experienced by students with the different topics'' in formal languages and automata theory. Of the pumping lemmas, Pillay states A majority of the students made logical errors when proving that a language is regular and using the Pumping Lemma to show that a language is nonregular. These could be attributed to a lack of problem-solving skills and an understanding of the Pumping Lemma.'' Devlin[18] observes that quantifiers can appear daunting to the uninitiated, and that statements containing multiple quantifiers can be difficult to understand. \subsection{ Symbols} H\"uttel and N{\o}rmark[45] described a successful method for improving both student activity level in the course and final grades, which combines peer assessment with creation of notes that can be used during the exam. (The incentive was that their answers to text (CHECK) questions would be available for them to use at the written exam. No other textual aids would be allowed at the exam.''[p. 4]) The better performance on the exam is welcome; whether it is due to having notes compared to closed book, or having performed the review might not be certain. According to Arnoux and Finkel[46], it is not unusual for students to acquire mathematical knowledge without attaching meaning to it, and leaving them unable to solve some problems. They go on to report that Paivio proved that double coding (verbal and visual)'' facilitated remembering. They also report that different parts of the brain are used to process verbal and visual information, and therefore more of the brain is involved when both verbal and pictorial communication is used. They prefer multi-modal representations. Xing[47] writes about aiding students comprehension of proofs being aided by graphs. She reports students feel that Pumping Lemma(PL) is so abstract to grasp that using it to prove that a language is non-regular is a daunting task.'' She shows a graphically laid out proof that a given language is not regular. This graph has the advantage over a traditional proof, i.e., a sequence of statements, that the dependencies of states on axioms or intermediate results are plainly shown by graph edges. Simon et al.[43] ask Is it possible that students plug and chug in computing, not really understanding the concepts as we would like them to?'' and go on to say We posit that the need exists for computing instructors to design assessments more directly targeting understanding, not just doing, computing. And, of course, to adopt teaching approaches that support student development of these skills.'' Mazur[25] developed peer instruction to address students' propensity to practice a plug-and-chug approach to problems. This approach has been applied to computer science teaching, including theory of computation, by several researchers including Simon, Zingaro, Porter, Bailey-Lee and others[48, 49, 50, 51, 27]. \subsection{Teaching Pumping Lemmas} In 2003 Weidmann[39] wrote a dissertation on teaching Automata Theory to students at the college level. She found that past performance in prerequisite theory courses was a statistically significant indicator for success in their college level course. She described a theoretical framework called pedagogical positivism'', a stance between logical positivism and constructivism, allowing the notion of a teaching method best suited to a group of students to learn Automata Theory. She interviewed a teacher with several'' years of experience teaching this course (p. 5), who admitted that she did not have a better way to teach abstract thinking other than repeated exposure'' (p. 98). In chapter 5, Discussion, Conclusions and Implications, of this dissertation[39], the suggestion Instead of simply providing the solution to a problem in class, or stating the intuitive leap that makes the problem easy to solve, the students should be exposed to the iterative thought process that lead to the intuition that created the solution.''(p. 201) appears. One suggestion is Learning objectives should be set to focus on familiarity with formalisms and rigorous mathematical notations” (p. 224) and another suggestion is “Include programming projects as part of the required coursework''(p. 224). The combination of these brings to mind the suggestion of Harel and Papert[40]: constructing personally designed pieces of instructional software'', and the thought that the students might dwell more effectively on the notion of abstraction as they tried to teach someone else about it. \section{ Proof by Induction} Kinnunen and Simon [7] describe an example applying phenomenography to computing education research, listing several recent examples, and also providing a detailed description of a mainly data- but also theory-driven refinement of categories. Berglund, Eckerdal and Thun\'e [16, 3, 4] have applied phenomenography to computing education research, obtaining classifications by judicious grouping of student conceptions derived from interview data. Eckerdal et al. [4] describe how the results using phenomenography showed additional insights beyond other methods. Jones and Herbst [6] considered which theoretical frameworks might be most useful for studying student teacher interactions in the context of learning about proofs. Bussey et al. [2] illustrated student teacher interactions in the space of learning, and the objects of learning, in variation theory, modified from the model of Rundgren and Tibell [13]. Reid and Petocz [12] used phenomenography to study students' conceptions of statistics. Their purposes included to enable teachers to develop curricula that focus on enhancing the student learning environment and guiding student conceptions of statistics.'' They asked students to describe how they understood statistics and then organised student responses into a hierarchy of conceptions. They used interviews to understand individual students, and the group of interviews to show the variations they found. They found the students with the most superficial understanding to be carrying out steps without knowing their meaning. Krantz [8] describes proof by induction, giving several examples in this book of proof techniques for computer science. \section{Domain, Range, Mapping, Relation, Function, Equivalence Relation in Proofs} Marilyn Carlson, \cite{carlson1998cross}shows that we can easily expect too much from our students in terms of what they understand of functions. This has significance for what we think are adequate examples to use for proof by mathematic induction, for example. \section{ Definitions, Language, Reasoning in Proofs} Weber, Alcock, Knuth \subsection{Procedural vs. Understanding} Is it that Tall and bifurcation are about learning the procedural vs. understanding approach to dealing with proof? There are indications\cite[p. 18]{loewenberg2003mathematical} that mathematics teachers in grade school and high school who were mathematics majors themselves learned a procedural approach to mathematics and "lacked an understanding of the meanings of the computational procedures or of the solutions. Their knowledge was often fragmented, and they did not integrate ideasthat could have been connected (e.g., whole-number division, fractions, decimals, or division in algebraic expressions.)" \section{Abstraction} Kemmerer\cite{kemmerer} 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}). Miron-Spektor et al.\cite{Miron-Spektor20111065} have shown that observing anger communicated through sarcasm enhances complex thinking and solving of creative problems. \section{Diagrams in Proof} Gibson\cite{gibson1998students} examined students' use of diagrams in proofs, and found that diagrams helped link students' ideas to mathematization, namely, to representation in symbols, and also to support variation, in the sense of the critical difference between what Harel and Sowder\cite{harel1998students} call perceptual and transformational conceptualizations. \section{ Equivalence Class, Generic Particular, Abstraction in Proofs}\section{Educational Psychology} \subsection{What do students need to construct?} Archavi et al.\cite[p. 13]{arcavi1998teaching} I'd like to have you doing some mathematics and I will do everything I can --- including using grading --- as a device for having you do that.'' \section{Phenomenography, Variation Theory} \section{Constructivism} Vygotsky in Language and Thought said we do as individuals build up thoughts and then becoming socialized with shared language, some accommodation would need to be enforced onto the child. [p.17] the psychological problem is to become convinced that always, necessarily a given picture has to appear as one of a multiple of possible graphs of the same category (i.e. only as a representative of a class \ldots must be grasped not in a final fixed state but rather \textit{in construction} the point moving) \subsection{What do students need to construct?} Archavi et al.\cite[p. 10]{arcavi1998teaching} To be successful, students must know both the appropriate heuristics and the mathematics required to solve the problem. \subsection{Intuition} Students have some knowledge constructed already, and it is not all conscious. overconfidence --- counter by search for reasons it might be wrong'' Koriat et al., 1980\\ confidence --- doesn't correlate with correctness\\ be as to conform Tweney Doherty Mynatt 1981\\ renouce several of his funcamental beliefs with regard to reality'' [p. 39]\\ $$Combine desire to gain points having taken the place of desire to learn, with propensity to learn how to take tests rather than how to believe, and obtain How we answer the tests, or, what we really think''$$ Piaget (Piaget-Beth 1966 [p. 195] ) ontogenetic construction of evidence a new domain integrates former domain as subdomain\\ (n heast?) intuition tends to survive even when contradicted by systematic formal instruction [p. 47]\\ Polanyi 1969 [p. 143-144]\\ \ldots in the structure of tacit knowledge we have found a mechanism which can produce discoveries by steps we cannot specify this mechanism may then account for scientific intuition \ldots not the supreme immediate knowledge called intuition by Leibnitz or Spinoza or Husserl, but a work-a-day skill for scientific guessing with a chance of guessing right.\\ Polanyi sees a deep analogy between integrative capacity $$there we have unsupervised'' specialization network formation during consolidation, perhaps$$ Where to turn for a logic by which such tacit powers can achieve and uphold true conclusions'' Polanyi 1967 [p. 137]\\ $$I'm thinking about conscious / unconscious$$ $$When we do something consciously we can be checking, when we do something unconsciously, we might not be$$ Fischbein [p.59] Inferential affirmatory intuition may have an inductive or deductive structures. After one has found that a certain number of elements (objects, substances, individual, mathematical entities, etc.) have certain properties in common one tends \textit{intuitively} to generalize and to affirm that the \textit{whole} category of elements possesses that property. This is not a mere logical operation. The generalization appears more of less suddenly with a feeling of confidence. This is a fundamental source of hypotheses in science. According to Poincar\'e generalization by induction copied, so to speak, from the proedures of experimental sciences'' is one of the basic categories of intuition (Poincar\'e 1920 [p. 20]).\\ \cite[p. 67]{fischbein1987intuition}(check) One morning walking on the bluff, the idea came to me with just the same characteristics of brevity, suddenness and immediate certainty that the arithmetical transformation of indeterminate ternary indefinite forms were identical to those of the non-Euclidean geometry Poincar\'e 1913 [p. 388]''\\ David Tall mathematician and psychological analyst, moment of insight never felt that he made 'conjectures'; what he say were 'truths' evidenced by strong resonances in his mind. Even though they often later proved to be false, at the time he felt much emotion vested in their truth \ldots intense intuitive certainties. Yet at the same time his contact with them often seemed tenuous and trasient; initially he had to write them down, even though they might be imperfect, before they vanished like ghosts in the night (Tall 1980 [p. 33])\\ $$being unconscious seems to go with consolidation. Either unconscious because attending to something else like walking on a bluff, or asleep, being unconscious is relevant to these integration occurring and then moving into consciousness$$ $$''Really wants to know'' implies an openness to change the pre-determined ideas, and complyig with requriements'' does not imply readiness to revise$$\\ \cite[p. 68]{Fischbein} citing Feller experienced player absorbs a complicated situation at a glance and is unable to account rationally for his intuition''\\ $$it (the part of the brain doing the reasoning) is functioning without conscious oversight (the neurons that did that when the capability was new have been deemed extraneous and removed)$$) $$Brooks -- the first system is done carefully with all consciousness like the beginning chessplayer in Feller, the second system has some unwarranted conviction and the third system has mostly warranted$$\\ \cite[p. 69]{Fischbein} Felix Kelin (1898) trained intuition\\ Suppes 1966 train the intuition for finding an writing mathematical proofs''\\ \cite[p. 72]{Fischbein} categorical syllogism type AAA seems easiest for which, EAE, AII 65\%, EIO (These are categorical syllogism types. See www.philosophypages.com/lg/e07a.htm) \cite[p. 77]{Fischbein} AAA and modus ponens come earliest and are class inclusion, maybe $\bar{p} \rightarrow q$ never develop\\ re \cite[p. 81]{Fischbein}, $$people have data stored to supply their intuition and it our be wrong education involves opening it up to conscious inspection, fixing it, and restoring the rapid unconscious operation$$\\ \cite[p. 106]{Fischbein} citing \cite[p. 228]{Wertheimer 1961} These thoughts did not come in any verbal formulation. I very rarely think in words at all. T thought comes, and I may try to express it in words afterwards''. Einstein\\ \cite[p. 119]{Fischbein} A total fusion of the generality of a principle and a particular directly graspable (in this case figural) expression of it. It is this kind of fusion which is the essence of intuition.\\ \cite[p. 120]{Fischbein} specific, directly convincing example and the general principle derived through similarity and proportionality from the particular case.\\ \cite[p. 129]{Fischbein} Analogy frequently intervenes in mathematical reasoning, Polya writes about great analogies \\ 138 1985 software by David Tall\\ \cite[p. 144]{Fischbein}ways in which people process concepts Smith Meliss 1981\\ \cite[p. 147]{Fischbein} For many students the concepts of parallelogram, square and rectangle are not organized hierarchically. They represent classes of quadrilaterals of the same generality''. $$some programmers before unified method were guilty of writing code this way, with wasteful effect. Moreover, Liskov Substitutability Principle was enunciated to help people know how to populate hierarchies when they were learning to do so. Students who are not organizing their concepts hierarchically are similarly disadvantaged. As I believe this hierarchical situation is consolidated during sleep or relaxation, ti becomes a research question$$ \cite[p. 159]{Fischbein} analogies already similar diagram post concept\\ \cite[p. 165]{Fischbein} diagram relies on intervening structure (conceptual structure) else it does not communicate\\ stability of intuition, Ajzen? 1983 epistemic freezing\\ \cite[p. 214]{Fischbein} concept of intuitive loading --- have to know students first before knowing how to teach them Intuition in Science and Mathematics An Educational Approach Efraim Fischbein 1987 Reidel Publish Westcott combines theoretical analysis with experimental findings. Andrea di Sessa building a theory of intuition Bergson 1952 essence of lining changing phenomena Kant intellectual (does not exist) and sensible intuition 1980 [p. 268] Poincar\'e useful Hahn 1956 source of misconception to use or to eliminate? Berne professional quality work without awareness says Westcott 1968 [p.42-46] immediacy --- is that because crosses into conscious unconscious\\ consistency, brevity of expression $\rightarrow$ bearty if the result of the brain's consolidation of knowledge, info, into uncouscious knowledge, usable, available for recall going into intuition, what you taught us for the test, or what we really think Fischbein p. 9 One may not be aware of the existence of such an explicit representation but it continues to act tacitly and to influence ways of reasoning. Seymour Papert 1980 apparently says something about intuition Brouwer, Weil, Kline 1980 [pp 306-327] The sum of the angles of a triangle is equal to two right angles'' Connect bewteen intuitino and reasoning \subsection{Social Constructivism} Archavi et al.\cite[p. 6]{arcavi1998teaching} Students' mathematical activity takes place in an inherently social milieu.'' \subsection{Tall: Set-befores and Met-befores} McGowan and Tall \cite[p. 172]{ (2010 Jour. Math. Behav.)} If learning defaults to the goal of learning how, it can be successful. However, if it is accompanied by a lack of conceptual meaning so that mistakes occur, it can become fragile and more likely to fail in the longer term. At this stage the problems may proliferate as the student becomes confused as to which rule to use, where to use it, and how to interpret it. Tall and Mejia-Ramos \cite[p. 138]{2010, Explanation and Proof in Mathmatics, Springer} Here proof develops through generalized arithmetic and algebraic manipulation'', different kinds of warrants for truth $$so assess student by asking what kind of warrant$$ see Pinto and Tall (1999 and 2002) build on met-befores. \begin{figure}[tbph] \centering \includegraphics[width=0.7\linewidth]{chp7p1} \caption{How proof develops, Tall Mejia-Ramos} \label{fig:chp7p1} \end{figure} \subsection{Harel and Sowder} \cite[p. 237]{harel1998students}Rather than gradually refining students' conception of what constitutes evidence and justification in mathematics, we impose on them proof methods and implication rules that in many cases are utterly extraneous to what convinces them. Editors Schoenfeld et al.\cite{kaput1998research} describe that Harel and Sowder\cite{harel1998students} characterize students' cognitive schemes of proof''. The subdivisions in the 1998 version of categories of conceptualizations \cite{harel1998students}, specifically intuitive -- axiomatic, structural and axiomatizing, matter much in computer science, because intuitive -- axiomatic could be thought to be less used in computer science than in math, program's content could be less intuitive than Euclidean geometry, more subject to checking by assertion checking or debugger examination. \cite[p. 268]{harel1998students} contextual proof scheme: --students have learned to work in a context, e.g., $\mathbb{R}^n$, and so, interpret statements that have greater generality as restricted to be in the context they have learned he or shee has not yet abstracted the concept \ldots beyond this specific context''. Compare this with Pang's (is it Pong?) observation that for students who know only one language, speaking'' and speaking that language'' are concepts that are undifferentiated. \cite[p. 274]{harel1998students} An important distinction between the structured proof scheme and the intuitive proof scheme is the ability to separate the abstract statements of mathematics (e.g., $1+1=2$) from their corresponding quantitative observations (e.g., 1 apple + 1 apple = 2 apples) or the axiomatically -- based observations from their corresponding visual phenomena \ldots , axiomatic proof scheme is epistemologically an extension of transformational proof scheme. One might mistakenly think of the axiomatic proof scheme is the ability to reason formally \ldots . \subsection{Pirie and Kieren Model of Mathematical Understanding} \cite{meel1998honors} Verify these are due to Pirie and Kieren rather than to Meel. \paragraph{Primitive Knowing} This is brought by the student, and is also known as intuitive knowledge, situated knowledge, prior knowledge and informal knowledge. \paragraph{Image Making} any mental image not necessarily pictorial \paragraph{Image Having} mental picture / objects, concept image, frame, knowledge representation structure, students' alternative frameworks \paragraph{Property Noticing} unselfconscious knowing, can notice distinctions combinations connections between mental objects \paragraph{Formalizing} abstract (this is a verb) common qualities from classes of images, classlike mental objects built from noticed properties, description of these class-like mental objects results in production of full mathematical definitions \paragraph{Observing} ability to consider one's own formal thinking, organize personal thought processes, recognize ramifications, \paragraph{Structuring} axiomatic system, conceive proofs of properties associated with a concept \paragraph{Inventising} create new questions, develop new concepts \paragraph{folding back} reorganizing lower level understanding to accommodate new information \subsection{van Hiele Levels} Abstraction is before deduction \subsection{Performance Levels} Baranchik and Cherkas\cite{baranchik1998supplementary} found three levels of understanding in a population taking algebra exams: \begin{enumerate} \item Early skills --- arithmetic and elementary algebra \item Later Skills --- subsequent algebra and a variety of skills involving methematical abstraction, and \item Formalism --- either devising a solution strategy or reformulating a problem into a standard form that permits a solution using early or later skills \end{enumerate} \subsection{Student Centered} Carlson\cite{carlson1998cross} has concluded that \ldots an individual's view of the function concept evolves over a period of many years and requires an effort of 'sense making' to understand an orchestrate individual function components to work in concert.'' \subsection{Use of Diagrams} Gibson\cite{kaput1998research} states Diagrams aided students' thinking by corresponding more closely to the part of their understanding with which they were operating at the time and by reducing the burden that proving placed on their thinking.'' \cite[p. 205]{kaput1998research} The nature of internal representations, however, is unclear because they are not observable.: $$nature of internal representations can be broad, and we can perhaps influence the nature of internal representations, which are ultimately neural nets, by how we teach, and nature of internal representations is such that some, e.g., perceptual, are nto as helpful as others, e.g., transformational. Get the superior colliculus involved, see the motion Ties in with variation theory. Also visualization parts of brain.(B17?)$$ Winn, B, (get the citation from Gibson article in Kaput RCME 1998)( Charts, graphs and diagrams in Ed. materials Psych Illus Basic Research Vol 1 Springer 1987 pp. 152-198) has a spectrum for internal representations from pictures to works, the word end is called abstract. Zimmerman, Visual thinking in Calculus Visualization Teaching Learning Math 1991 Gibson\cite[p. 132]{kaput1998research} There is no doubt that diagrams play a heuristic role in motivating and understanding proofs'' Tall 1991 Intuition and rigor, role of visualization in teaching learning mathematics Gibson\cite[p. 288]{kaput1998research} When student used visual language I inferred that they were operating with the visual part of their understanding'' Gibson\cite[p. 289]{kaput1998research} Students indicated that diagrams helped them understand information by appealing to their natural thinking. They said that diagrams seemed to coincide with the way their 'minds work' and that information represented visually seemed easier or clearer than verbal/symbolic representations.'' more concrete than verbal/symbolic Gibson\cite[p. 290]{kaput1998research} used it to help me see what would be happening'' $$executive parts of brain is engaging visual parts of brain$$ easier than holding the mental image is look at the drawn image Gibson\cite[p. 291]{kaput1998research}''When I read the definitions you can't think about the whole thing at once, but when you have a picture you can'' Gibson\cite[p. 294]{kaput1998research}''Because students did not usually think of their criteria in terms of formal definitions, their ability to decide whether their criteria had been met was hindered when they worked with information represented in only verbal/symbolic form.'' They could obtain ideas more readily from diagrams than they could from verbal/symbolic representations'' Gibson\cite[p. 297]{kaput1998research} Why always keep the picture in your mind when you can have it on the paper, allowing you to focus more on how to get to the end of the proof instead of always having to recall the picture in each individual step?'' $$visual rather than mirror area is possible$$ Gibson\cite[p. 298]{kaput1998research} students sometimes used diagrams to help them express their ideas'' symbolically $$compare proofs without words$$ Gibson\cite[p. 298]{kaput1998research} diagrams helped Laura write out her ideas by helping her connect her ideas to verbal/symbolic representations of these ideas'' Gibson\cite[p. 299]{kaput1998research}''you need to down load that picture on here so that you can touch it and then allow your brain to think about the words you need to say'' visualization does not always help, Gibson quoted some sources Gibson\cite[p. 302]{kaput1998research} when attempting to solve unfamiliar problems, students can benefit from using diagrams'' Moore\cite[p. 262]{moore1994making} The students' ability to use the definitions in the proofs depended on their knowledge of the formal definitions, which in turn depended on their informal concept images. The students often needed to develop their concept images through examples, diagrams, graphs and others means before they could understand the formal verbal or symbolic definitions'' %[p. 262], Moore R Making the transition to formal proof Ed Studeies in Math 1994. Gibson\cite[p. 303]{kaput1998research}''That students would operate in this manner (with the visual part of their concept images) and that such behavior might be of benefit is reasonable when one considers the nature of the concepts in the proofs together with the students' experiences as visual beings and the physiology of their brains''. \section{Cognitive Science} Archavi et al.\cite[p. 6]{arcavi1998teaching} Mathematics requires abstraction, and problems should inspire generalization and specialization. Thagard and Stewart~\cite{thagard2011aha} consider combinations of mental representations at the level of cognitive neuroscience. They hypothesize a process of convolution. Thagard and Stewart~\cite[p. 12]{thagard2011aha} supply reasons why single neurons are thought insufficient to represent a concept, consistent with Valiant~\cite{valiant2000circuits}. \section{ Neuroscience} An odor present during learning \ldots enhances these memories when the participant is re-exposed to the odor during slow-wave sleep after learning[?]'' There is strong evidence that eIF2$\alpha$ phosphorylation and the integrated stress response are important regulators of hippocampal synaptic plasticity, with dual effects of inducing long-term depression while suppressing long-term potentiation and the different types of learning associated with each phenomenon. [?] Adult neurogenesis plays an important role in learning and memory (Clelland et al., 2009, Deng et al. 2010 and Mattson 2012)'' from Brandhorst et al., A Periodic Diet that Mimics Fasting Promotes Multi-System Regeneration, Enhanced Cognitive Performance, and Healthspan, Cell Metabolism 7 July 2015 Kowatari et al.'s\cite{kowatari2009neural} results supported the idea that training increases creativity via reorganized intercortical interactions''. \subsection{Intrinsic Reward} Archavi et al.\cite[p. 9]{arcavi1998teaching} Good problems are \ldots non-routine and interesting mathematical tasks, which students want and like to solve, and for which they lack readily accessible means to achieve a solution''. \subsection{What do students need to construct?} Archavi et al.\cite[p. 13]{arcavi1998teaching}  There were occasions later in the course in which the whole-class discussion also dealt with issues of mathematical elegance and aesthetics.'' \