Skip to content
Branch: master
Find file Copy path
Find file Copy path
Fetching contributors…
Cannot retrieve contributors at this time
1152 lines (819 sloc) 153 KB
\chapter{Related Work}
Maybe I want to put this in the order of definition, relationships, specialization, symbolization, analogy, generalization/abstraction
\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?
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
{\.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 ``.
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.)"
\subsection{Recall of Relevant Information vs. Inert Knowledge}
Bransford et al.\cite[p. 296]{bransford2000designs} attempt to address the problem identified by as inert knowledge (in the sense of Whitehead\cite{whitehead1959aims}). They situated class activity in a problem solving environment, and they showed\cite{van1992jasper} that this instruction had better results for students' ability to transfer skills to new word problems than traditional instruction.
Lehrer et al. \cite[p. 334]{lehrer2000inter} found that "at least in some circumstances, giving children models may be less helpful than fostering their propensity to construct, evaluate, and revise models of their own to solve problems that they consider personally meaningful."
Lesh and Doerr\cite{lesh2000symbolizing} used model-eliciting to engage students in the creation of a combination of meaningful descriptions, explanations and procedures. These models were recognized as tools to be shared and reused, consequently the idea of generalization was implicit. Lesh and Doerr argue that by illuminating the idea of a tool that may be reused, they divide the problem of teaching generalization into two parts: making a general model, and discerning its domain of applicability.
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.
(We want to know, do they understand deduction, abstraction, where are they on van Hiele levels )
According to Gray and Tall \cite[p. 117]{gray1994duality}, Hiebert and Lefevre observed ``a connected web \ldots a network in which the linking relationships are as prominent as the discrete pieces of information \ldots a unit of conceptual knowledge cannot be an isolated piece of information;
by definition is it part of conceptual knowledge only if the holder recognizes its relationship to other pieces of information ``\cite[p. 3-4]{hiebert2013conceptual}
conceptual knowledge is harder to assess than other kinds of knowledge.
\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{Computer Science Education}
Thun\'e and Eckerdahl\cite{thune2009variation} have applied variation theory has been applied to the teaching of computer science.
\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}
Marton and Booth\cite{marton1997learning} have written
{\aa}kerlind \cite{aakerlind2012variation} has written on how
Runesson\cite{runesson2005beyond} has applied variation theory to math
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)
Vygotsky\cite[p. 49]{vygotsky1978mind} noted that "one child selected a picture of an onion to recall the word 'dinner'. When asked why she chose the picture, she gave the perfectly satisfactory answer, 'Because i eat an onion'. However, she was unable to recall the word 'dinner' during the experiment. This example shows that the ability to form elementary associations is not sufficient to ensure that the associative relation will fulfill the \textit{instrumental} function necessary to produce recall."
\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.
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]\\
$<Isay>$ 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'' $</Isay>$
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
$<Isay>$ there we have ``unsupervised'' specialization network formation during consolidation, perhaps $</Isay>$ ``Where to turn for a logic by which such tacit powers can achieve and uphold true conclusions'' Polanyi 1967 [p. 137]\\
$<Isay>$I'm thinking about conscious / unconscious$</Isay>$
$<Isay>$ When we do something consciously we can be checking, when we do something unconsciously, we might not be $<Isay>$
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])\\
$<Isay>$ 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 $</Isay>$
$<Isay>$''Really wants to know'' implies an openness to change the pre-determined ideas, and ``complyig with requriements'' does not imply readiness to revise $</Isay>$\\
\cite[p. 68]{Fischbein} citing Feller ``experienced player absorbs a complicated situation at a glance and is unable to account rationally for his intuition''\\
$<Isay>$ 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)$</Isay>$)
$<Isay>$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 $</Isay>$\\
\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
\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}, $<Isay>$ 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 $</Isay>$\\
\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''.
$<Isay>$ 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$</Isay>$
\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 $<Isay>$ so assess student by asking what kind of warrant$</Isay>$ see Pinto and Tall (1999 and 2002) build on met-befores.
\caption{How proof develops, Tall Mejia-Ramos}
\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}
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
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
ability to consider one's own formal thinking, organize personal thought processes, recognize ramifications,
axiomatic system, conceive proofs of properties associated with a concept
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:
\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
\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.:
$<Isay>$ 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?) $</Isay>$
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''
$<Isay>$ executive parts of brain is engaging visual parts of brain$</Isay>$
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?''
$<Isay>$visual rather than mirror area is possible$</Isay>$
Gibson\cite[p. 298]{kaput1998research} ``students sometimes used diagrams to help them express their ideas'' symbolically
$<Isay>$compare proofs without words$</Isay>$
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.
\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.''
Leslie Valiant\cite[p. 103]{valiant2000circuits} 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)."
\textbf{This, schemas and abstractions, earned a double question mark. Maybe that means it should be reported in more length.}
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."
Lesh and Doerr\cite[p. 379]{lesh2000symbolizing} encourage students to construct models, that may include "a combination of spoken words, written symbols, pictures or diagrams, or references to other models or real-life experiences \ldots in any case, the representation tends to organize and simplify the situation so that additional information can be noticed, or so that attention can be direct toward underlying patterns and regularities, which may, in turn, drive changes in conceptions."
Bowers\cite[p 390]{bower2000postscript} summarizes ideas on generalizing saying: "Bransford et al.\cite{bransford2000designs} describe several studies to support the claim that 'people's representations of problems and experiences have strong effects on the degree to which they will transfer their knowledge to new settings'. Similarly, Lesh and Doerr\cite{lesh2000symbolizing} argue that the models students produce when engaging in model-eliciting problems are not just solutions to the problem at hand, but instead stand as more generalized conceptual tools that can be 'shared and reused in other situations' ".
Huth et al.\cite{huth2012continuous}
A continuous semantic space describes the representation
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
$<Isay>$ One reason we delve into this is that we want to know something about to what degree different factors that might possibly assist learning are significant. For example, how important is it to note the beauty of a proof, and what is the significance of the order in which a proof is presented (for example, lemmas first), and how quickly might we expect students to grasp a hierarchy of abstractions. We learn possibly surprising things, such as, if we provide a fragrance during learning a proof with ideas about spatial locations, and we provide that same fragrance during the early part of sleep, the memory will be consolidated more effectively, as demonstrated by subsequent recall during an awake state.$</Isay>$
Cerf et al. \cite{cerf2014studying} opine "early neuron activity observed here could represent the early stages of the formation of a thought or recollection, the states at which we may not yet be fully aware of the content of the thought"
\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}.
Kosslyn and K\"onig \cite[p. 56--57]{kosslyn1992wet} describe some layout of lower level functions in the brain, "The ventral (object-properties-encoding) system in the temporal lobes not only registers key properties of shapes, but also encodes color and texture; this information is matched to that of objects stored in visual memory. This temporal-lobe memory stores information in a visual code, which cannot be accessed by input from other sensory modalities. \ldots The outputs from the \ldots encoding systems come together at an \textit{associative memory} (which relies on tissue in various places in the brain) \ldots once the appropriate information is accessed, one knows the name of the object, categories to which it belongs, sounds it makes, and so forth. \ldots there is ample evidence that the frontal lobe plays a critical role in this process"
Kosslyn and K\"onig \cite[p. 78]{kosslyn1992wet} "For example, if a caterpillar is still against a twig, it may be very difficult to notice. But if it is moving, we may identify it immediately. Computing motion relations appears to be qualitatively distinct from computing the organization of portions of static images, and distinct regions of the brain apparently encode motion (particularly areas MT and MST). There appears to be a distinct \textit{motion relations subsystem}. The motion relations subsystem extracts key aspects of motion fields, and operations concurrently with the preprocessing subsystem."
$<Isay>$ knowing whether two (simultaneously presented) parts of a proposed presentation will require attention from two systems able to operate concurrently, or will make conflicting demands upon one attention window is useful$</Isay>$.
Kosslyn and K\"onig \cite[p. 102]{kosslyn1992wet} "the frontal lobe clearly has a role in setting up plans. More important, the frontal lobe clearly has a role in formulating and testing hypotheses."$<Isay>$needed for creating proofs$</Isay>$.
Kosslyn and K\"onig \cite[p. 104]{kosslyn1992wet}"a subsystem that \textit{engages} attention. \ldots the thalamus. The thalamus is a kind of switching station, connecting many parts of the cortex."
Kosslyn and K\"onig \cite[p. 112]{kosslyn1992wet} "Visual object agnosia is often divided into two types Patients who are diagnosed as having \textit{apperceptive} agnosia have difficulty putting together visual information to form an integrated perception on an object. Some such patients describe the world as fragmented or chaotic. These patients cannot determine whether two objects are the same or different, let alone identify an object they see. However, even these patients are not blind. Patients who are diagnosed as having \textit{associative agnosia}, in contrast, have difficulty associating the perceptual input with previously stored information. Patients who have 'pure' associative agnosia can discriminate between and properly compare shapes, even though they cannot identify the shapes \ldots could tell whether objects were the same or different, but not what they were. In short, a problem appreciating the shape of an object is apperceptive; a problem identifying the object while being able to distinguish its shape is associative."
$<Isay>$ it would be nice to diagnose student difficulties to this degree, especially if an treatment could be associated with the diagnosis. Moreover, there could be a strong analogy here. Students could recognize two applications of rules of inference as being the same without being able to identify which rule of inference it is. This suggests a clicker question about rule applications, as to whether they are the same or not, or, choose the rule application that is different. $</Isay>$.
Kosslyn and K\"onig \cite[p. 114]{kosslyn1992wet} "Prosopagnosia is particularly puzzling because at least some of these patients apparently recognize faces unconsciously, and never become aware that they have done so; patients with prosopagnosia showed changes in the electrical properties of their skin (i.e., increased electrodemal skin conductance responses) when they were viewing familiar faces, compared to unfamiliar ones, even when they claimed to have no idea whom they were looking at. The stimulus must be matching a stored memory of a face in the pattern activiation subsystem (at least to some degree), but processing sops early -- and the persons is never aware that a match was made.
Kosslyn and K\"onig \cite[p. 118]{kosslyn1992wet} There are processes that activate stored visual information to generate visual mental images.
Posner for systems involved in attention
Kosslyn and K\"onig \cite[p. 124]{kosslyn1992wet} "one of the most intriguing aspects of the neglect syndrome is that it appears to affect consciousness itself."
Kosslyn and K\"onig \cite[p. 132]{kosslyn1992wet} "Patricia Goldman-Rakic and her colleagues showed that the frontal lobes contain a short-term spatial memory."
Kosslyn and K\"onig \cite[p. 136]{kosslyn1992wet} "the (mental) image is not like a picture; it is facing almost as soon as it is generated."(Fink, Pinker, Farah, Chambers, Reisberg)
Kosslyn and K\"onig \cite[p. 147]{kosslyn1992wet} "In either case, instructions somehow must be provided to the attention shifting subsystems; if the image is truly novel, the instructions cannot be previously stored. Thus, it is of interest that there are rich connections from the frontal lobes (which presumably direct the process), not only to parts of the parietal lobes known to be involved in attention, but also to the subcortical structures involved in shifting attention." Goldman-Rakic, posner, 1987, 1988, petersen 1990
$<Isay>$Kosslyn et al. have inferred an organization of processing in the brain, and we as instructors could think about how we are deploying the learning task over this architecture. We could think about what path information has follow, to be learned to the extent that it can be called into future use. That is, while some instruction could produce inert knowledge, the goal is usable knowledge.$</Isay>$
Kosslyn and K\"onig \cite[p. 152--153]{kosslyn1992wet} "By inferring that images occur in the same visual buffer that is used in perception, we expect that the properties of the buffer that affect perception also should affect imagery. If so, then the inability to maintain patterns in images for very long may be another consequence of this common mechanism. That is, in perception one does not want an image to linger; one wants to 'clear the buffer' every time the eyes move. Indeed, if images did not fade rapidly, they would smear and become overlaid. \ldots Thus, patterns in the visual buffer are transient in both perception and imagery."
Kosslyn and K\"onig \cite[p. 154]{kosslyn1992wet} "A symbolic use of imagery involves the same subsystems used in other kinds of imagery reasoning; one must generate the image and retain it long enough to operate on it and 'see' the results. But there is one critical difference between this kind of symbolic imagery and the other sorts: One now must decide how to convert abstract material to particular patterns in the image. For example, one could visualize relative intelligence not only as dots on a line, but also as a set of circles"
Kosslyn and K\"onig \cite[p. 163]{kosslyn1992wet} both hemispheres can generate images (posterior of brain), instructions come from frontal lobes. Left hemisphere naturally, right hemisphere with training. Kosslyn and K\"onig \cite[p. 164]{kosslyn1992wet} both hemispheres can generate images, but in different ways.
Kosslyn and K\"onig \cite[p. 165]{kosslyn1992wet} image transformations require contributions from both hemispheres
Kosslyn and K\"onig \cite[p. 168]{kosslyn1992wet} "Learning to read these (already familiar) words is learning to use an additional route into associative memory to access the information that is stored with words. The problem is, that we initially learned to access the relevant memories on the basis of hearing sounds, \ldots How are we able to use a pattern of lines to access these memories?"
$<Isay>$subvocalization is an inferior way to accomplish this$</Isay>$
Kosslyn and K\"onig \cite[p. 193]{kosslyn1992wet} "With repeated use, the entire pattern of lines becomes an entry in the pattern activation subsystem and the pattern recognized there is associated with a word in associative memory"
$<Isay>$repetition of relationships between items allows us to claim a pattern, a pattern suggests the thought of generalization, the symbolic representation of the pattern makes generalization more efficiently notated, maybe increases awareness of the possibility of generalization, and the ability to generalize is used in transfer of knowledge to further applications, counteracting inert knowledge$</Isay>$
Kosslyn and K\"onig \cite[p. 204]{kosslyn1992wet} "the information that allows one to understand a word must be in associative memory, and possibly is stored only in the left hemisphere. (The left hemisphere is typically the site of the most language processing, and hence it would not be surprising if representations of word meaning were stored primarily in this hemisphere.) \ldots the left hemisphere associative memory \ldots presumably implements the word shape/sound associations."
Kosslyn and K\"onig \cite[p. 207]{kosslyn1992wet} writing about a form of dyslexia that has provided insight, describe ideas of Coltheart\cite{schmalz2015getting} "He hypothesized that only words that name visible objects or properties are stored in the right hemisphere, \ldots "
Kosslyn and K\"onig \cite[p. 214]{kosslyn1992wet} "Sounds are represented initially in a cortical structure in the inner part of the superior (upper) temporal lobes. Heschl's gyrus \ldots is the auditory analog to area V1; indeed, this structure is often called A1 (to indicate that it is the first cortical auditory area). As in vision, the raw auditory input is organized and represented in the auditory buffer prior to high-level processing. Ulrich Neisser posited that such a buffer serves as an 'echoic memory';" rather than being organized as visual input, where spatially adjacent cells have spatially adjacent receptive fields, instead it is organized so that spatially adjacent cells receive input at different sound pitches/frequencies.
Kosslyn and K\"onig \cite[p. 215--216]{kosslyn1992wet} "Recall that in vision there are two subcortical pathways from the eyes to the brain, the geniculo-striate and the tecto-pulvinar; the tecto-pulvinar pathway (eye, superior colliculus, pulvinar, cortext) draws attention to potential regions of interest. The inferior colliculus projects to the deep and superior colliculus play a similar role in audition. \ldots the auditory receptive fields of neurons in the superior colliculus shift with changes in eye position, allowing the auditory and visual maps to remain aligned. (see, for example Maddox et al.,\cite{maddox2014directing}) Hence, one tupically pays attention to a single object, registering its appearance and sounds at the same time."
Kosslyn and K\"onig \cite[p. 219--220]{kosslyn1992wet} "visual preprocessing subsystem becomes 'tuned' by experience to encode useful visual patterns \ldots the separate auditory areas are also connected with reciprocal connections \ldots the reasoning we used to infer than experience tunes the visual preprocessing subsystem also leads us to expect that stimulus properties that distinguish between words will be noted downstream, and feedback will reinforce the encoding of those properties in the auditory preprocessing subsystem"
Kosslyn and K\"onig \cite[p. 220--221]{kosslyn1992wet} "\textit {categorical perception} \ldots lies at the heart of what is accomplished by the auditory preprocessing subsystem \ldots the same categories are extracted when a word is spoken by different people -- and these categories help one to understand the words spoken under different circumstances."
Kandel talks about removing some extraneous (i.e., those that might be redundant, maybe less often on) connections, do I think this is having the effect of categorization/generalization?
Choosing consciously to de-emphasize consideration of details at a high level, and unconsciously removing less-often exercised distinctions could have a similar effect.
Kosslyn and K\"onig \cite[p. 225]{kosslyn1992wet} "We are led to infer than the representations of the sounds of individual words depend on temporal--parietal cortex"
Kosslyn and K\"onig \cite[p. 227]{kosslyn1992wet} "If we assume that unimodal memories are stored in the subsystem that encodes them", which is done by Squire (1987)\cite{squire1987memory}(check this, there is a google book)
David and Squire reviewed protein synthesis connected with memory formation\cite{davis1984protein} " Evidence from learning curves, examination of short-term retention, and posttraining drug injection indicate that initial acquisition is not dependent on such synthesis, but it appears that protein synthesis, during or shortly after training, is an essential step in the formation of long-term memory."
Kosslyn and K\"onig \cite[p. 230]{kosslyn1992wet} "it is possible that semantic information is organized along these lines, with appearance-based meanings segregated from use-based meanings".
Kosslyn and K\"onig \cite[p. 343]{kosslyn1992wet} "We inferred in Chapters 3 and 4 that unimodal visual information is probably stored in the inferior temporal lobe (in the object-properties-encoding subsystem), and we inferred in Chapter 6 that unimodal auditory information is probably stored in temporal-parietal cortex."
Kosslyn and K\"onig \cite[p. 344]{kosslyn1992wet} "information in associative memory can be activated by input from any perceptual modality". Associations can be established between representations in perceptual memory and in associative memory. "memory formation subsystems rely on anatomical structures \ldots the principal members of this set being the \textit{hippocampus} (and related cortex), the \textit{limbic thalamus}, and the \textit{basal forebrain}."
Kosslyn and K\"onig \cite[p. 345]{kosslyn1992wet} "the hippocampus receives input from a number of other structures (the septum and the hypothalamus, via the fornix; the anterior thalamic nucleus and the subcallosal area, via the cingulum; and the amygdala). \ldots The hypothalamus appears to be involved in motivation, and the amygdala appears to have a role in emotion; clearly both factors affect what we remember. \ldots The hippocampus plays a critical role in the storage of new perceptual representations. \ldots also plays a critical role in storing associations between representations. (Mishkin and Appenzeller 1987, Squire 1987)" Long term potentiation is a phenomenon of hippocampal cells in neural microcircuits involved in storing associations between representations.
Kosslyn and K\"onig \cite[p. 346]{kosslyn1992wet} "many of the thalamic nuclei appear to be involved in attentional processes (Posner and others) \ldots The basal forebrain \ldots in turn issues a signal that new representations and/or associations should be stored.(Mishkin and Appenzeller 1987) This is a biochemical signal, consisting of the release of \textit{acetylcholine}."
Kosslyn and K\"onig \cite[p. 347]{kosslyn1992wet} "we shall decompose the memory formation subsystem into two more precisely characterized subsystems, \ldots these subsystems are involved in initiating the learning sequence, and in changing selected connections strengths in particular neural networks, respectively."
Kosslyn and K\"onig \cite[p. 347]{kosslyn1992wet} "the \textit{striatum} plays a critical role in skill acquisition \ldots the striatum receives information from cortical perceptual areas".
Kosslyn and K\"onig \cite[p. 349--350]{kosslyn1992wet} "If novel stimuli are perceived, \ldots when a match is not obtained, this information is sent to the frontal lobes and provides input to the print-now sybsystem; recall that perceptual encoding subsystems have anatomical projects into the frontal lobe (Goldman-Rakic, 1987) Thus, the perceptual encoding subsystems send outputs to the associative memory \ldots Structural changes are initiated that will allow the systems later to reconstitute the pattern of activation evoked by the novel stimulus"
Kosslyn and K\"onig \cite[p. 351]{kosslyn1992wet} "The key to storing new information in memory is the ability to change the 'strengths' of connections among neurons in just the right way. This \ldots has been documented in actual neural networks (Kandel and Schwartz, 1985, Shepherd 1988)"
Kosslyn and K\"onig \cite[p. 353]{kosslyn1992wet} "These changes (in synaptic connections) apparently begin in the hippocampus \ldots this phenomenon is called long term potentiation. \ldots it can last hours, days, weeks, or even longer -- depending on \ldots as well as various properties of the stimulus."
$<Isay>$ It is the accumulation of surface area from vesicles delivering neurotransmitter at the presynaptic cell in the synaptic cleft that creates microenvironments that are more effective (NMDA activated/molecule of neurotransmitter delivered) in activating the postsynatic neuron, since the corresponding membrane that is used to restore the number of vesicles is taken from smooth area.$</Isay>$
Kosslyn and K\"onig \cite[p. 358--359]{kosslyn1992wet} "The idea that associative memory hooks back into unimodal perceptual representations is consistent with a range of clinical findings. \ldots Squire suggests that the perceptual systems that encode information may actually store much of it.(Squire 87) \ldots We speculate that associative memory depends in part on the superior, posterior temporal lobe, if only because patients with lesions in this area often appear to have disrupted associations."
Kosslyn and K\"onig \cite[p. 361]{kosslyn1992wet} "When an object is later perceived and input enters associative memory, the relevant associations are activated."
Kosslyn and K\"onig \cite[p. 370--371]{kosslyn1992wet} "We often store new information even if it has no obvious relevance to any goal or problem at hand. This is called \textit{incidental} memory.\ldots if one pays attention to a stimulus, it is likely to be stored 'automatically,' with no decision to do so. \ldots one tries to memorize the information, and is more likely to remembers it than if the effort were not made. This sort of memory is called \textit{intentional} memory. \ldots the longer the information is attended to, the more likely it is that the memory formation subsystems eventually will store it in memory \ldots if the property lookup subsystems access more information about the to-be-remembered material, memory will be improved. \ldots called a \textit{depth of processing} effect"
Kosslyn and K\"onig \cite[p. 372]{kosslyn1992wet} "one can store information relatively effectively by inventing distinctive \textit{retrieval cues}. \ldots Allan Paivio reviews a large amount of evidence that we remember information better when we use a 'dual code' (visual and verbal) than when we store it in only a single way (Paivio 1971)"
Kosslyn and K\"onig \cite[p. 373]{kosslyn1992wet} "when one becomes an expert in any domain, one often cannot report how one performs the task. Much, if not most, of the information in memory cannot be directly accessed and communicated"
$<Isay>$ it is not in conscious memory anymore$</Isay>$
Lisman and Sternberg\cite{lisman2013habit} talk about Habit and nonhabit systems for unconscious and conscious behavior: Implications for multitasking.
Peter Graf and Daniel Schacter implicit memory
Kosslyn and K\"onig \cite[p. 375]{kosslyn1992wet} "There is considerable evidence that priming tasks and explicit memory tasks rely on distinct processing subsystems \ldots certain drugs impair both recall and recognition in explicit memory tasks, but do not affect the magnitude of priming."
Kosslyn and K\"onig \cite[p. 376]{kosslyn1992wet} In priming, "The words apparently were processed to some level within the system, even when the patient was fully unconscious."
Kosslyn and K\"onig \cite[p. 377]{kosslyn1992wet} "priming has two components, only one of which is perceptual" There is an advantage to having the priming and the recall be in the same sensory modality, only for the right hemisphere. " Graf and Schacter also showed that implicit memory for associations between words is modality-specific.
$<Isay>$ to make the most of the priming effect, the students should have practice writing responses to stimuli, like test questions. For memorization type test questions, that are answered by handwriting, note-taking in class, of the same item, should help.$</Isay>$
Kosslyn and K\"onig \cite[p. 380]{kosslyn1992wet} "Mishkin suggests that a subcortical structure called the \textit{substantia nigra} is critically involved in the 'reinforcement' process that strengthens connections between perceptual states and responses."
Stimulus/response learning, striatum, dopamine. Is this what is used in flash cards? This kind of learning is restricted, [p. 383]"the response can only be evoked by the appropriate stimulus"
Kosslyn and K\"onig \cite[p. 387]{kosslyn1992wet} "material in \textit{working memory} is used to aid reasoning processes (Baddeley 1986). Reasoning processes only can operate on information in short-term memory, but relatively little information can be stored in short-term memory."
$<Isay>$ so lemmas are good$</Isay>$
Kosslyn and K\"onig \cite[p. 387-388]{kosslyn1992wet} " We interpret the short-term memory structures Patricia Goldman-Rakic reports in the frontal lobe (Goldman-Rakic 1987, 1988) as extension of the perceptual encoding subsystems, which serve to make perceptual information immediately available to the decision processes; according to this view, it is debatable whether one want to conceive of the information as actually being stored -- as opposed to monitored -- in the frontal lobe.
Working memory, then, corresponds to the activated information in the long-term memories, the information in short-term memories, and the decision processes that manage which information is activated in the long-term memories and retained in the short-term memories(Kosslyn 1991)"
Kosslyn and K\"onig \cite[p. 398]{kosslyn1992wet} "recall that Goldman-Rakic provides evidence that perceptual information ultimately projects to the frontal lobe(Goldman-Rakic 1987, 1988)"
Kosslyn and K\"onig \cite[p. 398--399]{kosslyn1992wet}explicit memory formation uses acetylcholine, modifying the strengths of connections in the appropriate networks, implemented in part in the hippocampus and related cortex
Squire and Dede\cite{}
Martin\cite{martin2015grapes}"some of the latest functional
neuroimaging findings on the organization of object concepts
in the human brain. I argue that these data provide strong
support for viewing concepts as the products of highly inter-
active neural circuits grounded in the action, perception, and
emotion systems. The nodes of these circuits are defined by
regions representing specific object properties (e.g., form, color, and motion) and thus are property-specific, rather than
strictly modality-specific. How these circuits are modified
by external and internal environmental demands, the distinction between representational content and format, and the
grounding of abstract social concepts are also discussed."
$<Isay>$we have modality specific long term, and we have association area for other than single modality, we have consolidation and reconsolidation, do we have spatial arrays of concepts$</Isay>$
\subsection{Brain Imaging}
Bachmann\cite{bachmann2015brain} discusses "brain-imaging markers of neural correlates of consciousness"
$<Isay>$when we are learning a skill we can be conscious of exercising that skill, and later, we can become less conscious of specifically how we exercise that skill. Is the part of the brain that is conscious moving? Is the location of the memory moving? Could be both. Don't some motor skills move to the cerebellum, and aren't there problems with this for dyslexics? What's going on when something suddenly appears in consciousness? I think the metabolic activity brings the consciousness to the place where the memory is. But in these kind of answers popping into consciousness (Poincar\'e?) isn't it a new synthesis that pops into consciousness?$</Isay>$
Luck and LeClerc \cite{luck2014potentiation} wrote about the Potentiation of Associative Memory by Emotions: An Event-Related FMRI Study,
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})
Squire tells us \cite{squire2015conscious} "findings suggest that fMRI activity in the medial temporal lobe reflects processes related to the formation of long-term memory"
Chambers et al.~\cite[p. 1045]{chambers2003developmental}, citing Gurden et al, Mulder et al., Robinson and Kolb, and Hyman and Melenka~\cite{gurden1999integrity,mulder1997short,hyman2001addiction} state "Dopamine transmission in nucleus accumbens and prefrontal cortex regions projecting to the nucleus accumbens has been implicated in mechanisms of learning and plasticity, including changes in long-term potentiation and morphology of neuronal dendritic trees."
\subsection{Brain structure}
Who can trace out the secret threads by which our concepts are united?\\
Hermann von Helmholtz
Takeuchi et al. \cite{takeuchi2010training} Training of working memory impacts structural connectivity
Melby-Lerv{\aa}g \cite{melby2013working} "current findings cast doubt on both the clinical relevance of working memory training programs and their utility as methods of enhancing cognitive functioning in typically developing children and healthy adults. "
Rutishauser et al.\cite[p. 104-105]{fried2014single} medial temporal lobe "lesion subjects showed reduced (but above chance) recognition memory performance but, strikingly, has a complete lack of performance improvement for items shown together with task-irrelevant novel attributes. In contrast, normal control subjects had a substantial gain in memory performance for these items (Kishiyama et al., 2004\cite{kishiyama2004restorff}). Together, this indicates that the 'von Restorff effect' (Wallace, 1965\cite{wallace1965review}; Kinsbourne \& George, 1974\cite{kinsbourne1974mechanism}; Hunt, 1995\cite{hunt1995subtlety}; Parker et al., 1998\cite{parker1998restorff}) is driven by neuronal mechanisms that reside in the MTL. \ldots Novelty responses are thus a sensitive measure to quantify learning and plasticity. \ldots We identified a subpopulation of single neurons in the hippocampus and the amygdala that showed striking differences in their spiking response. \ldots the same cell would indicate the novelty of a stimulus regardless of which category the stimulus was from. \ldots These cells \ldots also show changes in firing rate as a function of repeated presentation even when subjects are only passively viewing stimuli without an explicit memory task (Perdreira et al., 2010\cite{pedreira2010responses})"
Rutishauser et al.\cite[p. 106]{rutishauser2014single} "it seems that there are at least two distinct classes of novelty-sensitive single neurons in the human MTL: abstract and visually tuned. The first class of untuned general novelty detectors could serve to signal the significance of stimuli during the acquisition of new memories (Lisman \& Otmakova, 2001\cite{lisman2001storage}). It has been suggested that such neurons trigger dopaminergic release through projections to the ventral tegmental area (Lisman \& Grace, 2005\cite{lisman2005hippocampal})"
Mormann et al.\cite[p. 131--133]{mormann2014visual} reports "Another seminal study described neurons representing a specific semantic concept in the human MTL (Quian Quiroga et al., 2005). \ldots A problem in determining the precise turning curves of semantic neurons in the MTL is to find a suitable parameterization of the stimulus space \ldots compounded in human studies by the short duration of any one experiment in any one patient."
Ouchi et al.\cite{ouchi2013reduced} "Adult neurogenesis is known to be important in hippocampus-dependent memory"
Ojemann\cite[p. 262]{ojemann2014human} has demonstrated that lateral temporal neurons are involved in recent memory, making use of attention to increase the success of consolidation of memory of items competing with distractions. "The evolutionary changes in the brain orgainzation for recent memory involve an expansion of the cortical component with relative conservation of the medial temporal -- hippocampal component. \ldots A substantial literature has established the importance of the temporal lobe in learning. \ldots neurons with changes during \ldots memory were significantly more likely to have associative learning changes."
Kowatari et al.\cite{kowatari2009neural} state "In the experts, creativity was quantitatively correlated with the degree of dominance of the right prefrontal cortex over that of the left, \ldots Our results supported the hypothesis that training increases creativity via reorganized intercortical interactions."
Waisman et al.\cite{waisman2014brain} observe that "various studies demonstrate that when complexity of the (arithmetic) problems rises, more brain areas simultaneously support the solving process", citing Zamarian et al.\cite{zamarian2009neuroscience}.
Waisman et al.\cite{waisman2014brain} investigated "cortical activity associated with solving problems that require translation between symbolic and graphical representations".
Waisman et al.\cite[p.691]{waisman2014brain} stated that the "posterior parietal cortex is know to be activated when mental representations are manipulated (Zacks 2008)\cite{zacks2008neuroimaging}.
Deng et al.\cite{deng2010new} report that "Neurons born in the subventricular zone(SVZ) differentiate and integrate into the local neural network as granule cells of the dentate gyrus."
Knutson et al.\cite{knutson2001anticipation} state that a region in the nucleus accumbens codes for expected positive incentive value, and
\cite[p. 4]{knutson2001anticipation} it is "an apparently lateralized response of the right nucleus accumbens."
Chambers et al.~\cite{chambers2003developmental} explained that "Adolescent neurodevelopment occurs in brain regions associated with motivation, impulsivity, and addiction. Adolescent impulsivity and/or novelty seeking as a transitional trait behavior can be explained in part by maturational changes in frontal cortical and subcortical monoaminergic systems. These developmental processes may advantagiously promote learning drives"
$<Isay>$ these systems that we are trying to use to shepherd information from the place it begins (sensory to MTL) to the right prefrontal cortex, from where it can be retrieved for creative application to problem solving, are developing$</Isay>$
\caption{In this figure reprinted from Chambers\cite{chambers2003developmental}, (I plan to do something, either redraw or ask permission) we see how sensory input, new information, is delivered to hippocampus. Other sources have shown us that new neurons are created in or near hippocampus that are a response to new information arriving and related to memory for the new information. We have seen how presence of dopamine, from striatum and VTA assist the consolidation of new memory into longer term memory. We have seen that for monomodal information, the long term memory is stored in cortex near where input is provided by that modality (visual cortex, auditory cortex) and for multimodal information, the long term memory is stored in association cortex. We have seen that information stored in association cortex is more readily retrieved, as any of the associated modalities can help retrieve it. We have seen that this consolidation requires protein and is facilitated by sleep, (I forget which of REM or slow wave sleep.) We have seen how reconsolidation can occur, assisted by nucleus accumbens, and can result in information accessible to the prefrontal cortex, on the right side. We have seen how anticipation of positive reward activates the nucleus accumbens on the right side. Could it be that anticipation of positive reward occurs in REM sleep, I wonder.}
Anderson et al.~\cite[p. 53]{anderson2011cognitive} state "There is some reason to suspect that the angular gyrus (ANG) may also be engaged to serve the metacognitive activities of monitoring and reflecting." (on non-routine problem solving). They go on to say "Regions close to the right ANG have been found to play a variety of metacognitive functions, citing a review by Decety \& Lamm\cite{decety2007role}.
Anderson et al.~\cite[p. 54]{anderson2011cognitive} state "Another region that is potentially involved in metacognition is Brodmann Area 10 or frontopolar cortex (FPC), particularly its lateral portion, (citing Fletcher and Henson\cite{fletcher2001frontal}). A number of converging lines of research suggest that this region of the brain may be critical in the ability to extend knowledge."
Anderson et al.~\cite[p. 58]{anderson2011cognitive} state "The left ANG is often distinguished from the right in many theories including the triple code, but the pattern of ANG effects in this experiment is basically the same in the two hemispheres."
Anderson et al.~\cite[p. 62]{anderson2011cognitive} state "In every case (brain areas related to metacognition), the patterns are roughly bilaterally symmetric."
\subsection{Brain function}
R. Quian Quiroga\cite{quiroga2012concept} opines that concept cells are the building blocks of declarative memory functions.
Suthana et al.\cite{suthana2012memory} report on Memory enhancement and deep-brain stimulation of the entorhinal area.
Imamoglu (fix the accents) et al.\cite{imamoglu2012changes} discuss changes in functional connectivity support conscious object recognition.
Murayama and Kitagami\cite{murayama2014consolidation}dopaminergic memory consolidation effect can result from extrinsic reward.
so, give a little quiz at the end of lecture, covering the main points, and hand out tickets in exchange for handing in quizzes. Then, tickets can be handed in with homework to count for points. So they are paid at the time they are thinking about quiz contents, and that is expected to help with consolidation of the material on the quiz.$</Isay>$
Born and Wilhelm\cite{born2012system} discuss System consolidation of memory during sleep
Diekelmann et al.\cite{diekelmann2012offline} describe that Offline consolidation of memory varies with time in slow wave sleep and can be accelerated by cuing memory reactivations.
Taylor et al.\cite{tayler2013reactivation} describe Reactivation of neural ensembles during the retrieval of recent and remote memory.
Cowansage et al.\cite{cowansage2014direct} describe the Direct reactivation of a coherent neocortical memory of context
Lustenberger et al.\cite{lustenberger2012triangular} discuss a triangular relationship between sleep spindle activity, general cognitive ability and the efficiency of declarative learning.
Roux and Uhlhaas\cite{roux2014working} consider working memory and neural oscillations, questioning whether alpha--gamma versus theta--gamma codes for distinct WM information.
Walker and Stickgold\cite{walker2014sleep} consider Sleep, memory and plasticity.
Tonnoni and Cirelli\cite{tononi2014sleep} discuss Sleep and the price of plasticity: from synaptic and cellular homeostasis to memory consolidation and integration.
Rutishauser et al.\cite[p. 107]{rutishauser2014single} "Neurons coded this information very reliably: The decoder (a function of a population of neurons) could tell (for correct trials) whether the stimulus was new or old on a single trial basis with an accuracy of 75\% \ldots (using) a single previously identified novelty/familiarity neuron. Performance increased to 93\% if six neurons were considered. \ldots for 75\% of error trials, the decoder predicted what the correct response would have been (but which was not given by the subject). \ldots a simple decoder outperform(s) the patient \ldots The neurons have better memory than the patient demonstrated behaviorally,"
Rutishauser et al.\cite[p. 111]{rutishauser2014single} "Many factors modulate the probability that a memory for a stimulus will be formed. Examples include attention, motivation, and saliency of the stimulus (Paller \& Wagner, 2002\cite{paller2002observing}). Structurally, the modification of synaptic circuits by plasticity mechanisms is thought to underlie memory formation (Martin et al., 2000\cite{martin2000synaptic})
Rutishauser et al.\cite[p. 112]{rutishauser2014single} "findings from animal literature indicate that theta oscillations and the timing of neuronal activity relative to the ongoing theta oscillations have a strong influence on plasticity as well as learning, suggesting the possibility that the two are functionally linked by theta oscillations. \ldots The power of theta oscillations measured on the scalp \ldots can be predictive of whether a memory is formed or not (Klimesch et al., 1996\cite{klimesch1996theta}; Sederberg et al., 2003\cite{sederberg2003theta}) \ldots what is relevant for whether a memory was formed or not is whether the preferred phase of a particular neuron was followed faithfully and not the absolute phase \ldots what was predictive (of whether a memory was formed or not) was whether the spikes that were fired were phase locked to ongoing theta or not."
Rutishauser et al.\cite[p. 113]{rutishauser2014single} "the spike field coherence at the time of learning was already indicative of whether a memory was later strong or weak."
Mormann et al.\cite[p. 140]{mormann2014visual} state "The process of encoding episodic memories consists of associating pieces of semantic information (what happened where with whom involved and so on) in a defined temporal order. Lesion studies in humans have shown that structures in the MTL are essential for the encoding of episodic memories (Squire et al., 2004; Squire et al., 2007; Squire, 2009; Milner et al. 1968, Scoville \& Millner 1957). Representations of semantic information at the single unit level are frequently found in these very structures and thus might provide a unique opportunity to investigate how our brain links pieces of semantic information together into episodic memories (Quiroga, 2012). \ldots Episodic memories are always context dependent whereas semantic memories are context invariant and can emerge via generalization of recurring context-dependent experiences (Buzsaki, 2005)."
Mormann et al.\cite[p. 142--143]{mormann2014visual} state "Earlier notions that the amygdala might be specialized to elicit or mediate fear responses (LeDoux, 1996) have been supplemented by more abstract accounts whereby the amygdala processes ambiguity or unpredictability in the environment (Herry et al., 2007) and mediates an organism's vigilance and arousal (Davis and Whalen, 2001). \ldots In humans, neuroimaging studies of the amygdala argue for a broad role in processing stimuli that are strongly rewarding or punishing (Sander et al., 2003; Ohman et al., 2007) \ldots In human fMRI memory studies, increased blood flow in the amygdala during encoding was found to correlate with improved memory formation (e.g, Canli et al., 2000). In addition, the phase locking of human amygdala neurons to ongoing theta oscillations was found to predict memory formation (Rutishauser et al., 2011) \ldots there might be a distinction between the hippocampus and the amygdala in terms of the extent to which unconscious information reaches those areas. If indeed unconscious information can reach the amygdala but not the hippocampus and surrounding structures"
Paz and Pare\cite{paz2013physiological}" in emotionally
arousing conditions, whether positively or negatively valenced,
the amygdala allows incoming information to be processed
more efficiently in distributed cerebral networks."
$<Isay>$ This is just corroborating that infectious enthusiasm for a subject, on the part of the instructor, helps students learn.$</Isay>$
Schwabe et al. \cite{schwabe2014reconsolidation} there is reconsolidation
Patel et al.\cite[p. 205]{patel2014human} "Functions such as reward processing, motivations, and learning have since been attribted to basal ganglia circuits."
Patel et al.\cite[p. 207]{patel2014human} "Examined impact of emotional valance and cACC responsitveness to complext attnetion tasks \dots Examined reward properties of dopaminergic neurons using virtual financial reward"
Patel et al.\cite[p. 208--209]{patel2014human} "when there is a differentce in expected and actual outcome -- a prediction error signal -- midbrain dopaminergic neurons rapidly fire at the onset of the unexpected reward. \ldots This feature is thought to drive reward-based learning and adaptive behavior. \ldots It is thought that phasic dopaminergic activity is the neural substrate of this type (classical conditioning) of learning (Schultz, 1998) \ldots dopamine release in the striatum"
$<Isay>$ So, deliver unexpected rewards. Maybe some clicker questions have greater value.$</Isay>$
Patel et al.\cite[p. 210]{patel2014human} "the ventral striatum has become a focal point in studies of reinforcement learning, \ldots phasically active neurons, thought to be the medium spiny neurons that make up about 95\% of striatal neurons, are more relevant to strengthiening functional circuits diring instrumental conditioning and procedural learning (Graybiel 2008, Jog 1999)" anticipation of rewards, including secondary rewards (monetary)
Patel et al.\cite[p. 212]{patel2014human} "NAcc activity reliably encodes the anticipation of reward proportional to reward magnitude"
Patel et al.\cite[p. 213--214]{patel2014human} "each subregion of the ACC is predomintely involved with distinct roles, such as motivation and cognition (Paus, 2001). Notably these areas are not homogeneous and demonstrate some overlap in function. \ldots The portion of the ACC implicated in reward processing and cognition is the dACC. \ldots The vmPFC is strongly connected to the limbic system, which is a group of brain structures that are implicated in several functions such as emotion, motivation, and memory."
Ojemann\cite[p. 268]{ojemann2014human} states "there are also 'subconscious' or 'implicit' memory processes, processes of which the subject is unaware that change performance. One such process is repetition priming, a shortening of reaction time with repeated presentation of the same item. Changes in lateral temporal cortical single neuron activity related to this, \ldots that study included recordings throughout lateral temporal cortex, neurons with those significant implicit memory changes were all in superior temporal gyrus or superior portion of middle gyrus, significantly more superior and posterior than neurons with recent memory processes \ldots implicit memory then involves neural networks in posterior superior temporal cortex, largely separate from those for recent memory"
Rutishauser et al.\cite[p. 348]{rutishauser2014next} state "Different timescales of memory formation have been described at the neuronal level."
Rutishauser et al.\cite[p. 348]{rutishauser2014next} "It has remained very difficult to directly link the mechanisms of synaptic plasticity to memories"
%$<Isay>$ Hebbian by deposition of membrane, increasing the folds per volume, because membrane harvesting for vesicle retrieval is from flat$</Isay>$
Rutishauser et al.\cite[p. 348]{rutishauser2014next} "We are able to recall memories that were established years or even decades ago, but the processes by which such remote call(sic) works remain largely unknown."
%$<Isay>$ it is one cell activating another, and glial cells bringing in blood to the region in response to activated neuron. Remember, it was Wilder Penfield stimulating neurons that brought back remote memories$</Isay>$
Rutishauser et al.\cite[p. 348]{rutishauser2014next} "emotions are likely to play a key role in memory formation (Cahill et al. 1995, Fanselow \& Gale 2004, Phelps, 2004)
Rutishauser et al.\cite[p. 351]{rutishauser2014next} "There is ample evidence that multiple tasks show sleep-dependent enhancement (Stickgold, 2005)"
There is a journal Neurobiology of Learning and Memory.(single neuron book p 182)
Patel et al.\cite[p. 211]{patel2014human} "Knutson and colleagues further demonstrated that NAcc activation increases proportional to the magnitude of the anticipated monetary reward (Knutson et al, 2001)
According to Kowatari et al.\cite[p.1679]{kowatari2009neural}, Carlsson et al.\cite{carlsson2000neurobiology} "reported that both hemispheres were involved in highly creative subjects." Kowatari et al. \cite[p.1679]{kowatari2009neural} "found that professional training reorganized brain activation patterns, which was correlated with increased creativity."
Kowatari et al. \cite[p.1682]{kowatari2009neural} "In the expert group, a right and left hemispheric difference was obvious; only the right prefrontal cortex (PFC) and parietal cortex (PC) were activated in the expert group, whereas in the novice group, bilateral PFC and PC were activated."
Kowatari et al. \cite[p.1682]{kowatari2009neural} "This results indicated that the direct or indirect interaction between the right and left PFC might contribute to producing highly original designs in the expert group."
Kowatari et al. \cite[p.1683]{kowatari2009neural} "Based on these observations, we postulated that \ldots brain regions that are involved in yielding high creativity indices shifted from the PC to the PFC."
Deng et al.\cite[p. 341]{deng2010new} report that "the adult born dentate granule cells (DGCs) exhibit stronger synaptic plasticity than mature DGCs, as indicated by their lower threshold for the induction of long term potentiation (LTP) and their higher LTP amplitude. and, citing Toni et al.\cite{toni2008neurons}, say "structural modification of dendritic spines and axonal boutons continues to occur as the just-born DGCs become older" Toni et al.\cite{toni2008neurons} report that neurons born in the adult dentate gyrus form functional synapses with target cells.
Deng et al.\cite[p. 343]{deng2010new} report, citing Kee et al. \cite{kee2007preferential} that "learning by a mouse when a set of adult-born DGC was at least 4-6 weeks of age led to preferential activation of these cells during memory retrieval" during the learned task "when the DGCs were 10 weeks old."
Trouche et al. \cite{trouche2009recruitment} state that recruitment of adult-generated neurons into functional hippocampal networks contributes to updating and strengthening of spatial memory.
Deng et al.\cite[p. 343]{deng2010new} surmise that finding suggest that, compared with their mature counterparts, adult-born DGCs may be specifically activated by an animal's experiences and thus can make unique contributions to learning and memory.
Deng et al.\cite[p. 344]{deng2010new} opine that neurogenesis allows plasticity to be mostly localized to newborn immature DGCs, preserving the information that is represented by mature DGCs. %dentate granule cells
Deng et al.\cite[p. 348]{deng2010new} report, citing Kitamura et al.\cite{kitamura2009adult} "a recent study in mice suggested that adult neurogenesis facilitated memory reorganization that led to a gradual reduction of the hippocampus-dependence of memories and the permanent storage of these memories in extra-hippocampal regions.
Diekelmann and Born\cite{diekelmann2010memory} state that "Sleep has been identified as a state that optimizes the consolidation of newly acquired information in memory, depending on the specific conditions of learning and timing of sleep. Consolidation during sleep promotes both quantitative and qualitative changes of memory representations. Through specific patterns of neuromodulator activity and electric field potential oscillations, slow-wave sleep (SWS) and rapid eye movement (REM) sleep support system consolidation and synaptic consolidation, respectively. During SWS, slow oscillations, spindles and ripples -- at minimum cholinergic activity -- coordinated the re-activation and redistribution of hippocampus-dependent memories to neocortical sites, whereas during REM sleep, local increases in plasticity-related immediate-early gene activity -- at high cholinergic and theta activity -- might favour the subsequent synaptic consolidations of memories in the cortex."
Diekelmann and Born\cite{diekelmann2010memory} state that, among its functions, "sleep's role in the establishment of memories seems to be particularly important". promoting primarily the consolidation of memory.
They tell us that "Consolidation refers to a process that transforms new and initially labile memories encoded in the awake state into more stable representations that become integrated into the network of pre-existing long-term memories.", involving active re-processing of 'fresh' memories within the neuronal networks that were used for encoding them.
Diekelmann and Born\cite{diekelmann2010memory} , citing Stickgold et al., and Walker et al.\cite{stickgold2000visual.walker2003dissociable} state that "For optimal benefit on procedural memory consolidation, sleep does not need to occur immediately but should happen on the same day as initial training."
Chambers et al.~\cite[p. 1044]{chambers2003developmental}, when discussing events activating loops within primary motivation circuity, state "These events may also facilitate mechanisms of neuroplasticity among nucleus accumbens neurons, and their afferents."
Chambers et al.~\cite[p. 1045]{chambers2003developmental}, citing Masterman et al.~\cite{masterman1997frontal}, stated "dopamine release into the nucleus accumbens is associated with motivational stimuli, subjective reward, premotor cognition(thought), and learning of new behaviors"
Chambers et al.~\cite[p. 1045]{chambers2003developmental}, citing Waelit et al.~\cite{waelti2001dopamine}, stated "Rewards delivered in intermittent, random, or unexpected fashions have greater capacity over repeated trials to maintain dopamine cell firing and reward-conditioned behavior."
Chambers et al.~\cite[p. 1045]{chambers2003developmental} state that "well-learned motivated behaviors or habits performed under expected contingencies become less dependent on nucleus accumbens dopamine release."
Chambers et al.~\cite[p. 1045]{chambers2003developmental}, citing Yates~\cite{yates1990theories} "In adolescence, the prefrontal cortex has not yet maximized a variety of cognitive functions \ldots Measures of prefrontal cortex function, including working memory, complex problem solving, abstract thinking, and sustained logical thinking, improve markedly during adolescence.
Wittmann et al.~\cite{wittmann2005reward}show that activation of dopaminergic midbrain is associated with enhanced hippocampus-dependent long term memory formation.
Wittmann et al.~\cite{wittmann2005reward} state that reward anticipation reliably elicits a dopaminergic response. They hypothesize that the known improved dopamine driven synaptic plasticity and long-term potentiation is associated with better memory consolidation in the hippocampus. Their findings are consistent with they hypothesis that activation of dopaminergic midbrain regions enhances hippocampus-dependent memory formation, possibly by enhancing consolidation. They have shown that activity of the ventral tegmental area and medial substantia nigra accompanied hippocampal activity related to memory formation, in that both structures were activated by novelty and in relation to subsequent free recall performance. The areas that respond to signals related to reward are the dopaminergic areas when the response is reward prediction and the mesial frontal cortex after learning the contingency between the predicting stimulus and the reward; there is a shift with learning.\cite{knutson2003region}. Their findings support their hypothesis that the hippocampus is a major site for the neuromodulatory influence of reward on long-term memory formation.
Lee et al.~\cite{lee2007strategic} reported on neuroanatomical correlates of converting between symbolic algebra and a pictorial representation. They found that for conversion in either direction, the active areas include the both left frontal gyri, intraparietal sulci bilaterally, which are linked to working memory and quantitative processing. They also found that using the symbolic method activated the posterior superior parietal lobules and the precuneus, in contrast to the pictorial method. They conclude that the symbolic and pictorial strategies impose different attentional demands.
Lee et al.~\cite{lee2007strategic}, citing Anderson et al.~\cite{anderson2003information}, observe that algebraic transformation is subserved by the left posterior parietal region and the left dorsal lateral prefrontal cortex. Lee et al.~\cite{lee2007strategic}, citing Sohn et al.~\cite{sohn2004behavioral}, who found that "anterior prefrontal activation
was greater in the story condition and posterior parietal
activation was greater in the equation condition".
$<Isay>$ then creativity is found in the same or nearby location to the storytelling way, rather than the symbolic representation way$</Isay>$
Lee et al.~\cite[p. 167]{lee2007strategic} report the symbolic method was associated with activation in the left precuneus and bilateral posterior superior parietal lobules. This finding suggest the symbolic condition recruited attentional processes more extensively than did the model method. Also activated were various loci in the visual processing area and in the basal ganglia. The model condition did not activate any areas beyond those activated by the symbolic ". According to Lee et al.~\cite[p. 167]{lee2007strategic}, both the precuneus and the posterior superior parietal lobules are associated with attentional processes.
$<Isay>$so, maybe it takes more attention to work with symbols. After all, they are more concise$</Isay>$
Lee et al.~\cite[p. 168]{lee2007strategic} citing Owen et al.~\cite{owen2005n} say that dorsolateral prefrontal, overlapping middle frontal is involved in reorganizing material into pre-existing knowledge structures.
$<Isay>$ thinking about proof might use this $</Isay>$
Lee et al.~\cite[p. 169]{lee2007strategic} found that "occipital areas were activated in the algebraic condition. This suggests participants spent more time viewing the questions in the algebraic conditions.\ldots activation in the posterior superior parietal lobules might be related"
Wittmann et al.~\cite{wittmann2007anticipation} found that the "hippocampus differed from the response profile of SN/VTA in responding to expected and 'unexpected' novelty". Their results demonstrate parallels between the processing of novelty and reward in the SN/VTA. Their fMRI analysis revealed that cues predicting novel images elicited significantly higher SN/VTA activation than cues predicting familiar stimuli.[p. 198]
Wittmann et al.~\cite[p. 200]{wittmann2007anticipation} "Irrespective of whether the dopaminergic midbrain drives the hippocampus or vice versa, coactivation of the hippocampus and SN/VTA could be associated with increased dopaminergic input to the hippocampus during anticipation. This, in turn, could induce a state that enhances learning for upcoming novel stimuli"
Keller and Menon~\cite{keller2009gender} studied brain activation during mathematical cognition, and compared men and women. They found that the same brain areas were used: right intr-parietal sulcus areas and angular gyrus regions, ventral stream of right lingual and parahippocampal gyri. "Females had greater regional density and greater regional volume where males showed greater fMRI activation. \ldots Our findings provide evidence for gender differences in the functional and structural organization of the right hemisphere brain areas involved in mathematical cognition. Together with the lack of behavioral differences, our results point to more efficient use of neural processing resources in females."
Keller and Menon~\cite[p. 348]{keller2009gender}stated "Gender differences were all localized to the right posterior regions of the brain."
Diekelmann et al.~\cite[p. 116]{diekelmann2010memory} observes that "a great number of studies indicate that sleep supports consolidation of memory in all major memory systems \ldots There is growing evidence that explicit encoding, even in procedural tasks, involves a dialogue between the prefrontal cortex and the hippocampus, citing \cite{schendan2003fmri}, which also integrates intentional and motivational aspects \ldots Sleep changes memory representations quantitatively and qualitatively. \ldots a strengthening of associations \ldots qualitative changes in memory representations."
$Isay$ In order to encourage the prefrontal cortex involvement, we should be explicit about motivation for choosing one inference rule over another as the demonstration / pedagogical proofs are exhibited.$</Isay>$
Diekelmann et al.~\cite[p. 116]{diekelmann2010memory} observes that "subjects learned single relations between different objects which, unknown to the subject, relied upon an embedded hierarchy, citing \cite{ellenbogen2007human}. When learning was followed by sleep, subjects at a re-test were better at inferring the relationship between the most distant object, which had not been learned before. Likewise, after sleep subjects more easily solved a logical calculus problem that they were unable to solve before sleep or after corresponding intervals of wakefulness citing \cite{wagner2004sleep}. Of note, sleep facilitated the gain of insight into the problem only if adequate encoding of the task was ensured before sleep."
$<Isay>$ need the definition of adequate encoding. Does this use of encoding refer to placing the representation into, say, hippocampus?$</Isay>$
Diekelmann et al.~\cite[p. 116]{diekelmann2010memory} "sleep can re-organize newly encoded memory representations, enabling generation of new assoications and the extraction of invariant features"
$<Isay>$ here is generalization, abstraction, we want cs students to learn$</Isay>$
Diekelmann et al.~\cite[p. 116]{diekelmann2010memory} "from complex stimuli, and therby easing novel inferences and insights. Re-organization of memory representations during sleep also promotes the transformation of implicit into explicit knowledge \ldots procedural and declarative memory systems interact during sleep-dependent consolidation."
They also observe that once implicit memory has become explicit, subjects no longer showed improvement in implicit procedural skill.
$<Isay>$I'm connecting explicit with conscious. Once the subject is conscious of (interiorized after internalized, using Harel and Sowder's 1998 scheme, thinking about what you are doing as you do it) the procedure, they carry it out consciously, which could be more slowly. Maybe eventually they will become unconscious of how they carry out this skill, maybe the slowing process of conscious execution will drop away$</Isay>$
(what has Stickgold been learning about sleep, memory, lately?)
Diekelmann et al.~\cite[p. 117]{diekelmann2010memory} "It is assumed that the re-activations during system consolidation stimulate the redistribution of hippocampal memories to neocortical storage sites"
(probably there is more recent on redistribution)
Diekelmann et al.~\cite[p. 117]{diekelmann2010memory} "In addition to system consolidation, consolidation involves strengthening of memory representation at the synaptic level."
$<Isay>$Maybe this is so that when the memories are moved farther out from the hippocampus, there is a stronger trail to it$</Isay>$
Diekelmann et al.~\cite[p. 121--122]{diekelmann2010memory} In the active system consolidation view, "It is assumed that in the waking brain events are initially encoded in parallel in neocortical networks and in the hippocampus. During subsequent periods of SWS the newly acquired memory traces are repeatedly re-activated and thereby become gradually redistributed such that connections within the neocortex are strengthened, forming more persistent memory representations. Re-activation of the new representations gradually adapt them to pre-existing neocortical 'knowledge networks', thereby promoting the extraction of invariant repeating features and qualitative changes in the memory representations."
Diekelmann et al.~\cite[p. 122]{diekelmann2010memory}, citing Hasselmo et al.\cite{} suggest that "acetylcholine serves as a switch between modes of brain activity, from encoding during wakefulness to consolidation during SWS"
Diekelmann et al.~\cite[p. 122]{diekelmann2010memory}, citing Wagner and Born \cite{wagner2008memory} observes that glucocorticoids (cortisol in humans) block the hippocampal information flow to the neocortex, and if the level of glucocorticoids is artificially increased during SWS, the consolidation of declarative memories is blocked.
$Isay>$ so, we need to find out whether students who play video games on the way to retiring are decreasing or increasing their cortisol in the process.$</Isay>$
There is evidence that some games help reduce cortisol "The impact of playing computer games on cortisol concentration of saliva before and after the game showed that the amount of saliva plasma after playing the game has dropped significantly." \cite{aliyari2015effects}, casual video games decrease stress\cite{russoniello2009effectiveness} and there is evidence that Tetris in particular reduces stress\cite{mercer2015stress} and there is evidence that excessive use of violent video games by some young men \cite{eickhoff2015excessive} can negatively impact their cognitive effectiveness. Maass et al. conducted a study on 117 university students; Maass et al. state "The more time spent on media the poorer cognitive performance is. This association has mainly been found for general-audience, violent, and action-loaded contents but not for educational contents. \ldots A significant univariate difference was found for high- vs. low-arousing contents in general (independent of type of media), the high-arousing content leading to poorer ability to concentrate after media use. The expected mediating and moderating effects are not supported. The study yields evidence that short-term mechanisms might play a role in explaining the negative correlations between media use and cognitive performance." \cite{maass2015does}
$Isay>$ might wish to advise students who play video games on the way to retiring to, close to retiring, use those that reduce stress$</Isay>$
Diekelmann et al.~\cite[p. 122]{diekelmann2010memory} report "The concept of a redistribution of memories during sleep has been corroborated by human brain imaging studies (82,83,149,159,158) Interestingly, in these studies, hippocampus-dependent memories were particularly redistributed to medial prefrontal cortex regions (82,83,122)."
$<Isay>$ if it is the case that when students. trying to understand some math in computer science, but opting to memorize, given the time available (Is this what David Tall's bifurcation is about?) form memories that are not hippocampus-dependent (procedural memories are less hippocampus dependent)? If so, is it then also the case that they are less likely to be conveyed to the medial prefrontal cortex? Is the intervention then that instructors provide better explicit descriptions, give exercises to write something in code, which we hope is a bridge from internalized/implicit to interiorized/explicit, declarative, and avail ourselves of emotional support by evaluating the beauty, to aid in recruiting hippocampus-dependent memory formation? $</Isay>$
Diekelmann et al.~\cite[p. 122]{diekelmann2010memory} state that "These regions not only have a key role in the recall and binding of these memories once the are stored for the long term, citing Frankland and Bontempi \cite{frankland2005organization}, \ldots prefrontal-hippocampal system might provide a selection mechanism that determines which memory enters sleep-dependent consolidation."
Diekelmann et al.~\cite[p. 123]{diekelmann2010memory} state that the REM time interval upregulation of genes related to plasticity is dependent upon the learning experience in prior wakefulness, and is localized to the brain regions involved in prior learning, citing Ribeiro \cite{ribeiro2007novel,ribeiro2002induction}.
\item the input is a demonstration of a proof that explicitly describes the motivation for choosing each successive inference rule after another, and explicitly describes the motivation for choosing each lemma, i.e., why was that part of the proof handled separately, (e.g., we know we can, and we know it will be useful, but something about how we suspect it will be useful) Moreover, because we know emotional content is helpful in consolidation of memory during sleep, we remark upon such beauty as we may find in the proof.
Note that it is important to be, in the unfolding of time, orderly.
\item sensory input appears in sensory acquisition (accompany with pleasant scent\cite{born2012system}, and relevance to future plans\cite{born2012system})
\item sensory input conveyed to medial temporal lobe, where it is sometimes seen with single neuron instrumentation
\item unexpected novelty stimulates dopamine, differently from expected novelty (as in exploration)
\item reward stimulates dopamine, or, if prediction of reward is learned, the stimulus role is transferred to the prediction
\item dopamine can come from SN/VTA
\item dopaminergic midbrain is associated with enhanced hippocampus-dependent long-term memory formation \cite{wittmann2005reward}, i.e., rewards help form long term memory, and it occurs in some sleep phase, maybe REM or slow wave
\item LTP is divided into early and late, and dopamine contributes to late~\cite{wittmann2005reward}
\item dopamine is also used for long term depression, which is also learning
\item new neurons are generated~\cite{deng2010new} related to new hippocampus memories
\item explicit (vs. only implicit) learning favors access to sleep-dependent consolidation\cite[p. 115]{diekelmann2010memory}
\item motivational tagging of memories, might signal behavioral effort and relevance and mediate preferential consolidation\cite[p. 116]{diekelmann2010memory}
\item if video games are played just before retiring, they should probably not be violent. Conversely, if violent video games are played, something else, such as Tetris, should be used after the violent ones, prior to retiring.
\item sleep deprivation is expected to be detrimental to learning
\item sleep occurring 3 hours after learning was more effective than sleep delayed by more than 10 hours.\cite{gais2006sleep,talamini2008sleep,walker2003dissociable}
\item use same scent as in class, during SWS
\item slow oscillations typically seen in slow wave sleep (earlier part of sleep) have a causal role in the consolidation of hippocampus-dependent memories~\cite[p. 119]{diekelmann2010memory}
\item ripples typically seen in slow wave sleep (earlier part of sleep) have a causal role in the consolidation of memories~\cite[p. 119]{diekelmann2010memory}
\item it is not a particular sleep stage per se that mediate memory consolidation, but rather the neurophysiological mechanisms associated with those sleep stages~\cite[p. 116]{diekelmann2010memory}
\item re-activation of encoded memories occurs during slow wave sleep, in the order the remembered material was experienced Maquet\cite{maquet2000experience} cited in Diekelmann\cite[p. 117]{diekelmann2010memory}
\item theta oscillations, associated with REM sleep, have been found specifically over the right prefrontal cortex to be correlated with the consolidation of emotional memories\cite{nishida2009rem}
\item practice retrieval\cite{Bridge01082015}, better after sleep, can disrupt decoding if before sleep (check with Bridge 2015)
\item once attention to an item of knowledge has been rewarded, subsequent attention to that item is involuntary \cite{sali2014role}
\item consolidation occurs, differentiated by number of modalities, could be inert knowledge, as Whitehead~\cite{whitehead1959aims}, or retrievable, preferably. The number of related modalities, the more easily retrieved.
\item Retrieval allows for reconsolidation.\cite{sandrini2015modulating} (check it)\cite{schwabe2014reconsolidation}(check it) \cite{forcato2013role}(check it)
\cite{walker2003dissociable} (check it)
\item dopamine, nucleus accumbens helps reconsolidation occur with connection to right pre-frontal cortex \cite{knutson2001anticipation} and this is helped by anticipation of positive reward
\item right pre-frontal cortex is what experts use to be creative \cite{kowatari2009neural}, so we want to shepherd our taught material here, so that it is readily retrievable for inventing proofs
Chou et al.~\cite[p. 726]{chou2011sex} observes "the analytic brain for mathematical and logical cognition comprises the inferior frontal gyrus, parietal cortex and supramarginal gyrus", citing Dehaene et al, 1998k Goel et al., 1998, Zago et al., 2001. They measured, using fractional anisotropy (FA), microstructure of white matter that differed significantly in several areas, between men and women.
$<Isay>$Of these, at least the bilateral precuneus has been identified as of interest in mental activity related to mathematical proofs.$</Isay>$
They state ~\cite[p. 731]{chou2011sex} "the interaction analysis of dispositional measures by sex demonstrated that FA of the WM \ldots underlying occipital gyrus and postcentral gyrus was negatively associated with systematic quotient (SQ)in females.\ldots males exhibited larger FA in the WM of hippocampus whereas females showed larger FA in WM of parahippocampal gyrus \ldots females typically hold an advantage in tasks related to declarative memory, in which the parahippocampal gyrus has been implicated, such as in the retrieval and recognition of longterm \ldots memories."
Lisman et al.\cite{lisman2011neohebbian} report that "For novel information and motivational events such as rewards this signal at hippocampal CA1 synapses is mediated by the neuromodulator, dopamine." They summarize a consequence of the Hebb framework "if cell A represented object A and cell B represented object B, the co-occurrence of the two object would, by the Hebb rule, strengthen the synaptic linkage between these cells. This link would subsequently be evident when only object A was presented because it would lead to the firing of cell B, thus bringing object B to mind by association."
Lisman et al.\cite[p. 537]{lisman2011neohebbian} state "Two types of experiments demonstrate that dopamine can strengthen the synaptic potentiation produced by learning itself."
%PPT pedunculopontine tegmentum
Lisman et al.\cite[p. 540]{lisman2011neohebbian} state "Thus, novelty, reward stimuli and aversive stimuli are all able to activate the dopamine system \ldots in humans.
Lisman et al.\cite[p. 540]{lisman2011neohebbian} state "This reward-related memory enhancement was associated with a coactivation of SN/VTA, striatum, and hippocampus, as detected by fMRI (\cite{adcock2006reward,wittmann2005reward}) Memory enhancement after long retention intervals (e.g. 24h) has been consistently found (\cite{krebs2009personality,wittmann2011behavioral}). Moreover, the enhancement was greater at late timepoints than at early intervals (i.e. 3 wks vs 20 min)(47)
$<Isay>$ using novelty, reward (or punishment, i.e., if there are bad test grades, maybe they can be used), make these memories.$</Isay>$
Lisman et al.\cite[p. 540]{lisman2011neohebbian}point out that it is necessary to be able to encode and consolidate after a single exposure.
\caption{connectivity in medial temporal lobs and hippocampus ventral tegmentum loop processing info about object and spatial context. Allows perirhinal cortical info about novelity to general dopamine response}
Lisman et al.\cite[p. 542]{lisman2011neohebbian} summarize a model, citing Frey and Morris~\cite{frey1997synaptic} "weak stimulation induces on ly early long term potentiation (LTP). By contrast, stronger stimulation produces the dompamine-dependent protein synthesis that allows late LTP.\ldots the memory for event sthat occur before or after the dopamine release would depend not only on their own properties, but also on whether they fell within the penumbra of a dopamine-releasing stimulus."
$<Isay>$Music that gives "chills" gives dopamine, so if we could play something like the prelude to the StarWars IV before class, students would be primed. Some national anthems might be chill inducing, such as the French, Marseillaise (lyrics?) How long is this penumbra? It could be different for each person. for rodents, 1/2 hour$</Isay>$
Lisman et al.\cite[p. 542]{lisman2011neohebbian} "novel photographs of natural scenes ('strong events', such as those one would expect to see in the magazine National Geographic)" had a penumbra of at least 5 minutes.
$<Isay>$so, have some slides with nat geo pictures, at least every 5 minutes, new every time$</Isay>$
Lisman et al.\cite[p. 542]{lisman2011neohebbian} "Cholinergic\cite{sarter2005unraveling} and noradrenergic\cite{frey2008synaptic} projections to Medial temporal lobes can also modulate Long term potentiation and long-term memory. \ldots Reward-related SN/VTA activation improves memory for the rewarded stimulus but not for the non-rewarded stimuli given in close temporal proximity \cite{wittmann2011behavioral} implying either a very short or a very stimulus-specific penumbra. This is at odds with the observation that novelty-related activation of the SN/VTA has a long (ca 30 min) penumbra that affect memory for unrelated information (e.g., exposure to novel scenes can affect memory for words)\cite{fenker2008novel} One possible resolution is that the duration or stimulus-specificity of the penumbra depends on the type of motivational event that triggers dopamine release."
Lisman et al.\cite[p. 544]{lisman2011neohebbian} "the ability to recollect newly acquired information could be intrinsically rewarding. In fact, the study of human learning has revealed an interesting puzzle; long-term retention is not helped by simple re-exposure to recently learned material but is greatly helped by retesting even when subjects already know the answer \cite{karpicke2008critical}. One interesting possibility is that retesting provides an opportunity to generate intrinsic reward signals, thereby enhancing long-term persistence of newly learned material."
Mains et al.~\cite{Mains01072015}"Embedding three (clicker) questions within a
30 min lecture increased students' knowledge
immediately after the lecture and 2 weeks later. We
hypothesise that this increase was due to forced
information retrieval by students during the learning
process, a form of the testing effect."
Bridge and Voss \cite{Bridge01082015} "Cueing with actively retrieved objects facilitated memory of associated objects, which was associated with unique patterns of viewing behavior during study and enhanced ERP correlates of retrieval during test, relative to other reminder cues that were not actively retrieved. Active short-term retrieval therefore enhanced binding of retrieved elements with others, thus creating powerful memory cues for entire episodes."
$<Isay>$students may make their own flash cards to practice retrieval, but it appears a more effective strategy would have multiple differing cues$</Isay>$
Rottschy e al.\cite[p. 830]{rottschy2012modelling} define "working memory subsumes the capability to memorize, retrieve and utilize information for a limited period of time".
Rottschy e al.\cite[p. 836]{rottschy2012modelling} "experiments using non-verbal material showed significantly higher convergence in the left (pre-)motor area and bilateral dorsal pre motor cortex."
Rottschy e al.\cite[p. 843]{rottschy2012modelling} "selective attention system (Shulman et al.) \ldots is right-dominant and \ldots includes the temporo-parietal junction. \ldots apparent overlap between a distributed central executive for working memory, (and) the attention system"
$<Isay>$we can get their attention, for example by music that gives chills, and it will bring blood circulation to those areas, so, working memory will be supplied with circulation$</Isay>$
Wittman et al.\cite{wittmann2011behavioral}"Recent functional imaging studies link reward-related activation of the midbrain substantia nigra –ventral tegmental area
(SN/VTA), the site of origin of ascending dopaminergic projections, with improved long-term episodic memory. Here,
we investigated in two behavioral experiments how (1) the contingency between item properties and reward, (2) the magnitude of reward, (3) the uncertainty of outcomes, and (4) the contextual availability of reward affect long-term memory.
We show that episodic memory is enhanced only when rewards are specifically predicted by the semantic identity of the
stimuli and changes nonlinearly with increasing reward magnitude. These effects are specific to reward and do not occur in
relation to outcome uncertainty alone. These behavioral specifications are relevant for the functional interpretation of how
reward-related activation of the SN/VTA, and more generally dopaminergic neuromodulation, contribute to long-term
Abraham et al.\cite[p. 1906]{abraham2012creativity} investigates a specific aspect of creativity that they call conceptual expansion. They found that this activity selectively involved the anterior inferior frontal gyrus, the temporal poles and the lateral frontopolar cortex. These findings "go against \ldots dominance of the right hemisphere during creating thinking, and indicate \ldots anterior cingulate cortex \ldots (for) abstract facets of cognitive control."
Abraham et al.\cite[p. 1907]{abraham2012creativity} explain that conceptual expansion "refers to the ability to widen the conceptual structures of acquired concepts, a process that is especially critical in the formulation of novel ideas", citing Ward~\cite{ward1994structured}.
Abraham et al.\cite[p. 1910 -- 1911]{abraham2012creativity} found that the regions involved were left anterior inferior frontal gyrus, lateral frontopolar cortex, temporal poles, posterior regions in the inferior frontal gyrus, the middle frontal gyrus, the anterior cingulate cortex, the dorsomedial prefrontal cortex and the inferior parietal lobule. The "activation pattern is strongly lateralized to the left hemisphere." For working memory, "the overall brain activation pattern as a function of working memory was stronger in the right hemisphere"
\caption{left frontal polar cortex playing a particularly relevant role in concept expansion, is thought to mediate cognitive control at the most abstract level of information processing. The left dorsolateral pre frontal cortex, and the dorsomedial prefrontal cortex (BA8/9 and 8) also showed stronger brain activity}
\caption{Conceptual expension was associated with greater brain activity in left anterior inferior frontal gyrus (right is pictured)}
\caption{The inferior frontal gyrus, temporal poles and frontopolar cortex are involved in coceptual expansion. The roles of the anterior cingulate cortex and the dorsolateralprefrontal cortex were found to be most responsive in conceptual expansion, and active in divergent thinking.}
Abraham et al.\cite[p. 1912]{abraham2012creativity} In a measure intended to isolate cognitive control processes, results showed the dorsolateral prefrontal cortex and superior parietal lobule bilaterally, and "only the right dorsolateral prefrontal cortex and anterior cingulate were found to be involved."
$<Isay>$we are seeing that being creative calls upon left side structures, and when we get specifically to being creative with symbolic and diagrammatic representations, we may call upon also the right sides. $<Isay>$
Abraham et al.\cite[p. 1913]{abraham2012creativity} "The roles of the anterior cingulate cortex and the dorsolateral pre-frontal cortex are particularly noteworthy given the patterns of activation in these regions in the current study. Not only were they found to be more activated during divergent thinking compared to working memory, more importantly, they were also found to be most responsive as a function of conceptual expansion. \ldots the posterior aspect of the dorsomedial pre frontal cortex was also activated as a function of conceptual expansion. \ldots As this region has been discussed with reference to concepts that are central to hypothetical reasoning, such as constructive processes in cognition~\cite{abraham2008thinking} which involve flexible recombination of representations from memory~\cite{schacter1998cognitive} and evaluative judgment~\cite{zysset2003functional}, the dorsomedial prefrontal cortex may prove to be highly relevant structure for select aspects of creative thinking. \ldots speaks against the ubiquitous idea the right brain is more 'creative' than the left. \ldots in the current study, we have explored the deliberate problem solving mode of creating thinking under time constraints. There is, however, another vast dimension of creative thinking where idea generation occurs spontaneously, effortlessly, and/or in a state of defocused attention~\cite{|} In fact, creating idea generation is far less likely to result from deliberate cogitation during real everyday problem solving, but instead, it occur spontaneously and unpredictably. This unconscious non-deliberate spectrum of creating thinking \ldots is less amenable to well-controlled investigation. "
Born and Wilhelm~\cite[p. 192]{born2012system} "Experimental evidence for these three central implication is provided: It has been shown that reactivation of memories during slow-wave sleep(SWS) plays a causal role for consolidation, that sleep and specifically SWS consolidates preferentially memories with relevance for future plans, and that sleep produces qualitative change in memory representations such that extraction of explicit and conscious knowledge from implicitly learned materials is facilitated."
$<Isay>$In procedural memory we don't need to know why, there might not be any, (for example, remember the melody) but for some things they are accompanied by why. Is declarative everything other than procedural? What about implicit vs. explicit? These both are compatible with "why". There are times when we can use knowing why to save on what would otherwise need to be remembered. Does it have a name? Is it named in David Tall's article with bi-furcation in the name?$</Isay>$
Born and Wilhelm~\cite[p. 195]{born2012system} "Via the olfactory system odour stimulation acquires immediate access to the hippocampus. \ldots we found that the odour when re-exposed during SWS after learning induced a distinct activation of the left hippocampus, i.e. the odour served as a cue that reactivated the new memories for the card locations encoded in the left hippocampus, thereby enhancing these memories \ldots hippocampal networks are particularly sensitive in SWS to inputs capable of reactivating memories."
Born and Wilhelm~\cite[p. 197]{born2012system} "explicit encoding favours access to sleep-dependent memory consolidation (\cite{robertson2004awareness}). Involvement of the prefrontal-hippocampal system underlying explicit encoding has been proposed as prerequisite for consolidation to occur during sleep(\cite{marshall2007contribution}). \ldots emotionality of the encoded events can increase the memory benefit from sleep (\cite{kuriyama2004sleep,wagner2006brief})."
Wagner et al. \cite{wagner2006brief} investigated memory after a four hour interval of sleep. "Sleep following learning compared with wakefulness enhanced memory for emotional texts after 4 years (p = .001). No such
enhancement was observed for neutral texts (p = .571)."
Born and Wilhelm~\cite[p. 197]{born2012system} "Processing of anticipatory aspects of behaviour such as expaectancies and plans is particularly linked to executive functions of the prefrontal cortex that regulates activation of memory representations during anticipated retrieval and accommodates specifically the intentional and prospective aspects of a memory representation~\cite{polyn2008memory} \ldots prefrontal tagging of memories explicitly encoded under control of the prefrontal-hippocampal system could be decisive for the selectivity in off-line memory consolidation"
Born and Wilhelm~\cite[p. 199]{born2012system} "there is convergent evidence \ldots that the system consolidation process during sleep supports the extraction of invariant and repeating features in newly encoded memories, and in this way, the conversion of implicit into an explicit and conscious form of memory \ldots more than twice as many subjects of the sleep group gained insight into the hidden structure as compared with the wake control group \ldots subjects who had slept after \ldots training were distinctly more able to deliberately generate the sequence underlying \ldots than the subjects who had stayed awake"
Born and Wilhelm~\cite[p. 201]{born2012system} "sleep appears to prime the transformation of implicitly encoded information into explicit knowledge, i.e., something that is not conscious before sleep enters consciousness through sleep".
Dudai~\cite[p. 229]{dudai2012restless} defines declarative memory as that which requires conscious awareness for retrieval (facts, events), and nondeclarative can be retrieved in absence of conscious awareness (habit, skill).
Dudai~\cite[p. 231]{dudai2012restless} reminds "university students can improve their memory bye practicing self-testing, because retrieval practice is a powerful mnemonic enhancer", citing Karpicke and Roediger~\cite{karpicke2008critical}.
Forcato et al.~\cite[p. 1]{forcato2013role} observe "the reconsolidation process alllows new information to be integrated into the background of the original memory; second it strengthens the original memory. \ldots at least one labilization-reconsolidation process strengthens a memory via evaluation 5 days after its re-stabilization. \ldots this effect is not triggered by retrieval only. \ldots repeated labilization-reconsolidation processes made the memory more resistant to interference during re-stabilzation."
Forcato et al.~\cite[p. 1]{forcato2013role} "reconsolidation does not represent recapitulation of initiali consolidation, but rather, if refers to the functional role of this process: to stabilize memories."
Forcato et al.~\cite[p. 2]{forcato2013role} "when the reminder only included contextual cues (context reminder), the memory was evoked but not labilized."
A reminder has the effect of labilization which allows reconsolidation. "We found that just one labilization-reconsolidation process was enough to strengthen a memory that was evaluated 5 days following its re-stabilization. \ldots Memory persistence is increased by repeated triggering of labilization-reconsolidation."
Wirebring et al. \cite{}"Repeated testing is known to produce superior long-term retention of the to-be-learned material compared with repeated encoding and
other learning techniques, much because it fosters repeated memory retrieval. This study demonstrates that repeated memory retrieval
might strengthen memory by inducing more differentiated or elaborated memory representations in the parietal cortex, and at the same
time reducing demands on prefrontal-cortex-mediated cognitive control processes during retrieval. The findings contrast with recent
demonstrations that repeated encoding induces less differentiated or elaborated memory representations. Together, this study suggests
a potential neurocognitive explanation of why repeated retrieval is more beneficial for long-term retention than repeated encoding, a
phenomenon known as the testing effect."
\subsection{Educational Neuroscience}
\section{Systems Biology}
be sure to get the Kandel, Dudai 2014 review
\section{Physiologically Informed Constructivism}
Research has shown the progression of an idea, from a short term representation in the medial temporal lobe, to a consolidated memory in an area related to the single sensory modality with which that idea was received, to a representation in association cortex when multiple sensory modalities are involved, to a reconfiguration in right prefrontal cortex when an idea is used creatively.
How we as instructors shepherd these representations in the minds of our students can be suggested by cognitive neuroscience. One example is a deeper understanding of the utility, for memory and attention, to intrinsically rewarding learning activities. Another example is the application of multiple sensory modalities, such as representations in figures, text and mathematical symbols, and also pseudocode. Yet another is the deliberate construction of pseudo-monetary rewards, of unpredictable value, in the immediate aftermath of
instruction of a new idea. (This derives its effect from the dopamine response to the monetary reward, which is augmented by not knowing the amount.)
We can expect information stored in the cortex near a single sensory modality to be retrieved when a similar situation recurs. When multiple modalities are involved there are more ways to arrive at increased metabolism reactivating this memory, bringing it into consciousness.
Music is rewarding. The areas of the brain that are rewarded by memory are the same ones that help us remember. (Blood Zatorre salimpoor) We can reward the students / administer dopamine to help them remember. We have from Blood and Zatorre~\cite{blood2001intensely} that music so intensely pleasurable that it creates "chills" is correlated with activity in nucleus accumbens, which we also know, from (whom?) advances consolidation of memory representations in medial temporal cortex into longer term memory. %Here I am thinking how REM sleep is seen to be helpful to consolidation of memory, and I am thinking about the "subject matter" of REM sleep, and wondering whether a method more time-efficient than sleep could be developed
$<Isay>$ bagpipe music generates chills$</Isay>$
We want the students to construct an understanding of the material, for example, the correctness or resource utilization of an algorithm, and to be able to creatively apply the information they have to construct a proof of it.
The material first arrives in the student's awareness through some sensory modality, and we wish the information to be remembered, not only as inert knowledge, but as information that enters consciousness in response to a situation in which it is usefully employed in a proof.
Lisman and Grace\cite{lisman2005hippocampal} describes the hippocampal--ventral-tegmental area loop, controlling the entry of information into long term memory, facilitated by dopamine(DA). One stimulus for production of DA is the arrival of an unexpected reward \cite{schultz2000neuronal}, another is novelty\cite{kafkas2015striatal} (check this). Instructors can provide unexpected rewards with, for example, increased points for clicker questions. Other ways to provoke DA release, in the part of the brain that produces memories, is with music.\cite{salimpoor2015predictions} (check that this is the right article). Lisman and Grace\cite{lisman2005hippocampal} describe reactions that they state "provide a basis for the dopaminergic modulation of early long term potentiation (LTP)", which is a change to the signaling behavior of neurons in the brain. They continue "It follows that novelty itself should enhance LTP." Moreover they claim\cite[p. 707]{lisman2005hippocampal} "There is reasonable evidence from animal experiments that DA enhances learning, as would be expected from its enhancement of LTP." and \cite[p. 708]{lisman2005hippocampal}"The spiny cells in the accumbens are a likely site for combining novelty signals and goal-dependent motivational signals." and \cite[p. 709]{lisman2005hippocampal} "reasonable working hypothesis that \ldots combines novelty signals with information about saliency and goals \ldots thereby enhance the entry of the information into memory." \cite[p. 709]{lisman2005hippocampal} "without dopamine, late LTP does not occur and early LTP decays within about an hour."
Wittmann et al.\cite{wittmann2005reward}"Long=term potentiation in the hippocampus can be enhanced and prolonged by dopaminergic inputs from midbrain structures such as the substantia nigra. This improved synaptic plasticity is hypothesize to be associated with better memory consolidation in the hippocampus." They say that reward anticipation reliably elicits a dopaminergic response. They hypothesize that "activation of dopaminergic midbrain regions enhances hippocampus-dependent memory formation, possibly by enhancing consolidation." Wittmann et al.\cite[p. 464--465]{wittmann2005reward}"supports our hypothesis that the hippocampus is a major site for the neuromodulatory influence of reward on long-term memory formation \ldots supports the hypothesis that dopaminergic neuromodulation enhances hippocampus-dependent memory formation \ldots It is likely that a greater proportion of subsequently recallable items will undergo hippocampus-dependent consolidation than of subsequently recognizable items. \ldots these results provide evidence for a relationship between activation of dopaminergic areas and hippocampus-dependent long-term memory formation."
Here I want to diagram a sequence of activities mapped onto the brain areas, showing the progression of memory from the medial temporal lobes with new dentate granular cells, followed by consolidation in one sensory cortext or association cortext followed by reconsolidation into prefrontal and parietal followed by prefrontal from where it can be used creatively in the construction of proofs. It would be good if I could cite references for each milestone, and associate with conceptualizations, such as those in Harel and Sowder and Tall.
% % % % % % % % % % % % % % % %
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.
Lindquist \cite{lindquist2013mind} describes Mind wandering during lectures: Observations of the prevalence and correlates of attentional lapses, and their relationships with task characteristics and memory.
Brosch et al.\cite{brosch2013impact} examined the impact of emotion on perception, attention, memory, and decision-making.
Sali et al.\cite{sali2014role} show that "Previously rewarded stimuli involuntarily capture attention."
Ojemann\cite[p. 257]{ojemann2014human} " \textit{episodic, explicit} or \textit{declarative} memory, memory for the specific event (specific name, word) that occurs at encoding "
Ojemann\cite[p. 257]{ojemann2014human} "Our memory paradigm requires storage over a short period and thus a measure of \textit{recent} or \textit{short-term} memory. In the functional neuroimaging literature memory for this duration is often referred to as \textit{working} memory.
Ojemann\cite[p. 257--258]{ojemann2014human} "Lateral temporal cortex \ldots models of brain stubstrate for recent verbal memory \ldots during the encoding phase."
\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]
Though it is easy to assume that communication between practitioners is carried out verbally, there are examples of proofs without words \cite{nelson1993proofs}.
D\"orfler\cite[p. 129]{dorfler2000means} proposes that by discussion, abstraction can be promoted "the abstract terms might serve the purpose of talking about a variety of concrete, even physical experiences such as describing observations \ldots This abstracted manner of talking then acquires some independence from the experiences and experiential phenomena referred to, so that the abstract objects gain discursive existence. First, the abstract description lends 'meaning' to the experiences. Later, the abstract objects derive their 'meaning' from their taken-to-be representations or applications."
D\"orfler\cite[p. 129]{dorfler2000means} summarizes "our experiences with material objects are schematized in an image schema that is projected metaphorically to terms such as \textit{abstract object}. \ldots At times, these image schemas might even be exteriorized by symbolic expressions that, in turn, can serve as generic terms. \ldots A cognitively oriented explanation for the failure to be inducted into the mathematical discourse is, therefore, the lack of image schemata on which to base the discursive extension \ldots this lack might results from an absence of pertinent experiences \ldots there can be differences in the abilities of each individual to schematize his or her experiences and to make metaphoric use of words."
\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.
Van Oers\cite[p. 133--134]{van2000appropriation} emphasizes the role of the adoption of symbol use, as students learn mathematics, stating "Mathematics as a discipline is now generally conceived of as an activity in which constructive representation, with the help of symbols, plays a decisive role" citing Bishop, Freudenthal and Kaput.
Van Oers\cite[p. 133--134]{van2000appropriation} summarizes: "According to Freudentahl, mathematics is basically an activity of mathematizing: that is, organizing a (concrete empirical or abstract mental) domain, representing it with the hop of symbols \ldots experimenting with symbolic means". Freudenthal\cite[p. 10]{freudenthal1973mathematics} describes some of the history of symbol use: "Another algebraic idea is symbolism, the use of signs which do not belong to everyday language, to indicate variables. 'Think of a number' is how the problems are introduced in old narrative algebra. In Diaphantus' work the word 'number' becomes more and more a computation symbol. This continues in Indian and Arabian mathematics. The \textit{cossists} of the late Middle Ages has a whole system of symbols for the unknown and its powers \ldots "
Van Oers\cite[p. 135]{van2000appropriation} states that van Hiele "emphasized the importance of symbols in mathematics for grasping the mathematical meaning. \ldots appropriating the meaning of symbols is primarily a communicative process" requiring a mutual understanding of the meaning.
Van Oers\cite[p. 136]{van2000appropriation} states: "Symbols are indispensable as means for coding the results of thinking. More importantly, however, symbols also function as ways of organizing in the course of thinking". He goes on to report " According to empirical investigations of the development of mathematical thinking in pupils, the failure of meaningful appropriation of mathematical symbols has turned out to be one of the main problems in mathematics learning.", citing Hughes, 1986 and Walkerdine, 1988, also Miles and Miles 1992.
Van Oers\cite[p. 170]{van2000appropriation} states: "The analysis of symbolizing in a mathematical context has led us to the realization that symbol use is intrinsically related to meaning, negotiation of meaning, and communication."
Nemirovsky and Monk described symbolizing\cite[p. 177--178]{nemirovskymonk} "Conceiving of symbolizing as the creation of a space in which the absent is made present and ready at hand elicits at least two major issues: (a) the nature of such a space, and (b) the ways in which the absent is made present and ready at hand".
They observe \cite[p. 178]{nemirovskymonk} "our play as children is a crucial activity through which each one of us has practiced and learned to symbolize", citing Piaget 1962, Slade and Wolf 1994 and Winnicott, 1971/1992.
Nemirovsky and Monk give an example \cite[p. 204]{nemirovskymonk} "Lin shifted her attention from being immersed in creating something \ldots to reflecting on it as a particular manner of doing things". They go on to say, "Symbolizing is making possible the sudden and unanticipated encounter with past experience that can radically transform the 'here and now' of the symbolizer.
$<Isay>$ sudden, unanticipated, coming into consciousness is a relevant idea $</Isay>$.
Nemirovsky and Monk state\cite[p. 212]{nemirovskymonk} "Insights that 'come' or 'happen' to us is a way of saying that we often experience what we become as surprising and unexpected."
Bransford et al.\cite{bransford2000designs} mention "a common problem of expertise, namely, that things become so intuitively obvious that one forgets the difficulties that novices have in grasping new ideas".
$<Isay>$ consider that procedures to which we have become so accustomed that we do not need conscious attention (e.g., shifting gears for a car or bike) could be represented more efficiently in neurons, see Kandel and Squire on Memory and Attention\cite{squire2000memory} and Squire \cite{squire2015conscious}$</Isay>$
You can’t perform that action at this time.