BISOCIATION of Arthur Koestler in the

ACT OF CREATION (1964)

as the foundation of

HUMAN and COMPUTER CREATIVITY

Bronislaw Czarnocha Napoli, May 13, 2014

Arthur Koestler, The Act of Creation, 1964 • “I have coined the term ‘bisociation’ in order

to make a distinction between the routine skills of thinking on a single ‘plane’ as it were, and the creative act, which…always operates on more than one plane” p. 36

• for Koestler, bisociation represents a “spontaneous flash of insight...which connects previously unconnected matrices of experience” (p.45)

!Aha! Moment Eureka Experience

Albert Einstein (1949) Autobiographical Notes. P.7

• “What exactly is thinking? When at the reception of sense impressions, a memory picture emerges, this is not yet thinking, and when such pictures form series, each member of which calls for another, this too is not yet thinking. When however, a certain picture turns up in many of such series then – precisely through such a return – it becomes an ordering element for such series, in that it connects series, which in themselves are unconnected, such an element becomes an instrument, a concept.”

Progress of Understanding and Exercise of Understanding

(Koestler, p.619) “...it is necessary to distinguish between progress

in understanding - the acquisition of new insights, and the exercise

of understanding at any given stage of development. Progress in

understanding is achieved by the formulation of new codes

through the modification and integration of existing codes by

methods of empirical induction, abstraction and discrimination,

bisociation. The exercise or application of understanding the

explanation of particular events then becomes an act of subsuming

the particular event under the codes formed by past experience.

To say that we have understood a phenomenon means that we

have recognized one or more of its relevant relational features as

particular instances of more general or familiar relations, which

have been previously abstracted and encoded”.

Associative and Bisociative Thinking and Pattern Finding

[Koestler] distinguishes

associations that work within a given domain

(called a matrix by Koestler) and are limited to

repetitiveness (here, in Computer Creativity:

finding other/new occurrences of already identified

patterns)

and

bisociations representing novel connections

crossing independent domains (matrices).

Two Aha moments of Sultan, the genius among Koehler’s chimpanzees (1914)

• (I7.2.1914) Beyond some bars, out of arm's reach, lies an objective [a banana]; on this side, in the background of the experiment room, is placed a sawn-off castor-oil bush, whose branches can be easily broken off. It is impossible to squeeze the tree through the railings, on account of its awkward shape; besides, only one of bigger apes could drag it as far as the bars. Sultan is let in, does not immediately see the objective, and, looking about him indifferendy, sucks one of the branches of the tree. But, his attention having been drawn to the objective, he approaches the bars, glances outside, the next moment turns round, goes straight to the tree, seizes a thin slender branch, breaks it off with a sharp jerk, runs back to the bars, and attains the objective. From the turning round upon the tree up to the grasping of the fruit with the broken-off branch, is one single quick chain of action, without the least 'hiatus', and without the slightest movement that does not, objectively considered, fit into the solution described.• (p.103)

• The chimpanzee Sultan first of all squats indifferently on the box which has been left standing a little back from the railings; then he gets up, picks up the two sticks, sits down again on the box and plays carelessly with them. While doing this, it happens that he finds himself holding one rod in either hand in such a way that they lie in a straight line; he pushes the thinner one a little way into the opening of the thicker, jumps up and is already on the run towards the railings, to which he has up to now half turned his back, and begins to draw a banana towards him with the double stick. I call the master: meanwhile, one of the animal's rods has fallen out of the other, as he has pushed one of them only a little way into the other; whereupon he connects them again

Had Sultan known Greek he would certainly have shouted Eureka! (p.103)

an Aha moment from 5000 years ago:The

Hymns of Humble Appar

Ero così ignorante (pieno di cecità indotta dal Malam), che non conoscevo il Chaste Tamil di versi illuminanti e non componevo poesie e testi con essi. Non sapevo come apprezzare le grandi arti e scienze portati alla perfezione attraverso riflessioni ripetute e continue su di esse. A causa di tali incompetenze non ero in grado di apprezzare la presenza dell’ ESSERE e della Sua essenza. Ma come una madre e un padre pieno di amore e di cura, l’ESSERE dischiuse su Sua spontanea volontà la Sua presenza ed essenza e continuò a stare con me durante la mia evoluzione tenendomi sempre come Suo proprio soggetto. Ora, pieno di vera comprensione dell'ESSERE, salgo su per la collina di ERunbiyuur e testimonio l’ESSERE come Luce benevola. COMMENT: Uno degli oggetti dell’Ontologia Fondamentale che è stato portato in parole dai giganti della spiritualità Tantrica come Tirumular Namazvar e così via è quella di MALAM, il Buio Metafisico che rende le anime CIECHE e quindi incapaci di vedere qualsiasi cosa. Questa nozione metafisica è antica quanto il Sumerico NeRi di Suruppak (3000 a.C.). I filosofi sumeri hanno anche notato che qualunque competenza umana, comprese le competenze tecniche come inventare un alfabeto per scrivere il linguaggio, è lì solo perché ESSENDO emerge nelle profondità dell'anima come il Sole Interiore che viola il buio interiore e lascia che ci sia la luce dell’Intelligenza (Utu ude-a aAM Uru iGanamee - 505 Enmerkar e Araata)

Appar interpreta la sua intelligenza contro una comprensione metafisica fondata da questa Ontologia Fondamentale e in questo registra anche una continuità con i filosofi Sumeri.

Examples of Eureka moment through a bisociation.

Poincare: “Then I wanted to represent these functions by the quotient

of two series; this idea was perfectly conscious and deliberate, the

analogy with elliptic functions guided me….Just at this time I left

Caen, where I was then living, to go on a geologic excursion under the

auspices of the school of mines. The changes of travel made me forget

my mathematical work. Having reached Coutances, we entered an

omnibus to go some place or other. At the moment when I put my

foot on the step the idea came to me, without anything in my former

thoughts seeming to have paved the way for it, that the

transformations I had used to define the Fuchsian functions were

identical with those of non-Euclidean geometry. I did not verify the

idea;…but I felt a perfect certainty” (p.115)

Examples of bisociation; Darwin

wherein lies Darwin's greatness, the originality of his contribution? In picking up, one might say, the disjointed threads, plaiting them into a braid, and then weaving an enormous carpet around it. The main thread was the evolutionist's credo that the various species in the animal and vegetable kingdom had not been independently created, but had descended, like varieties, from other species…but it gave no explanation of the reasons which caused the common ancestor to transform itself gradually into serpents, walruses, and giraffes. The second thread that he picked up was of almost as trivial a nature for a country-bred English gentleman as Archimedes's daily bath: domestic breeding. The improvement of domestic breeds is achieved by the selective mating of favourable variations…

He had found the third thread…In …Malthus's An Essay on the Principle of Population. When Darwin read the he saw in a flash the 'natural selector', the causative agent of evolution, for which he had been searching:…” (p.1`40)

Examples of bisociation: Guttenberg’s

Printing Press

Here, then, we have matrix or skill No. I: the printing from wood blocks by means of rubbing. It

leads Gutenberg, by way of analogy, to the seal: 'When you apply to the vellum or paper the seal

of your community, everything has been said, everything is done, everything is there. Do you not

see that you can repeat as many times as necessary the seal covered with signs and characters?'

• Yet all this is insufficient.

I took part in the wine harvest. I watched the wine flowing, and going back from the effect to the

cause, I studied the power of this press which nothing can resist....

At this moment it occurs to him that the same, steady pressure might be applied by a seal or

coin-preferably of lead, which is easy to cast on paper, and that owing to the pressure, the lead

would leave a trace on the paper - Eureka!

• A simple substitution which is a ray of light.... To work then! God has revealed to me the secret

that I demanded of Him•••. (p.123)

Bisociation: When two habitually independent matrices of perception or reasoning interact with each other the result is either: a collision ending in laughter.A Smullyan joke:

Un visitatore che vuole conoscere come vivono i carcerati viene condotto in giro dal direttore. Passano per i corridoi e guardano non visti, nelle celle attraverso certi spioncini chiamati "sportelli di Giuda". In una delle celle 4-5 prigionieri sono seduti sulle brande e ogni tanto uno dice un numero (per esempio sedici) e gli altri ridono. Dopo avere osservato per un po' la scena, il visitatore chiede al direttore che cosa accade, che cosa sono quei numeri e perché i carcerati ridono. "Semplice - risponde il direttore - raccontano barzellette. Ne hanno fatto un elenco, ognuna con il suo numero. Le hanno sentite così tante volte che le conoscono a memoria"I due continuano ad osservare mentre molti numeri vengono lanciati. A un certo punto uno dice 72, e nessuno ride. "E adesso, che sta succedendo?" chiede il visitatore. "Oh, - risponde il direttore - quel tipo le barzellette non le sa proprio raccontare!"

A fusion of intellectual synthesis. Poincare fusion:

For fifteen days I strove to prove that there could not be any functions like those I have since called Fuchsian functions. I was then very ignorant; every day I seated myself at my work table, stayed an hour or two, tried a great number of combinations, and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination.

confrontation in an aestetic experience Golden lads and girls all must, As chimney-sweepers, come to dust. (Cymbeline)

Computational Creativity, from M.R. Berthold (ed)

Bisociative Knowledge Discovery, LNAI 7250, 2012

Along with other essentially human abilities, such as intelligence, creativity has long been viewed as one of the unassailable bastions of the human condition. Since the advent of the computer age this monopoly has been challenged. A new scientific discipline called computational creativity aims to model, simulate or replicate creativity with a computer (Boden,1999;Dubitsky et al, 2012). Boden(1994) distinguishes three types of creative discoveries: Combinatorial, Exploratory and Transformational.

Computational Creativity – bases.

Essential Distinction: …bisociation can be defined as sets of concepts that bridge two otherwise not – or very sparcely – connected domains (viz progress in understanding; finding patterns across domains) whereas an association bridges concepts within a given domain (viz. exercise of understanding, finding patterns in individual domains). Definition 1 Creativity is the ability to come up with ideas or artifacts that are new, surprising, and valuable. Example: “in 1996 Akihiro invented a “digital pet” called Tamagotchi which soon became a best seller

Computational Creativity – types of bisociation

1 Bridging Concept 3.Bridging by Graph Structural Similarity

Computational Creativity/Human Creativity

2. Bridging graphs

Human Creativity: R. Catanuto, Everest Academy, Switzerland LEARNING ROUTES METHOD LEARNING ROUTES METHOD

Teaching-Research Questions

1. Given their common origin, what are the ways in which human creativity and computer creativity can mutually positively reinforce each other?

2. What are the essential parameters of difference between human and computer creativity?

My hypothesis (TRQ2):

• …Non è un quadretto ma una finestra, in cui si può mettere un numero.

• B: Come sarebbe?

• P: Due finestre sono uguali a 64, una finestra è uguale a 32. Infatti, se sottrai 12 da entrambi I lati, vedrai che le due finestre sono uguali a 64.

• B: Ma ci sono numeri nelle finestre?

• P: Due finestre sono 64, perciò una finestra è 32.

• B: Finestra!?

• P: Proprio così: una finestra. Guarda: un elefante più un elefante fa 64. Allora, a che cosa è uguale un elefante? Due elefanti sono uguali a 64. Allora, un elefante a che cosa è uguale?

• B: Un elefante? Uhm, sì. Un elefante è uguale a 32. Ora capisco… dunque l’equazione…

• P: Se due elefanti sono uguali a 60, a che cosa è uguale un elefante?

• B: Un elefante?, ok, un elefante è uguale a 30. Ora lo vedo. Ora l’equazione…………..aaaaaaa

COMMENTARY of the observer:

Riflettendo su questo dialogo, si pongono diverse domande: Perché a Przemek è venuto in mente un elefante? Perché per Bartek funziona un elefante, dove non avevano funzionato né un quadretto né un segmento? Da dove viene fuori l’elefante?

C’erano sulla mensola due statuine, un maialino e un elefante. Il maialino non può funzionare per i significati che vi si associano (almeno in lingua polacca), ma l’elefante è neutro, pronto per essere preso come simbolo di un qualche oggetto mentale. Così l’elefante è stato usato come simbolo adeguato di un oggetto mentale, che spesso viene indicato con ics ma senza che ce ne sia necessità. Non si tratta di un episodio accidentale. L’uso fatto è ciò che si chiama metonimia. Quando risolviamo problemi in matematica, specialmente in algebra, usiamo spesso metonimie

BISOCIATION-AS- EU GRANT IDEA ???PDTR in bisociation / Aha moment???

• The situation is interesting: we have one precise principle underlying

both Human and Computer creativity. Each domain is in the beginning of its development. A collaboration in the investigation of both and their mutual impact promises, in the spirit of Bisociation to bring a wealth of new results and discoveries for both. On one hand “The ability of humans to perform creative reasoning like bisociative thinking outstrips that of machines by far”, and on the other hand, the generality of computer creativity offer wealth of applications to the classroom.

• .The re-introduction of creativity into mathematics classroom might be the only way through which our students will get back interest in, and enjoyment with mathematics. Our respective tasks, facilitation of the discovery in the classroom and sensitizing computers to “habitually separate domains”, although different bear, a similarity, which could be the basis of collaboration leading to the next EU grant, the follower to the Commenius Programme grant 2005-2008 PDTR and to the Bisonet grant of EU 2009-2011.

HUMAN AND COMPUTER

CREATIVITY

Bisociation of Koestler (1964) as the

Theory of the !Aha!-moment.

Chinese National Association of

Mathematics Education Conference

Lanzhow, Gansu, China

June 2014

Bronislaw Czarnocha

Hostos Community College

City University of New York

NYC,USA

Arthur Koestler, The Act of Creation,

1964

• “I have coined the term ‘bisociation’ in order to make a

distinction between the routine skills of thinking on a

single ‘plane’ as it were, and the creative act,

which…always operates on more than one plane” p. 36

• for Koestler, bisociation represents a “spontaneous

flash of insight...which connects previously

unconnected matrices of experience” (p.45)

Reminder from CTRAS 5: Wu Zhipeng The

Construction of High School Mathematics

Wisdom Class

• Professor Cheng Shangrong

synthesized research about

intelligence in and out of China, and

came to the conclusion that

intelligence is a typical character

produced and expressed in education

situations, and it is directed by virtue

and creativity. The purpose of

obtaining intelligence is to cultivate

and develop the students’ capability,

mathematics sensitivity and sudden

enlightenment.

Mao Tse Tung: On Practice, July 1937, p.68

As social practice continues, things that give rise to

man’s perception and impressions in the course of his

practice are repeated many times; then a sudden

change, (a leap) takes place in the brain in the process

of cognition, and concepts are formed. Concepts are

not longer the phenomena, the separate aspects and

external relations of things; the grasp the essence, the

totality and the internal relations of things.

Albert Einstein (1949) Autobiographical Notes,

p.7 Notes. P.7

lbert. P.7 • What exactly is thinking? When at the reception of sense

impressions, a memory picture emerges, this is not yet

thinking, and when such pictures form series, each

member of which calls for another, this too is not yet

thinking. When however, a certain picture turns up in

many of such series then – precisely through such a

return – it becomes an ordering element for such series,

in that it connects series, which in themselves are

unconnected, such an element becomes an instrument, a

concept.”

Bisociation: When two habitually independent matrices

of perception or reasoning interact with each other the

result is either: a collision ending in laughter

A fusion of intellectual synthesis.

Poincare fusion

For fifteen days I strove to prove that there could not be any functions like those I have since called Fuchsian functions. I was then very ignorant; every day I seated myself at my work table, stayed an hour or two, tried a great number of combinations, and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination

A visitor who wants to investigate life of prisoners

comes to a prison and is led around by the prison’s

chief. They walk along the prison corridor and look,

unnoticed, into cells through small, so called Judas

windows placed within the doors. In one of the cells,

4,5 cellmates are sitting on their beds, from time to

time one of them states a number, like for example

sixteen (16) and the rest are laughing. After observing

it for a while, the visitor asks the prison’s chief:

What’s happening here? What are those numbers and

why they laugh?

The chief says, it’s simple. They tell jokes; they made

a list of jokes, each joke has a number and they heard

it so many times they know it by heart.

hey continue watching as several numbers are shouted

by different prisoners. At some point one of them says

56, they laugh and stop for awhile, but one of them

laughs so much, he falls under the table laughing.

The visitor is asking, what happened to this one?

Oh – the chief answers- he heard it for the first time!

A Raymond Smullyan joke:

A confrontation through

esthetic experience

Examples of bisociation: Guttenberg

Press (Koestler, p.123)Printing Press

Here, then, we have matrix or skill No. I: the printing from wood blocks by

means of rubbing. It leads Gutenberg, by way of analogy, to the seal:

'When you apply to the vellum or paper the seal of your community,

everything has been said, everything is done, everything is there. Do you not

see that you can repeat as many times as necessary, the seal covered with

signs and characters?'

Yet all this is insufficient.

I took part in the wine harvest. I watched the wine flowing, and going back

from the effect to the cause, I studied the power of this press which nothing

can resist....

At this moment it occurs to him that the same, steady pressure might be

applied by a seal or coin-preferably of lead, which is easy to cast on paper,

and that owing to the pressure, the lead would leave a trace on the paper -

Eureka!

A simple substitution which is a ray of light.... To work then! God has

revealed to me the secret that I demanded of Him•••.

Koestler’s Triptych

an example of the triptych assignment used by V. Prabhu in her Introductory Statistics class:

Trailblazer Outlier Originality Sampling Probability

Confidence Interval Law of Large Numbers

Lurker/Lurking Variable Correlation Causation The triptych below is an example of student work:

Trailblazer OUTLIER Original Random SAMPLING Gambling Chance PROBABILITY Lottery

Lurking Variable CORRELATION Causation Testing CONFIDENCE INTERVALS Results

Sample Mean LAW OF LARGE NUMBERS Probability

Classroom of Algebra: 0 The teacher asked the students during the review: “Can all real values of be used for the domain of the function 𝑋 + 3?” 1 Student: “No, negative X’s cannot be used.” (The student habitually confuses the general rule which states that for the function only positive-valued can be used as the domain of definition, with the particular application of this rule to .) 2 Teacher: “How about -5 ?” 3 Student: “No good.” 4 Teacher: “How about -4 ?” 5 Student: “No good either.” 6 Teacher: “How about -3 ?” 7 Student, after a minute of thought: “It works here.” 8 Teacher: “How about -2?” 9 Student: “It works here too.” 10 A moment later Student adds:” Those X’s which are smaller than -3 can’t be used here.” (Elimination of the habit through original creative generalization.) 11 Teacher: “How about 𝑋 − 1?” 12 Student, after a minute of thought: “Smaller than 1 can’t be used.” 13 Teacher: “In that case, how about 𝑋 − 𝑎 ?” 14 Student: “Smaller than “a” can’t be used.”(Second creative generalization)

Act of Creation is the defeat of habit by

originality. (Koestler,p 96)

That means that bisociation not only is the cognitive

reorganization of the concept by “an immediate perception of

relations”, but also it can be an affective catalyzer of the

transformation of habit into originality. Liljedahl (2004) meta-

findings: “…Aha experience has a helpful and strongly

transformative effect on student’s beliefs and attitudes towards

mathematics…” (p.213). The presence of this cognitive/affective

duality of creativity, of the Aha moment, can provide the intrinsic

motivation to bridge the Achievement Gap in US (Prabhu, 2014)

and in other centres of educational inequality, such as Poland

(PISA in Focus,36).

How to facilitate bisociation- Aha! Moment

in the classroom?

The dual character of bisociation as cognitive restructuring and as an

affective act of liberation makes it an excellent classroom tool for

classroom teaching in the contemporary era. Since “…minor,

subjective bisociative processes do occur on all levels, and are the

main vehicles of untutored learning.”(p.658).

Therefore, in order to approximate the conditions of “untutored

learning” in the mathematics classroom necessary for Koestler’s

bisociation we, as teacher-researchers are led to the “guided inquiry

leading to discovery” method, which allows us to find, within the

classroom discourse, the space for intellectual freedom within which

these conditions are met.

Why and how utilize the bisociation in the

mathematics classroom?

WHY “Students in remedial mathematics at community colleges are at risk.

Their success in higher education depends on overcoming obstacles to

learning, many of which are attitudinal, related to affect perception, and

detrimental to cognition. “ (Prabhu, 2014)

WHY Mathematical creativity may be the only gate through which to

reactivate the interest and the value of mathematics among contemporary

youth whose engagement in the field is hampered by disempowering habits

expressed as “I can’t do it,” “I am not good in math,” ”thinking tires me”

(Czarnocha et al, 2011).

WHY Habits are indispensable core of stability and ordered behavior; they

also have a tendency to become mechanized and to reduce a man to the

status of conditioned automaton. The creative act, by connecting unrelated

dimensions of experience, enables him [the man, or her, the woman] to

attain to a higher level of mental evolution. (Koestler, p641)

Achievement gap is the persistence difference in

student scholastic achievement between different

social groups, classes, genders.

Analysis of PISA 2012 exam revealed that whereas many participatnts

solidified its lead (Shanghai, China) in overall achievement or have

made significant advances (Poland from 25th to 14th position), at the

same time equally significant Achievement Gaps between children of

different social classes were observed. (PISA 2012 News #36). The

degree of achievement gap was measured in the table below by the

difference in the average scores of the top social class (professionals,

managers) and the average scores of the lowest social class

(elementary professions, blue collar workers).

In US, teaching with the help of the Act of Creation that is facilitation of

the Aha moment is one of the routes through which the Achievement

Gap can be closed. The second route comes from Texas and is

described in http://www.nytimes.com/2014/05/18/magazine/who-gets-to-

graduate.html?_r=0 (NYTimes, 5/15/14)

Achievement Gaps revealed by PISA 2012 1 Korea 585-540 45 5

2 Japan 565-520 45 7

3 Norway 515--465 50 30

4 Canada 545—490 55 13

5 Estonia 555-500 55 11

6 Latvia 530- 435 65 28

7 Russia 520-455 65 34

8 Finland 565-460 65 12

9 UK 525-460 65 26

10 Lithuania 510-445 65 37

11 Ireland 565-475 70 20

12 USA 520-450 70 36

13 Spain 530-460 70 33

14 Sweden 510-440 70 38

15 Italy 525-455 70 32

16 Denmark 535-460 75 22

17 Netherland 560-480 80 10

18 Vietnam 575-490 85 17

19 Malayasia 460-405 55 52

20 Mexico 455-400 55 53

21 Costa Rica 450-405 45 56

22 Kazachstan 450-425 25 49

23 Poland 575-485 90 14

23 Germany 570-480 90 16

23 France 545-455 90 25

23 Luxemburg 545-455 90 29

24 Portugal 555-455 100 31

24 Israel 515-415 100 41

25 Belgium 570-465 105 15

26 Shangai 650-540 110 1

26 Taipei 625-515 110 4

26 Slovakia 555-445 110 35

Achievement Gap in USA

Computational Creativity, from M.R. Berthold (ed)

Bisociative Knowledge Discovery, LNAI 7250, 2012

Along with other essentially human abilities, such as intelligence, creativity has long been viewed as one of the unassailable bastions of the human condition. Since the advent of the computer age this monopoly has been challenged. A new scientific discipline called computational creativity aims to model, simulate or replicate creativity with a computer (Boden,1999;Dubitsky et al, 2012). Computational Creativity = Bisociation (minus) leap of insight.

Computational Creativity – types of bisociation

1 Bridging Concept

3.Bridging by Graph Structural Similarity

Computational Creativity/Human Creativity

Possibility of the new type of collaboration between

Mathematics Education and Informatics

• The situation is interesting: we have one precise principle underlying both Human and Computer creativity. Each domain is in the beginning of its development. A collaboration in the investigation of both and their mutual impact promises, in the spirit of Bisociation to bring a wealth of new results and discoveries for both. On one hand “The ability of humans to perform creative reasoning like bisociative thinking outstrips that of machines by far”, and on the other hand, the generality of computer creativity offer wealth of applications to the classroom.

• .The re-introduction of creativity into mathematics classroom might be the only way through which our students will get back interest in, and enjoyment with mathematics. Our respective tasks, facilitation of the discovery in the classroom and sensitizing computers to “habitually separate domains”, although different bear, a similarity, which could be the basis of new type collaboration between Math Ed and Informatics.

Bisociation of Koestler (The Act of

Creation,1964) as the Theory of the

!Aha!-moment.

The basis for the mathematical creativity in

the mathematics classroom, and beyond.

Bronislaw Czarnocha

Napoli, Italia

9 of May, 2014

Plan of the conversation

• Elephant, the resonance and bisociation.

• Bisociation and discourse.

• Examples of bisociation in the history of

science.

• Use of bisociation in the classroom.

• Teaching-Research as bisociative

framework.

Il momento di “Un Elefante”

Discussion:

What is happening in the Elephant episode?

a)how many different approaches to the

problem are evidenced?

b) how many different frames (frameworks) are

involved in the final Aha moment?

D. Iannece, M. Mellone, R.Tortora (2006) New Insights

Into Learning Processes from Some Neuroscience

Issues,

p.1 A model of cognitive dynamics with a distinctive

feature of a basic resonance dynamics

…which implies that a cognitive shifting from one

cognitive dynamics to another…is a specific feature

and a specific goal of the learning process.

p.5 …the monitoring of subjects engaged in the task

by means of BIT (brain imaging techniques) reveals

that the active brain areas are respectively: a zone in

the back part of the brain, specialized in the

processing of perceptive information..; and the

frontal zone involved in logical reasoning.

Arthur Koestler, The Act of Creation, 1964

• “I have coined the term ‘bisociation’ in order

to make a distinction between the routine

skills of thinking on a single ‘plane’ as it were,

and the creative act, which…always

operates on more than one plane” p. 36

• for Koestler, bisociation represents a

“spontaneous flash of insight...which

connects previously unconnected matrices

of experience” (p.45)

Therefore:

• in my opinion, the elephant incident represents “the

spontaneous flash of insight” which connected there

two previously unconnected “matrices of

experience”: the perceptual experience of an

elephant and the logical experience of solving a

linear equations.

• I hypothesize, that “basic resonance dynamics

• …[of] a cognitive shifting from one cognitive

dynamics to another” is the manifestation of

Koestler’s bisociation.

an Aha moment from 5000 years ago:The Hymns

of Humble Appar

• I was so ignorant (full of blindness induced by the

Malam) that I did not know the Chaste Tamil of

illuminating verses and compose poems and lyrics

with the same. I did not know how to appreciate the

great arts and sciences brought to perfection through

repeated and continuous reflections on them.

Because of such incompetencies I was not able to

appreciate the presence of BEING and His

essences. But like a mother and father full of love

and care, BEING disclosed on His own accord His

presence and essences and continued to be with me

along with my developments always keeping me as

His own subject. Now full of true understanding of

BEING, I climb up the hill of ERunbiyuur and witness

BEING as the Benevolent Light .

Is it true that learning and in particular learning of mathematics is

primarily the question of the discourse? (Anna Sfard)

A Stick and a banana outside of the cage with the

chimpanzee

• :

• The chimpanzee Sultan first of all squats indifferently on the box which

has been left standing a little back from the railings; then he gets up,

picks up the two sticks, sits down again on the box and plays carelessly

with them. While doing this, it happens that he finds himself holding one

rod in either hand in such a way that they lie in a straight line; he pushes

the thinner one a little way into the opening of the thicker, jumps up and

is already on the run towards the railings, to which he has up to now half

turned his back, and begins to draw a banana towards him with the

double stick. I call the master: meanwhile, one of the animal's rods has

fallen out of the other, as he has pushed one of them only a little way into

the other; whereupon he connects them again

Personal recollection:

• A long time ago I met a friend who prepared a special problem for me to

solve. I guess he wanted to check me out mathematically and gave one

of the little metal puzzles which are constructed out of two pieces,

connected in a mysterious, definitely not obvious way. This is a typical

problem to solve. I had to do it to keep his respect and my status as “the

crazy math professor” or “gypsy scholar”. Although I was anxious about

the challenge, I was 100% committed to finding the solution and I knew

it would take me some time. I also knew that attempts at rational solution

are not my forte so decided to let the intuition to discover the way, and I

kept this little mathematical puzzle in my hand in the pocket of my jacket

and played with it, while we took a walk. After a long while, I felt

suddenly that the two pieces somehow started disengage themselves

and the goal was for my fingers to understand and retrace the steps they

have done in my hand. I caught the motion and managed to put it back

into their mutual lock and tried again from my conscious attention.

Koestler’s examples of bisociation from the history

of science and mathematics

1. Poincare (p.115) And now follows one of the most lucid

introspective accounts of Eureka act by a great scientist:

For fifteen days I strove to prove that there could not be any functions like

those I have since called Fuchsian functions. I was then very ignorant;

every day I seated myself at my work table, stayed an hour or two, tried a

great number of combinations, and reached no results. One evening,

contrary to my custom, I drank black coffee and could not sleep. Ideas rose

in crowds; I felt them collide until pairs interlocked, so to speak, making a

stable combination. By the next morning I had established the existence of a

class of Fuchsian functions, those which come from the hyper-geometric

series; I had only to write out the results, which took but a few hours.

2. August von Kekule (1865). Discovery of organic

molecular rings

p.118 I turned my chair to the fire and dozed, he relates.

Again the atoms were gamboling before my eyes. This rime

the smaller groups kept modestly in the background. My mental

eye, rendered more acute by repeated visions of this kind,

could now distinguish larger structures, of manifold

conformation; long rows, sometimes more closely fitted

together; all twining and twisting in snakelike motion. But look!

What was that? One of the snakes had seized hold of its own

tail, and the form whirled mockingly before my eyes. As if by a

flash of lightning I awoke ... Let us learn to dream, gentlemen.

Why and how utilize the bisociation in the

mathematics classroom?

WHY “Students in remedial mathematics at community colleges are at risk.

Their success in higher education depends on overcoming obstacles to

learning, many of which are attitudinal, related to affect perception, and

detrimental to cognition. “ (Prabhu, 2014)

WHY Mathematical creativity may be the only gate through which to

reactivate the interest and the value of mathematics among contemporary

youth whose engagement in the field is hampered by disempowering habits

expressed as “I can’t do it,” “I am not good in math,” ”thinking tires me”

(Czarnocha et al, 2011).

WHY Habits are indispensable core of stability and ordered behavior; they

also have a tendency to become mechanized and to reduce a man to the

status of conditioned automaton. The creative act, by connecting unrelated

dimensions of experience, enables him [the man, or her, the woman] to

attain to a higher level of mental evolution. (Koestler, p641)

How?

• The dual character of bisociation as cognitive restructuring

and as an affective act of liberation makes it an excellent

classroom tool for classroom teaching in the contemporary

era. Since “…minor, subjective bisociative processes do

occur on all levels, and are the main vehicles of untutored

learning.”(p.658).

• Therefore, in order to approximate the conditions of

“untutored learning” in the mathematics classroom necessary

for Koestler’s bisociation we, as teacher-researchers are led

to the “guided inquiry leading to discovery” method, which

allows us to find, within the classroom discourse, the space

for intellectual freedom within which these conditions are met.

How? – Vrunda Prabhu (2014): Koestler Tripych.

How? Vrunda Prabhu classroom triptychs:

• an example of the triptych assignment used by V. Prabhu in her

Introductory Statistics class:

Trailblazer Outlier Originality

Sampling

Probability

Confidence Interval

Law of Large Numbers

Lurker/Lurking Variable Correlation Causation

The triptych below is an example of student work:

Trailblazer OUTLIER Original

Random SAMPLING Gambling

Chance PROBABILITY Lottery

Lurking Variable CORRELATION Causation

Testing CONFIDENCE INTERVALS Results

Sample Mean LAW OF LARGE NUMBERS Probability

How? Broni Czarnocha guided inquiry:

0 The teacher asked the students during the review: “Can all real values of

be used for the domain of the function 𝑋 + 3?”

•1 Student: “No, negative X’s cannot be used.” (The student habitually

confuses the general rule which states that for the function only positive-

valued can be used as the domain of definition, with the particular

application of this rule to .)

•2 Teacher: “How about -5 ?”

•3 Student: “No good.”

•4 Teacher: “How about -4 ?”

•5 Student: “No good either.”

•6 Teacher: “How about -3 ?”

•7 Student, after a minute of thought: “It works here.”

•8 Teacher: “How about -2?”

•9 Student: “It works here too.”

• 10 A moment later Student adds:” Those X’s which are smaller than -3

can’t be used here.” (Elimination of the habit through original creative

generalization.)

•11 Teacher: “How about 𝑋 − 1?”

•12 Student, after a minute of thought: “Smaller than 1 can’t be used.”

•13 Teacher: “In that case, how about 𝑋 − 𝑎 ?”

•14 Student: “Smaller than “a” can’t be used.”(Second creative

generalization)

How? Roberto Catanuto, Everest Academy, Lugano, Swiss

Teaching – Research as the bisociative framework • Teaching and Research are two, habitually not very compatible

frameworks. When integrated into one activity it is the source of

teachers’ creativity.

• The connection between TR and Koestler’s bisociation was during

Vrunda Prabhu’s collaborative teaching experiment involving 3

different faculty in one classroom of mathematics.

“Three instructional approaches emerged, each arising from the natural

inclination toward mathematics and problem-solving of each teacher-

researcher on the team. There, of course, were differences in individual

approaches, one being more procedural, another more conceptual,

however, the commonality across instructional approaches, is the

commitment and intent for learners to discover the underlying

mathematical structures called for in each problem situation. The

instructional approaches can all be explained using the theoretical

perspective created by Arthur Koestler. Bisociation was facilitated, as the

creative leap that occurs when several frames of reference are held in

simultaneous scrutiny and insight” (Prabhu, 2014).

Fairy Tales Digression: The Triad of Piaget and Garcia

What’s happening with

Mathematics Education in US

and

in New York City?

Mooc (massive Online Open Course) – elite universities

Common Core – public education nationwide

Real work – community colleges in the Bronx and…

MOOC – Massive Open Online Courses • Stanford University: • 3 2 , 0 0 0 students i n " Writing in the Sciences”

• 2 1 , 0 0 0 i n " Statistics in Medicine” 4 1 , 0 0 0 in " How to Learn Math”

Harvard University: Introduction to Computer Science. Estimated effort: 9 problem sets (10 to 20 hours each), 1 final project.

Ways to take it: a) Audit this Course free

b) Earn a Verified Certificate of Achievement ($90 USD)

c) Earn Harvard Credit ($2050 USD)

Results: About 10% of the students who sign up typically complete the course.

Reasons: a)…missing, many students complained, was a human connection beyond the streamed lecture.(NPR,12/31/14

b) the amount of peer-graded homework on the course,

c) that teacher involvement in a thread seems to accelerate the decline

Common Core State Standards/Race to the Top

Common Core State Standards Initiative is an educational initiative in the United States that details what K-12 students should know in English language arts and mathematics at the end of each grade. It seeks to establish consistent educational standards across the states as well as ensure that students graduating from high school are prepared to enter credit-bearing courses at two- or four-year college programs or enter the workforce. Race to the Top is a $4.35 billion United States Department of Education contest created to spur innovation and reforms in state and local district K-12 education… States were awarded points for satisfying certain educational policies, such as performance-based standards (often referred to as an Annual professional performance review) for teachers and principals, complying with Common Core standards, lifting caps on charter schools, turning around the lowest-performing schools, and building data systems.

Common Core Issues

1. Computer based standardized exam 2. Teacher evaluation based on these exams. 3. Research based Learning Trajectories Alaska, North Dakota, Texas, and Vermont did not submit Race to the Top applications for either round; Texas Governor Rick Perry stated, "we would be foolish and irresponsible to place our children’s future in the hands of unelected bureaucrats and special interest groups thousands of miles away in Washington.”

M. Common Core Standards are based on Learning Trajectory Framework

• Learning Trajectory for a concept is a hypothetical path in the space of relevant concepts that needs to be traversed by a learner to grasp and master a concept.

• For (Clements, D. and Sarama, J., 2009), Learning Trajectory (LT) of a particular mathematical concept consists of three components:

• a specific mathematical goal,

• a developmental path along which students’ thinking and comprehension develops and

• a set of instructional activities that help students move along that path.

Learning Trajectory for Linear Equations

From: Learning Trajectories from the Arithmetic/Algebra Divide, NA PME 12

Rational Number Sense to Algebraic Thought.

Place Value Concept Map

Real work – community colleges in the Bronx, and…

…at the national statistics on college graduation rates, there are two big trends that stand out right away. The first is that there are a whole lot of students who make it to college but never get their degrees. More than 40 percent of American students who start at four-year colleges haven’t earned a degree after six years. If you include community-college students in the tabulation, the dropout rate is more than half, worse than any other country except Hungary. (NYT

magazine, May 15, 2014)

At Hostos Community College in the Bronx, altogether only 27% of students graduate after 6 years (73% DO NOT)

The second trend is that whether a student graduates or not seems to depend today almost entirely on just one factor — how much money his or her parents make.

(NYT magazine, May 15, 2014 )

It is NOT the ability that matters, but income and class background.

The basic question of mathematics education in US:

How, precisely, do you motivate students to take the steps they need to take in order to succeed?

In the Bronx:

• Peer leaders in the classroom-math department wide

• Student coaches- college wide

• student involvement: Handshake, Didactic Contract, Students as Partners in Learning (Vrunda Prabhu)

• collaborations with creative expression (with poetry, drama, psychologist support, (Vrunda Prabhu);

• facilitating bisociativity, “the spontaneous leap of insight” (Koestler, 1964),

University of Texas

• If you want to help low-income students succeed, it’s not enough to deal with their academic and financial obstacles. You also need to address their doubts and misconceptions and fears. …you first need to get inside the mind of a college student.

• Texas Interdisciplinary Plan, or TIP. Students in TIP were placed in their own, smaller section of Chemistry 301, taught by Laude. But rather than dumb down the curriculum for them, Laude insisted that they master exactly the same challenging material as the students in his larger section. In fact, he scheduled his two sections back to back

• … he supplemented his lectures with a variety of strategies: He offered TIP students two hours each week of extra instruction; he assigned them advisers who kept in close contact with them and intervened if the students ran into trouble or fell behind; he found upperclassmen to work with the TIP students one on one, as peer mentors. He conveyed to the TIP students a new sense of identity: They weren’t subpar students who needed help; they were part of a community of high-achieving scholars…. And when the course was over, this group of students who were 200 points lower on the SAT had exactly the same grades as the students in the larger section.

• Laude: “My bet is that the vast majority of them will make it. And they will, because nobody will give them the chance to simply give up.”

David Yeager, a 32-year-old assistant professor who is emerging as one of the world’s leading experts on

the psychology of education

In the experiment, 288 community-college students enrolled in developmental math were randomly assigned, at the beginning of the semester, to read one of two articles. The control group read a generic article about the brain. The treatment group read an article that laid out the scientific evidence against the entity theory of intelligence. “When people learn and practice new ways of doing algebra or statistics,” the article explained, “it can grow their brains — even if they haven’t done well in math in the past.” After reading the article, the students wrote a mentoring letter to future students explaining its key points. The whole exercise took 30 minutes, and there was no follow-up of any kind. But at the end of the semester, 20 percent of the students in the control group had dropped out of developmental math, compared with just 9 percent of the treatment group. …a half-hour online intervention, done at almost no cost….!!!

Back to the Bronx: Creativity. Vrunda Prabhu: The discovery of Bisociation-the !AHA! moment as the basis for the Creativity in mathematics

classrooms

• “I have coined the term ‘bisociation’ in order to make a distinction between the routine skills of thinking on a single ‘plane’ as it were, and the creative act, which…always operates on more than one plane” p. 36

• for Koestler, bisociation represents a “spontaneous flash of insight...which connects previously unconnected matrices of experience” (p.45)

Why and how to utilize the bisociation in the mathematics classroom?

WHY? “Students in remedial mathematics at community colleges are at risk. Their success in higher education depends on overcoming obstacles to learning, many of which are attitudinal, related to affect perception, and detrimental to cognition. “ (Prabhu, 2014)

WHY? Mathematical creativity may be the only gate through which to reactivate the interest and the value of mathematics among contemporary youth whose engagement in the field is hampered by disempowering habits expressed as “I can’t do it,” “I am not good in math,” ”thinking tires me” (Czarnocha et al, 2011).

WHY? Habits are indispensable core of stability and ordered behavior; they also have a tendency to become mechanized and to reduce a man to the status of conditioned automaton. The creative act, by connecting unrelated dimensions of experience, enables him [the man, or her, the woman] to attain to a higher level of mental evolution.

Creative act is “an act of liberation – the defeat of habit by originality.” Koestler, (p.149)

HOW? Poincare (p.115) And now follows one of the most lucid introspective accounts of Eureka act by a great scientist:

For fifteen days I strove to prove that there could not be any functions like those I have since called Fuchsian functions. I was then very ignorant; every day I seated myself at my work table, stayed an hour or two, tried a great number of combinations, and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination. By the next morning I had established the existence of a class of Fuchsian functions, those which come from the hyper-geometric series; I had only to write out the results, which took but a few hours.

How? Broni Czarnocha guided inquiry:

The teacher asked the students during the review: “Can all real values of be used for the domain of the function 𝑋 + 3?”

•1 Student: “No, negative X’s cannot be used.” (The student habitually confuses the general rule which states that for the function only positive-valued can be used as the domain of definition, with the particular application of this rule to .) •2 Teacher: “How about -5 ?” •3 Student: “No good.” •4 Teacher: “How about -4 ?” •5 Student: “No good either.” •6 Teacher: “How about -3 ?” •7 Student, after a minute of thought: “It works here.” •8 Teacher: “How about -2?” •9 Student: “It works here too.” • 10 A moment later Student adds:” Those X’s which are smaller than -3 can’t be used here.” (Elimination of the habit through original creative generalization.) •11 Teacher: “How about 𝑋 − 1?” •12 Student, after a minute of thought: “Smaller than 1 can’t be used.” •13 Teacher: “In that case, how about 𝑋 − 𝑎 ?” •14 Student: “Smaller than “a” can’t be used.”(Second creative generalization)

How? Roberto Catanuto, Everest Academy, Lugano, Swiss

How? Un elefante

• Przemek scrive l’equazione: x + (x + 12) = 76. Un bel problema risolvere quest’equazione, ma lui non si perde d’animo. Disegna così un intervallo; e poi ha luogo il dialogo seguente:

• • P: Eccolo il numero: poi prolunga l’intervallo all’incirca della stessa lunghezza, e fa lo stesso con il

secondo. E questo qui è il numero più 12. • B: E tutto insieme fa 76… • P: No, questa è un’equazione, capisci… • B non si convince… • B: Perché hai tracciato questo segmento, se ancora non sai quanto deve essere lungo? • P: Non è importante. • B: Perché 76? • P: Perché sta nel problema • B: C’è una ics, poi c’è “ics più 12”, e il tutto deve fare 76..? • P: Aspetta, guarda che sul libro al posto della ics c’è un quadretto – P mostra il quadretto sul libro. • B: Aha, ma qui c’è scritta un’altra cosa. • P: Ma possiamo fare come sta qui. Adesso io metto un numero in questo quadretto. • B: Un numero?! E perché proprio nel quadretto? • P: Anzi, in questa finestrella. Metto il numero che esce in questa finestrella. • B: Ma qui c’è un quadretto. – B insiste. • P: Non è un quadretto ma una finestra, in cui si può mettere un numero. • B: Come sarebbe? • P: Due finestre sono uguali a 64, una finestra è uguale a 32. Infatti, se sottrai 12 da entrambi I lati,

vedrai che le due finestre sono uguali a 64. • B: Ma ci sono numeri nelle finestre? • P: Due finestre sono 64, perciò una finestra è 32. • B: Finestra!? • P: Proprio così: una finestra. Guarda: un elefante più un elefante fa 64. Allora, a che cosa è

uguale un elefante? Due elefanti sono uguali a 64. Allora, un elefante a che cosa è uguale? • B: Un elefante? Uhm, sì. Un elefante è uguale a 32. Ora capisco… dunque l’equazione… • P: Se due elefanti sono uguali a 60, a che cosa è uguale un elefante? • B: Un elefante?, ok, un elefante è uguale a 30. Ora lo vedo. Ora l’equazione…………..aaaaaaa

DEMOCRATIZATION OF MATHEMATICAL CREATIVITY

• Focus on bisociation as the theory of !Aha! moment introduces democratization into facilitation of creativity in mathematics classrooms:

Hadamard: Between the work of the student who tries to solve a

problem in geometry or algebra and a work of invention, one can say that there is only the difference of degree, the difference of a level, both works being of similar nature (1945, p.104).

Koestler:

THE AIM: To introduce new,

“authoritative definition of creativity”

in mathematics

In the light of frequently met assertion:

There is no single, authoritative perspective or

definition of creativity (Mann, 2006; Sriraman,

2005; Leikin, 2011, Kattou et al., 2011)

Vrunda Prabhu

Bronx CC

Bronislaw Czarnocha

Hostos CC, CUNY

PME 30, July,

Vancouver, CA

Bisociation

of Arthur Koestler in The Act of Creation,(1964) as

Theory of the !Aha!-moment

THE STATE OF THE FIELD

1. Two Definitions of Creativity: Wallas, (1926) originating in Gestalt

approach, and Thorrance (1966) closer to the behaviorist way of

thinking.

2. The Thorrance definition based on the Thorrance Tests of Creative

Thinking in 1966[47] involved simple tests of divergent thinking

and other problem-solving skills, which were scored on:

Fluency – The total number of interpretable, meaningful and relevant

ideas generated in response to the stimulus.

Originality – The statistical rarity of the responses among the test

subjects.

Elaboration – The amount of detail in the responses. Leikin (2007) and Silver (1996) transformed it to fluency, flexibility and

originality making the definition one of the foundation for understanding

creativity in mathematics education

CREATIVITY: The State of the Field • The Wallace definition based in the Gestalt theories postulates:

• (i) preparation (preparatory work on a problem that focuses the

individual's mind on the problem and explores the problem's

dimensions),

• (ii) incubation (where the problem is internalized into the

unconscious mind and nothing appears externally to be

happening),

• (iii) intimation (the creative person gets a "feeling" that a solution is

on its way),

• (iv) illumination or insight (where the creative idea bursts forth

from its preconscious processing into conscious awareness); and

• (v) verification (where the idea is consciously verified, elaborated,

and then applied).

The stage of intimation was later subsumed under the stage of

incubation leaving four stages which survived to contemporary

times.(Wikipedia,2104)

CREATIVITY: The State of the Field • Neither approach, however, addresses itself directly to the act

of creativity nor to the structure of the “Aha moment” as the

commonly recognized site of creativity itself (Sriraman, 2005).

And, neither of them suggest what are the cognitive and

affective environments in which the creative act can take

place. Moreover, recent research by (Leikin, 2009) indicates

that use of Torrance test of Creative Thinking actually lowers

the creativity. The authors point correctly to the fluency and

flexibility as the carriers of the habit which diminished the

originality of student subjects:”…when students become more

fluent they have less chance to be original”. This apparently

complementary relationship between fluency and creativity

dictates an utmost care in conducting the research into

creativity with the help of the definition which includes fluency,

because it may result in undesired lowering of creativity

CREATIVITY: The State of the Field. Conflation of

research into creativity with the research into

giftedness There are two recently published excellent collections of papers,

dealing with creativity in mathematics education, (Sriraman and Lee,

2011; Leikin et al, 2009). Both collections join the issue of creativity

with the education of gifted students, indicating that the interest in

creativity of all learners of mathematics is not the central focus of the

field. There can be several reasons for so restrictive a focus on

creativity: it could be due to the efforts of globalization so that “the

winds are changing” (Sriraman and Lee, p.2) or it could be that our

understanding of the creative process is not sufficiently sharp to allow

for the effective focus of research on the mathematical creativity by all

students including, of course, the gifted. This observation raises the

issue of democratization of creativity in mathematics research and

teaching. However, a clear understanding of the cognitive and

affective conditions for the creative act is important at present to be as

the jumpstart for bridging the Achievement Gap in the US or start the

numerical literacy campaign among the Tamilian Dalits of India

(Prabhu, Czarnocha, 2008). ).

Arthur Koestler: The Act of Creation, 1

“I have coined the term ‘bisociation’ in order to make

a distinction between the routine skills of thinking on a

single ‘plane’ as it were, and the creative act,

which…always operates on more than one plane” p.

36

• for Koestler, bisociation represents a “spontaneous

flash of insight...which connects previously

unconnected matrices of experience” (p.45)

Fortunately, the theory developed by Arthur Koestler in his

1964 work, Act of Creation, does exactly that. It builds our

understanding of creativity on the basis of a thorough

inquiry into the Aha moment, which Koestler calls a

bisociative leap of insight.

We are proposing bisociation as the authoritative

definition of creativity in the field of mathematics.

Its relationship to two basic definitions (1) coming from Gestalt

approach as well as (2) from a more behavioristic school

depending on fluency, is clear. In the first case it focuses on the

stage of illumination, the actual stage of creativity; in the second

case, it suggest that fluency, which can correlate well with

creativity, can undermine it at the same time. Clearly fluency

does not measure nor defines creativity but instead some

composition of creativity with a habit. Bisociation, on the other

hand, is the “pure” act of creation in the making. Its

disassociation from fluency is very important for the facilitation

of mathematical creativity in the remedial and elementary

mathematics classrooms of community colleges, where it is

exactly fluency that’s missing. It is the definition of creativity for

everyone, because “everyone” knows Aha moment.

The Unity of Cognitive Reorganization with

Affective Liberation- the characteristic quality of

the Act of Creation.

Bisociation is a very strong concept, it also has the power to transform a habit

into originality in agreement with Koestler’s battle cry:

“The act of creation is the act of liberation – it’s the defeat of habit by

originality!”. (p.96)

Hence, creativity in mathematics may be the only gate through which to

reactivate the interest and the value of mathematics among contemporary

youth whose engagement in the field is hampered by disempowering habits

expressed as “I can’t do it,” “I am not good in math,” ”thinking tires me” (xxxx et

al, 2011).

Thus bisociation plays a dual role, that of a cognitive reorganizer and that of an

effective liberator from a habit – it is planting a double root for creativity.

The confirmation of the role as the affective liberation can be glimpsed from

the research of Liljedahl: (2009)“…Aha experience has a helpful and strongly

transformative effect on a student’s beliefs and attitudes towards

mathematics…” (p.213).

Two remarks from creative thinkers:

EINSTEIN & POINCARE:

For fifteen days I strove to prove

that there could not be any

functions like those I have since

called Fuchsian functions. I was

then very ignorant; every day I

seated myself at my work table,

stayed an hour or two, tried a great

number of combinations, and

reached no results. One evening,

contrary to my custom, I drank

black coffee and could not sleep.

Ideas rose in crowds; I felt them

collide until pairs interlocked, so to

speak, making a stable com-

bination. By the next morning I had

established the existence of a class

of Fuchsian functions, those which

come from the hyper-geometric

series; I had only to write out the

results, which took but a few hours.

• “What exactly is thinking? When

at the reception of sense

impressions, a memory picture

emerges, this is not yet thinking,

and when such pictures form

series, each member of which

calls for another, this too is not

yet thinking. When however, a

certain picture turns up in many

of such series then – precisely

through such a return – it

becomes an ordering element for

such series, in that it connects

series, which in themselves are

unconnected, such an element

becomes an instrument, a

concept.” (Einstein,1949)

The Democratization of Creativity in Math. Education

is based on three principles:

1. General familiarity with Aha moment (or Eureka experience).

2. Statement of Hadamard: Between the work of the student

who tries to solve a problem in geometry or algebra and a

work of invention, one can say that there is only the

difference of degree, the difference of a level, both works

being of similar nature (1945, p.104).

3. Koestler: “minor subjective bisociation processes…are the

vehicle of untutored learning”(p.658).

Since minor subjective bisociations are the standard vehicle of self-learning experienced by everyone, and since their nature is similar to that of the mature mathematical inventor, therefore we can view bisociation as the process that underlies any creativity in mathematics for all - therefore it’s an excellent concept to define it.

Why and how utilize the bisociation in

the mathematics classroom?

WHY “Students in remedial mathematics at community colleges are at

risk. Their success in higher education depends on overcoming

obstacles to learning, many of which are attitudinal, related to affect

perception, and detrimental to cognition. “ (Prabhu, 2014)

WHY Mathematical creativity may be the only gate through which to

reactivate the interest and the value of mathematics among

contemporary youth whose engagement in the field is hampered by

disempowering habits expressed as “I can’t do it,” “I am not good in

math,” ”thinking tires me” (Czarnocha et al, 2011).

WHY Habits are indispensable core of stability and ordered behavior;

they also have a tendency to become mechanized and to reduce a

man to the status of conditioned automaton. The creative act, by

connecting unrelated dimensions of experience, enables him [the

man, or her, the woman] to attain to a higher level of mental evolution.

(Koestler, p641)

Why and how utilize the bisociation in the

mathematics classroom? How? Since “minor subjective bisociation processes…are the

vehicle of untutored learning”(p.658).

To facilitate creativity in the classroom our task is to

approximate the classroom environment to the conditions of

untutored learning, when bisociation processes are

encountered. Methods of discovery or inquiry in the classroom

are therefore the best teaching approximation to the condition

of untutored learning.

The definition of bisociation tells us how to structure our

facilitation: work across at least two different frames of

discourse (initial examples provided in Proceedings PME 30).

The simplicity of bisociative facilitation through the discovery &

creative problem solving in the context of a triptych approaches

provides us with ready pedagogical techniques of teaching and

researching it.

Dichotomy between Progress of Understanding

and Exercise of Understanding

Koestler, p.619) “...it is necessary to distinguish between progress

in understanding - the acquisition of new insights, and the exercise

of understanding at any given stage of development. Progress in

understanding is achieved by the formulation of new codes through

the modification and integration of existing codes by methods of

empirical induction, abstraction and discrimination, bisociation.

The exercise or application of understanding to the explanation of

particular events then becomes an act of subsuming the particular

event under the codes formed by past experience.

To say that we have understood a phenomenon means that we

have recognized one or more of its relevant relational features as

particular instances of more general or familiar relations, which

have been previously abstracted and encoded”.