Five presentations on Koestler and Human and Computer Creativity

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
• What exactly is thinking?
When
lbert.
P.7 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 Raymond Smullyan joke:
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 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 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
Introductory Statistics class:
Trailblazer  Outlier  O
 Sampling 
 Probability 
 Confidence Interva
 Law of Large Numbe
Lurker/Lurking Variable  Correlat
The triptych below is an example of student wo
Trailblazer  OUTLIER 
Random  SAMPLING 
Chance  PROBABILITY 
Lurking Variable  CORRELATIO
Testing  CONFIDENCE INTERV
Sample Mean  LAW OF LARGE NUM
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) metafindings: “…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-tograduate.html?_r=0 (NYTimes, 5/15/14)
Achievement Gaps revealed by PISA 2012
1
2
3
4
Korea
Japan
Norway
Canada
585-540
565-520
515--465
545—490
45
45
50
55
5
7
30
13
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
23
23
23
24
24
25
26
26
26
Estonia
Latvia
Russia
Finland
UK
Lithuania
Ireland
USA
Spain
Sweden
Italy
Denmark
Netherland
Vietnam
Malayasia
Mexico
Costa Rica
Kazachstan
Poland
Germany
France
Luxemburg
Portugal
Israel
Belgium
Shangai
Taipei
Slovakia
555-500
530- 435
520-455
565-460
525-460
510-445
565-475
520-450
530-460
510-440
525-455
535-460
560-480
575-490
460-405
455-400
450-405
450-425
575-485
570-480
545-455
545-455
555-455
515-415
570-465
650-540
625-515
555-445
55
65
65
65
65
65
70
70
70
70
70
75
80
85
55
55
45
25
90
90
90
90
100
100
105
110
110
110
11
28
34
12
26
37
20
36
33
38
32
22
10
17
52
53
56
49
14
16
25
29
31
41
15
1
4
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 positivevalued 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 ?”
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 teacherresearcher 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 performancebased 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 communitycollege 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
Arthur Koestler: The Act of Creation, 1
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.
“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)
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 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
• “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 selflearning 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”.

Download Report