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