This post is in collaboration with Prof. Vrunda Prabhu from BCC who discovered Koestler’s Act of Creation for the teaching of mathematics.
One of the central problems to be solved by us in the courses of algebra is the absence of interest of students in mathematics itself. “Mathematics is not cool” one often hears from students, from the community and sometimes, from colleagues of faculty. The reasons, as evidenced by students essays describing their attitudes to mathematics are, generally, earlier exposure to it, the sense of failure and “cannot do” attitude. Students are often aware of their situation and despair about their inability to break those habits. Mathematical creativity maybe one of the tools (if not the only one) we, the mathematics instructors have to reverse the trend of failure and “cannot do” attitude and transform it into enjoyment of the subject, and successful mastery of the concepts under consideration.
That raises the question, what is mathematical creativity and how to facilitate it in the course of elementary algebra or arithmetic.
The Act of Creation of Koestler (1964) formulates it by defining “bisociation” that is “the creative leap [of insight], which connects previously unconnected frames of reference and makes us experience reality on several planes at once.” Consequently, the creative leap of insight or bisociation can take place only if we are considering at least two different frames of reference, within a discourse. Note the similarity of Koestler’s definition of bisociation to Einstein’s description in “What is Thinking?” in the previous post: “…When, however, a certain picture turns up in many of such series, the precisely through such a return, it becomes an ordering element for such series, in that it connects series which by themselves are unconnected, such an element becomes an instrument, a concept.”
The similarity suggests that mathematical creativity is closely related to understanding. In fact, Koestler ”…distinguish[es] 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”.
How to facilitate that process? Koestler offers a suggestion in the form of a triptych, which consists of “three panels…indicating three domains of creativity which shade into each other without sharp boundaries: Humor, Discovery and Art.” Each such row of a triptych stands for a pattern of creative activity which is represented on them; for instance:
Comic comparison <–> objective analogy <–> poetic image.
The first is intended to make us laugh, the second to make us understand, the third to make us marvel. The creative process to be initiated in our classes of developmental and introductory mathematics urgently needs to address the emotional climate of learners, and here is where the first panel of the triptych comes into play, Humor. Having found humor and the bearings of the concept in question, the connections within it have to be explored further to “discover” the concept in detail, and finally to take the discovery to a form that discovery is sublimated to Art.
An example of the triptych assignment used by V. Prabhu in the class of Introductory Statistics consists of students completing the given skeletal triptych below and adding a sentence or two for each completed row to indicate the connections between the words used (the assignment was staggered and repeated several times during the semester):
Trailblazer <———-> outlier <————–> original/ity
<———–> sampling <————->
<————-> confidence interval <————->
<————-> Law of Large Numbers <————->
Lurker <————->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
Triptych assignments facilitate student awareness of connections between relevant concepts and thus they facilitate understanding. However, what maybe even more important, the accompanying discussions help to break the “cannot do” habit and transform it into original creativity. Below is the triptych (with a student’s completion) from a developmental algebra class:
Number<—————> ratio <—————->division
Part-whole <———-> fraction <————-> decimal
Particularity <———–>abstraction <——–>generality
<———————> variable <———————>
multiplication<———-> exponent <———->power
The triptychs of Prof. Prabhu are being refined and their utility assessed with every new semester cycle of classroom Teaching-Research.
Use of triptychs in the mathematics class, bring back the puzzle inherent in mathematics. What is the connection between the stated concepts? What could be concepts connected to the given concepts? Given the largely computational nature of the elementary classes, and students’ habit of remembering pieces of formulas from previous exposures to the subject, a forum for meaning making is created in connecting prior knowledge, with synthesized, reasoned exploration. The question “how”, answered by the computations is augmented with the “why” through the use of mathematical triptychs.