**2. Koestler theory of bisociation**

#### **2.1 Elements of the Koestler theory of bisociation**

The work presented here on the creative moments of insight popularly called Eureka experience or Aha! Moment is based on [2] where the author introduced a new concept/term of bisociation, which is the spontaneous act of thought which combines or dialectally synthesizes the information from two different, generally unconnected domains. Bisociation is seen here as distinct from association, which produces knowledge within a single domain. Koestler makes a clear distinction between more routine or habitual thinking (association) operating within a single plane or a matrix of thought, referred to here as the exercise of understanding and the more creative bisociative mode of thinking that connects independent autonomous matrices called progress in understanding [6].

What is a matrix of thought also called a frame of reference?

*The matrix is the pattern before you, representing the ensemble of permissible moves. The code which governs the matrix can be put into simple mathematical equations … or it can be expressed in words. The code is the fixed invariable factor in a skill or*

*habit, the matrix is the variable part. The two words do not refer to different entities, they refer to different aspects of the same activity.* ([2], p. 40).

A clear example of a matrix and a code is given in chess where the matrix is the full collection of available moves, while the code are the rules for the movement and interaction among the chess pieces. In high school algebra, the matrix is the full set of polynomials and power series in one or two variables; the code are the established rules of operations on these mathematics objects.

In our effort to apply Koestler's theoretical framework in the creativity process within learning mathematics, we focus on two related questions: (1) How can you describe the genesis of a new code-matrix? and (2) How can you characterize moments of creative insight that lead to new structure-matrices and ultimately new codes? Koestler considered the formation of new structure-matrices, as well as the hierarchy of matrices serving the organism as "[s]ymbolic models of the external world" (p. 506) that govern how we interpret and react to a situation in a predictable manner, as the central role of cognition. For Koestler, an important component in characterizing moments of insight is expressed in his notion of the degree of originality, which is inherent in our discussion of the depth of knowledge acquired during a moment of insight. In learning theory, an individual's development of structure is a focal point of constructivist research and cognitive science.

Koestler translates the term matrix very broadly into such diverse domains as literature, art, drama, motor skills and humor. He notes that matrix in cognitive psychology would be termed a schema.

*The concept of matrices with fixed codes and adaptable strategies, proposed as a unifying formula, appears to be equally applicable to, perceptual, cognitive, and motor skills and to the psychological structures variously called, 'frames of reference', 'associate contexts', 'universes of discourse', 'mental sets', or 'schemata' etc.* (p. 96)

Cognitive psychologists use the term schema to describe mental structures that guide our response to life situations; thus, there is a schema for work, family relationships, commuting, religious services, etc. The particular schema that interests us is an individual's schema for mathematics problem-solving.

Constructivists use the term scheme to describe a mental process for resolving problem situations. We loosely translate schema as a hierarchical structure of schemes. Thus, the building of schema begins in the second stage of the Piaget Garcia Triad [7] through the connection of schemes into hierarchical collections (mental toolboxes for a given domains of math). The third stage begins connections between schemata. The term code translates into the constructivist notion of an invariant relationships, which can be understood as the automated principle of a conceptual relationship that underlies an activity–effect relationship in a problem situation (scheme).

Creativity or a Eureka moment occurs when an idea is suddenly understood to exist simultaneously in two previously unrelated frame (**Figure 1**).

*The perceiving of a situation, or idea, L, in two self-consistent but previously incompatible frames of reference (fig. 1) The event L in which the two intersect is made to vibrate simultaneously on two different wavelengths, as it were. While this unusual situation lasts, L, is not merely linked to one associative context, but bisociated with two*. ([2], p. 35).

**Figure 1.** *Bisociated concept L between matrices M1 and M2.*

#### **2.2 Koestler: progress in understanding new codes**

In the first stage of the Piaget Garcia Triad, the birth of a conceptual relationship begins with interiorization. For Koestler the genesis of a new code is referred to as progress in understanding; the equivalent constructivist term is accommodation. In constructivist theory, all acts of accommodation are the result of what Piaget refers to as reflective abstraction (abstraction of processes). Interiorization is understood as the foundational type or process of such abstraction that translates externally directed activity into an internal process. Koester describes the genesis of codes, or progress in understanding, as due to moments of insight, within the learning process, or moments of creativity insight that lead to the synthesis of two matrices or, "bisociation"

*[p]rogress in understanding – the acquisition of new insights … is achieved by the formulation of new codes by … empirical induction, abstraction, and discrimination, bisociation*. (p. 619).

The Transition from Empirical to Abstract reasoning.

Concepts are born in the first stage of the Piaget & Garcia Triad. Constructivist research based on Piaget's work and social constructivism based on Vygotsky; both view the development of a child's cognition as transitioning from empirical to more abstract reasoning. Constructivists view the transition from empirical reasoning, during which one's intuitive solution activity is directed by the situation, as interiorization. This process leads to an abstract activity–effect relationship based on understanding the relevant concept and thus no longer dependent on the situation. They view this first-stage transition as the way human cognition evolved historically as well as the pathway for child development.

Vygotsky refers to the transition from empirical to abstract thought as one from spontaneous to scientific concepts and considers it essential for the dawning of the child's ability to engage in self-reflection. Vygotsky considers such internalization of exterior activity as social-based, primarily guided by adult communication, and in large part based on imitation of adult behavior.

In this debate, Koestler is strictly a constructivist. Indeed in his chapter on science and emotion, he titled one of the sections "The Boredom of Science" lamenting how direct instruction of theorems and repetition have made math and science "antihuman". In contrast to the social-cultural approach, in his view individual moments of insight lead to a transcendence experience, and only from this can we begin to appreciate math and science. Like Vygotsky, Koestler understands objective abstract thought as essential for human self-reflection, but like constructivists he situates such reflection, as based upon transcend moments of insight that are individualistic. Thus,

Koestler like constructivist researchers is primarily focused on bisociation within the learning process of an individual and not on moments of insight within social discourse.

The phenomenon under investigation is moments of insight within the learning process, which lead to progress in understanding, or concept development. As teacher researchers our contention is that such moments occur both during social discourseinternalization as well as during reflection on our own solution activity. Furthermore, the novel concepts and structure developed through the connections established at this moment of insight (progress in understanding) can be assessed through the Piaget-Garcia Triad. Before continuing with these themes, we review Eureka or Aha! Moments and creativity theory.

### **2.3 Eureka experience/Aha! Moment**

The phenomenon under investigation of the proposed theory is the act of creation in mathematics and science called Eureka experience or Aha! Moment. It is that moment when suddenly, after a long period of trying to solve a problem or understand a new concept without success, the solution comes in a flash, generally with a good doze of satisfaction. It is a very particular form of creativity that appears as an insight, as a discrete insight in that it appears instantaneously at separate moments of time.

Gestalt creativity approaches the Eureka experience as the stage of illumination within the sequence of preparation, incubation, illumination and verification stages suggested by [8, 9]. The sequence represents the stages through which the formation of the creative idea takes place. While the first three stages came from psychological research, the fourth stage was added as the necessary part of the creative process at the Poincare insistence [8]. Why would Poincare insist on verification as the components of creativity? Because Poincare in [10] says: *It never happens that unconscious work supplies a ready-made result of a lengthy calculation in which we have only to apply the fixed rules … All that we can hope from these inspirations, which are the fruits of unconscious work, is to obtain points of departure for such calculations. As for calculations themselves, they must be made in the second period of conscious work, which follows the inspiration and in which the results of inspiration are verified and the consequences deduced* (pp. 62–63).

The more so, of course, because sometimes the Aha! Moment is false. Consequently, the verification stage plays a dual role: On one hand as a possible completion of the creative act and as the check on the correctness of the logical-causal structure of its content. Recent examination of Wallas's work suggests in [11] the fifth stage of intimation to be placed between incubation and illumination. Intimation is, for Wallas, the

*"fringe of consciousness" which surrounds our "focal" consciousness as the Sun's 'corona' surrounds the disk of full luminosity", Wallas continues: "This fringe consciousness may last up to the flash instance, may accompany it, and in some cases may continue beyond that."*

In such a case intimation can be an excellent "point of departure" for calculations in the verification stage with certain though unclear anticipations for its results. The three-process theory formulated by [12] helps in identifying different types of insights. Selective combination takes place when someone suddenly puts together elements of the problem situation in a way that previously was not obvious to the

individual. Selective encoding occurs when a person suddenly sees … one or more features that previously have not been obvious [12]. Selective comparison occurs when a person suddenly discovers a nonobvious relationship between new and old information. Koestler's Act of Creation occupies the central illumination stage of those 4 or 5 stages pathway. His definition of bisociation, as a generalization of Aha! Moment and Eureka experience, allows to deepen our knowledge of the illumination stage into cognitive (and affective) aspects of the insight.

The second approach to creativity is via [13, 14] who were interested in the development of creative characteristic human attributes centered on divergent thinking. It is the creative product theory in distinction to the process-oriented Gestalt approach. In mathematics education its central qualities have been established to be fluency and flexibility of thinking assessed by speed and precision of thinking and the number of different solutions to the same problem. The third quality is originality measured by comparing individual solutions with those of the whole set of participants.

Originality of thought or action displayed through an Aha! Moment is the quality that joins Koestler's bisociation theory with Guilford's approach. However, this spontaneous originality is presented within the tension of automatization of a habit. In fact, Koestler sees in [2] the creative act of the insight as "an act of liberation—the defeat of habit by originality". He of course realizes that "habits are indispensable core of stability and ordered behavior, [yet] they also have a tendency to become mechanized and reduce man to state of conditions automata" (p. 96). This tension has a direct bearing on the mathematics classroom. Till recently learning of mathematics involved learning the procedures or codes and, actually, trying to make them automatic exactly as Koestler describes, because it increases the fluency. The habits are condensation of codes (procedures) learning. Recent curricular changes with the emphasis on problem solving have as a goal precisely lessening the grip of habits to create the conditions for creative solutions. If as Czarnocha in [15] suggests, creativity should be the foundation of learning, in particular of learning in mathematics, then we have a serious philosophical didactical problem to solve: What should be route of integration of creativity with the necessity of knowing and mastering procedures?

An important issue that arises in the debate between conceptual or relational instruction versus a procedural orientation is the nature of the concept that arises during a moment of insight. In social-constructivist theory, this issue can be understood within the context of Vygotsky's statement that unlike procedural knowledge, conceptual knowledge cannot be taught. Thus, it is the individual, guided by a mentor, who gives meaning to the cultural artifact being presented. In constructivist theory the concept-process is first interiorized and then reflected upon. This reflection upon the interiorized activity leads, through the emerging action scheme, to further structural development, which can be assessed with the help of Piaget and Garcia Triad. This structural development is called here the depth of knowledge (DoK) gained in the moment of insight.
