**3. The development of theoretical framework underlying the assessment of the depth of knowledge reached during the insight**

#### **3.1 The Triad of Piaget and Garcia**

The Koestler-based definition of an Aha! Moment or Eureka experience insight of bisociation as a spontaneous leap of insight that connects unconnected frames of

#### *Bisociation: Creativity of an Aha! Moment DOI: http://dx.doi.org/10.5772/intechopen.110694*

reference, matrices of thought is very informative. This suggests that the creativity of an Aha! Moment is in building new connections between different matrices of thought or more precisely between different schemas of thinking (that is, between networks of concepts and connections/relationships between them). Consequently, to understand a so defined creative process, we need to use a theory of schema development. The Triad of Piaget and Garcia developed in [7] turns out to be an excellent constructivist model of schema development whose three stages, -intra, -inter and -trans, allow us to characterize the cognitive levels of different Aha! Insights.

The -intra stage of the development of a concept in the mind of a learner consists of isolated instances, concrete manifestations of the concept as actions, rules or operations, whose performance requires an external help. The process of making connections between isolated instances of the concept means entering the -inter stage of the development. The transition between -intra and -inter case that is from isolated instances to connected instances of the developing concept is one of the sources of Aha! Moments because the transition between the two creates the necessary bisociative context. The -inter stage is rich in possibilities of different connections, from two concepts, single connection schema to multi-connections schemas, and each new connection made can be the effect of the creative insight.

The second important transition in the development of a concept is from the -inter stage connectivity to the -trans stage of generality or abstraction, finding the unifying principle or unifying structure of the concept. This transition is also good source for Aha! Moments. The thinking mechanism which propels the learner along these developmental pathways is called a reflective abstraction, that is, a learner's reflection on its own solution activity. Baker [16] has shown that a bisociative frame can be identified within the reflective abstraction process composed of two parts: projection and constructive generalization indicating a close relationship between bisociation and reflective abstraction.

Our aim is to create assessment of the depth of knowledge reached during the particular insight by the analysis of the nature of created schema of thinking. We take for that purpose numbers of connections made, the conceptual distance between the involved concepts and the quality of connections made. As the measure of that quality, we will take verbal or written description of discovered relationships.

#### **3.2 Koestler/Prabhu theory of creativity**

The presentation of the Koestler/Prabhu theory follows Redford, who in [17] formulates requirements for a theory in mathematics education more precisely by suggesting that "a theory can be seen as a way of producing understanding and ways of action based on:


The phenomenon under investigation is moments of insight in the learning process: those leading to progress in understanding, or concept development. As teacher researchers our contention is that such moments occur both during social discourse, internalization, as well as during reflection on solution activity.

The basic principle underlying the Koestler/Prabhu theory of learning through creativity of Aha! Moment is the conviction that creativity is the cornerstone of that learning, which leads to progress in understanding. Aha! Moment as a *spontaneous leap of insight, which connects unconnected matrices of thought,* termed by Koestler *bisociation,* is the insight, which leads to progress in understanding through creation of the new conceptual connection, the new entities that is new (for the learner) knowledge.

The second principle of the theory is the *bisociative frame:* that is, those originally disconnected matrices of experience within which the Aha! Moment insight takes, or may take, place. Such moments of insight help transcend the learner or solver to a more universal plane within their community of peers and the mathematics classroom, producing positive affect. Since, as we show below, bisociative frame can be identified within different theories of learning, creativity can be seen as an irreducible component of learning. The bisociative frame is the possible creativity detector; it is the bisociative frame that is responsible for the coordination of creativity with different theories of learning, both on the practical level of the classroom as well as on the theoretical research level. The ease and the method with which a bisociative frame can be identified within the "host" theory is described by Czarnocha and Baker in ([4], Introduction). This process of identification has become the tool with the help of which we can indicate the areas of heightened creativity within the cognitive structure of the "host" theory; see examples in [18, 19]. The strength of this tool can be seen in the process of identification of creativity within Vygotsky's sociocultural theory within the process of internalization [20].

Koestler/Prabhu's theory of learning investigates the nature of creativity of Aha! Moment insight in several dimensions paradigmatic research horizons:


The third principle of the theory is measurability of creativity, whose aim is to establish the depth of knowledge (DoK) reached during Aha! Moment insight; however, that needs to be done in a way that does not disturb creativity itself, or as [21] suggests, it is measured by internal variables. We use the theory of the Piaget and Garcia Triad [7], which allows us to assess the dynamic development of the new schema from the learner's or a scientist's description of the Aha! Moment insight coordinated with the researcher's view of the mathematical situation. Important

#### *Bisociation: Creativity of an Aha! Moment DOI: http://dx.doi.org/10.5772/intechopen.110694*

disclaimer: Although we use a constructivist Piagetian concept here, we are not necessarily ascribing to the full scope of constructivist approach, although Koestler's point of view in his Act of Creation is not positioned very far. What is important, however, is that we can undertake an analysis of creativity within the sociocultural approach with the help of the same concept of the theory, the bisociative frame, which has been developed in the individual context. We see that Koestler/Prabhu theory has a consistent, independent relationships with both the Piagetian and the Vygotskian approaches. Investigation of that strange fact and its consequences is the research theme for a next investigation.

While the central paradigmatic question of the theory is two-fold


each of the horizons of the theory has its own paradigmatic questions, and the results of recent inquiries into the horizon of learning are presented in Section 4.

The methodology of the theory can be viewed through several angles, including the angle of teaching, of research and of assessment. Within teaching, we attempt to create mathematical situations that contain gaps in student understanding. For instance, we frame the questions, hints or assessment process using bisociative frames, which we suspect will be unconnected in student minds. Or sometimes we add a component that we suspect is needed for the student to gain an insight. In its research aspect, the aim of our methodology is to identify bisociative frames and hence to identify creative possibilities within the "host" theories of learning. Having done that, we have the possibility of using the methodology of a host theory to express the new forms of creativity within it. In its assessment aspect, we assess the change in the development of the relevant schema with the help of PG Triad.

An excellent example of such a coordination between bisociation and modes of reflective abstraction is presented by [18], who demonstrates the presence of a bisociative frame in two critical stages of his framework called "reflection on activity– effect relationship," the Ref\* AER. The framework Ref\*AER is the elaboration of a Piaget mechanism of reflective abstraction, within which Tzur [18] identifies two different stages of development, participatory and anticipatory. In the participatory stage, he finds six different categories of reflection, each equipped with a bisociative frame, suggesting the presence of sufficient cognitive conditions for facilitation of six, cognitively different Aha! Moments. That work immediately suggested to him a host of new paradigmatic research question. It is instructive how identification of bisociative frame within Ref\*AER theory has broadened and deepened the theory itself.

Similar investigations have also been undertaken in [19] where identifying bisociation within different modes of attention allowed the introduction of dynamics within its structure. This bisociative dynamic results in a shift of attention from dyadic to triadic mode of attention, that is, from two components of the bisociative frame to both components together with the new relationship provided by the Aha! Moment.

This unusual capacity of a bisociative frame to be identified within different theories of learning and therefore in corresponding teaching practices provides an important opportunity to conceptually unify the constructivists and the sociocultural theories of learning. That pathway of inquiry has been initiated by Baker in [16] where he identified bisociative frames within the constructive generalization of Piaget, and in [20], he explored creativity within central sociocultural concepts of internalization and appropriation. One major effort of the theory is clarification of the relationship between the concept of interiorization arising in Piagetian theories and internalization of the sociocultural theory to which we devote Section 4.
