METHOD OF COGNITIVE SYMMETRY: MECHANISM OF INTERRELATIONSHIP BETWEEN EXTERNAL AND INTERNAL REPRESENTATIONS

Mourat A. Tchoshanov

University of Texas at El Paso, TX, USA

Kostroma State Technological University, Russia

mouratt@utep.edu

We consider representational thinking as the learner’s ability to interpret, construct, and operate (communicate) effectively with both forms of multiple representations, external and internal, individually and within social situations. The key research problem of this paper is how can we develop students’ representational thinking more effectively and what would be a sequence of instructional steps that we can follow to improve students’ mathematical understanding using multiple representations? Our theoretical and practical findings show, that this problem might be resolved using method of cognitive symmetry, a mechanism of designing a holistic process of developing students’ representational thinking, which involves a symmetrical interaction of internalization and externalization of multiple representations.

It is now well accepted that use of particular modes of representation (for example, visual/spatial) leads to improvement of students’ mathematical abilities and development of their advanced problem solving and reasoning skills (Krutetskii, 1976; Presmeg, 1997; Tchoshanov, 1997; Yakimanskaya, 1991; Zimmerman, Cunningham, 1990). At the same time, in the fields of psychology, pedagogy, and mathematics education there is an ongoing debate about how the mind operates with representations. On one hand, some researchers (Arnheim, 1969; Mc Kim, 1972) believe that the problem of development of students’ thinking is directly connected to the problem of operating with mental images (e.g., seeing, imagining, and idea-sketching). On the other hand, the advocates of a “picture” theory of representation (Mitchell, 1994; Wileman, 1980) argue that there is no difference between external and internal (mental) representations: a mental representation is equivalent to what it represents. Furthermore, based on their critique of the “picture” assumption, constructivists reject the “representational view of mind” (Cobb, Yackel, & Wood, 1992) claiming that, translated into instructional practices, a representational view begins with experts’ conceptions and attempts to reproduce these ideas by students rather than allowing them socially co-construct meanings of mathematical representations.

All this debate does not contribute to the clarification of the relationship between external (e.g., pictures, diagrams, graphs, charts, 3-D models, computer graphics, etc.) and internal (e.g., mental images, mind maps, webs and hierarchies, schemata, semantic nets, etc.) representations and their role in the development of students’ representational thinking. In this paper we advocate the position that the development of students’ representational thinking is a two-sided process, an interaction of internalization of external representations and externalization of mental images. We also assume that there is a mutual influence between the two forms of representations: the nature of an external representation influences the nature of the internal one and vice versa. Finally, we argue that representation is not an end in itself but a vehicle for understanding; and the construction of an understanding of representations is an individual as well as a social activity. Therefore, we consider representational thinking as the learner’s ability to interpret, construct, and operate (communicate) effectively with both forms of representations, external and internal, individually and within social situations.

The educational significance of this conceptualization is in presenting an alternative holistic approach to representational thinking development through construction of students’ understanding (internalization) and improvement of their creativity (externalization). Unlike previous studies (Herscovics, 1996; Hiebert, & Carpenter, 1992), which paid attention primarily to the internalization stage, the role of representations in the development of students' understanding, this approach is characterized by its completeness and orientation towards creativity through understanding. We firmly believe that, in the development of students' representational thinking, internalization without externalization is non-holistic and incomplete. The process of interrelationship of internalization and externalization we call cognitive representation. The important point here is that despite a tacitly accepted one-sided view of internal representation as a cognitive one (Seeger, 1998), we consider cognitive representation as a zone of interaction of external and internal representations. From our perspective, cognitive representation reflects both the process (internalization) and the products (externalization) of representational thinking.

This conceptualization is further based on recent findings in theory of cognition and brain investigation (Caine, & Caine, 1994; Chabris, & Kosslyn, 1998; et al.). According to these studies the brain works more effectively while making representational patterns for encoding (internalizing) and decoding (externalizing) information. Unfortunately, as opposed to the varied and complex patterns generated in the human brain, most mathematical content offered to students is basically presented in linear abstract/symbolic forms. However, the patterning capacity of the human brain more closely resembles non-linear representational network forms. It seems reasonable that the language of the brain is a multiple representational language. Therefore, the improvement of students’ brainpower requires a development of their representational thinking.

This approach is also built on a number of previous studies done in the field of using multiple representations in teaching and learning of mathematics (Greeno, & Hall, 1997; Lesh, Post, & Behr, 1987; Presmeg, 1997; Wheatley, 1997; et al.). However, there are still a lot of open questions in the theory and practice of development of students’ representational thinking. Some of them are:

· How may different modes of representations, sequences and translations among/within them support the development of students’ mathematical understanding?

· What relationships and connections exist within/between external and internal representations?

· Why are particular modes/types of representation sometimes ineffective in developing students’ mathematical understanding?

These open questions led us to formulate and solve the key problem: how can we develop students’ representational thinking more effectively and what would be a sequence of instructional steps that we can follow to improve students’ mathematical understanding using multiple representations? Our theoretical and practical findings show, that this problem might be resolved using the method of cognitive symmetry. In the context of this problem, the method of cognitive symmetry is a way of designing a holistic process of developing students’ representational thinking. The holistic process, which involves a symmetrical interaction of internalization and externalization of representations, we call a cognitive cycle.

In understanding the nature of the internalization process, we adhere to the Vygotskian conception of mediation. L. Vygotsky and his supporters argue that determination of individual cognition might be presented by the following scheme: collective (social) activity – culture signs/symbols – individual activity – individual cognition (Vygotsky, 1996; Leont’iev, 1978, et al.). In the framework of developing learners’ representational thinking we consider the Vygotskian scheme as a basis for designing a cognitive cycle. The importance of the method of cognitive symmetry is that it gives a clue for designing the structure of externalization process based on the scheme of the internalization one. So, the holistic cycle of students’ representational thinking development includes the sequence of the following stages: first, potential representation transfers to standard/conventional representation through communication and interpretation; then students construct their internal representation and express it externally as non-standard/idiosyncratic representation; and finally, they create/generalize meta-representation through reflection and abstraction (Fig. 1). We must make it clear that if internalization is primarily a guided zone, externalization is basically an independent domain of students’ activity. Furthermore, if internalization aims at understanding (e.g., seeing, comprehension, interpretation, etc.), externalization tends toward creativity (e.g., construction, generalization, abstraction, etc.). Therefore the key proposition of our vision is that effective methodology for developing students’ mathematical representational thinking might be designed using the method of cognitive symmetry which reflects an interaction of the internalization and externalization processes of representations.

Below we briefly discuss some outcomes of the pilot experiment we conducted in 1995-96 with Russian high school students on trigonometric problem solving and proof using method of cognitive symmetry along with multiple representations. The first comparison group of students (“pure-analytic”) was taught by traditional analytic (algebraic) way of trigonometric problem solving and proof. The second comparison group (“pure-visual”) was taught by visual (geometric) way using enactive (manipulative aids) and iconic (pictorial) representations. The third - experimental group (“representational”) was taught by combination of analytic and visual ways using translations among different representational modes. The representational group scored 26% higher than the visual and 43% higher than the analytic groups.

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This experiment also showed that students in the “pure” (analytic and/or visual) groups “stick” to one particular mode of representation, they were reluctant to use different representations. For instance, students in the pure-visual group (even though there were random distribution of students with abstract/analytic and visual/geometric preferences of reasoning) tried to avoid any analytic solutions: they were “comfortable” if and only if they could use visual (geometric) problem solving and proof techniques. It means that students are getting used to the particular mode of representation, which they were previously taught. While students in the representational group were much more flexible “switching” from one mode of representation to another in search of better understanding of mathematical concepts. Therefore, we realized that any intensive use of only one particular mode of representation does not improve students’ conceptual understanding and representational thinking. Another important outcome of this experiment is that students in the representational group gained more conceptual understanding while using analytic representation before visual one, as well as using more generalized representation before concrete one. The final key issue of the experiment is that the development of students’ representational thinking cannot be effective enough if it is not supported by using generalization processes (meta-representations) in mathematical problem solving and reasoning.

The main focus on conducting this experiment using method of cognitive symmetry was improvement of students’ representational thinking in the context of:

· students’ exploration of alternative ways of mathematical problem solving, inquiry and reasoning;

· students’ construction and co-construction (social aspect) of non-standard multiple representations of problem solving and proof techniques;

· involvement of students in a variety of hands-on and minds-on activities (e.g., modeling, drawing, imagining, mapping, etc.) in the process of generalization of mathematical concepts and ideas;

· students’ understanding of harmonic relationship between different forms of multiple representation of mathematical knowledge.

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