> HJGa (jbjbA]A] $:+?+?".......::::V3~,Ro.....BD,......rQ>:x203cc. TEACHING MATHEMATICS WITH AND FOR CREATIVITY:
AN INTERCULTURAL PERSPECTIVE1
Roza Leikin, University of Haifa, Israel
Creativity and giftedness
Creativity is typically used to refer to the act of producing new ideas, approaches or actions, while innovation is the process of both generating and applying such creative ideas in some specific context (Horowitz & OBrien, 1985; Piirto, 1999; Davis & Rimm, 2004; Sternberg, 1999). Creativity is manifested in the production of a creative work (for example, a new work of art or a scientific hypothesis) that is both novel and useful.
Unlike many phenomena in science, there is no single, authoritative perspective or definition of creativity. The association between high IQ and giftedness and creativity is not simple. O'Hara and Sternberg (1999) provided some support to the suggestion of Torrance (1974) which is termed "the threshold hypothesis" which holds that a high degree of intelligence appears to be a necessary but not sufficient condition for high creativity. An alternative perspective, Renzulli's three-ring hypothesis, sees giftedness as based on both intelligence and creativity (e.g. Davis & Rimm, 2004; Renzulli, 2002; Sternberg, 1999).
Though most researchers agree that giftedness and creativity are associated, clearly not all gifted children are creative. Clearly, cultures vary considerably in how they view these issues, the importance they ascribe to creativity, the measures they use to identify it, the domains which seem to them important, and the ways they employ to foster creativity in the different domains. The following questions remain quite obscure to this date:
How to identify and assess creativity, and especially how to foster creativity?
How creativity, giftedness, and relationships among them are perceived in different cultural contexts?
Creativity as related to mathematics education.
Research literature distinguishes between general and specific giftedness, and general and specific creativity (e.g., Piirto, 1999). Specific giftedness refers to clear and distinct intellectual ability in a given area, for example, mathematics. It is usually reflected in socially recognized performance and accomplishment. Specific creativity is expressed in clear and distinct ability to create in one area, for example, mathematics.
Torrance (1974) defined fluency, flexibility, and novelty as the main components of creativity. Krutetskii (1976), Ervynck (1991), and Silver (1997) explored the concept of creativity in mathematics in the context of multiple-solution tasks. In this context (Silver, 1997, Ervynck, 1991, Leikin & Lev, 2007), flexibility refers to the number of different solutions generated by a solver, novelty refers to the conventionality (relative to a specific curriculum) of suggested solutions, and fluency refers to the pace of solving procedure and switches between different solutions.
There are many open questions about the development of creativity which arise in at least two contexts in mathematics education.
1. Fostering Student Creativity
It has become commonplace for educators to accept that the goal of mathematics education is more than the mastery of a body of algorithms and methods, and that mathematics is an ideal training ground for the development of logical reasoning in students. It is less widely accepted in
the field that mathematics is also a training ground for creativity. Some questions associated with the notion of creativity in mathematics might be:
Can creativity be actively fostered in the classroom?
What classroom techniques lead students to exercise their minds creatively? What areas of mathematics lend themselves most to students independent exploration?
What must a teacher know, or know how to do, in order to support students in developing their creativity in mathematics?
These are questions that can be asked with regards to students and classrooms on any ability level and any age level.
2. Fostering creativity in teaching
A second context in which we may discuss the phenomenon of creativity within mathematics education is in the context of teaching. The act of teaching may itself be ordinary or creative, standard, or novel. A teacher may simply follow the textbook. A more creative teacher might give her own examples that illustrate the points in a textbook. A still more creative teacher might invent his or her own explanations or activities to convey a concept or method. And there are certainly levels of creativity above this one. Questions about creativity in teaching might be the same as those about creativity among students. But additional issues come up in this wider context:
How does creative teaching interact with the need to standardize, to measure, to be held accountable?
How do we train teachers to be creative?
How do we manage teachers or schools, to allow for creativity to emerge?
What is the role of creative teacher in the classroom? In the school? In the profession?
Creativity and Culture
The phenomenon of culture, of ethnicity, of belonging to a group, runs very deep. All humans have culture, just as all humans have language or music or dance. However, the specific nature of culture varies widely. The multicultural approach to creativity is especially relevant in multicultural countries which are characterized by great diversities of languages, cultures,
traditions, mentalities, and educational achievements. The cultures in different countries have characteristics which can be used to help develop the creative abilities.
To increase students' creative potential, a variety of methodologies may be employed, based on the ideas of Vygotsky (1984), Freudenthal (1977), Davydov (1996), and Polya (1945/1973). An example of the important source for creative mathematics education is the experience of schools
established approximately 30 years ago in Russia at the initiative of the famous Soviet mathematician Kolmorogov (1965). In this context we may ask the following questions:
How artifacts of local culture may be used in stimulating creativity?
How is the creative student (or teacher) regarded in different cultural contexts by peers? By authorities? What role does such a person play in the society?
How is cultural borrowing viewed by the host culture? Is there resistance? Attraction to the exotic? When cultural practices change, what are the forces at work to change it?
Acknowledgements
The paper addresses some issues discussed at the international workshop "Intercultural aspects of creativity " held in Haifa, March, 2008 with support of the Templeton Foundation. The author thanks Dr. Mark Soul for his contribution in preparation of the earlier version of the article.
References
Davis, G. A. & Rimm, S. B. (2004). Education of the gifted and talented (5th ed.). Boston, MA: Pearson Education Press.
Davydov V. V. (1996). Theory of Developing Education, Moscow: Intor (in Russian).
Ervynck, G. (1991). Mathematical creativity. In Tall, D. (Ed), Advanced Mathematical Thinking (pp. 42-53). Netherlands: Kluwer
Freudenthal H. (1977). Mathematik als padagogische aufgabe, Stuttgart: Ernst Klett verlay (in German).
Horowitz, F. D. & M. OBrien (Eds.), (1985). The gifted and talented: Developmental perspectives. Washington, DC: American Psychological Association.
Kolmorogov A. N. (1965). On the content of mathematics curricula in the school: Mathematics in the School, 4 (in Russian).
Krutetskii, V.A. (1976). The Psychology of Mathematical Abilities in Schoolchildren. (Translated by Teller, J.; edited by J. Kilpatrick and I. Wirszup). Chicago, IL: The University of Chicago Press.
Leikin, R. & Lev, M. (2007). Multiple solution tasks as a magnifying glass for observation of mathematical creativity. In the Proceedings of the 31st International Conference for the Psychology of Mathematics Education. Plya, G. (1945/1973). How to solve it. Princeton, NJ: Princeton University.
O'Hara, L. A., & Sternberg, R. J. (1999). Learning styles. In M. Runco & S. R. Pritzker (Eds.), Encyclopedia of creativity (Vol. II) (pp. 147-153). San Diego: Academic Press
Piirto, J. (1999) Talented children and adults: Their development and education. Upper Saddle River, N.J.: Prentice Hall.
Renzulli, J. (2002). Expanding the Conception of Giftedness to Include Co-Cognitive Traits and to Promote Social Capital. Phi Delta Kappan, 84(1), 33-58.
Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM, 3, 75-80.
Sternberg, R. J. (Ed.). (1999) Handbook of creativity. New York: Cambridge University Press
Torrance, E. P. (1974). Torrance tests of creative thinking. Bensenville, IL: Scholastic Testing Service.
Vygotsky L. (1984). Psychology of Children. Collected works, v. 4, Moscow: Pedagogy (in Russian).
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