NITSA MOVSHOVITZ-HADAR AND ISRAEL KLEINER

3. INTELLECTUAL COURAGE AND MATHEMATICAL 1 CREATIVITY

This paper considers examples from the history of mathematics in which intellectual courage played an important role in mathematical creativity. In many cases social courage is also involved. Lessons are drawn about the education of mathematically gifted students.

Intellectual courage, intellectual honesty, and wise restraint are the moral qualities of the scientist. George Polya (1954)

INTRODUCTION In the past two decades research into the characteristics of creativity and giftedness in young people, and in particular of mathematically talented students, has yielded new dimensions and instruments for identifying them (e.g., Mann, 2005, Renzulli, 2002, Sternberg et al., 2002). Researchers’ jargon speaks much about curiosity and challenge but very little about courage – an important element in doing mathematics. Unlike a handful of mathematicians, such as Henkin (1996), Halmos (1985), Polya (1954), Poincaré (1952), Hadamard (1945) and Hamilton (1844, van der Warden 1976), the majority of practicing mathematicians are not concerned with describing the creative processes that are involved in doing mathematics. It is the establishment of results that largely occupies the minds and the writings of the mathematics research community. Very few leave a record of the process of their discoveries and of the emotional investment they had put into them.2 Mainstream philosophy of mathematics focuses, in general, on mathematical foundations, dealing mainly with the nature of mathematics as an intellectual discipline and its logic-based development rather than with mathematics as a human endeavor. Only a few contemporary philosophers of mathematics, such as Hersh, Kitcher, Lakatos, and Wilder recognized that the philosophy of mathematics cannot be based on logic alone, that it must explain mathematics as a human activity, including its cultural origins and the motivation involved in its creation. Thus, the study of the acquisition of mathematical knowledge, and the psychology underlying mathematics-in-the-making, remain, almost solely the interest of mathematics educators and cognitive scientists. These communities embrace a

R. Leikin, A. Berman and B. Koichu (eds.), Creativity in Mathematics and the Education of Gifted Students, 31–50. © 2009 Sense Publishers. All rights reserved. MOVSHOVITZ-HADAR AND KLEINER postmodern, humanist form of constructivism, tending to focus on epistemological issues, which do not always apply to the needs of practicing educators. Teacher-education literature (e.g., Lampert, 1990; Silver, 1997) includes some pointers to the need to encourage students to form conjectures and take risks by permitting them to make mistakes, particularly in calculations and in problem solving. For, much can be learned from mistake analysis, and moreover, discouraging risk-taking may inhibit the development of mathematical creativity. Historians of mathematics usually address the historical development of mathematical ideas, focusing on mathematicians’ work. Here and there they praise a specific result as courageous, not explaining what they mean. (e.g., according to Dunham (1990, p. 57), “…with a boldness to match his brilliance”, Euler discovered and proved that the sum of the reciprocals of squares of all positive integers converges to π 2 / 6 ). Most recently Byers (2007) stated in a more general way: Any great quest demands courage. It is a voyage into the unknown with no guaranteed results. What is the nature of this courage? It is the courage to open oneself up to the ambiguity of the specific situation. (ibid, p. 57) This paper considers examples of works in the history of mathematics, searching for indications of intellectually courageous acts, often leading to new findings. Consequently, some lessons are drawn about the education of the mathematically inventive young generation.3

A FEW WORDS ABOUT COURAGE Generally speaking, courage is linked to taking risks and overcoming fear. Let us distinguish between three kinds of courage: physical courage, social courage and intellectual courage: Physical courage usually involves taking bodily risks and coping with fear of pain, injury or even death; for example, the courage needed to save endangered human life in a burning house, flood or war, the courage needed to jump into cold water in winter, to merge in heavy traffic or to jump off an airplane with a packed parachute. Social courage is usually a part of some kind of conscience-driven action that involves taking the risk of losing one’s reputation or social standing. It involves handling the fear of a personal penalty such as being fired from a job or sent to jail; for example, the courage to express atheistic beliefs in a religious society, the courage to stand up for one’s principles against the majority, or the courage to expose corruption in one’s workplace. Intellectual courage involves motivation and persistence to expand one’s understanding of an issue, or to illuminate it from a new point of view. It entails acting on one’s intuition and following a path, knowing that it may not lead to any desired results. This activity usually involves neither physical pain nor social penalty (unless its publication brings about social ostracism). Intellectually courageous acts require cognitive self confidence and insight, and have nothing to do with

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