BORIS KOICHU AND AGNIS ANDŽƖNS MATHEMATICAL CREATIVITY AND GIFTEDNESS IN OUT-OF-SCHOOL ACTIVITIES This chapter describes the development of out-of-school activities designed to foster creativity and giftedness in the area of mathematics, and analyses the social and historical circumstances that have affected these activities. The discussion extends to current problems, acute tasks, and possible future activities. INTRODUCTION One of the most important missions of any society is educating young generations in a way that ensures continuous development of the existing body of knowledge and experience. This implies that children must assimilate and accommodate some sub-set of knowledge and experience produced by past generations to become scientifically literate citizens. It also implies that children must be provided with a broad range of opportunities to surpass their parents and teachers in exploring the world and making it a better place. The first implication concerns primarily the fulfilment of the children's intellectual potential; and the second one, of their creative potential. By creative potential we mean the developing capability to act effectively not only in accordance with what has been taught but also by inventing new methods, refining existing algorithms and rules, discovering new knowledge, and finding new applications within the existing body of knowledge and connected with it (Ervynck, 1991; Silver, 1997; Shye & Yuhas, 2004). According to this definition, creativity cannot be taught directly 1 and should be promoted indirectly in learning environments that support risk taking, challenging the rules and algorithms, inventiveness, and the free circulation of ideas (Davis & Rimm, 2004). This dual objective – to study what is known and to prepare for creating new knowledge – applies to any educational system, 2 but cannot be fully achieved within formal, in-school, education alone. This is especially true for mathematically promising and gifted individuals. Indeed, the regular mathematics education system must pay attention to all students, and often the special needs of mathematically promising and gifted students remain unaddressed. But out-of- school education tends to address the needs of particular groups of students. –––––––––––––– 1 Indeed, a taught "creative" problem-solving approach stops being creative for a person who merely applies it. This observation is associated with the so called "Creativity Paradox" (e.g., Hoffman et al., 2005). 2 The need to promote creativity has not only philosophical or ethical justifications but also a mathematical one. It derives from theorists in the computer sciences who have proved that there will always be room for optimizing knowledge inference algorithms (see, e.g., Barzdin & Frejvald, 1974).