1 GETTING AWARE: Secondary School Teachers Learn about 'Mathematics Challenges in Education' Roza Leikin Faculty of Education, University of Haifa, Haifa, 31905 rozal@construct.haifa.ac.il ABSTRACT This paper addresses one of the questions raised by the discussion documents of the ICMI STUDY 16: 'How can teachers be made aware of the existence of different types of challenges in school mathematics?' The paper attends to the complexity of the teachers' knowledge structure and suggests several principles for the design of the courses for mathematics teachers. The paper suggests that a Big Idea for the course should be chosen and describes several modes of work with secondary school mathematics teachers. A description of one particular course focusing symmetry exemplifies the principles as applied to one of the modes. RATIONAL As mentioned in the Discussion Document of ICMI Study 16 (Barbeau & Taylor, 2005) word 'challenge' has different meanings. In Cambridge Advanced Learner's Dictionary 1 we find that 'challenge' may be defined as 'difficult job', namely, something needing great mental (or physical) effort in order to be done successfully and which therefore tests a person's ability. Thus 'to challenge' may mean 'to test someone's ability or determination'. Correspondingly teachers' challenging role in Mathematics classroom is to stimulate students' mathematical reasoning, to set up their participation in mathematical explorations, to design situations in which students are required to prove mathematical statements, to let students participate in competitions. In order to develop pupil's mathematical understanding, and allow their successful problem-solving performance a teacher must create situations that demand from the students great mental effort. "In this endeavor, the role of the teacher is critical" (Barbeau & Taylor, 2005; p. 2). Teachers' choices of mathematical tasks for their classes and the ways in which these tasks are introduced determine the quality of mathematics in the classroom (Stigler & Hiebert, 1999). However many teachers choose "stereotypical tasks" for their lessons and guide students towards "stereotypical solutions" (Barbeau & Taylor, 2005; Leikin, 2003). The word 'stereotypical' denotes kind of tasks and solutions, which are included in a particular textbook or prescribed by educational authorities. One of the ways that can help teachers apply challenging mathematics in their classes is changing the textbooks so that series of challenging tasks will be available for the teachers (Barbeau & Talor, 2005). However, simply providing teachers with ready-to-use challenging math activities is not sufficient for their implementation (Leikin et al., in press). The following conditions appear to be important in employing challenging mathematics in school: Teachers should be aware and convinced of the importance of such implementation; Teachers should 'feel safe' when dealing with such kind of mathematics (mathematically and pedagogically); Teachers should have autonomy (Krainer, 2001) in employing this kind of mathematics in their classes. Additionally the teachers themselves have to be able to choose mathematical tasks from "outside the textbook", to create those tasks, to change stereotypical tasks so that they will become challenging and stimulating and finally – which is most relevant for this paper – solve stereotypical problems in non-stereotypical ways. TEACHERS' KNOWLEDGE AND BELEIFS Teachers' knowledge (Shulman, 1986), and beliefs (Cooney, 2001) determine their decision making when planning, performing, and reflecting. Teachers' knowledge and beliefs are interrelated and both have very complex structure. Leikin (in press) suggested using 3D model of teacher's knowledge, which describes this complexity (e.g., Scheffler, 1965; Shulman, 1986). In the context of the ICMI Study -16, I present this model in relation to challenging mathematics (see Figure 1). 1 http://dictionary.cambridge.org/define.asp?key=12394&dict=CALD