Learning to Teach Mathematics for Understanding: The Role of Reflection Joanne Goodell Cleveland State University This paper explores how a group of 11 preservice secondary mathematics teachers developed their understandings of what it means to teach for understanding. The paper begins with a discussion of what teaching for understanding might look like in practice. Next, the activities the students took part in are discussed, along with their reactions to these activities as outlined in the course evaluation and follow-up questionnaire. Next, evidence of their learning, as presented by the students themselves in their final paper is discussed. Finally, directions for similar courses in future are explored. The purpose of this paper is to report the results of a study which explored how a group of 11 preservice secondary mathematics teachers developed their understandings of what it means to teach for understanding. This investigation is set in the context of the secondary mathematics education methods course 1 and practicum teaching experience which all 11 participants were undertaking. The paper begins with a discussion of the theoretical framework underpinning the course and its goals. Then some background to the study is provided. Next, the design and methodology of the study which is the subject of this paper is presented, followed by the results of the study. Finally, a discussion of the implications of the findings of this study for mathematics teacher education in general are discussed. Background and Theoretical Framework At Cleveland State University, the mission of the College of Education is to produce a teacher candidate who is “a responsive, reflective professional: a partner in learning”. As developer of this course, the author had additional goals in mind, based on her own beliefs derived through 13 years of high school teaching and 5 years of doctoral research: that mathematics teaching must become more connected to the real world and be devoted to developing students’ understanding of concepts as opposed to focusing on rote memorization (Goodell, 1998; Goodell & Parker, 2001). Thus, there were two foci for this course. The first was on developing the preservice teachers’ understanding of the art of “teaching for understanding”. The second was on facilitating their reflective thinking, and learning from that process. Teaching for Understanding Teaching for understanding has always been important in mathematics education, with recent publications on the subject by Hiebert, et al. (1997), Carpenter and Lehrer (1999), Fennema, Sowder and Carpenter (1999), and Sierpinska (1994). Current notions of teaching for understanding are largely based on constructivist 1 In this paper, a course is a single subject, unit, or class. A group of courses which lead to a degree or certification will be referred to as a program. Mathematics Teacher Education and Development 2000, Vol. 2, 48-60