The idea of teachers Learning through Teaching (LTT) ¿ when presented to a naïve bystander ¿ appears as an oxymoron. Are we not supposed to learn before we teach? After all, under the usual circumstances, learning is the task for those who are being taught, not of those who teach. However, this book is about the learning of teachers, not the learning of students. It is an ancient wisdom that the best way to ¿truly learn¿ something is to teach it to others. Nevertheless, once a teacher has taught a particular topic or concept and, consequently, ¿truly learned¿ it, what is left for this teacher to learn? As evident in this book, the experience of teaching presents teachers with an exciting opp- tunity for learning throughout their entire career. This means acquiring a ¿better¿ understanding of what is being taught, and, moreover, learning a variety of new things. What these new things may be and how they are learned is addressed in the collection of chapters in this volume. LTT is acknowledged by multiple researchers and mathematics educators. In the rst chapter, Leikin and Zazkis review literature that recognizes this phenomenon and stress that only a small number of studies attend systematically to LTT p- cesses. The authors in this volume purposefully analyze the teaching of mathematics as a source for teachers¿ own learning.

Introduction Part I: Theoretical and Methodological Perspectives on Teachers¿ Learning through Teaching Chapter 1 Roza Leikin and Rina Zazkis Teachers¿ opportunities to learn mathematics through teaching Chapter 2 John Mason Attention and intention: Learning about teaching through teaching Chapter 3 Ron Tzur What and How might mathematics teachers learn via teaching: Contributions to closing an unspoken gap Chapter 4 Roza Leikin Learning through teaching though the lens of multiple solution tasks Part II: Examples of Learning through teaching: Pedagogical mathematics Interlude 1 Chapter 5 Rina Zazkis What have I learned: Mathematical insights and pedagogical implication Chapter 6 Marcelo Borba and Rúbia B. A. Zulatto Dialogical education and learning mathematics online from teachers. Chapter 7 Carolyn Kieran and José Guzmán Role of task and technology in provoking teacher change: A case of proofs and proving in high school Algebra Chapter 8 Nick Jackiw and Nathalie Sinclair Learning through teaching when teaching machines: Discursive interaction design in Sketchpad Chapter 13 Robin Marcus and Daniel Chazan What experienced teachers have learned from helping students think about solving equations in the one-variable-first algebra curriculum Part III: Examples of Learning through teaching: Mathematical pedagogy Interlude 2 Chapter 9 Michal Yerushalmy and Shulamit Elikan Exploring reform ideas of teaching Algebra: Analysis of videotaped episodes and conversations about them Chapter 10 Peter Liljedahl On rapid professional growth: cases of learning through teaching Chapter 11 Lara Alcock Interactions between teaching and research:Developing pedagogical content knowledge for Real Analysis Chapter 12 Helen Doerr and Stephen Lerman Teachers learning from their teaching: The case of communicative practices Chapter 14 Dave Hewitt Feedback: Expanding a repertoire and making choices Conclusion.