Pat Herbst is Professor of Education and Mathematics at the University of Michigan, USA. Pat is a former high school mathematics teacher in Argentina and his research focuses on the nature of the mathematical work that students do in secondary classrooms and the work that teachers do to manage knowledge development. Taro Fujita is a lecturer in mathematics education at the University of Exeter, UK. Originally a mathematics teacher in Japan, Taro currently teaches the learning and teaching of mathematics in primary schools and higher mathematics, and he is also editorial assistant for the International Journal for Technology in Mathematics Education.Stefan Halverscheid is Professor of Mathematics Education at the University of Göttingen, Germany. His background is in complex and differential geometry in the presence of symmetries and in research on the teaching and learning of mathematics. He has experience as a high school teacher, has lectured in Teacher Education at Münster, Oldenburg and Bremen Universities, and was Dean of Studies and Dean of the Faculty of Mathematics and Computer Science at Göttingen. Michael Weiss is currently on the faculty of the Department of Mathematics at the University of Michigan, USA. His background is in mathematics education and pure mathematics, and he was formerly a high school mathematics teacher in the United States.

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• IMPACT—Series ForewordAcknowledgmentsIntroductionChapter 1: The Discourse of Teaching and Learning Secondary Geometry through History1.1. Introduction1.2. Overview of This Chapter1.3. The Development of Geometry up to the So-Called Foundational Crisis1.4. The Shaping of Geometry Curricula in the Nineteenth and Twentieth Centuries1.5. Recent Developments1.6. ConclusionChapter 2: Geometric Figures and Their Representations2.1. Introduction2.2. Conceptions of Figure: What Ee Mean by Conception2.3. Initial Conceptions of Geometric Figures2.4. The Geometric Diagram in the Literature2.5. A Modeling Perspective in the Study of Figures2.6. ConclusionChapter 3: Students’ Thinking and Learning in Geometry3.1. Introduction3.2. Conceptions of Figure and Students’ Cognition3.3. Geometric Figures and Students’ Learning as Progressive Change in Cognition3.4. Enriching Semiotic Registers, Operations, and Control Structures with DGS3.5. Theoretical Underpinnings for Learning Trajectories of Geometric Figures3.6. ConclusionChapter 4: Teaching Practice and Teacher Knowledge in Geometry Instruction4.1. Introduction4.2. Teaching Practice in Geometry4.3. Teacher Knowledge of Geometry4.4. Studies of Preservice Teachers’ Knowledge of Geometry4.5. Another Look at Elementary and Middle Grades Teachers4.6. Beliefs of Secondary Geometry Teachers4.7. ConclusionChapter 5: Improving the Teaching and Learning of Geometry in Secondary School Classrooms5.1. Introduction5.2. Communication Tasks: A Contribution of the Theory of Didactical Situations to the Design of Interventions5.3. Secondary Geometry in the Service of Modeling the Experience with Shape and Space5.4. Communication Tasks in the Teaching and Learning of Geometry5.5. Concluding this ChapterChapter 6: A Conclusion and a Beginning: Doing Research on The Teaching and Learning of Secondary Geometry6.1. Introduction6.2. Research questions6.3. ConclusionReferences

Pat Herbst is Professor of Education and Mathematics at the University of Michigan, USA. Pat is a former high school mathematics teacher in Argentina and his research focuses on the nature of the mathematical work that students do in secondary classrooms and the work that teachers do to manage knowledge development. Taro Fujita is a lecturer in mathematics education at the University of Exeter, UK. Originally a mathematics teacher in Japan, Taro currently teaches the learning and teaching of mathematics in primary schools and higher mathematics, and he is also editorial assistant for the International Journal for Technology in Mathematics Education.Stefan Halverscheid is Professor of Mathematics Education at the University of Göttingen, Germany. His background is in complex and differential geometry in the presence of symmetries and in research on the teaching and learning of mathematics. He has experience as a high school teacher, has lectured in Teacher Education at Münster, Oldenburg and Bremen Universities, and was Dean of Studies and Dean of the Faculty of Mathematics and Computer Science at Göttingen. Michael Weiss is currently on the faculty of the Department of Mathematics at the University of Michigan, USA. His background is in mathematics education and pure mathematics, and he was formerly a high school mathematics teacher in the United States.

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• IMPACT—Series ForewordAcknowledgmentsIntroductionChapter 1: The Discourse of Teaching and Learning Secondary Geometry through History1.1. Introduction1.2. Overview of This Chapter1.3. The Development of Geometry up to the So-Called Foundational Crisis1.4. The Shaping of Geometry Curricula in the Nineteenth and Twentieth Centuries1.5. Recent Developments1.6. ConclusionChapter 2: Geometric Figures and Their Representations2.1. Introduction2.2. Conceptions of Figure: What Ee Mean by Conception2.3. Initial Conceptions of Geometric Figures2.4. The Geometric Diagram in the Literature2.5. A Modeling Perspective in the Study of Figures2.6. ConclusionChapter 3: Students’ Thinking and Learning in Geometry3.1. Introduction3.2. Conceptions of Figure and Students’ Cognition3.3. Geometric Figures and Students’ Learning as Progressive Change in Cognition3.4. Enriching Semiotic Registers, Operations, and Control Structures with DGS3.5. Theoretical Underpinnings for Learning Trajectories of Geometric Figures3.6. ConclusionChapter 4: Teaching Practice and Teacher Knowledge in Geometry Instruction4.1. Introduction4.2. Teaching Practice in Geometry4.3. Teacher Knowledge of Geometry4.4. Studies of Preservice Teachers’ Knowledge of Geometry4.5. Another Look at Elementary and Middle Grades Teachers4.6. Beliefs of Secondary Geometry Teachers4.7. ConclusionChapter 5: Improving the Teaching and Learning of Geometry in Secondary School Classrooms5.1. Introduction5.2. Communication Tasks: A Contribution of the Theory of Didactical Situations to the Design of Interventions5.3. Secondary Geometry in the Service of Modeling the Experience with Shape and Space5.4. Communication Tasks in the Teaching and Learning of Geometry5.5. Concluding this ChapterChapter 6: A Conclusion and a Beginning: Doing Research on The Teaching and Learning of Secondary Geometry6.1. Introduction6.2. Research questions6.3. ConclusionReferences

Pat Herbst is Professor of Education and Mathematics at the University of Michigan, USA. Pat is a former high school mathematics teacher in Argentina and his research focuses on the nature of the mathematical work that students do in secondary classrooms and the work that teachers do to manage knowledge development. Taro Fujita is a lecturer in mathematics education at the University of Exeter, UK. Originally a mathematics teacher in Japan, Taro currently teaches the learning and teaching of mathematics in primary schools and higher mathematics, and he is also editorial assistant for the International Journal for Technology in Mathematics Education.Stefan Halverscheid is Professor of Mathematics Education at the University of Göttingen, Germany. His background is in complex and differential geometry in the presence of symmetries and in research on the teaching and learning of mathematics. He has experience as a high school teacher, has lectured in Teacher Education at Münster, Oldenburg and Bremen Universities, and was Dean of Studies and Dean of the Faculty of Mathematics and Computer Science at Göttingen. Michael Weiss is currently on the faculty of the Department of Mathematics at the University of Michigan, USA. His background is in mathematics education and pure mathematics, and he was formerly a high school mathematics teacher in the United States.

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• IMPACT—Series ForewordAcknowledgmentsIntroductionChapter 1: The Discourse of Teaching and Learning Secondary Geometry through History1.1. Introduction1.2. Overview of This Chapter1.3. The Development of Geometry up to the So-Called Foundational Crisis1.4. The Shaping of Geometry Curricula in the Nineteenth and Twentieth Centuries1.5. Recent Developments1.6. ConclusionChapter 2: Geometric Figures and Their Representations2.1. Introduction2.2. Conceptions of Figure: What Ee Mean by Conception2.3. Initial Conceptions of Geometric Figures2.4. The Geometric Diagram in the Literature2.5. A Modeling Perspective in the Study of Figures2.6. ConclusionChapter 3: Students’ Thinking and Learning in Geometry3.1. Introduction3.2. Conceptions of Figure and Students’ Cognition3.3. Geometric Figures and Students’ Learning as Progressive Change in Cognition3.4. Enriching Semiotic Registers, Operations, and Control Structures with DGS3.5. Theoretical Underpinnings for Learning Trajectories of Geometric Figures3.6. ConclusionChapter 4: Teaching Practice and Teacher Knowledge in Geometry Instruction4.1. Introduction4.2. Teaching Practice in Geometry4.3. Teacher Knowledge of Geometry4.4. Studies of Preservice Teachers’ Knowledge of Geometry4.5. Another Look at Elementary and Middle Grades Teachers4.6. Beliefs of Secondary Geometry Teachers4.7. ConclusionChapter 5: Improving the Teaching and Learning of Geometry in Secondary School Classrooms5.1. Introduction5.2. Communication Tasks: A Contribution of the Theory of Didactical Situations to the Design of Interventions5.3. Secondary Geometry in the Service of Modeling the Experience with Shape and Space5.4. Communication Tasks in the Teaching and Learning of Geometry5.5. Concluding this ChapterChapter 6: A Conclusion and a Beginning: Doing Research on The Teaching and Learning of Secondary Geometry6.1. Introduction6.2. Research questions6.3. ConclusionReferences