Acknowledgements Introduction Chapter 1: The Discourse of Teaching and Learning Secondary Geometry through History 1. Introduction 1.1. Overview of this Chapter 1.2.1. The History of Reasoning and Figural Concepts 1.2.

2. The Mathematical Emancipation of Euclid: From the Enlightenment to the Foundational Crisis 1.3. The Shaping of Geometry Curricula in the 19th and 20th Centuries 1.3.1. Geometry as an Opportunity for Students to Experience the Work of Mathematicians 1.3.

2. Geometry as an Opportunity for Students to Learn Logic 1.3.3. Geometry as an Opportunity for Students to Formalize their Intuitive Experiences in the World 1.3.4. Geometry as an Opportunity for Students to Learn Practical Applications 1.

4. Recent Developments 1.4.1. The Impact of Computers 1.4.2. Changes Due to Developments in Mathematics 1.

4.3. Greater Emphasis on Modeling and Applications 1.4.4. The "Fundamental Ideas" Discussion 1.4.5.

Constructivist Ideas on Learning 1.4.6. Mathematics as a Human Activity 1.4.7. Geometry as Empirical Theory 1.5.

Figural Concepts, Linking History with the Future of Geometry Instruction Chapter 2: Geometric Figures and their Representations 2.1. Introduction 2.2. Conceptions of Figure: What we Mean by Conception 2.3. Initial Conceptions of Geometric Figures 2.3.

1. The Figure as Navigation of the Macrospace 2.3.2. The Figure as Capture of Objects and Actions in the Mesospace 2.3.3. Conceptions of Figure in the Microspace 2.

3.3.1. The Figure as Construction of Objects int he Microspace 2.3.3.2. The Figure as Description of Objects int he Microspace 2.

3.4. Conceptions of Figure and the Secondary Geometry Curriculum 2.3.4.1. Modeling the Experiential World 2.4.

The Geometric Diagram in the Literature 2.4.1. The Geometric Diagram in Mathematics Education and Cognitive Science 2.4.1.1. Seeing vs.

Knowing: How Diagrams Preserve and Lose Information 2.4.1.2. Spatio-Graphical vs. Theoretical Properties: How Diagrams Add Information 2.4.1.

3. The Epistemological Need for Representations 2.4.1.4. Diagrams as Representations of Many Concepts 2.4.1.

5. Many Diagrams as Representations of Single Concepts 2.4.2. The Crucial Role of Diagrams in the History of Geometry 2.4.3. The Role of Diagrams and Logic in Geometry Instruction 2.

4.4. Diagrams and other Representations, and the Conceptions of Figure 2.5 A Modeling Perspective in the Study of Figures 2.5.1. The Modeling Approach and the Four Conceptions of Figure 2.5.

2. The Modeling Approach and Informal Proof 2.5.3. Conclusion of chapter 2 Chapter 3: Students'' Thinking and Learning in Geometry 3.1. Introduction 3.2.

Conceptions of Figure and Students'' Cognition 3.2.1. Connecting Conceptions of Figure and Modeling to Students'' Cognition 3.2.2. Geometric Thinking as Interactions Between Figural and Conceptual Aspects 3.2.

2.1. Fischbein''s Figural Concepts of Geometric Figures 3.2.2.2. Roles of Definitions in Geometric Figures 3.2.

2.3. Different Conceptions of Geometric Figures in Geometric Thinking 3.2.2.4. Dimensional Deconstruction of Figures 3.2.

2.5. Different Conceptions of 3D Figures 3.2.3. Conceptions of Geometric Figures from a Cognitive Perspective 3.3. Geometric Figures and Students'' Learning as Progressive Change in Cognition 3.

3.1. Research into van Hiele''s Levels of Geometric Thinking 3.3.1.1. Van Hiele''s Levels of Geometric Thinking 3.3.

1.2. Studies around van Hiele Levels of Thinking 3.3.2. Extensions to the van Hiele Levels of Geometric Thinking 3.4. Enriching Semiotic Registers, Operations, and Control Structures with DGS 3.

4.1. The Use of DGS in the Learning of Geometry 3.4.2. Students'' Thinking of the Conceptions of Geometric Figure with DGS 3.4.2.

1. DGS and the Conception of Figure as Description and Manipulation of Small Objects in the Microspace 3.4.2.2. Modeling Navigational Experiences and Large (Mesospace) Objects in 2D 3.4.2.

3. Modeling Large Objects in 3D 3.4.2.4. DGS and Geometric Figures as Construction of Small Microspace Objects 3.5. Theoretical Underpinnings for Learning Trajectories of Geometric Figures 3.

5.1. Hypothetical Learning Trajectories 3.5.2. Hypothetical Learning Trajectories for Geometric Figure 3.5.2.

1. A Learning Trajectory for Figure as Navigation of the Macrospace 3.5.2.2. A Learning Trajectory for Figure as Capture of Objects in the Mesospace 3.5.2.

3. A Learning Trajectory for Figure as Description and Manipulation of Objects in the Microspace 3.5.2.3. A Learning Trajectory for Figure as Construction of Objects in the Microspace 3.5.3.

What Needs to be Done to Research these Learning Trajectories 3.6. Conclusions of this Chapter Chapter 4: Teaching Practice and Teacher Knowledge in Geometry Instruction 4.1. Introduction 4.2. Teaching Practice in Geometry 4.2.

1. The Didactical Contract and Instructional Exchanges 4.2.2. Instructional Situations in Geometry 4.2.2.1.

Instructional Situation as a Fractal Concept 4.2.2.2. Instructional Situations as Frames for Work 4.2.2.3.

Different Instructional Situations in Geometry 4.2.2.4. Dynamic Geometry Software in the Teaching of Geometry in Secondary School 4.2.2.5.

Is the Work of Teaching Geometry Subject Specific? 4.3. Teacher Knowledge of Geometry 4.3.1. Teacher knowledge of geometry from professional preparation programs 4.3.2.

The Role and Nature of Teachers'' Knowledge in the Context of Geometry 4.3.2.1. The Emergence of an Interest on What Teachers Know 4.3.2.2.

Knowledge Development: Connecting Opportunity to Learn and Knowledge Use 4.3.2.3. Extending the Epistemology: Mathematical Knowledge for Teaching (MKT) 4.3.2.4.

Mathematical KNowledge for Teaching Geometry 4.4. Studies of Preservice Teachers'' Knowledge of Geometry 4.5. Another Look at Elementary and Middle Grades Teachers 4.5.1. Prospective Primary Teachers'' Memories and Conceptions of Geometry Instruction 4.

5.2. Primary Teachers'' Knowledge of Geometry for Teaching 4.5.3. Prospective Middle-Grades Teachers'' Understanding of Geometry 4.5.4.

Prospective Middle School Teachers'' Conceptions of Translations 4.6. Beliefs of Secondary Geometry Teachers 4.6.1. Beliefs on the Nature of Geometric Objects 4.6.2.

Teachers'' Beliefs about the Role of Proof in Geometry (and the Role of Geometry in Teaching Proof) 4.6.3. The Practical Rationality of Geometry Instruction 4.7. To Conclude the Chapter Chapter 5: Improving the Teacher and Learning of Geometry in Secondary School Classrooms 5.1. Introduction 5.

2. Communication Tasks: A Contribution of the Theory of Didactical Situations to the Design of Interventions 5.3. Secondary Geometry in the Service of Modeling the Experience with Shape and Space 5.3.1. Using Existing Conceptions of Figure to Represent Navigation of the Macrospace 5.3.

2. Using Existing Conceptions of Figure to Anticipate the Capture of Objects in the Mesospace 5.3.3. Improving the Control of Microspace Conceptions 5.4. Communication Tasks in the Teaching and Learning of Geometry 5.4.

1. Description or Prescription of a Navigation of a Space 5.4.2. To Describe or Prescribe the Capture of an Object 5.4.3. To Create or Discover a Code or Language for the Description of Geometric Diagrams and Small Objects 5.

4.3.1. From Names and Prototypes to Properties 5.4.3.2. From Seeing Diagrams to Stating Properties 5.

4.3.3. From Dynamic Explorations of Diagrams to Properties of Figures 5.4.3.4. Summarizing the Three Cases 5.

4.4. Identifying the Controls Needed to Describe an Accomplished Construction or Prescribe a Construction Procedure 5.2.4.1. Examining Reasons for Features of a Procedure 5.2.

4.2. Gaining Theoretical Control of Initial Conditions 5.2.4.3. Turning Declarative Statements into Construction Operators 5.2.

4.4. A Declarative Statement and its Associated Action Rules 5.2.4.5. Summing Up this Section 5.3.

Concluding this Chapter Chapter 6: A Conclusion and a Beginning: Doing Research on The Teaching and Learning of Secondary Geometry 6.1. Introduction 6.2. Research questions 6.2.1. Communication Tasks as Design Research Projects 6.

2.2. Communication Tasks as Instructional Experiments 6.2.2.1. Researching the Behavior of Activity Systems such as Contract, Situation, and Task 6.2.

2.2. Researching teachers'' individual assets such as beliefs and knowledge 6.2.2.3. Researching the Sources of Justification for actions in teaching 6.3.

In Conclusion References.