[eBook Code] Interactions on Digital Tablets in the Context of 3D Geometry Learning (eBook Code, 1st)

Preface ix

Introduction  xi

Chapter 1. Construction of Spatial Representation and Perspective in Students 1

1.1. Spatial representation in children according to Piaget  3

1.1.1. From perception to representation 3

1.1.2. Projective space  8

1.1.3. Euclidean space  13

1.1.4. Summary  14

1.2. The representation of geometric objects: the status of drawings 15

1.2.1. Status of drawings in mathematics: drawings versus figures 15

1.2.2. Use of geometrical representations  18

1.2.3. The three main functions of drawings in geometry 25

1.3. From the physical shape to its planar representation 25

1.3.1. The institutional perspective  25

1.3.2. Teaching 3D geometry  27

1.3.3. Different representations of 3D objects  29

1.3.4. The conflict between the SEEN and the KNOWN in children 34

1.4. Benefits of new technologies and dynamic 3D geometry  37

1.4.1. Advantages of 3D geometry programs  38

1.4.2. Limits of 3D geometry programs and consequences 40

1.4.3. Partial conclusions and initial hypotheses  46

Chapter 2. Mobile Devices and 3D Interactions 49

2.1. Why mobile devices? 50

2.1.1. A long-standing tradition in mathematics 51

2.1.2. Interest from the educational community 54

2.1.3. A field reality 56

2.2. Mobile devices 57

2.2.1. Different types of mobile devices 58

2.2.2. Entry systems of mobile terminals 61

2.3. Interactions on mobile devices and physiology  70

2.3.1. Specificities of mobile devices 70

2.3.2. Limitations due to physiologic characteristics  71

2.4. 3D interaction techniques  74

2.4.1. Mathematical reminders 74

2.4.2. 3D selection/manipulation and navigation interactions 77

2.5. “Language” of interactions and classifications  88

2.5.1. Language and grammar of gestures  89

2.5.2. Classifications 92

Chapter 3. Elaboration and Classification of Interactions  95

3.1. Human-centered design  95

3.1.1. A definition  96

3.1.2. Principles of the user-based approach 97

3.2. Study of the needs and behaviors of users 98

3.2.1. Study of pre-existing 3D geometry software 98

3.2.2. Study of users’ behaviors and needs  103

3.3. Our grammar and interaction language  109

3.3.1. Classification of tactile movement interactions 109

3.3.2. Definition of the grammar  111

3.3.3. The prototype: FINGERS (Find INteractions for GEometry leaneRS) 112

3.3.4. Our gestural language of interactions 114

3.4. Evaluation of the acceptance of interactions (selection, translation and rotation) 133

3.4.1. Experimental challenges and constraints 133

3.4.2. Preliminary evaluation of the acceptance of rotation and point of view change interactions  134

3.4.3. Comparison between gyroscope, face-tracking and multi-touch 140

3.4.4. Student learning of prototype interactions  147

3.5. Conclusion and perspectives 152

Chapter 4. Evaluation of the Educational Benefits for 3D Geometry 155

4.1. Partnerships 156

4.1.1. The schools in the field 156

4.1.2. The ESPÉ 157

4.1.3. Mathematics teachers’ associations  157

4.2. Limits  157

4.2.1. Ethical: the equality of chances for students 157

4.2.2. Practical: progression of the concepts throughout the year . 158

4.3. Evaluation of problem solving aids 158

4.3.1. In the field 159

4.3.2. Laboratory (EEG) 166

4.4. Evaluation of the benefits in learning 3D geometry 174

4.4.1. Participants  174

4.4.2. Material and experimental conditions 174

4.4.3. Experimental plan  176

4.4.4. Results and discussion  178

4.5. Partial conclusions 185

Conclusion 187

Bibliography  191

Index 203