Interactions on Digital Tablets in the Context of 3D Geometry Learning – Contributions and Assessments

<p>Preface ix</p>
<p>Introduction&nbsp; xi</p>
<p>Chapter 1. Construction of Spatial Representation and Perspective in Students 1</p>
<p>1.1. Spatial representation in children according to Piaget&nbsp; 3</p>
<p>1.1.1. From perception to representation 3</p>
<p>1.1.2. Projective space&nbsp; 8</p>
<p>1.1.3. Euclidean space&nbsp; 13</p>
<p>1.1.4. Summary&nbsp; 14</p>
<p>1.2. The representation of geometric objects: the status of drawings 15</p>
<p>1.2.1. Status of drawings in mathematics: drawings versus figures 15</p>
<p>1.2.2. Use of geometrical representations&nbsp; 18</p>
<p>1.2.3. The three main functions of drawings in geometry 25</p>
<p>1.3. From the physical shape to its planar representation 25</p>
<p>1.3.1. The institutional perspective&nbsp; 25</p>
<p>1.3.2. Teaching 3D geometry&nbsp; 27</p>
<p>1.3.3. Different representations of 3D objects&nbsp; 29</p>
<p>1.3.4. The conflict between the SEEN and the KNOWN in children 34</p>
<p>1.4. Benefits of new technologies and dynamic 3D geometry&nbsp; 37</p>
<p>1.4.1. Advantages of 3D geometry programs&nbsp; 38</p>
<p>1.4.2. Limits of 3D geometry programs and consequences 40</p>
<p>1.4.3. Partial conclusions and initial hypotheses&nbsp; 46</p>
<p>Chapter 2. Mobile Devices and 3D Interactions 49</p>
<p>2.1. Why mobile devices? 50</p>
<p>2.1.1. A long–standing tradition in mathematics 51</p>
<p>2.1.2. Interest from the educational community 54</p>
<p>2.1.3. A field reality 56</p>
<p>2.2. Mobile devices 57</p>
<p>2.2.1. Different types of mobile devices 58</p>
<p>2.2.2. Entry systems of mobile terminals 61</p>
<p>2.3. Interactions on mobile devices and physiology&nbsp; 70</p>
<p>2.3.1. Specificities of mobile devices 70</p>
<p>2.3.2. Limitations due to physiologic characteristics&nbsp; 71</p>
<p>2.4. 3D interaction techniques&nbsp; 74</p>
<p>2.4.1. Mathematical reminders 74</p>
<p>2.4.2. 3D selection/manipulation and navigation interactions 77</p>
<p>2.5. Language of interactions and classifications&nbsp; 88</p>
<p>2.5.1. Language and grammar of gestures&nbsp; 89</p>
<p>2.5.2. Classifications 92</p>
<p>Chapter 3. Elaboration and Classification of Interactions&nbsp; 95</p>
<p>3.1. Human–centered design&nbsp; 95</p>
<p>3.1.1. A definition&nbsp; 96</p>
<p>3.1.2. Principles of the user–based approach 97</p>
<p>3.2. Study of the needs and behaviors of users 98</p>
<p>3.2.1. Study of pre–existing 3D geometry software 98</p>
<p>3.2.2. Study of users behaviors and needs&nbsp; 103</p>
<p>3.3. Our grammar and interaction language&nbsp; 109</p>
<p>3.3.1. Classification of tactile movement interactions 109</p>
<p>3.3.2. Definition of the grammar&nbsp; 111</p>
<p>3.3.3. The prototype: FINGERS (Find INteractions for GEometry leaneRS) 112</p>
<p>3.3.4. Our gestural language of interactions 114</p>
<p>3.4. Evaluation of the acceptance of interactions (selection, translation and rotation) 133</p>
<p>3.4.1. Experimental challenges and constraints 133</p>
<p>3.4.2. Preliminary evaluation of the acceptance of rotation and point of view change interactions&nbsp; 134</p>
<p>3.4.3. Comparison between gyroscope, face–tracking and multi–touch 140</p>
<p>3.4.4. Student learning of prototype interactions&nbsp; 147</p>
<p>3.5. Conclusion and perspectives 152</p>
<p>Chapter 4. Evaluation of the Educational Benefits for 3D Geometry 155</p>
<p>4.1. Partnerships 156</p>
<p>4.1.1. The schools in the field 156</p>
<p>4.1.2. The ESP&Eacute; 157</p>
<p>4.1.3. Mathematics teachers associations&nbsp; 157</p>
<p>4.2. Limits&nbsp; 157</p>
<p>4.2.1. Ethical: the equality of chances for students 157</p>
<p>4.2.2. Practical: progression of the concepts throughout the year . 158</p>
<p>4.3. Evaluation of problem solving aids 158</p>
<p>4.3.1. In the field 159</p>
<p>4.3.2. Laboratory (EEG) 166</p>
<p>4.4. Evaluation of the benefits in learning 3D geometry 174</p>
<p>4.4.1. Participants&nbsp; 174</p>
<p>4.4.2. Material and experimental conditions 174</p>
<p>4.4.3. Experimental plan&nbsp; 176</p>
<p>4.4.4. Results and discussion&nbsp; 178</p>
<p>4.5. Partial conclusions 185</p>
<p>Conclusion 187</p>
<p>Bibliography&nbsp; 191</p>
<p>Index 203</p>