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Articulations Between Tangible Space, Graphical Space and Geometrical Space
Resources, Practices and Training
Claire Guille-Biel Winder (Edited by), Guille–Biel Win (Author), Teresa Assude (Edited by)
9781786308405, Wiley
Hardback, published 24 August 2023
352 pages
23.5 x 15.6 x 2.2 cm, 0.758 kg
This book aims to present some of the latest research in the didactics of space and geometry, deepen some theoretical questions and open up new reflections for discourse. Its focus is as much on the approach of geometry itself and its link with the structuring of space as it is on the practices within the classroom, the dissemination of resources, the use of different artefacts and the training of teachers in this field. We study how spatial knowledge, graphical knowledge and geometric knowledge are taken into account and articulated in the teaching of space and geometry in compulsory schools, teaching resources (programs and textbooks) and current teacher training. We question how the semiotic dimension (language, gestures and signs) of geometric activity can be taken into account, and we identify the role of artefacts (digital or tangible) in the teaching and learning of geometry. This book brings together some fifteen contributions from Frenchspeaking researchers from different countries (France, Switzerland and Canada).
Preface xv Part 1 Articulations between Tangible Space, Graphical Space and Geometric Space 1 Chapter 1 The Geometry of Tracing, a Possible Link Between Geometric Drawing and Euclid’s Geometry? 3 1.1 Introduction 3 1.2 Geometry in middle school 5 1.2.1 What underlying axiomatics? 5 1.2.2 An example 6 1.2.3 The current lack of consistency 8 1.3 Geometry of tracing, a possible link between material geometry and Euclid’s geometry? 8 1.3.1 Figure visualization and figure restoration 9 1.3.2 The geometrical use of tracing instruments, a first step to make sense to an axiomatic 10 1.3.3 Distinguishing between the hypothesis and the conclusion 12 1.3.4. Restoration, description, construction of figures and geometric language 14 1.4 Dialectics of action, formulation and validation with regards to the reproduction of figures with instruments 15 1.4.1 Formulation situations and possible variations 15 1.4.2 Validation situations 17 1.5. From tracing to the characterization of objects and geometric relationships 18 1.5.1 On the concepts of segments, lines and points 18 1.5.2 On the notion of perpendicular lines 21 1.6 Towards proof and validation situations in relation to figure restoration 27 1.6.1 Equivalence between two construction programs and the need for proof 27 1.6.2 Validation situations involving programs for the construction of a square and introducing a proof process 29 1.7 Conclusion 31 1.8 References 32 Chapter 2 How to Operate the Didactic Variables of Figure Restoration Problems? 35 2.1 Introduction 35 2.2 Theoretical framework 35 2.2.1 Studying a specific type of problem: figure restoration 35 2.2.2 Studying the concepts involved in figure restoration problems 37 2.3 Values of the didactic variables of the first problem family 39 2.3.1 Values of the didactic variables for the “figure” and the “beginning of the figure” 39 2.3.2 Value for the didactic variable “instruments made available” 40 2.3.3 Rules of action and theorems-in-action associated with development on the geometrical usage of the ruler 41 2.4 Conclusion 44 2.5 References 44 Chapter 3 Early Geometric Learning in Kindergarten: Some Results from Collaborative Research 47 3.1 The emergence of the first questions 47 3.2 Theoretical insights 48 3.2.1 Global understanding and visual perception of geometric shapes 48 3.2.2 Operative understanding and visual perception of geometric shapes 49 3.2.3. Topological understanding and visual perception of geometric shapes 50 3.2.4 Haptic perception 51 3.2.5 Association of visual and haptic perceptions: towards a sequential understanding of geometric shapes 52 3.3 The role of language in early geometric learning 53 3.3.1 But which lexicon? 54 3.3.2 Verbal and gestural language 58 3.4 Assembling shapes 60 3.4.1 Free assembly of shapes 60 3.4.2 Assembling triangles 62 3.5 Gestures to learn 68 3.6 Conclusion 69 3.7 References 71 Chapter 4 Using Coding to Introduce Geometric Properties in Primary School 73 4.1 Coding in geometry 73 4.2 Two examples of communication activities requiring the use of coding 75 4.2.1 A co-constructed coding 75 4.2.2 Personal coding 77 4.3 Conclusion: perspectives on the introduction of coding in geometry 78 4.4 References 79 Chapter 5 Freehand Drawing for Geometric Learning in Primary School 81 5.1 Introduction 81 5.2 Drawings in geometry and their functions 82 5.3 Freehand drawing in research 83 5.4 Exploring the milieu around a freehand reproduction task of the Mitsubishi symbol on a blank white page 84 5.4.1 Freehand drawing reveals a reasoning between spatial knowledge and geometric knowledge 87 5.4.2 Freehand drawing as a dynamic process to build and transform knowledge 88 5.5 Conclusion 89 5.6 References 90 Part 2 Resources and Artifacts for Teaching 93 Chapter 6 Use of a Dynamic Geometry Environment to Work on the Relationships Between Three Spaces (Tangible, Graphical and Geometrical) 95 6.1 Added value with a dynamic geometry environment: the ecological and economical point of view 95 6.2 Tangible space, graphical space and geometric space 100 6.3 Designing situations for first grade primary school 103 6.3.1 Our choices for designing situations 104 6.3.2 Presentation of situations 104 6.4 Analysis of the situations for the first-grade class 105 6.4.1 Instrumental dimension: perceptive–gestural level 105 6.4.2 Instrumental dimension: spatial–geometric relationships 106 6.4.3 Instrumental dimension: exploration and graphical space 107 6.4.4 Instrumental dimension: tool-geometric space symbiosis 108 6.4.5 Praxeological dimension 109 6.4.6 Praxeological dimension: observe and describe 111 6.5 Conclusion 113 6.6 References 115 Chapter 7 Robotics and Spatial Knowledge 119 7.1 Introduction 119 7.2 Theoretical framework and development for a categorization of spatial tasks 120 7.2.1 Spatial knowledge 120 7.2.2 Types of spatial tasks 121 7.2.3 Types of tasks and techniques 121 7.3 Research methodology 122 7.4 Analysis: reproducing an assembly 123 7.4.1 Test item 123 7.4.2 Test results 124 7.4.3 Analysis of the results 125 7.5 Conclusion 126 7.6 References 127 Chapter 8 Contribution of a Human Interaction Simulator to Teach Geometry to Dyspraxic Pupils 129 8.1 Introduction 129 8.2 General research framework 130 8.2.1 Teaching geometry 130 8.2.2 Dyspraxia and consequences for geometry 131 8.3 What alternatives are there for teaching geometry? 132 8.3.1 Using tools in a digital environment 132 8.3.2 Dyadic work arrangement 135 8.4 Designing the human interaction simulator 138 8.4.1 General considerations 138 8.4.2 Choice of instrumented actions 139 8.4.3 Interaction choices 140 8.4.4 Ergonomic considerations 142 8.5 Initial experimental results 143 8.5.1 Data collected 144 8.5.2 Jim’s diagnostic evaluation 144 8.5.3 Analysis of the first experimentation 146 8.5.4 Conclusion 150 8.6 References 152 Chapter 9 Research and Production of a Resource for Geometric Learning in First and Second Grade 155 9.1 Presentation of the ERMEL team’s research on spatial and geometric learning from preschool to second grade 155 9.1.1 Origins of the research 156 9.1.2 Introduction to the chapter 156 9.2 Learning to trace straight lines 157 9.2.1 Significance of the straight line 157 9.2.2 Initial hypotheses 157 9.2.3 The RAYURE situation 159 9.2.4 Using straight lines 160 9.2.5 A few summary elements 161 9.3 Plane and solid figures 162 9.3.1 Findings and assumptions 162 9.3.2 The SQUARE AND QUASI-SQUARE situation 163 9.3.3 The emergence of criteria for comparing solids: the IDENTIFYING A SOLID situation 165 9.3.4 Identification of cube properties: the CUBE AND QUASI-CUBE situation 166 9.3.5 Progression on solids and plane figures 167 9.4 The appropriation of research results by the resource 168 9.5 Conclusion 169 9.6 References 170 Chapter 10 Tool for Analyzing the Teaching of Geometry in Textbooks 171 10.1 General framework and theoretical tools 172 10.1.1 Didactic co-determination scale, mathematical and didactic organizations 172 10.1.2 Reference MO and theoretical tools for analysis 174 10.2 Analysis criteria: definition and methodology 181 10.2.1 Institutional conformity 181 10.2.2 Educational adequacy 182 10.2.3 Didactic quality 182 10.3 Introducing the analysis grid 183 10.3.1 Analysis of tasks and task types 183 10.3.2 Analysis of techniques 184 10.3.3 Analysis of knowledge 185 10.3.4 Analysis of ostensives 186 10.3.5 Analysis of organizational and planning elements 189 10.3.6 Summary 191 10.4 Conclusion 191 10.5 References 192 Part 3 Teaching Practices and Training Issues 197 Chapter 11 Study on Teacher Appropriation of a Geometry Education Resource 199 11.1 Introduction 199 11.2 Research background 200 11.2.1 Study on dissemination possibilities in ordinary education 200 11.2.2 Resource design approach 201 11.2.3 A working methodology based on assumptions 202 11.2.4 Designing a situation using the didactic engineering approach for development 205 11.3 Focus on the adaptability of this situation to ordinary education 206 11.3.1 Details about the theoretical framework and the research question 206 11.3.2 Presentation on the follow-up of teachers, details of the research question and the methodology 207 11.3.3 Presentation of the analysis methodology 208 11.4 Elements of the analysis 209 11.4.1 Analysis a priori of the situation and anticipatory analysis of the teacher’s activity 209 11.4.2 Analysis of practices 211 11.5 Conclusion 217 11.6 References 219 Chapter 12 Geometric Reasoning in Grades 4 to 6, the Teacher’s Role: Methodological Overview and Results 221 12.1 Introduction 221 12.2 Theoretical choices and the problem statement 221 12.2.1 Geometrical paradigms 222 12.2.2 The different spaces 223 12.2.3 Study on reasoning 223 12.2.4 The role of the teacher 225 12.2.5 Problem statement 225 12.3 Methodology 225 12.3.1 General principle 225 12.3.2 The situations 226 12.3.3 Analysis methodology 226 12.4 Conclusion 227 12.5 References 229 Chapter 13 When the Teacher Uses Common Language Instead of Geometry Lexicon 231 13.1 Introduction 231 13.2 An attempt to categorize the uses of common vernacular terms in place of geometry lexicon terms within teacher discourse 232 13.2.1 The phenomenon of didactic reticence 232 13.2.2 The phenomenon of semantic analogy: comparison with common concepts to construct meaning for mathematical knowledge 233 13.2.3 The phenomenon of lexical competition: use of common vernacular terms to designate common concepts 234 13.2.4 The phenomena of repeating pupil formulations 235 13.2.5 The phenomenon of didactic repression 236 13.3 Conclusion 237 13.4 References 238 Chapter 14 The Development of Spatial Knowledge at School and in Teacher Training: A Case Study on 1, 2, 3… imagine! 241 14.1 Introduction and research question 241 14.2 Conceptual framework 243 14.2.1 Components set to address SK in primary school 244 14.2.2 Levels of abstraction that value SK 245 14.2.3 Main variables in situations where SK is valued 246 14.3 Presentation of the activity 1, 2, 3 … imagine! 247 14.4 Experiments with this activity in primary school and in teacher training in Quebec 251 14.4.1 Teaching sequence experimented in primary school 251 14.4.2 Teaching sequence tested in teacher training 254 14.5 Experiment results 255 14.5.1 Experiment results of the teaching sequence in primary school 255 14.5.2 Experiment results of this teaching sequence in teacher training 257 14.6 Conclusion 259 14.7 References 260 Chapter 15 What Use of Analysis a priori by Pre-Service Teachers in Space Structuring Activities? 265 15.1 Introduction – an institutional challenge of transposing didactic knowledge 265 15.1.1 Choice of external transposition: institutional constraints 265 15.2 Theoretical framework 267 15.2.1 Choice of internal transposition: the moments of the study of the analysis a priori 268 15.3 Research questions 269 15.4 Methodology 269 15.4.1 Selection of activities and brief analysis 270 15.5 Results 272 15.6 Conclusion 273 15.7 References 273 Part 4 Conclusion and Implications 275 Chapter 16 Questions about the Graphic Space: What Objects? Which Operations? 277 16.1 Semiotic tools of geometric work and graphic space 277 16.2 Graphic space: graphic expressions, denotation and meaning 280 16.2.1 How can we define the graphic space? 280 16.2.2 Which objects in the graphic space? 280 16.2.3 Graphic expressions: which operations? 282 16.3 References 285 Chapter 17 Towards New Questions in Geometry Didactics 289 17.1 Current questions in geometry didactics 289 17.2 Continuities and breaks in the teaching of geometry 291 17.2.1 Institutional continuity? 291 17.2.2 Theoretical continuity from “geometry of tracing” to “abstract geometry”? 291 17.2.3 Praxis continuity from the “geometry of tracing” to “abstract geometry” 294 17.3 Articulation between resources, practices and teacher training 297 17.4 References 299 Appendices 303 Appendix 1 305 Appendix 2 309 Appendix 3 311 Appendix 4 313 List of Authors 315 Index 317
Claire GUILLE-BIEL WINDER and Teresa ASSUDE
Anne-Cécile MATHÉ and Marie-Jeanne PERRIN-GLORIAN
Karine VIÈQUE
Valentina CELI
Sylvia COUTAT
Céline VENDEIRA-MARÉCHAL
Teresa ASSUDE
Emilie MARI
Fabien EMPRIN and Edith PETITFOUR
Jacques DOUAIRE, Fabien EMPRIN and Henri-Claude ARGAUD
Claire GUILLE-BIEL WINDER and Edith PETITFOUR
Christine MANGIANTE-ORSOLA
Sylvie BLANQUART
Karine MILLON-FAURÉ, Catherine MENDONÇA DIAS, Céline BEAUGRAND and Christophe HACHE
Patricia MARCHAND and Caroline BISSON
Ismaïl MILI
Teresa ASSUDE
Claire GUILLE-BIEL WINDER and Catherine HOUDEMENT
Subject Areas: Psychology [JM]
