Articulations Between Tangible Space, Graphical Space and Geometrical Space

Claire Guille-Biel Winder is a lecturer in mathematics education at AixMarseille Université (ADEF team), France. Her research focuses on the teaching of geometry in primary school, as well as on the practices and training of elementary school teachers.Teresa Assude is a full professor in mathematics education at AixMarseille Université (ADEF team), France. Her research focuses on the study of preventive aid systems, the use of digital technologies and the teaching of mathematics to disabled students.

• Preface xvClaire GUILLE-BIEL WINDER and Teresa ASSUDEPart 1 Articulations between Tangible Space, Graphical Space and Geometric Space 1Chapter 1 The Geometry of Tracing, a Possible Link Between Geometric Drawing and Euclid’s Geometry? 3Anne-Cécile MATHÉ and Marie-Jeanne PERRIN-GLORIAN1.1 Introduction 31.2 Geometry in middle school 51.2.1 What underlying axiomatics? 51.2.2 An example 61.2.3 The current lack of consistency 81.3 Geometry of tracing, a possible link between material geometry and Euclid’s geometry? 81.3.1 Figure visualization and figure restoration 91.3.2 The geometrical use of tracing instruments, a first step to make sense to an axiomatic 101.3.3 Distinguishing between the hypothesis and the conclusion 121.3.4. Restoration, description, construction of figures and geometric language 141.4 Dialectics of action, formulation and validation with regards to the reproduction of figures with instruments 151.4.1 Formulation situations and possible variations 151.4.2 Validation situations 171.5. From tracing to the characterization of objects and geometric relationships 181.5.1 On the concepts of segments, lines and points 181.5.2 On the notion of perpendicular lines 211.6 Towards proof and validation situations in relation to figure restoration 271.6.1 Equivalence between two construction programs and the need for proof 271.6.2 Validation situations involving programs for the construction of a square and introducing a proof process 291.7 Conclusion 311.8 References 32Chapter 2 How to Operate the Didactic Variables of Figure Restoration Problems? 35Karine VIÈQUE2.1 Introduction 352.2 Theoretical framework 352.2.1 Studying a specific type of problem: figure restoration 352.2.2 Studying the concepts involved in figure restoration problems 372.3 Values of the didactic variables of the first problem family 392.3.1 Values of the didactic variables for the “figure” and the “beginning of the figure” 392.3.2 Value for the didactic variable “instruments made available” 402.3.3 Rules of action and theorems-in-action associated with development on the geometrical usage of the ruler 412.4 Conclusion 442.5 References 44Chapter 3 Early Geometric Learning in Kindergarten: Some Results from Collaborative Research 47Valentina CELI3.1 The emergence of the first questions 473.2 Theoretical insights 483.2.1 Global understanding and visual perception of geometric shapes 483.2.2 Operative understanding and visual perception of geometric shapes 493.2.3. Topological understanding and visual perception of geometric shapes 503.2.4 Haptic perception 513.2.5 Association of visual and haptic perceptions: towards a sequential understanding of geometric shapes 523.3 The role of language in early geometric learning 533.3.1 But which lexicon? 543.3.2 Verbal and gestural language 583.4 Assembling shapes 603.4.1 Free assembly of shapes 603.4.2 Assembling triangles 623.5 Gestures to learn 683.6 Conclusion 693.7 References 71Chapter 4 Using Coding to Introduce Geometric Properties in Primary School 73Sylvia COUTAT4.1 Coding in geometry 734.2 Two examples of communication activities requiring the use of coding 754.2.1 A co-constructed coding 754.2.2 Personal coding 774.3 Conclusion: perspectives on the introduction of coding in geometry 784.4 References 79Chapter 5 Freehand Drawing for Geometric Learning in Primary School 81Céline VENDEIRA-MARÉCHAL5.1 Introduction 815.2 Drawings in geometry and their functions 825.3 Freehand drawing in research 835.4 Exploring the milieu around a freehand reproduction task of the Mitsubishi symbol on a blank white page 845.4.1 Freehand drawing reveals a reasoning between spatial knowledge and geometric knowledge 875.4.2 Freehand drawing as a dynamic process to build and transform knowledge 885.5 Conclusion 895.6 References 90Part 2 Resources and Artifacts for Teaching 93Chapter 6 Use of a Dynamic Geometry Environment to Work on the Relationships Between Three Spaces (Tangible, Graphical and Geometrical) 95Teresa ASSUDE6.1 Added value with a dynamic geometry environment: the ecological and economical point of view 956.2 Tangible space, graphical space and geometric space 1006.3 Designing situations for first grade primary school 1036.3.1 Our choices for designing situations 1046.3.2 Presentation of situations 1046.4 Analysis of the situations for the first-grade class 1056.4.1 Instrumental dimension: perceptive–gestural level 1056.4.2 Instrumental dimension: spatial–geometric relationships 1066.4.3 Instrumental dimension: exploration and graphical space 1076.4.4 Instrumental dimension: tool-geometric space symbiosis 1086.4.5 Praxeological dimension 1096.4.6 Praxeological dimension: observe and describe 1116.5 Conclusion 1136.6 References 115Chapter 7 Robotics and Spatial Knowledge 119Emilie MARI7.1 Introduction 1197.2 Theoretical framework and development for a categorization of spatial tasks 1207.2.1 Spatial knowledge 1207.2.2 Types of spatial tasks 1217.2.3 Types of tasks and techniques 1217.3 Research methodology 1227.4 Analysis: reproducing an assembly 1237.4.1 Test item 1237.4.2 Test results 1247.4.3 Analysis of the results 1257.5 Conclusion 1267.6 References 127Chapter 8 Contribution of a Human Interaction Simulator to Teach Geometry to Dyspraxic Pupils 129Fabien EMPRIN and Edith PETITFOUR8.1 Introduction 1298.2 General research framework 1308.2.1 Teaching geometry 1308.2.2 Dyspraxia and consequences for geometry 1318.3 What alternatives are there for teaching geometry? 1328.3.1 Using tools in a digital environment 1328.3.2 Dyadic work arrangement 1358.4 Designing the human interaction simulator 1388.4.1 General considerations 1388.4.2 Choice of instrumented actions 1398.4.3 Interaction choices 1408.4.4 Ergonomic considerations 1428.5 Initial experimental results 1438.5.1 Data collected 1448.5.2 Jim’s diagnostic evaluation 1448.5.3 Analysis of the first experimentation 1468.5.4 Conclusion 1508.6 References 152Chapter 9 Research and Production of a Resource for Geometric Learning in First and Second Grade 155Jacques DOUAIRE, Fabien EMPRIN and Henri-Claude ARGAUD9.1 Presentation of the ERMEL team’s research on spatial and geometric learning from preschool to second grade 1559.1.1 Origins of the research 1569.1.2 Introduction to the chapter 1569.2 Learning to trace straight lines 1579.2.1 Significance of the straight line 1579.2.2 Initial hypotheses 1579.2.3 The RAYURE situation 1599.2.4 Using straight lines 1609.2.5 A few summary elements 1619.3 Plane and solid figures 1629.3.1 Findings and assumptions 1629.3.2 The SQUARE AND QUASI-SQUARE situation 1639.3.3 The emergence of criteria for comparing solids: the IDENTIFYING A SOLID situation 1659.3.4 Identification of cube properties: the CUBE AND QUASI-CUBE situation 1669.3.5 Progression on solids and plane figures 1679.4 The appropriation of research results by the resource 1689.5 Conclusion 1699.6 References 170Chapter 10 Tool for Analyzing the Teaching of Geometry in Textbooks 171Claire GUILLE-BIEL WINDER and Edith PETITFOUR10.1 General framework and theoretical tools 17210.1.1 Didactic co-determination scale, mathematical and didactic organizations 17210.1.2 Reference MO and theoretical tools for analysis 17410.2 Analysis criteria: definition and methodology 18110.2.1 Institutional conformity 18110.2.2 Educational adequacy 18210.2.3 Didactic quality 18210.3 Introducing the analysis grid 18310.3.1 Analysis of tasks and task types 18310.3.2 Analysis of techniques 18410.3.3 Analysis of knowledge 18510.3.4 Analysis of ostensives 18610.3.5 Analysis of organizational and planning elements 18910.3.6 Summary 19110.4 Conclusion 19110.5 References 192Part 3 Teaching Practices and Training Issues 197Chapter 11 Study on Teacher Appropriation of a Geometry Education Resource 199Christine MANGIANTE-ORSOLA11.1 Introduction 19911.2 Research background 20011.2.1 Study on dissemination possibilities in ordinary education 20011.2.2 Resource design approach 20111.2.3 A working methodology based on assumptions 20211.2.4 Designing a situation using the didactic engineering approach for development 20511.3 Focus on the adaptability of this situation to ordinary education 20611.3.1 Details about the theoretical framework and the research question 20611.3.2 Presentation on the follow-up of teachers, details of the research question and the methodology 20711.3.3 Presentation of the analysis methodology 20811.4 Elements of the analysis 20911.4.1 Analysis a priori of the situation and anticipatory analysis of the teacher’s activity 20911.4.2 Analysis of practices 21111.5 Conclusion 21711.6 References 219Chapter 12 Geometric Reasoning in Grades 4 to 6, the Teacher’s Role: Methodological Overview and Results 221Sylvie BLANQUART12.1 Introduction 22112.2 Theoretical choices and the problem statement 22112.2.1 Geometrical paradigms 22212.2.2 The different spaces 22312.2.3 Study on reasoning 22312.2.4 The role of the teacher 22512.2.5 Problem statement 22512.3 Methodology 22512.3.1 General principle 22512.3.2 The situations 22612.3.3 Analysis methodology 22612.4 Conclusion 22712.5 References 229Chapter 13 When the Teacher Uses Common Language Instead of Geometry Lexicon 231Karine MILLON-FAURÉ, Catherine MENDONÇA DIAS, Céline BEAUGRAND and Christophe HACHE13.1 Introduction 23113.2 An attempt to categorize the uses of common vernacular terms in place of geometry lexicon terms within teacher discourse 23213.2.1 The phenomenon of didactic reticence 23213.2.2 The phenomenon of semantic analogy: comparison with common concepts to construct meaning for mathematical knowledge 23313.2.3 The phenomenon of lexical competition: use of common vernacular terms to designate common concepts 23413.2.4 The phenomena of repeating pupil formulations 23513.2.5 The phenomenon of didactic repression 23613.3 Conclusion 23713.4 References 238Chapter 14 The Development of Spatial Knowledge at School and in Teacher Training: A Case Study on 1, 2, 3… imagine! 241Patricia MARCHAND and Caroline BISSON14.1 Introduction and research question 24114.2 Conceptual framework 24314.2.1 Components set to address SK in primary school 24414.2.2 Levels of abstraction that value SK 24514.2.3 Main variables in situations where SK is valued 24614.3 Presentation of the activity 1, 2, 3 … imagine! 24714.4 Experiments with this activity in primary school and in teacher training in Quebec 25114.4.1 Teaching sequence experimented in primary school 25114.4.2 Teaching sequence tested in teacher training 25414.5 Experiment results 25514.5.1 Experiment results of the teaching sequence in primary school 25514.5.2 Experiment results of this teaching sequence in teacher training 25714.6 Conclusion 25914.7 References 260Chapter 15 What Use of Analysis a priori by Pre-Service Teachers in Space Structuring Activities? 265Ismaïl MILI15.1 Introduction – an institutional challenge of transposing didactic knowledge 26515.1.1 Choice of external transposition: institutional constraints 26515.2 Theoretical framework 26715.2.1 Choice of internal transposition: the moments of the study of the analysis a priori 26815.3 Research questions 26915.4 Methodology 26915.4.1 Selection of activities and brief analysis 27015.5 Results 27215.6 Conclusion 27315.7 References 273Part 4 Conclusion and Implications 275Chapter 16 Questions about the Graphic Space: What Objects? Which Operations? 277Teresa ASSUDE16.1 Semiotic tools of geometric work and graphic space 27716.2 Graphic space: graphic expressions, denotation and meaning 28016.2.1 How can we define the graphic space? 28016.2.2 Which objects in the graphic space? 28016.2.3 Graphic expressions: which operations? 28216.3 References 285Chapter 17 Towards New Questions in Geometry Didactics 289Claire GUILLE-BIEL WINDER and Catherine HOUDEMENT17.1 Current questions in geometry didactics 28917.2 Continuities and breaks in the teaching of geometry 29117.2.1 Institutional continuity? 29117.2.2 Theoretical continuity from “geometry of tracing” to “abstract geometry”? 29117.2.3 Praxis continuity from the “geometry of tracing” to “abstract geometry” 29417.3 Articulation between resources, practices and teacher training 29717.4 References 299Appendices 303Appendix 1 305Appendix 2 309Appendix 3 311Appendix 4 313List of Authors 315Index 317