Articulations Between Tangible Space, Graphical Space and Geometrical Space E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Geisteswissenschaft
• Sprache: Englisch

Seitenzahl: 540

Veröffentlichungsjahr: 2023

Beschreibung

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).

Leseprobe

Table of Contents

Cover

Title Page

Copyright Page

Preface

References

Part 1: Articulations between Tangible Space, Graphical Space and Geometric Space

1 The Geometry of Tracing, a Possible Link Between Geometric Drawing and Euclid’s Geometry?

1.1. Introduction

1.2. Geometry in middle school

1.3. Geometry of tracing, a possible link between material geometry and Euclid’s geometry?

1.4. Dialectics of action, formulation and validation with regards to the reproduction of figures with instruments

1.5. From tracing to the characterization of objects and geometric relationships

1.6. Towards proof and validation situations in relation to figure restoration

1.7. Conclusion

1.8. References

2 How to Operate the Didactic Variables of Figure Restoration Problems?

2.1. Introduction

2.2. Theoretical framework

2.3. Values of the didactic variables of the first problem family

2.4. Conclusion

2.5. References

3 Early Geometric Learning in Kindergarten: Some Results from Collaborative Research

3.1. The emergence of the first questions

3.2. Theoretical insights

3.3. The role of language in early geometric learning

3.4. Assembling shapes

3.5. Gestures to learn

3.6. Conclusion

3.7. References

4 Using Coding to Introduce Geometric Properties in Primary School

4.1. Coding in geometry

4.2. Two examples of communication activities requiring the use of coding

4.3. Conclusion: perspectives on the introduction of coding in geometry

4.4. References

5 Freehand Drawing for Geometric Learning in Primary School

5.1. Introduction

5.2. Drawings in geometry and their functions

5.3. Freehand drawing in research

5.4. Exploring the milieu around a freehand reproduction task of the Mitsubishi symbol on a blank white page

5.5. Conclusion

5.6. References

Part 2: Resources and Artifacts for Teaching

6 Use of a Dynamic Geometry Environment to Work on the Relationships Between Three Spaces (Tangible, Graphical and Geometrical)

6.1. Added value with a dynamic geometry environment: the ecological and economical point of view

6.2. Tangible space, graphical space and geometric space

6.3. Designing situations for first grade primary school

6.4. Analysis of the situations for the first-grade class

6.5. Conclusion

6.6. References

7 Robotics and Spatial Knowledge

7.1. Introduction

7.2. Theoretical framework and development for a categorization of spatial tasks

7.3. Research methodology

7.4. Analysis: reproducing an assembly

7.5. Conclusion

7.6. References

8 Contribution of a Human Interaction Simulator to Teach Geometry to Dyspraxic Pupils

8.1. Introduction

8.2. General research framework

8.3. What alternatives are there for teaching geometry?

8.4. Designing the human interaction simulator

8.5. Initial experimental results

8.6. References

9 Research and Production of a Resource for Geometric Learning in First and Second Grade

9.1. Presentation of the ERMEL team’s research on spatial and geometric learning from preschool to second grade

9.2. Learning to trace straight lines

9.3. Plane and solid figures

9.4. The appropriation of research results by the resource

1

9.5. Conclusion

9.6. References

10 Tool for Analyzing the Teaching of Geometry in Textbooks

10.1. General framework and theoretical tools

10.2. Analysis criteria: definition and methodology

10.3. Introducing the analysis grid

10.4. Conclusion

10.5. References

Part 3: Teaching Practices and Training Issues

11 Study on Teacher Appropriation of a Geometry Education Resource

11.1. Introduction

11.2. Research background

11.3. Focus on the adaptability of this situation to ordinary education

11.4. Elements of the analysis

11.5. Conclusion

11.6. References

12 Geometric Reasoning in Grades 4 to 6, the Teacher’s Role: Methodological Overview and Results

12.1. Introduction

12.2. Theoretical choices and the problem statement

12.3. Methodology

12.4. Conclusion

12.5. References

13 When the Teacher Uses Common Language Instead of Geometry Lexicon

13.1. Introduction

13.2. An attempt to categorize the uses of common vernacular terms in place of geometry lexicon terms within teacher discourse

13.3. Conclusion

13.4. References

14 The Development of Spatial Knowledge at School and in Teacher Training: A Case Study on

1, 2, 3... imagine!

14.1. Introduction and research question

14.2. Conceptual framework

14.3. Presentation of the activity 1, 2, 3 ... imagine!

14.4. Experiments with this activity in primary school and in teacher training in Quebec

14.5. Experiment results

14.6. Conclusion

14.7. References

15 What Use of Analysis a priori by Pre-Service Teachers in Space Structuring Activities?

15.1. Introduction - an institutional challenge of transposing didactic knowledge

15.2. Theoretical framework

15.3. Research questions

15.4. Methodology

15.5. Results

15.6. Conclusion

15.7. References

Part 4: Conclusion and Implications

16 Questions about the Graphic Space: What Objects? Which Operations?

16.1. Semiotic tools of geometric work and graphic space

16.2. Graphic space: graphic expressions, denotation and meaning

16.3. References

17 Towards New Questions in Geometry Didactics

17.1. Current questions in geometry didactics

17.2. Continuities and breaks in the teaching of geometry

17.3. Articulation between resources, practices and teacher training

17.4. References

Appendices

Appendix 1: Appendix 1Four Situations

Situation 1: blue and red circles

Situation 2: little ants

Situation 3: doll

Situation 4: compositions

Appendix 2: Appendix 2FOLDING AND SYMMETRY Situation

Appendix 3: Appendix 3Support Systems

Appendix 4: Appendix 4Triangles within Quadrilateral

List of Authors

Index

Other titles frominInnovations in Learning Sciences

End User License Agreement

List of Tables

Chapter 6

Table 6.1.

Summary of situations

Table 6.2.

Types of tasks

Table 6.3.

Types of tasks by types of space

Chapter 7

Table 7.1.

Type of spatial tasks

Chapter 10

Table 10.1.

Types of elementary tasks involving the relations of perpendicu

...

Chapter 14

Table 14.1.

Possible techniques for the Yackel and Whealtey (1990) activity

Table 14.2.

Didactic variables that can be used for this activity

Table 14.3.

Pupil success rate a priori and a posteriori for 1, 2, 3 ... im

...

Table 14.4.

Sixth-grade average test pass rates for each category of the Ra

...

Table 14.5.

Number of observations needed for pupils to reproduce the Tangr

...

Table 14.6.

Success rates of teacher education students on the Ramful et al

...

Chapter 15

Table 15.1.

Evaluation criteria for the 3rd-semester internship

Chapter 16

Table 16.1.

Summary of the different transformations

Appendix 1

Table A1.1.

Situation 1

Table A1.2.

Situation 2

Table A1.3.

Situation 3

Table A1.4.

Composition of figures

List of Illustrations

Chapter 1

Figure 1.1.

A preliminary false figure

Figure 1.2.

A completed figure to solve with symmetry

Figure 1.3.

A figure (a) to be replicated and (b) its starting figure

Figure 1.4.

Adding traces on the model figure

Figure 1.5.

Tracing on the starting figure

Figure 1.6.

Variation of formulation situations

Figure 1.7.

Example of support for a situation where the formulation to othe

...

Figure 1.8.

Example of pupil productions

Figure 1.9.

Example of figure restoration involving perpendicularity

Figure 1.10.

Drawing a line or a segment perpendicular to a specified line

Figure 1.11.

A situation of verbal or written formulation communicated to ot

...

Figure 1.12.

Reproduction of the figure (model and starting figure completed

...

Figure 1.13.

An example of a pupil’s explanation

Figure 1.14.

An example of restoration

Figure 1.15.

A second example of restoration

Figure 1.16.

An invalid use of the set square

Figure 1.17.

A possible counterexample

Figure 1.18.

A freehand-drawn figure

Chapter 2

Figure 2.1.

Model figure and the beginning of the figure (Perrin-Glorian and

...

Figure 2.2.

a) See the juxtapositions and superimpositions of figurative uni

...

Figure 2.3.

A complex figure prepared for our experimentation

Figure 2.4.

Example of mobilizing surface vision: production of a point as t

...

Figure 2.5.

Example of mobilizing the line vision: production of a point as

...

Figure 2.6.

Example of mobilizing the point vision: production of a point as

...

Figure 2.7.

Rule of action no. 1

Figure 2.8.

Rule of action no. 2

Figure 2.9.

Rule of action no. 3

Figure 2.10.

Rules of action to be mobilized for the family of problems bein

...

Chapter 3

Figure 3.1.

A pupil tilts his head to recognize a square

Figure 3.2.

A shape classification problem

Figure 3.3.

With the help of the adult, the pupil’s finger runs along the co

...

Figure 3.4.

Assortment of interlocking shapes: the lighter shapes fit into t

...

Figure 3.5.

Describe a shape by relying only on haptic perception

Figure 3.6.

The hollow shape corresponding to the full shape described by Li

...

Figure 3.7.

Polygonal shapes, rounded edge shapes and mixed edge shapes

Figure 3.8.

An example of a free assembly made by a pupil

Figure 3.9.

Two ways of placing two triangular shapes on a table

Figure 3.10.

Materials made available to pupils.

Figure 3.11.

Difficulties encountered by pupils to obtain a given shape and

...

Figure 3.12.

Find in an assortment of shapes the one that is isometric to th

...

Chapter 4

Figure 4.1.

Example of a figure corresponding to a card association.

Figure 4.2.

Explanation of properties using shared coding.

Figure 4.3.

Cabri-drawing model

Figure 4.4.

Examples of pupil messages

Chapter 5

Figure 5.1.

Extract from a notebook on Swiss francophone teaching resources

...

Figure 5.2.

Symbol to be reproduced freehand on a blank sheet of paper

Figure 5.3.

Three ways of reproducing the Mitsubishi symbol: according to th

...

Figure 5.4.

Two ways to by which reproduce the Mitsubishi symbol from a tria

...

Figure 5.5.

Reproduction of the symbol by creating a triangular network insi

...

Figure 5.6.

Two methods of reproduction by creating a triangular network

Figure 5.7.

Freehand production of the Mitsubishi symbol by a pupil

Figure 5.8.

First freehand production by a pupil

Figure 5.9.

Second pupil production (A) freehand and then (B) corrected

Chapter 6

Figure 6.1.

Example of an analysis task

Figure 6.2.

Ostensives

Figure 6.3.

Relationships between the four dimensions

Chapter 7

Figure 7.1.

BeeBot® programmable floor robot

Figure 7.2.

Conduct of the experiment

Figure 7.3.

Reproduce an assembly: test item

Figure 7.4.

Example of pupils’ achievements

Figure 7.5.

Results for the item: reproduce an assembly

Chapter 8

Figure 8.1.

Compass reproduction by a 5th grade dyspraxic pupil

Figure 8.2.

Virtual instruments

Figure 8.3.

Virtual instruments

Figure 8.4.

Working in dyad to trace a right angle from a side

Figure 8.5.

Deictic gestures that can accompany technical language

Figure 8.6.

The “Least Likely” rule followed by the constructor

...

Figure 8.7.

The Embodied Conversational Agent (ECA) in the simulator

Figure 8.8.

Instrumented actions by the simulator.

Figure 8.9.

Technical terms

Figure 8.10.

Interactions with the human interaction simulator.

Figure 8.11.

Interactions with the human interaction simulator.

Figure 8.12.

Positioning of instruments by Jim

Figure 8.13.

Positioning the ruler in response to: “Place the ruler on point

...

Figure 8.14.

Statement 1 and feedback from the avatar

Figure 8.15.

Software result summary

Chapter 9

Figure 9.1.

The cones.

Figure 9.2.

The student worksheet

Figure 9.3.

(a) Freehand traces; (b) mixed traces; (c) straight line traces;

...

Figure 9.4.

(a) Stacked rectangles and (b) trace to be completed.

Figure 9.5.

Broken rectangle

Figure 9.6.

Square, rhombus and rectangles

Figure 9.7.

(a) Assemblies and teaching aids; (b) an assembly layout

Figure 9.8.

The assembly of squares and rectangles

Figure 9.9.

The assembly of squares and rectangles

Figure 9.10.

Grouping solid shapes

Figure 9.11.

Gestures associated with the formulations

Figure 9.12.

Introducing quasi-cubes

Figure 9.13.

Activity progression

Chapter 10

Figure 10.1.

Extract from the didactic co-determination scale

Figure 10.2.

Presentation of the different meanings intrinsic to perpendicul

...

Figure 10.3.

Presentation of the different meanings intrinsic to the paralle

...

Figure 10.4.

(a) Perpendicular lines and (b) non-parallel lines

Figure 10.5.

Four categories of objects through which to teach geometric rel

...

Figure 10.6.

Activities from two textbooks: (top) La Tribu des maths CM1 (Du

...

Figure 10.7.

Teacher’s guide of Nouveau Cap Maths Fourth-Grade (Charnay and

...

Figure 10.8.

The fourth-grade Singapore Method (Tek Hong 2009, p.83) (transl

...

Figure 10.9.

Synoptic tables of the year-long programming in the geometric d

...

Figure 10.10.

Summary of the different analysis points

Chapter 11

Figure 11.1.

Three different visions of the same figure

Figure 11.2.

Figure restoration situations

Figure 11.3.

Model figure

Figure 11.4.

Questioning and methodology implemented

Figure 11.5.

A sequence organized into four phases.

Figure 11.6.

A student destabilized by another student’s proposal

Figure 11.7.

Céline’s intervention

Figure 11.8.

A student’s procedure (phase 4)

Chapter 14

Figure 14.1.

Diagram of the activity generating structure (AGS)

Figure 14.2.

Examples of images that can be processed in this activity.

Figure 14.3.

Tangram from the 1st cycle teaching sequence (6–7 years).

Figure 14.4.

Tangram from the cycle 2 teaching sequence (ages 8–9).

Figure 14.5.

Compositions of objects from the 3rd cycle teaching sequence (1

...

Figure 14.6.

Tangram from the teacher training sequence

Chapter 15

Figure 15.1.

Excerpt from MER 8H (pupils aged 12)

Figure 15.2.

Training situation where the pupil playing the role of the teac

...

Appendix 1

Figure A1.1.

Figure given at the start of situation 1.

Figure A1.2.

Type of figure targeted in situation 2

Figure A1.3.

Figure given at the start of situation 3

Figure A1.4.

Figure to observe and reproduce in the first stage of situation

...

Guide

Cover Page

Title Page

Copyright Page

Preface

Table of Contents

Begin Reading

Appendix 1 Four Situations

Appendix 2 FOLDING AND SYMMETRY Situation

Appendix 3 Support Systems

Appendix 4 Triangles within Quadrilateral

List of Authors

Index

Other titles from in Innovations in Learning Sciences

WILEY END USER LICENSE AGREEMENT

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Education Setcoordinated byAngela Barthes and Anne-Laure Le Guern

Volume 14

Articulations between Tangible Space, Graphical Space and Geometrical Space

Resources, Practices and Training

Edited by

Claire Guille-Biel WinderTeresa Assude

First published 2023 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd27-37 St George's RoadLondon SW19 4EUUK

John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd 2023The rights of Claire Guille-Biel Winder and Teresa Assude to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.

Library of Congress Control Number: 2022950715

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-78630-840-5

Preface

Geometry is one of the oldest branches of mathematics. According to Brousseau, it “appears, through its aims, through its statements, through its methods, and through its multiple representations across the many branches of Mathematics and Science, sometimes in unexpected ways” (2000, p. 67, translated by author). Brousseau identifies in the teaching of geometry, on the one hand, a didactic means for “introducing mathematics”, in that it offers, contrary to other domains, the possibility for teachers “to elicit in their students an activity recognized as authentically mathematical by most mathematicians themselves” (ibid.), and on the other hand, as a means by which to represent space. Geometry thus appears to be an area within teaching that is still relevant today. The report of the commission on the teaching of mathematics, directed by Jean-Pierre Kahane, goes a little further and highlights four advantages to teaching geometry (Kahane 2002): to appropriate a vision of space and its representations; to learn geometrical reasoning; to be initiated into aesthetic and cultural aspects; and to have access to certain acumen that is useful across many trades. Numerous works realized over these last few years testify to the importance that the didacticians of mathematics accord with it in compulsory education.

The teaching of geometry in elementary school refers to two fields of knowledge that are intimately linked, but not to be confused (Berthelot and Salin 1993): spatial knowledge that allows us to control our relationship to the surrounding space; and geometric knowledge that allows us to solve problems involving objects in the physical, graphical or geometric space. Due to the importance of these two fields, one of the questions addressed in this book is: How is this knowledge taken into account and/or articulated in the teaching of space and geometry in compulsory education, in teaching resources (curricula, textbooks, etc.), or in current teacher training?

Moreover, the teaching of geometry in elementary school is often associated with the manipulation of instruments. However, teaching cannot be limited to a game with tangible objects, but must allow for mediation with the world of theoretical objects. This semiotic mediation is at the heart of mathematical activity, and geometry is a domain wherein the question of this mediation inevitably arises. Moreover, a geometric activity brings into play the language register and the graphical register (in particular that of figures), which must be articulated (Duval 2005). The language activity is also coordinated alongside a physical activity, for figures that must be traced or modified, with instruments or by freehand. Gestures also have a special place in the geometric activity itself. Therefore, the second line of questioning is as follows: How to take into account this semiotic dimension (language, gestures, signs, etc.) of geometric activity? What is the role of artifacts (digital or tangible) in the teaching and learning of geometry?

This book aims to present some of the latest research in the didactics of space and geometry, to deepen some theoretical questions and to open up new reflections for discourse, as much on the approach of geometry itself and its connection with the structuring of space, as on practices within the classroom, the dissemination of resources, the use of different artifacts or the training of teachers on this subject. It mobilizes about 15 contributions from French-speaking researchers based in different parts of the world (France, Switzerland, Quebec). It is organized into three main parts, which we present in the following.

Part 1 deals with the articulations between tangible space, graphic space and geometrical space. The contribution of Anne-Cécile Mathé and Marie-Jeanne Perrin-Glorian explores possible continuities between physical geometry and theoretical geometry through a figure reproduction task for pupils at the end of French elementary school (9–11-year-olds). Continuing on from this, Karine Viéque's text deals with the question of choosing the values for the didactic variables (geometrical and physical) in the elaboration of shape reproduction problems at the beginning of French elementary school (6–8-year-old students), in order to develop different understandings of plane shapes, in terms of visualization and deconstruction Valentina Celi focuses on early geometric learning in kindergarten (3–6-year-olds). She presents the first results of an ongoing research-action, in which problems concerning geometric shapes are tested, from manipulation to graphical tracing, from the global understanding of shapes to a more analytical understanding of them, by articulating visual and haptic modalities and by progressively introducing an appropriate lexicon. Sylvia Coutat's work deals with the representation of geometric properties in the graphical register by proposing avenues for reflecting through the introduction of coding geometric properties in elementary school. Finally, Céline Vendeira-Maréchal questions the impact and relevance of using freehand drawing in construction/reproduction tasks with 8- to 10-year-olds, in particular, to free them from the manipulative constraints implied by geometric instruments.

Part 2 is devoted to resources and artifacts for teaching. Three chapters deal with digital resources, two others with teaching proposals. Teresa Assude shows how a dynamic geometry software can be a tool through which to work on the relations between tangible space, graphical space and geometric space. Emilie Mari's contribution deals with the impact of using programmable floor robots on the development of spatial and geometrical knowledge in French elementary school, among 6–8-year-old students. Fabien Emprin and Edith Petitfour present, within the framework of instrument construction problems, a possible exploitation for dyspraxic pupils and the possibilities offered by a human interaction simulator for geometric learning. Concerning teaching resources, Jacques Douaire, Fabien Emprin and Henri-Claude Argaud propose a presentation on the evolution of the research questions by the ERMEL team, concerning the drawing of straight lines and discovering the characteristics of plane shapes and solids, by explaining an analysis of the knowledge of 6- to 8-year-old pupils as well as the proposed problems. Elements on the implementation of situations as well as on the progression and structuring of the resource are also discussed. Finally, Claire Guille-Biel Winder and Edith Petitfour try to determine what could enlighten, from a didactic point of view, the choice of textbooks in the framework of geometry teaching, by analyzing the proposals for teaching the notions of perpendicularity and parallelism in fourth grade (9–10-year-olds).

Part 3 of the book focuses on teaching practices and training issues. At the intersection between resource development, practice and training, Christine Mangiante-Orsola studies the process of appropriation, by three teachers, of a situation developed during a research project on the teaching of geometry from third grade to fifth grade (8–11-year-olds), and then questions the possibilities for enriching teaching practices. Next, two contributions focus on teaching practices. Sylvie Blanquart conducts a clinical analysis of the same sequence of shape reproduction in fifth grade (10–11-year-olds) and in sixth grade (11–12-year-olds), with the aim of identifying how, in the progress of their teaching project in plane geometry, teachers integrate (or not) the valid or erroneous reasoning implemented explicitly or implicitly by the students. Karine Millon-Fauré, Catherine Mendonca Dias, Céline Beaugrand and Christophe Hache are interested in the discourse of the mathematics teacher, when they employ a term used by vernacular language instead of the appropriate term from the geometry lexicon. In their contribution, Patricia Marchand and Caroline Bisson describe and analyze a teaching sequence aimed at developing spatial knowledge that is deployed over the first three cycles of schooling. They then address the issues of initial training, since this same sequence was adapted and experimented with students in training, with both mathematical and didactic objectives. Finally, Ismail Mili is interested in the professional knowledge mobilized by teachers in training, and provides an example concerning the mobilization of a priori analysis in the implementation of a space structuring activity.

January 2023

References

Berthelot, R. and Salin, M.-H. (1993). L'enseignement de la géométrie à l'école primaire.

Grand N,

53, 39–56.

Brousseau, G. (2000). Les propriétés didactiques de la géométrie élémentaire : l'étude de l'espace et de la géométrie.

Actes du séminaire de didactique des mathématiques,

Rethymnon, 67–83, hal-00515110.

Duval, R. (2005). Les conditions cognitives de l'apprentissage de la géométrie : développement de la visualisation, différenciation des raisonnements et coordination de leurs fonctionnements.

Annales de didactique et de sciences cognitives,

10, 5–53.

Kahane, J.-P. (ed.) (2002).

L'enseignement des sciences mathématiques : commission de réflexion sur l'enseignement des mathématiques.

CNDP, Odile Jacob, Paris.

Notes

Preface written by Claire Guille-Biel Winder and Teresa Assude.

Part 1Articulations between Tangible Space, Graphical Space and Geometric Space