Articulations Between Tangible Space, Graphical Space and Geometrical Space

Name: Articulations Between Tangible Space, Graphical Space and Geometrical Space | Resources, Practices and Training
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Resources, Practices and Training

Claire Guille-Biel Winder Teresa Assude(Herausgeber*in)

Wiley (Verlag)

1. Auflage

Erschienen am 17. Juli 2023

352 Seiten

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Inhalt

Cover
Title Page
Copyright Page
Contents
Preface
Part 1. Articulations between Tangible Space, Graphical Space and Geometric Space
Chapter 1. The Geometry of Tracing, a Possible Link Between Geometric Drawing and Euclid's Geometry?
1.1. Introduction
1.2. Geometry in middle school
1.2.1. What underlying axiomatics?
1.2.2. An example
1.2.3. The current lack of consistency
1.3. Geometry of tracing, a possible link between material geometry and Euclid's geometry?
1.3.1. Figure visualization and figure restoration
1.3.2. The geometrical use of tracing instruments, a first step to make sense to an axiomatic
1.3.3. Distinguishing between the hypothesis and the conclusion
1.3.4. Restoration, description, construction of figures and geometric language
1.4. Dialectics of action, formulation and validation with regards to the reproduction of figures with instruments
1.4.1. Formulation situations and possible variations
1.4.2. Validation situations
1.5. From tracing to the characterization of objects and geometric relationships
1.5.1. On the concepts of segments, lines and points
1.5.2. On the notion of perpendicular lines
1.6. Towards proof and validation situations in relation to figure restoration
1.6.1. Equivalence between two construction programs and the need for proof
1.6.2. Validation situations involving programs for the construction of a square and introducing a proof process
1.7. Conclusion
1.8. References
Chapter 2. How to Operate the Didactic Variables of Figure Restoration Problems?
2.1. Introduction
2.2. Theoretical framework
2.2.1. Studying a specific type of problem: figure restoration
2.2.2. Studying the concepts involved in figure restoration problems
2.3. Values of the didactic variables of the first problem family
2.3.1. Values of the didactic variables for the "figure" and the "beginning of the figure"
2.3.2. Value for the didactic variable "instruments made available"
2.3.3. Rules of action and theorems-in-action associated with development on the geometrical usage of the ruler
2.4. Conclusion
2.5. References
Chapter 3. Early Geometric Learning in Kindergarten: Some Results from Collaborative Research
3.1. The emergence of the first questions
3.2. Theoretical insights
3.2.1. Global understanding and visual perception of geometric shapes
3.2.2. Operative understanding and visual perception of geometric shapes
3.2.3. Topological understanding and visual perception of geometric shapes
3.2.4. Haptic perception
3.2.5. Association of visual and haptic perceptions: towards a sequential understanding of geometric shapes
3.3. The role of language in early geometric learning
3.3.1. But which lexicon?
3.3.2. Verbal and gestural language
3.4. Assembling shapes
3.4.1. Free assembly of shapes
3.4.2. Assembling triangles
3.5. Gestures to learn
3.6. Conclusion
3.7. References
Chapter 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
4.2.1. A co-constructed coding
4.2.2. Personal coding
4.3. Conclusion: perspectives on the introduction of coding in geometry
4.4. References
Chapter 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
5.4.1. Freehand drawing reveals a reasoning between spatial knowledge and geometric knowledge
5.4.2. Freehand drawing as a dynamic process to build and transform knowledge
5.5. Conclusion
5.6. References
Part 2. Resources and Artifacts for Teaching
Chapter 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
6.3.1. Our choices for designing situations
6.3.2. Presentation of situations
6.4. Analysis of the situations for the first-grade class
6.4.1. Instrumental dimension: perceptive-gestural level
6.4.2. Instrumental dimension: spatial-geometric relationships
6.4.3. Instrumental dimension: exploration and graphical space
6.4.4. Instrumental dimension: tool-geometric space symbiosis
6.4.5. Praxeological dimension
6.4.6. Praxeological dimension: observe and describe
6.5. Conclusion
6.6. References
Chapter 7. Robotics and Spatial Knowledge
7.1. Introduction
7.2. Theoretical framework and development for a categorization of spatial tasks
7.2.1. Spatial knowledge
7.2.2. Types of spatial tasks
7.2.3. Types of tasks and techniques
7.3. Research methodology
7.4. Analysis: reproducing an assembly
7.4.1. Test item
7.4.2. Test results
7.4.3. Analysis of the results
7.5. Conclusion
7.6. References
Chapter 8. Contribution of a Human Interaction Simulator to Teach Geometry to Dyspraxic Pupils
8.1. Introduction
8.2. General research framework
8.2.1. Teaching geometry
8.2.2. Dyspraxia and consequences for geometry
8.3. What alternatives are there for teaching geometry?
8.3.1. Using tools in a digital environment
8.3.2. Dyadic work arrangement
8.4. Designing the human interaction simulator
8.4.1. General considerations
8.4.2. Choice of instrumented actions
8.4.3. Interaction choices
8.4.4. Ergonomic considerations
8.5. Initial experimental results
8.5.1. Data collected
8.5.2. Jim's diagnostic evaluation
8.5.3. Analysis of the first experimentation
8.5.4. Conclusion
8.6. References
Chapter 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
9.1.1. Origins of the research
9.1.2. Introduction to the chapter
9.2. Learning to trace straight lines
9.2.1. Significance of the straight line
9.2.2. Initial hypotheses
9.2.3. The RAYURE situation
9.2.4. Using straight lines
9.2.5. A few summary elements
9.3. Plane and solid figures
9.3.1. Findings and assumptions
9.3.2. The SQUARE AND QUASI-SQUARE situation
9.3.3. The emergence of criteria for comparing solids: the IDENTIFYING A SOLID situation
9.3.4. Identification of cube properties: the CUBE AND QUASI-CUBE situation
9.3.5. Progression on solids and plane figures
9.4. The appropriation of research results by the resource
9.5. Conclusion
9.6. References
Chapter 10. Tool for Analyzing the Teaching of Geometry in Textbooks
10.1. General framework and theoretical tools
10.1.1. Didactic co-determination scale, mathematical and didactic organizations
10.1.2. Reference MO and theoretical tools for analysis
10.2. Analysis criteria: definition and methodology
10.2.1. Institutional conformity
10.2.2. Educational adequacy
10.2.3. Didactic quality
10.3. Introducing the analysis grid
10.3.1. Analysis of tasks and task types
10.3.2. Analysis of techniques
10.3.3. Analysis of knowledge
10.3.4. Analysis of ostensives
10.3.5. Analysis of organizational and planning elements
10.3.6. Summary
10.4. Conclusion
10.5. References
Part 3. Teaching Practices and Training Issues
Chapter 11. Study on Teacher Appropriation of a Geometry Education Resource
11.1. Introduction
11.2. Research background
11.2.1. Study on dissemination possibilities in ordinary education
11.2.2. Resource design approach
11.2.3. A working methodology based on assumptions
11.2.4. Designing a situation using the didactic engineering approach for development
11.3. Focus on the adaptability of this situation to ordinary education
11.3.1. Details about the theoretical framework and the research question
11.3.2. Presentation on the follow-up of teachers, details of the research question and the methodology
11.3.3. Presentation of the analysis methodology
11.4. Elements of the analysis
11.4.1. Analysis a priori of the situation and anticipatory analysis of the teacher's activity
11.4.2. Analysis of practices
11.5. Conclusion
11.6. References
Chapter 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
12.2.1. Geometrical paradigms
12.2.2. The different spaces
12.2.3. Study on reasoning
12.2.4. The role of the teacher
12.2.5. Problem statement
12.3. Methodology
12.3.1. General principle
12.3.2. The situations
12.3.3. Analysis methodology
12.4. Conclusion
12.5. References
Chapter 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
13.2.1. The phenomenon of didactic reticence
13.2.2. The phenomenon of semantic analogy: comparison with common concepts to construct meaning for mathematical knowledge
13.2.3. The phenomenon of lexical competition: use of common vernacular terms to designate common concepts
13.2.4. The phenomena of repeating pupil formulations
13.2.5. The phenomenon of didactic repression
13.3. Conclusion
13.4. References
Chapter 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
14.2.1. Components set to address SK in primary school
14.2.2. Levels of abstraction that value SK
14.2.3. Main variables in situations where SK is valued
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
14.4.1. Teaching sequence experimented in primary school
14.4.2. Teaching sequence tested in teacher training
14.5. Experiment results
14.5.1. Experiment results of the teaching sequence in primary school
14.5.2. Experiment results of this teaching sequence in teacher training
14.6. Conclusion
14.7. References
Chapter 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
15.1.1. Choice of external transposition: institutional constraints
15.2. Theoretical framework
15.2.1. Choice of internal transposition: the moments of the study of the analysis a priori
15.3. Research questions
15.4. Methodology
15.4.1. Selection of activities and brief analysis
15.5. Results
15.6. Conclusion
15.7. References
Part 4. Conclusion and Implications
Chapter 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
16.2.1. How can we define the graphic space?
16.2.2. Which objects in the graphic space?
16.2.3. Graphic expressions: which operations?
16.3. References
Chapter 17. Towards New Questions in Geometry Didactics
17.1. Current questions in geometry didactics
17.2. Continuities and breaks in the teaching of geometry
17.2.1. Institutional continuity?
17.2.2. Theoretical continuity from "geometry of tracing" to "abstract geometry"?
17.2.3. Praxis continuity from the "geometry of tracing" to "abstract geometry"
17.3. Articulation between resources, practices and teacher training
17.4. References
Appendices
Appendix 1
Appendix 2
Appendix 3
Appendix 4
List of Authors
Index
EULA

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