Mathematics Education in the Age of Artificial Intelligence: How Artificial Intelligence can Serve Mathematical Human Learning

Foreword
General Introduction
Contents
Contributors
Creation of AI Milieus to Work on Mathematics
Evolution of Automated Deduction and Dynamic Constructions in Geometry
	1 Introduction
	2 Automated Deduction in Geometry
		2.1 Synthetic Methods
		2.2 Algebraic Methods
		2.3 Semi-Synthetic Methods
		2.4 Generic First-Order Provers
		2.5 Other Approaches
	3 Dynamic Geometry
	4 Other Lines of Research
	5 Conclusions
	References
Automated Reasoning Tools with GeoGebra: What Are They? What Are They Good For?
	1 Introduction
	2 GeoGebra Automated Reasoning Tools
		2.1 The Relation Tool and Command
		2.2 The Prove and ProveDetails Commands
		2.3 The LocusEquation Command
		2.4 The Discover Tool and Command
	3 Toward an Automated Geometer
	4 Discussion and Conclusions
	References
Intelligence in QED-Tutrix: Balancing the Interactions Between the Natural Intelligence of the User and the Artificial Intelligence of the Tutor Software
	1 Context
		1.1 Symbiosis Between the Mathematical Work in Schools and Computer Science
		1.2 Existing Tutor Softwares
	2 Genesis of the QED-Tutrix Project
		2.1 Task Description
		2.2 Software Overview
		2.3 The Core Layers of QED-Tutrix
	3 The Need for Automated Proof Generation
		3.1 Existing Theorem Provers
		3.2 The Choice of Logic Programming to Generate Inferences
		3.3 Available Data
		3.4 Encoding of a Problem
		3.5 Generation of the Complete Set of Proofs
		3.6 Validation
		3.7 Limitations
	4 Conclusion
	References
A Decision Making Tool for Mathematics Curricula Formal Verification
	1 Introduction
	2 A Theoretical Proposal
	3 Design and Implementation of the Theoretical Proposal
		3.1 First Approach (Rule Based Expert System)
		3.2 Second Approach (Graph Theory)
		3.3 Case Study
	4 Conclusions
	References
A Classification of Artificial Intelligence Systems for Mathematics Education
	1 Introduction
		1.1 Artificial Intelligence and Machine Learning
	2 A Glimpse of the Present
		2.1 A New Breed of Calculators
		2.2 Blueprint of a Data-Driven Intelligent Tutoring System
	3 A Taxonomy of AI Techniques for Mathematics Education
		3.1 Information Extractors
		3.2 Reasoning Engines
		3.3 Explainers
		3.4 Data-Driven Modeling
	4 The Present, Revisited
		4.1 AI-Based Calculators
		4.2 Intelligent Tutoring Systems
	5 Modeling the Mathematical Learner: a Most Ambitious Goal
	6 Conclusions and Discussion
	References
AI and Mathematics Interaction for a New Learning Paradigm on Monumental Heritage
	1 Introduction
		1.1 Mathematics Education: Reality, Models, Computation and Solutions
		1.2 Computational Support in Mathematics Education
		1.3 The Mathematics in Monuments: A Unique Form of Mathematics Education
	2 Forms of Mathematical Reasoning: Deduction, Induction, and Abduction
		2.1 Classical Forms of Reasoning
		2.2 AI and Its Forms of Reasoning
	3 MonuMAI: An AI-Driven Environment for Monumental Analysis
		3.1 MonuMAI Dissection: How to Classify Monuments Using Deep Learning
		3.2 The Citizen Science Methodology on MonuMAI
		3.3 Learning and Mistakes
		3.4 A Critical Look at MonuMAI: How to Troll the System
	4 Toward the Construction of an Automatic Geometric Model (AGM) of Architectural Façades
		4.1 The Problem of Perspective in Photographies
		4.2 Correcting Perspective Issues in Façades
		4.3 The Hough Transform to Identify Basic Geometries in an Image
		4.4 Straight Line Classification and Optimization of the Camera Angles
		4.5 Constructing an Automatic Geometrical Model for the Façade
	5 Experiences in Education
	6 What\'s Next for the Future?
	References
AI-Supported Learning of Mathematics
Using Didactic Models to Design Adaptive Pathways to Meet Students’ Learning Needs in an Online Learning Environment
	1 Introduction
	2 Theoretical Elements, Foundations of Didactic Modelling
		2.1 An Example to Address the Roles of the Knowledge Model and of the Learner Model
		2.2 Praxeological Model of Knowledge
		2.3 Praxeological Model of Algebraic Knowledge
		2.4 Didactic Modelling of Tasks
		2.5 Didactic Modelling of the Learner
		2.6 In Conclusion: An Ontology to Establish the Links Between the Knowledge Model and the Learner Model
	3 Pépite Online Learning Environment Around Assessment and Regulation of Algebra in the Classroom
		3.1 Knowledge Model and Diagnostic Tasks
		3.2 Analysis of Test Responses
		3.3 Student’s Profile
		3.4 Groups and Exercises Path
		3.5 Computer Model and Partial Ontology
		3.6 Limits
	4 The MindMath Online Learning Environment Around Adaptive Paths in Algebra and Geometry
		4.1 Relations Between Praxeologies in the Didactic Model of Knowledge
		4.2 Didactic Model of Task Families
		4.3 Learner Model and Student’s Profile
		4.4 Didactic Path Model
		4.5 Computer Representation of Mathematical Knowledge and Activity Through an Ontology
	5 Conclusion and Perspectives
	Appendix: Algorithm for Calculating Justification Modes for UA
	References
Combining Pencil/Paper Proofs and Formal Proofs, A Challenge for Artificial Intelligence and Mathematics Education
	1 Introduction
	2 The Sum of Angles of a Triangle is Two Right
		2.1 Some Questions and Issues Raised by the Proof by Pythagoras
		2.2 The Formal Proof That the Sum of Angles of a Triangle is Two Right Angles
		2.3 Some Questions and Issues Raised by the Formalization of the Proof in Coq
	3 Varignon\'s Theorem
		3.1 Logical Analysis of a Classical Proof of Varignon\'s Theorem
		3.2 Issues and Challenges Raised by the Formalization of the Classical Proof of Varignon\'s Theorem
		3.3 Logical Analysis of Alternative Proofs of Varignon\'s Theorem
		3.4 Didactical Implication
	4 A Teacher Training Session on Varignon\'s Theorem
		4.1 Context, Motivation, Description and a Priori Analysis of the Session
		4.2 Account of Naturalist Observation Along Several Years
		4.3 Evolution of the Teacher Training Session by Introducing a Proof Assistant
	5 Conclusions
	References
Interaction Between Subject and DGE by Solving Geometric Problems
	1 Introduction
	2 Theoretical Background
		2.1 The Toulmin Model
	3 The Model: Classification of Experimental Facts Attained with the Help of DGE
	4 Illustrations of the Model
		4.1 The First Example
		4.2 The Second Example
	5 Conclusions
	References
Creative Use of Dynamic Mathematical Environment in Mathematics Teacher Training
	1 Introduction
	2 Historical Context
	3 Problem 36
	4 Problem 36 from the Contemporary Perspective
		4.1 GeoGebra
		4.2 OK Geometry
	5 Vaňaus\' Trisectrix
	6 Conclusion
	References
Experimental Study of Isoptics of a Plane Curve Using Dynamical Coloring
	1 Tradition Versus Experimental Mathematics
	2 Isoptic Curves of Plane Curves—A Short Historical Survey
	3 Experiments with Dynamical Coloring
		3.1 First Experiments
		3.2 Automated Way of Obtaining Dynamical Colored Plots
	4 Final Remarks and Directions for Future Work
		4.1 What Has Been Earned with the New Approach
		4.2 What We Are Still Missing
		4.3 Conclusions
	References
Teaching Programming for Mathematical Scientists
	1 Background
	2 Introduction to Numerical Analysis
		2.1 Choice of Programming Language
		2.2 Pedagogical Methods
		2.3 Assessment
	3 Computational Discovery/Experimental Mathematics
		3.1 Choice of Programming Language
		3.2 Pedagogical Methods
		3.3 Assessment
	4 Programming and Discrete Mathematics
		4.1 Choice of Programming Language
		4.2 Pedagogical Methods
		4.3 Assessment
	5 Artificial Intelligence and Programming
	6 Outcomes
	7 Ethics, Teaching, and Eudaemonia
	8 Concluding Remarks
	References
The present and future of AI in ME: Insight from empirical research
CAS Use in University Mathematics Teaching and Assessment: Applying Oates’ Taxonomy for Integrated Technology
	1 Introduction
	2 Literature Review
	3 A Model for Quantifying Technology Integration in University Mathematics Courses
	4 Methods
	5 Research Findings
		5.1 Initial Findings Elaborating on the Context of Our Case Study
		5.2 Data Analysis Using the Oates Taxonomy Model
	6 Discussion
		6.1 Technology in Mathematics Teaching, Learning, and Assessment
		6.2 Observations from the Oates Model Analysis
		6.3 Future Recommendations
	Appendix 1: MATH 373 Assignment 2 (Spring 2015)
	Appendix 2: MATH 373 Assignment 3 (Spring 2015)
	Appendix 3: MATH 373 Mid-Term Examination (Spring 2015)
	References
Modeling Practices to Design Computer Simulators for Trainees’ and Mentors’ Education
	1 Introduction
	2 Theoretical Framework
		2.1 Choices About Student Learning
		2.2 Twofold Approach to Define and Model Teaching Practices
		2.3 Instrumental Genesis
	3 How to Create a Model of Mentor–Teacher Interactions?
		3.1 From the Design of a Mentoring Dialogue Simulator (MDS) to a More General Model
	4 Collecting Data in a Classroom Simulator
		4.1 Highlights in the Implementation Analysis
		4.2 Highlights in the Design of the CSS Simulator
		4.3 Main Concepts from the Didactics at Stake
		4.4 Collecting Data on Teachers’ Practices
	5 Discussion
	References
Exploring Dynamic Geometry Through Immersive Virtual Reality and Distance Teaching
	1 Introduction
	2 Levels of Object Manipulation
	3 Neotrie VR
		3.1 How to Implement Neotrie in Face-to-Face and Online Teaching
		3.2 A Short Start in Neotrie VR and Their Tools
	4 A Real Case: Parametric Equations of Surfaces
		4.1 Contents Contextualization and Pedagogical Methodology
		4.2 Justification of the Used Technologies
		4.3 Observing Learning Session with Neotrie VR
		4.4 Manipulating Learning Task with GeoGebra
	5 Opinions on the Use of Various Supports
	6 Stereoscopic Videos and Extra VR Session
		6.1 Stereoscopic Videos
		6.2 Extra Session in the VR Room
	7 Future Development Plans
	8 Conclusions
	References
Historical and Didactical Roots of Visual and Dynamic Mathematical Models: The Case of “Rearrangement Method” for Calculation of the Area of a Circle
	1 Introduction
	2 Historical Roots of the Rearrangement Method
	3 The Rearrangement Method in Nineteenth–Early Twentieth-Century Western Textbooks
	4 Looking into Modern Mathematical Working Spaces Through the Lens of Historical, Didactical, and Instrumental Genesis
		4.1 Rearrangement of the Circle in Modern Middle School Geometry Lessons: Possible Teaching Scenarios and Didactical Challenges
		4.2 Novel Approaches to Area Investigation Using Dynamic Digital Tools
		4.3 Discussion and Conclusions
	References
Implementing STEM Projects Through the EDP to Learn Mathematics: The Importance of Teachers’ Specialization
	1 Introduction
	2 Teaching Mathematics Through Technology
	3 STEM Projects and the Engineering Design Process for Learning Mathematics
	4 Design and Implementation of STEM Activities in Secondary Education
	5 The Study
		5.1 Sample
		5.2 Guidelines for Project Development
		5.3 Data Analysis
	6 Results
		6.1 Identification: Star Wars Robot
		6.2 Reasoning: Rubik’s Cube
		6.3 Modelling: Astrolabe
	7 Discussion
	8 Conclusions and Implications for Further Research
	References
Digital Technology and Its Various Uses from the Instrumental Perspective: The Case of Dynamic Geometry
	1 Introduction: Role of Digital Technology in Education
	2 Theoretical Framework
		2.1 Instrumental Approach
		2.2 SAMR Model
	3 Various Uses of Dynamic Geometry
		3.1 Substitution Level Tasks
		3.2 Augmentation Level Tasks
		3.3 Modification Level Tasks
		3.4 Redefinition Level Tasks
	4 Concluding Remarks
	References
Conclusions
Appendix  Epilogue
Appendix  Appendix: Photographs of the Book Project and Some of the Authors
Index