Mathematics and Its Connections to the Arts and Sciences (MACAS): 15 Years of Interdisciplinary Mathematics Education

Contents
Introduction: The History of MACAS
Introduction: 15 Years of Mathematics and Its Connections to the Arts and Sciences
	1 ICME Topic Study Group 21, Copenhagen 2004
	2 MACAS 2005, Schwäbisch Gmünd
	3 MACAS 2007, Odense
	4 MACAS 2009, Moncton
	5 MACAS 2015, Schwäbisch Gmünd
	6 MACAS 2017, Copenhagen
	7 MACAS 2019, Montréal
	8 15 Years of MACAS
	References
The Community of MACAS
MACAS as a Collaboration Hub
	1 International Projects
	2 Interdisciplinary Models and Concepts for Pre-Service and In-Service Teacher Training
		2.1 Models for Interdisciplinary Teaching
		2.2 The “MathBio in the Study Package” Program
		2.3 The ScienceMath Professional Development Concept
		2.4 “Modeling and Interdisciplinary Teaching”—A Course for Pre-service Teachers
	3 International Networks
	References
Mathematics in a Pedagogical Context and from an Educational and Historical Perspective
Mathematics Education in Different Contexts
	1 Introduction: Construction of a School Mathematics Curriculum
	2 Mathematics as a Human Activity
	3 Mathematics Education in Different Contexts
		3.1 Mathematics and Science
		3.2 Mathematics and Art
		3.3 Mathematics and Technology
		3.4 Mathematics and Literature
		3.5 Mathematics and Citizenship
	4 Outlook: Mathematics Education as a Design Science
	References
Selected Views on Mathematics Education
Developing Historical Awareness Through the Use of Primary Sources in the Teaching and Learning of Mathematics
	1 Introduction
	2 On the Notion of Historical Awareness
	3 Context and Setting of the First Case
		3.1 Primary Source Project: Rigorous Definition of Derivative
		3.2 Student Reflections: Challenging Previous Understanding
	4 Context and Setting of the Second Case
		4.1 HAPh-Module on Boolean Algebra and Shannon Circuits
		4.2 Persistent Historical Awareness in Students’ Reflections
	5 Potentials for Developing Historical Awareness
	6 Conclusion
	Appendix 1
	Appendix 2
	References
Rethinking the 21st-Century School: New Citizens’ Skills for the Digital Era and Their Interaction with Mathematics Teaching and Learning
	1 Introduction
	2 From the Big Data Phenomenon to the Birth of Artificial Intelligence, Machine Learning and Data Science
	3 A New World of Big Data: What Citizen’s Skills?
	4 New Approaches to Teaching and Learning Related to the Digital Transformation: Three Education Forces
	5 Discussion and Conclusion
	References
In Dialogue with Planet Earth: Nature, Mathematics, and Education
	1 Planet Earth
	2 Nature
	3 Mathematics
	4 Mathematics Education
	5 Concluding Remarks
	References
Mathematics and STEM
STEM Education and Its Connection to Mathematics
	1 Introduction
	2 The Origins of STEM
	3 STEM and Modelling
	4 STEM and STEAM
	5 STEM and Technology
	6 STEM and Mathematics
	7 Conclusions: Mathematics in STEM Education
	References
Problematizing STEM: What It Is, What It Is Not, and Why It Matters
	1 Introduction
	2 Origins and Motivation Behind the STEM Movement
		2.1 Origins of STEM Movement
		2.2 Brief Overview of STEM-Related Educational Reforms
	3 Epistemologies of STEM Disciplines: Commonalities and Differences
		3.1 How is New Knowledge Judged, Scrutinized, and Validated in STEM Disciplines?
		3.2 Summary of Key Epistemological Commonalities of STEM Disciplines
	4 Connections Between Epistemology and Pedagogy: Implications for Teacher Education
		4.1 Why STEM Epistemology Matters: Its Role in Content and Pedagogy
		4.2 Examples of Epistemologically Sound Models of STEM Teacher Education
		4.3 Challenges of STEM Integration in Teacher Education Programmes
	5 Conclusions and Implications for Theory and Practice
	References
Interdisciplinary Mathematical Modeling
	1 Introduction
	2 Modeling for the Elementary Grades
	3 Mathematical Modeling in a Social Science Context: Early Settlements
		3.1 Participants and Background
		3.2 Procedures
	4 The Early Settlement Problem
	5 Samples of Students’ Models and Modeling Processes
		5.1 Marking and Tallying
		5.2 Assigning Positive and Negative Scores
		5.3 Initial Ranking of Variables and Tallying
		5.4 Categorizing Variables and Tallying
		5.5 Applying a Rating System
	6 Discussion
	References
From STEm to STEM: Learning from Students Working in School Makerspaces
	1 Introduction
	2 Objectives
	3 Theoretical Perspectives
		3.1 Mathematics and STEM Education
		3.2 Learning Environments for STEM Education—Makerspaces
		3.3 Problem Solving
		3.4 K-12 New Brunswick Mathematics Curriculum
	4 Methods and Data Sources
	5 Results and Discussion
	6 Conclusions and Other Considerations
	References
Towards Proportionality within Developmental Instruction Approach: The First Steps
	1 Introduction
	2 Developmental Instruction Approach
	3 Proportionality: Assembling Sets as the First Step
	4 “Inks” Module
	5 The “Paint-Splitting” Task
	6 The “Paint-Comparison” Task
	7 Discussion
	8 Conclusion
	References
STEAM: Considering Possibilities and Barriers for STEM Education
	1 Introduction
	2 Possibilities for STEM to STEAM
	3 Linking STEM Processes to the Arts
		3.1 Creativity
		3.2 Visualization
	4 Barriers for STEM to STEAM
		4.1 The Lack of a Clear Vision
		4.2 Difficulties with Integrative Education
	5 Conclusion
	References
Techno-creative Problem-Solving (TCPS) Framework for Transversal Epistemological and Didactical Positions: The Case Studies of CreaCube and the Tower of Hanoi
	1 Introduction
	2 Three Transversal Competencies
		2.1 Problem-Solving
		2.2 Creativity
		2.3 Computational Thinking
	3 Intersections of Problem-Solving, Creativity, and Computational Thinking
		3.1 Intersections of Problem-Solving and Creativity
		3.2 Intersections of Computational Thinking and Problem-Solving
		3.3 Intersections of Computational Thinking and Creativity
	4 TCPS in Problem-Solving Tasks Using Tangible and Interactive Tools
		4.1 Analysis of the Tower of Hanoi (ToH) Task
		4.2 Analysis of the CreaCube Task
	5 Learning Tasks Within the Techno-creative Problem-Solving (TCPS)
	6 Discussion
	References
Navigating in a Complex World Using Mathematics: The Role Played by Financial Numeracy
	1 Background
	2 Numeracy as Social Practice
	3 Financial Numeracy
	4 Three Dimensions of Teaching Financial Numeracy
	5 Financial Numeracy in School Mathematics
	6 Financial Numeracy and Citizenship
	7 Financial Numeracy and STEM Education
	8 Concluding Remarks
	References
Mathematics and the Sciences
Mathematics and the Sciences—A Special Connection
	1 About Mathematics
	2 Obtaining Findings and Forming Concepts in Mathematics and the Sciences
	3 Phenomena
	4 Experimentation
	5 Modeling and Mathematization
	References
Structural Skills of Students in Solving Physical–Mathematical Tasks
	1 Introduction
	2 Technical and Structural Skills of Students
	3 Empirical Studies in Lower Secondary School
		3.1 Preparatory Interview Study
		3.2 Laboratory Study
	4 Discussion and Implications
	References
Developing Basic Principles of Calculus and Motion in Lower Secondary Education
	1 Introduction
	2 Example 1: Patterns in Change
	3 Example 2: The Slide
	4 Example 3: Interacting with Graphs of Motion
	5 Discussion
	References
Calculus Between Ancient Times and Covid Pandemic
	1 Archimedes Geometric Approach to Calculus
	2 Calculus and Real-Life Measurements
	3 From Motion to Calculus
	4 Calculus and Covid Pandemic
	References
Some Didactical Issues About the Teaching of Vectors and Translations in Mathematics and Physics Based on a Historical Approach
	1 Introduction
	2 History of Vectors: Some Comments
	3 Naïve Situations from Physics in Mathematics Teaching of Vectors
	4 A Non-conventional Example
	5 Movement of Translation and Translation: An Impossible Dialogue Between Mathematics and Physics?
	6 Conclusion
	References
Conceptual Understanding and Mathematical Literacy Through Interdisciplinary Activities Between Mathematics and the Sciences—Findings with Physics, Chemistry, and Biology
	1 Contemporary and Future-Oriented Mathematics Education
		1.1 Mathematical Literacy
		1.2 Conceptual Understanding
		1.3 Expanding the Domain
		1.4 Interdisciplinary Teaching
	2 Interdisciplinarity as a Promoter of Conceptual Understanding and Mathematical Literacy—Findings from Tested Teaching Examples with Physics, Biology, and Chemistry
		2.1 Interdisciplinarity for Contemporary and Future-Oriented Mathematical Education
		2.2 Modeling and Experiments for a Comprehensive Understanding of Mathematics Concepts
		2.3 Conceptual Understanding and Mathematical Literacy in the Digital Era
	3 Concluding Remarks
	References
Mathematics, Aesthetics and Arts
Mathematics, Aesthetics, and the Arts
	1 Introduction
	2 Mathematics and Aesthetics
	3 Mathematics and Music
	4 Mathematics and Art
	5 Summary
	References
Pluralising Mathematics Through Aesthetic Criticism
	1 Introduction
	2 On Making Distinctions: Take 1
	3 On Making Distinctions: Take 2
	4 Wrapping up and Making Space
	References
Aesthetic Mathematics Experiences that Travel
	1 Introduction
	2 Mathematics that Travels
	3 Ideas that Travel Through Curriculum Implementation
	4 Which Curriculum?
	5 How is Our Approach Different?
	6 The Case of Algebra and Coding in Grade 4
	7 Algebra as “Finding the Missing Number”
	8 Algebra as “Relationships Among Quantities that Vary”
	9 Using Computational Representations
	10 Discussion
	11 Revitalizing Aesthetic Aspects of Mathematics
	References
Wonders of Mathematics Through Technology and Music Creativity in a School Setting
	1 Context
		1.1 Research Project Context
	2 Conceptual Framework
	3 Methodology
		3.1 General Description
		3.2 Description of the Data Collection Environment for Each Case
	4 Results
		4.1 Teachers’ Perceptions of Students’ Learning: Data from Post-Project Interviews
	5 Discussion and Conclusion
	References
The Sound of Mathematics—Summary of International Research on Interdisciplinary Educational Work Between Mathematics and Music
	1 Why to connect Music and Mathematics?
	2 A Geometry of Music
	3 Mathematics and Arts
	4 The Well-Known Problem
	5 Firstly, the Interdisciplinary Basis: The Concept of “Pattern”
	6 Secondly, the Exploratory Study
	7 Thirdly, Theoretical and Research Work
	8 Conclusion
	References
Magic Symmetry
	1 Example
	2 A Puzzle
	3 Symmetry
	4 Construction of a Magic Square
	5 Generalization
	6 Integers Without a Minus Sign
	7 Composition of Two Magic Squares
	8 Composition of Two or Three Equal Squares
	References
Skirt Making as an Open School, Interdisciplinary Material Activity in Mathematics, Arts and Crafts
	1 Introduction
	2 ‘Open School’ as Field of Research
	3 Methodology and Empirical Data
	4 Case: Designing and Sewing a Skirt Using Mathematics and Arts and Craft
	5 Taking Measures
	6 Drawing the Arch
	7 Analysis: From a Mathematics Point of View
	8 From 2D to 3D and Back again—and Again, Again
	9 Analysis: From an Arts and Crafts Point of View
	10 Analysis: From an Open School Perspective
	11 Conclusion: An Interdisciplinary, Material Activity
	References
Mathematics and Language and Literature
Mathematics and Language and Literature
	1 Mathematics and Literature
	2 Mathematics and Language
	3 Mathematics and Literary Analysis
	4 The Articles in the Chapter
	References
Mathematics, Language, and Literature in Interdisciplinary Education—Theoretical Approach and Practical Examples
	1 Introduction
	2 Theoretical Approach
		2.1 Literature
		2.2 Interpretation, Esthetic Communication, and Emotions
		2.3 Writing and Language
		2.4 Argumentation
	3 Practical Implementation
		3.1 Fruitful Convergence Between the Subjects
		3.2 The Significance of Mathematics as a Stylistic Device
		3.3 Using the Work as Inspiration for Mathematical Activity and/or to Support the Acquisition of Mathematical Concepts
	4 Conclusion
	References
Let’s Put Mathematics into Comics! A Didactic Analysis of the Comics and Science Workshops
	1 Introduction
	2 Comics and Mathematics Education: A Literature Review
	3 Context: The Comics and Science Workshops
	4 Let’s Create a Math Comic Strip! Analysis of the Panels Created by Young, Non-Professional Authors
		4.1 The Comics and Cryptography Workshops: Conceptual Content and General Organization
		4.2 Analyses of the Math-Panels
		4.3 Analysis of the Math-Panels
	5 Results
	6 Discussion and Conclusion
	References
Ambiguities of Several Chinese Mathematical Vocabularies
	1 Common Questions
	2 Ambiguity
	3 Transformation
	References
Mr. Frog Challenged by Mathematical Conjectures: Fermat’s Last Theorem and Twin Primes
	1 Introduction
	2 A Challenge for Mr. Frog
	3 Yet Another Challenge for Mr. Frog
	4 Two Picture Books Positioned Between Fiction and Non-Fiction
	5 A Potential for Developing children’s Overview and Judgement
	6 Discussion of the Books’ Mathematics Educational Potential
	References