Learning Mathematics in the Context of 3D Printing: Proceedings of the International Symposium on 3D Printing in Mathematics Education

Contents
3D Printing in Mathematics Education—A Brief Introduction
	1	Motivation
	2	The Technical Basics of 3D Printing
		2.1	Computer-Aided Design—Building 3D Models
			2.1.1 Direct Modeling
			2.1.2 Parametric Modeling
		2.2	Slicing—Preparation for Printing
		2.3	3D Printer—Manufacturing Objects
	3	3D Printing in Mathematics Education
	4	This Book
	References
3D Transformations for Architectural Models as a Tool for Mathematical Learning
	1	Introduction
		1.1	Dynamic Lesson Plan
	2	Literature Review
	3	Methodology
	4	Theoretical Framework
		4.1	TDS
		4.2	TPACK
		4.3	AMOEBA
		4.4	Program Development
	5	Upper Austria Intervention
		5.1	Upper Austria Teacher Interviews
		5.2	Upper Austria Workshop
		5.3	Students’ Final Projects
			5.3.1 Students’ work in AR
			5.3.2 Students’ work in 3D Software
			5.3.3 Students’ work in 3D Print
			5.3.4 Reflections on the 3D Printing process
		5.4	Students’ 3D Printing Reflections
		5.5	Reflections on the Intervention’s Data
		5.6	Upper Austria Summary
	6	Discussion and Future Steps
	7	Conclusions
	References
Increasing the Skills on Occupationally Relevant Digital Technologies Among Students in Southern Denmark and Northern Germany: 3D Printing as a Learning Context in Regular Mathematics Classes
	1	Introduction
	2	Digitalization of the Workplace
		2.1	Additive Manufacturing as an Example of a Digital Technology
		2.2	The Mathematics Behind 3D Printing
			2.2.1 Modeling
			2.2.2 Tessellation or Triangulation
			2.2.3 Slicing
			2.2.4 Printing
	3	Digitalization of German Schools
		3.1	Digitalization in Mathematics Education
	4	3D Printing as a Learning Context in Regular Mathematics Teaching
		4.1	Conceptual Framework
		4.2	3D Printing as a Learning Context in Mathematics Education
			4.2.1 Modeling
			4.2.2 Triangulation
			4.2.3 Slicing
			4.2.4 Printing Process
			4.2.5 Digital Competences in Mathematics Education
		4.3	Example of a Teaching Idea
	5	Conclusion
	References
Vignettes of Research on the Promise of Mathematical Making in Teacher Preparation
	1	Introduction
		1.1	Chapter Structure
		1.2	Making in Mathematics Teacher Preparation
		1.3	Curriculum Context
		1.4	Practical and Theoretical Rationales
		1.5	The Pilot Study
			1.5.1 A Teacher Knowledge Analysis of Pedagogical Content Knowledge in Interaction with Design Activity
			1.5.2 Findings
			1.5.3 Implications
	2	Knowledge Interacts with Design
		2.1	The Interwoven Discourses Associated with Learning to Teach Mathematics in a Maker Context
			2.1.1 A Discourse Analysis of Identity in Interaction with Mathematical and Pedagogical Design Activity
			2.1.2 Findings
			2.1.3 Implications
		2.2	Making as a Window into the Process of Becoming a Teacher: The Case of Moira
			2.2.1 Framing Making as Mediated Learning
			2.2.2 Findings
				2.2.2.1 Moment 1: “I Should not Be Allowed in This Class. I’m Having Too Much Fun.”
				2.2.2.2 Moment 2: “Ugh, You’re Right. We can’t Flip the First One. I’m Gonna Work on This.”
				2.2.2.3 Moment 3: “Do I Know What a Torus is? No. Am I Using It? Yep!”
				2.2.2.4 Moment 4: “I just Figured Out so Much Math!”
			2.2.3 Implications
		2.3	The Nature of Prospective Mathematics Teachers’ Design Activity as They Make Original Manipulatives
			2.3.1 Theoretical Framing
			2.3.2 Findings
				2.3.2.1 Student-Centered Design
				2.3.2.2 The Nature of the Tools
				2.3.2.3 The Role of Aesthetics
			2.3.3 Implications
	3	Knowledge Interacts with Practice
		3.1	Prospective Mathematics Teachers’ Designed Manipulatives as Anchors for Their Pedagogical and Conceptual Knowledge
			3.1.1 Research Design
			3.1.2 Findings
				3.1.2.1 Reasoning About the Unit Whole
				3.1.2.2 Noticing in Action
			3.1.3 Implications
		3.2	Dare to Care: A Case Study of a Caring Pedagogy on Mathematical Making, Teaching, and Learning
			3.2.1 Theoretical Framing and Methods
			3.2.2 Findings
			3.2.3 Implications
		3.3	Harmony and Dissonance: An Enactivist Analysis of the Struggle for Sense Making in Problem Solving
			3.3.1 An Enactivist and Semiotic Analysis of Emergent Problem-Solving Activity Involving Multiple Representations of Fraction Division
			3.3.2 Findings
				3.3.2.1 Embarking on a Path of Problem Solving
				3.3.2.2 A Crowning Achievement
			3.3.3 Implications
	4	Conclusion
	References
Plane Tessellation
	1	Introduction
	2	Platonic tessellation
	3	Archimedian tessellation
	4	3D printing of tessellations
	5	Application of Archimedian tessellations at school
	6	Laves Lattices
	7	Application of Laves Lattices at school
	8	Quadrilateral (4gons)
	9	Art and Mathematics
		9.1	Vasarely
		9.2	Escher
		9.3	The Alhambra
	10	Conclusion
The Platonic Solids as Edge-Models
	1	Introduction
	2	The Platonic Solids
	3	The Vertex-connector
		3.1	Determination of α
		3.2	Construction of the Vertex-connector
	4	Application of the Vertex Connectors
		4.1	Derivation of Icosahedron Formulas
		4.2	The Kraul Model
		4.3	Other Models with Vertex-connectors
		4.4	Application for Geometry at School
		4.5	Conclusion
Doing Mathematics with 3D Pens: Five Years of Research on 3D Printing Integration in Mathematics Classrooms
	1	Motivation
	2	Affordances of Diagramming with a 3D Pen
		2.1	Support Visualization of 3D Geometrical Objects
		2.2	New Modes of Thinking
		2.3	The Dual Nature of Diagramming and Manipulating
	3	Theoretical Background
	4	Lesson Designs with 3D Pens
		4.1	Example 1: Primary School Geometry (Ng & Ferrara, 2020)
		4.2	Example 2: Functions and Calculus (Ng & Sinclair, 2018)
	5	Empirical Studies with 3D Pens
		5.1	Study 1: Ng et al. (2020)
		5.2	Study 2: Ng and Ye 2022
	6	Conclusion and Future Direction
	References
Possibilities for STEAM Teachers Using 3D Modelling and 3D Printing
	1	Introduction
		1.1	The Role of 3D Modelling and 3D Printing in STEAM Education
		1.2	Obstacles for Teachers to Use 3D Modelling and Printing
	2	Approach
	3	Reflections on the Workshops
	4	Possible Options and Opportunities
	5	A First Developed and Tested Course Concept
		5.1	Course Evaluated by Bloom’s Taxonomy
		5.2	Course Looked at from a TPACK View
	6	Discussion
	7	Outlook
	References
“I Cannot Simply Insert Any Number There. That Does not Work” — A Case Study on the Insertion Aspect of Variables
	1	Introduction and Motivation
	2	Theoretical Background
		2.1	Empirical-Oriented Mathematics Classes with Self-Made Empirical Objects
		2.2	The Concept of Empirical Theories
	3	Methodology
		3.1	Data Collection and the Learning Environment
		3.2	Context of the Case Study – Interviews with the Two Students Luke and Harry
		3.3	Case Study Research – Luke and Harry
	4	Luke and Harry
		4.1	Description
		4.2	Results and Discussion
			4.2.1 How Do Students Develop Mathematical Knowledge of the Concept of Variables in Geometrical Empirical Contexts with 3D Printing?
			4.2.2 How Can Students Be Supported in Their Mathematical Concept Formation Process with a Specially Designed Learning Environment Based on the Usage of 3D Printed Objects?
		4.3	Conclusion and Outlook
	References
Coding in the Context of 3D Printing
	1	Introduction
	2	Coding and Algorithms—A Learning Opportunity for Problem Solving Skills in Mathematics and Computer Science Education
		2.1	Problem Solving and Algorithms in Mathematics and Computer Science Education
		2.2	Problem Solving at the Intersection of Mathematics and Computer Science Education
	3	Coding in the Context of 3D Printing
	4	The Analysis of Three Example Tasks
		4.1	Example 1: The Cube Building
		4.2	Example 2: The building block generator
		4.3	Example 3: The Spiral Staircase
	5	Conclusion and Outlook
	References
Modelling and 3D-Printing Architectural Models—a Way to Develop STEAM Projects for Mathematics Classrooms
	1	Introduction
		1.1	The Uruguayan Context
		1.2	Opportunities for Better Utilisation of Technologies
	2	Theoretical Background
	3	The Proposed Work Process
		3.1	A Working Example
	4	Results and Reflexion about the Students’ Process
	5	Discussion
	References
Interfaces in Learning Mathematics—Challenging and Encouraging Visualizations Switching from 3D to 2D and 2D to 3D
	1	Introduction
	2	The Issue
	3	Theoretical Perspective
	4	Research Question
	5	Design
		5.1	Context and Situation
		5.2	Methodological Decisions and Analysis
	6	Analysis
		6.1	Description Level 1—Visualization and Inscription
		6.2	Description Level 2—Explicative Content Analysis
			6.2.1 Analysis of the Teaching Situation
			6.2.2 Determination of the Evaluation Unit
			6.2.3 Lexical-grammatical Definition of the Text Passage
			6.2.4 Determination of Acceptable Explicative Material and Collection of Material (Close Contextual Analysis)
			6.2.5 Formulation of the Explicative Paraphrase
			6.2.6 Evaluation of Explicative Paraphrase
	7	Conclusion & Discussion
	References
Mathematical Drawing Instruments and 3D Printing—(Re)designing and Using Pantographs and Integraphs in the Classroom
	1	Introduction
	2	Learning Mathematics with Drawing Instruments
		2.1	Mathematical Knowledge as Empirical Theories
		2.2	Mathematical Knowledge and Drawing Instruments
	3	The Pantograph—Drawing Scaled Up and Scaled Down Figures
		3.1	Functionality of a Pantograph
		3.2	3D Printing of a Pantograph
	4	The Integraph—Drawing Integral Functions
		4.1	Functionality of an Integraph
		4.2	3D Printing of an Integraph
	5	Conclusion and Outlook
	References
3D-Printing in Calculus Education—Concrete Ideas for the Hands-On Learning of Derivatives and Integrals
	1	Introduction
	2	Subject-Matter Didactics—Concepts in Calculus Education
	3	3D Printing Technology in Calculus
		3.1	Function Graphs
		3.2	Periodic Functions
		3.3	Upper and Lower Sums
	4	Integraph
	5	Solids of Revolution
	6	Mean Values
	7	Conclusion and Outlook
	References
Maistaeder—on the Evolution of a Versatile Polyhedron
	1	Material Mathematical Models
		1.1	The Collection of Mathematical Models at the Tu Dresden
		1.2	Digital Archive of Mathematical Models
		1.3	Integration of Models into Teaching
	2	Maistaeder
	3	Mathematics Behind Maistaeder
		3.1	Combinatorics
		3.2	Geometry
		3.3	Graph Theory
	4	Conclusion