Comparison of Mathematics and Physics Education I: Theoretical Foundations for Interdisciplinary Collaboration (MINTUS – Beiträge zur mathematisch-naturwissenschaftlichen Bildung)

Preface
	References
Contents
List of Figures
List of Tables
Authors of this Volume
Part I Interdisciplinarity in School
and Teacher Training
Programs
	1
Introduction
		The Inter TeTra project as a Contribution to Integrated Teacher Training
		Identification of Suitable Contents for a Scientific Exchange Between Mathematics and Physics Education
		Structure of this Volume
		References
	2
Interdisciplinarity in School and Teacher Training Programs
		1 Interdisciplinarity Teaching and Learning in STEM
		2 Advantages and Challenges to Teach Mathematics and Physics in an Integrated Way
			2.1 Exemplary Physical PCK for Teaching Math
			2.2 Exemplary Mathematical PCK for Teaching Physics
		3 Relevant Contents from the Perspective of Mathematics and Physics Didactics
		4 Conclusion
		References
Part II Physics and Mathematics as
Interwoven Disciplines
	Introduction to Part II
		3
Mathematics in History – From an Empirical to a Formalistic Conception
			1 Introduction
			2 Euclid
			3 Projective Geometry
			4 Non-euclidean Geometry
			5 Geometry á la Hilbert: the modern formalistic approach
			References
		4
Abstraction as an Essential Characteristic of Modern Mathematics in the Paradigmatic Example of Fermat’s Little Theorem
			1 Introduction
			2 Fermat’s Little Theorem – Historical Approach and Statement
			3 Fermat‘s Little Theorem as a Statement in Group Theory
			4 Homomorphisms
			5 Generator and Order
			6 Coset
			References
		5
Mathematics From the Pupils’ Point of View
			1 Introduction
			2 An Empirical Investigation of Alan H. Schoenfeld
			3 High-School-Students’ Mathematical World-View
			4 A Look at School-Lessons
			References
		6
The Mathematization of Physics Throughout History
			1 Introduction
			2 About the Role of Mathematics in Ancient Natural Theory
			3 The Role of Deduction in the Experimental Method of Galilei
				3.1 Galilei’s Derivation of the Laws of Uniformly Accelerated Movement
				3.2 Galilei’s examination of the theoretically derived laws
				3.3 Discussion of Galilei’s approach and resulting didactic conclusions
			4 Classical Mechanics – A Theory “More Geometrico”
			5 About the Implementation of Calculus in Physics by Analytical Mechanics
			6 The Role of Mathematics in Modern Physics: Non-Euclidean Geometries and the General Theory of Relativity
			7 Conclusion
			References
		7
The Nature of Science
			1 Introduction
			2 What is Nature of Science?
			3 Why Should We Consider Nature of Science in Class?
			4 Misconceptions on Nature of Science
			5 Aspects of Relevance for the Comparison of Mathematics and Physics
				5.1 Hypotheses, Theories and Laws
				5.2 Models and their Relation to Reality
				5.3 Models of Knowledge Acquisition
			6 Conclusion
			References
		8
On the Relationship between Mathematics and Physics according to Günther Ludwig
			1 Introduction
			2 The Three Main Parts of a Physical Theory
			3 Introductory Example: Physical Pre-Theories for Distance and Time Measurement
			4 The Fundamental Domain of the Real Conditions
			5 The Structure of a Mathematical Theory
			6 The Recording Process
			7 Inaccuracies in the Recording Process
			8 Conclusion
			References
		9
Example 1 – The Cardanos Formula and the Van der Waals Gas
			1 Cardano’s method for the solution of a cubic equation
			2 The Cardanic solution formula for complex roots
			3 Van der Waals equation
		10
Example 2 – Linear Differential Operators, Fourier-Series and the RCL-Circuit
			1 Linear differential operators
				1.1 Linear differential equations
				1.2 Linear differential operators with constant coefficients
				1.3 Operator rules
				1.4 Finding particular solutions to inhomogeneous equations
				1.5 Higher order homogeneous linear ODE’s with constant coefficents
			2 Fourier-Series
			3 Impulse response of a system being modeled by a linear second order ODE
			4 Modelling a series RCL circuit
Part III Comparison of Educational
Theories
	Introduction to Part III
		11
Individual Concepts in Physics and Mathematics Education
			1 Introduction
				1.1 Definition of terms
				1.2 Reasons for the development of individual concepts – Historical overview
				1.3 Why do we have to take individual concepts into account?
			2 Individual Concepts in Physics Education
				2.1 Examples
				2.2 Explore students’ ideas
			3 Individual Concepts in Mathematics Education
				3.1 Misconceptions in mathematics education
				3.2 “Subjective Domains of Experiences” as description of the learning process
				3.3 Basic Mental Models (“Grundvorstellungen”) as descriptive, normative and constructive dimensions of specification for the learning process
			4 Synthesis
			References
		12
Models and Modeling
			1 Models and Modeling in Science
				1.1 Models in Science
			2 Models and Modeling in Physics Education
				2.1 Models in Physics Education
				2.2 Functions of Models in Physics Education
				2.3 Modeling in Physics Education
			3 Models and Modeling in Mathematics Education
				3.1 Modeling at school
				3.2 Modeling Cycles in Mathematics Education
				3.3 Result of a Modeling Process
				3.4 Models in Mathematics Education
			4 Synthesis
			References
		13
Development of Knowledge in Mathematics and Physics Education
			1 Introduction
			2 Arguing, Proving and Concept Building in Mathematics Class
				2.1 The Terms Arguing, Proving and Concept Building
				2.2 The Toulmin Method of Reasoning
				2.3 Levels of Argumentation
				2.4 Methods of Proving
				2.5 Functions of Proving
				2.6 Concept Building in Mathematics Class
			3 Experimenting and Arguing in Physics Class
				3.1 Gathering Insights in Physics and the Didactical Consequences for Physics Education
				3.2 Knowledge Acquisition in Physics Education
				3.3 The Function of Experiments in Physics Education
				3.4 The Relationship between Experimentation and Arguing in Physics Education
			4 Comparison
			References
		14
Problem Solving
			1 What is a Problem?
			2 Problem Solving in Mathematics Education
				2.1 Problem solving in the curricular requirements for teaching mathematics
				2.2 Pólyas circle of problem solving
				2.3 Heurisms concerning Bruder and Collet
				2.4 Schoenfelds Categories of Problem Solving
			3 Problem Solving in Physics Education
				3.1 Problem solving in the curricular requirements for teaching physics
				3.2 Knowledge-centered problem solving according Friege
				3.3 Basic Ideas as a Problem-Solving Tool in Physics
				3.4 Problem solving in learning new concepts
			4 Comparison of Theories About Problem Solving in Mathematics and Physics Education
			References
		15
Conclusive Remarks
Index