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Table of Contents
Intro
Preface
The Book Structure
Contents
Part I: Conceptual Background
Chapter 1: Physics Constructs Viewed Through the Prism of Mathematics
1.1 Mathematics as an Indispensable Part of Physics Inquiry
1.2 Laws of Physics and Their Mathematical Embodiments
1.3 Principles and Their Relations to Laws
1.4 Theories and Laws
1.5 Theories and Theorems
References
Chapter 2: The Interface Between the Contents of Physics and Mathematics
2.1 Mathematics as a Language in Physics Classroom
2.2 Philosophy and the Substance of the Knowledge of Mathematics
2.3 Procedural and Conceptual Mathematical Knowledge
2.4 Unifying Classification of Math Knowledge Used in Physics Education
2.5 Arrays of Applying Mathematics in Physics
2.6 Search for Tools and Methods
2.7 Mathematical and Scientific Reasoning
Are These Mental Actions Equivalent?
2.8 Synthesis of Students' Challenges with Math Knowledge Transfer
References
Part II: Designing Learning Environments to Promote Math Reasoning in Physics
Chapter 3: Modeling as an Environment Nurturing Knowledge Transfer
3.1 Scientific Modeling and Models
3.2 Modeling Cycles in Physics Education
3.3 Merging Mathematics and Physics Representations
References
Chapter 4: Proposed Empirical-Mathematical Learning Model
4.1 Didactical Underpinnings of the Design
4.2 Description of the Learning Phases
4.3 Hypotheses as Learners' Proposed Theories
4.4 Mainstream of the Inquiry and Its Confirmation
4.5 Methods of Enacting Mathematical Structures
4.6 Concluding Phases of the Learning Process
References
Chapter 5: Covariational Reasoning
Theoretical Background
5.1 Quantities, Parameters, and Variables
5.2 Formulas in Science and Mathematics
5.3 Covariational Reasoning in Mathematics Education
5.4 Covariational Reasoning in Physics Education
5.4.1 Viewing Phenomena as Covariations of Their Parameters
5.4.2 Proposed Categories of Covariations Embedded in Physics Formulas
5.4.3 Discussing Covariations of Parameters in Experiments
5.5 Limiting Case Analysis
5.5.1 Evaluating Limits when the Variable Parameter Is Getting Very Large
x→∞
5.5.2 Evaluating Limits when the Variable Parameter Is Close to a Specific Value
x→a
5.5.3 Is Limiting Case Analysis Really "Limiting"?
References
Part III: From Research to Practice
Chapter 6: Extending the Inquiry of Newton's Second Law by Using Limiting Case Analysis
6.1 Limits
Tools for Extending Scientific Inquiry
6.2 Research Methods
6.2.1 Research Questions, Logistics, and Participants
6.2.2 Criteria for the Study Content Selection
6.2.3 Discussion of the Applied Algebraic Tools
6.3 Description of the Instructional Unit
6.3.1 Analyzing Acceleration of the System in the Function of Mass m2
Preface
The Book Structure
Contents
Part I: Conceptual Background
Chapter 1: Physics Constructs Viewed Through the Prism of Mathematics
1.1 Mathematics as an Indispensable Part of Physics Inquiry
1.2 Laws of Physics and Their Mathematical Embodiments
1.3 Principles and Their Relations to Laws
1.4 Theories and Laws
1.5 Theories and Theorems
References
Chapter 2: The Interface Between the Contents of Physics and Mathematics
2.1 Mathematics as a Language in Physics Classroom
2.2 Philosophy and the Substance of the Knowledge of Mathematics
2.3 Procedural and Conceptual Mathematical Knowledge
2.4 Unifying Classification of Math Knowledge Used in Physics Education
2.5 Arrays of Applying Mathematics in Physics
2.6 Search for Tools and Methods
2.7 Mathematical and Scientific Reasoning
Are These Mental Actions Equivalent?
2.8 Synthesis of Students' Challenges with Math Knowledge Transfer
References
Part II: Designing Learning Environments to Promote Math Reasoning in Physics
Chapter 3: Modeling as an Environment Nurturing Knowledge Transfer
3.1 Scientific Modeling and Models
3.2 Modeling Cycles in Physics Education
3.3 Merging Mathematics and Physics Representations
References
Chapter 4: Proposed Empirical-Mathematical Learning Model
4.1 Didactical Underpinnings of the Design
4.2 Description of the Learning Phases
4.3 Hypotheses as Learners' Proposed Theories
4.4 Mainstream of the Inquiry and Its Confirmation
4.5 Methods of Enacting Mathematical Structures
4.6 Concluding Phases of the Learning Process
References
Chapter 5: Covariational Reasoning
Theoretical Background
5.1 Quantities, Parameters, and Variables
5.2 Formulas in Science and Mathematics
5.3 Covariational Reasoning in Mathematics Education
5.4 Covariational Reasoning in Physics Education
5.4.1 Viewing Phenomena as Covariations of Their Parameters
5.4.2 Proposed Categories of Covariations Embedded in Physics Formulas
5.4.3 Discussing Covariations of Parameters in Experiments
5.5 Limiting Case Analysis
5.5.1 Evaluating Limits when the Variable Parameter Is Getting Very Large
x→∞
5.5.2 Evaluating Limits when the Variable Parameter Is Close to a Specific Value
x→a
5.5.3 Is Limiting Case Analysis Really "Limiting"?
References
Part III: From Research to Practice
Chapter 6: Extending the Inquiry of Newton's Second Law by Using Limiting Case Analysis
6.1 Limits
Tools for Extending Scientific Inquiry
6.2 Research Methods
6.2.1 Research Questions, Logistics, and Participants
6.2.2 Criteria for the Study Content Selection
6.2.3 Discussion of the Applied Algebraic Tools
6.3 Description of the Instructional Unit
6.3.1 Analyzing Acceleration of the System in the Function of Mass m2