Understanding Physics Using Mathematical Reasoning: A Modeling Approach for Practitioners and Researchers

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
			6.3.2 Analyzing Acceleration of the System in the Function of Mass m1
		6.4 Data Analysis
			6.4.1 Analysis of the Pretest Results
			6.4.2 Analysis of the Posttest Results
		6.5 Conclusions
		References
	Chapter 7: Reconstructing Newton’s Law of Universal Gravity as a Covariate Relation
		7.1 Prior Research Findings
		7.2 Theoretical Framework
			7.2.1 Historical Perspective
			7.2.2 Contemporary Presentations of the Law of Universal Gravity
		7.3 Methods
		7.4 Didactical Underpinnings of the Instructional Unit
		7.5 The Lecture Component
			7.5.1 Gravitational Field Intensity and the Effects of Covariate Quantities
			7.5.2 Reconstructing the Formula to Calculate Mutual Gravitational Force
		7.6 Analysis of Pretest - Posttest Results
			7.6.1 Analysis of the Pretest Results
			7.6.2 Analysis of the Posttest Results
		7.7 Conclusions and Suggestions for Further Research
		References
	Chapter 8: Parametrization of Projectile Motion
		8.1 Prior Research Findings
		8.2 Theoretical Framework
			8.2.1 Categories of Motion Studied in High School and Undergraduate Physics Courses
			8.2.2 Why Parametric Equations?
			8.2.3 Foundations of Constructivist Learning Theory
		8.3 Methods
			8.3.1 Study Description and the Research Question
			8.3.2 The Participants
			8.3.3 Lecture Component Sequencing
			8.3.4 Topics Embedded within the Curriculum to Enhance the Treatment
		8.4 General Lab Description
			8.4.1 Lab Logistics
			8.4.2 Gathering Data to Construct Positions Functions for a Projected Object
			8.4.3 Constructing Representations of the Position Functions
			8.4.4 Finding Velocities and Acceleration Functions
			8.4.5 Verification Process
		8.5 Treatment Evaluation
		8.6 Summary and Conclusions
		References
	Chapter 9: Reimaging Lens Equation as a Dynamic Representation
		9.1 Introduction
		9.2 Prompts Used for the Instructional Unit Design
			9.2.1 Mathematical Background
			9.2.2 Lab Equipment
			9.2.3 Conversion of Lens Equation into a Covariational Relation
			9.2.4 Sketching and Scientifically Interpreting the Graph of the Lens Function
			9.2.5 Formulating Magnification Function
			9.2.6 Merging Mathematical and Experimental Representations into One Inquiry
		9.3 Suggested Independent Student Work
		9.4 Summary
		References
	Chapter 10: Embracing the Mole Understanding in a Covariate Relation
		10.1 Introduction and Prior Research Findings
		10.2 Theoretical Framework
			10.2.1 Weaknesses of the Mole Understanding
			10.2.2 Proportional Reasoning, Rates, and Ratios
		10.3 Methods
		10.4 The Lecture Component
			10.4.1 The Mole as a Fundamental Unit of the Substance Amount
			10.4.2 Converting the Number of Atoms to the Units of Moles
			10.4.3 Converting Mass of Substance to Moles
			10.4.4 Converting Mass of a Substance to the Number of Atoms
		10.5 Pretest Posttest Analysis
			10.5.1 Analysis of the Pretest Results
			10.5.2 Comparisons of the Pretest and Posttest Results
		10.6 Summary and Conclusions
		References
	Chapter 11: Enabling Covariational Reasoning in Einstein’s Formula for Photoelectric Effect
		11.1 Prior Research
		11.2 Theoretical Background
		11.3 Embracing the PE into the Framework of Covariational Representation
			11.3.1 Weaknesses of the Graph of KMAX Versus Photons’ Frequency Presented in Physics Resources
			11.3.2 Covariation of Photon’s Energy and Frequency as a Linear Function
			11.3.3 Electrons’ Binding Energy as a Function of Photons Threshold Frequency
			11.3.4 Maintaining a Minimum Number of Covariational Parameters During the Inquiry
		11.4 Reassembling the PE Formula to Assure a Coherence of Representations
			11.4.1 Graph Constructing
			11.4.2 Finding Algebraic Representation of the Graph
			11.4.3 Linking the Photons Threshold Frequency and the Work Function hfo = Wo
		11.5 Summary and Conclusions
		References
	Chapter 12: Are Physics Formulas Aiding Covariational Reasoning? Students’ Perspective
		12.1 Introduction and Prior Research Findings
		12.2 Theoretical Background and Methods
			12.2.1 Foundations of Covariation Reasoning
			12.2.2 Study Description, Participants, Research Questions, and Evaluation Instrument
		12.3 Data Analysis
		12.4 Summary and Conclusions
			12.4.1 Traditional Formula Notation Does Not Aid Covariational Reasoning in Physics
			12.4.2 Physics Depends on the Mathematical Rules and Notation
		References
	Chapter 13: Adaptivity of Mathematics Representations to Reason Scientifically Students’ Perspective
		13.1 Prior Research Findings
		13.2 Theoretical Framework, Research Questions, and Study Logistics
		13.3 Study Instrument
			13.3.1 General Characteristics of the Treatment: How Did Covariational Reasoning Emerge?
			13.3.2 Actions Taken to Exercise Covariation Model Using Laboratory
		13.4 Data Analysis
		13.5 Summary and Conclusions
		References
Teaching Physics Using Mathematical Reasoning
	Research and Practice
Index