Secondary Mathematics for Mathematicians and Educators: A View from Above

Cover
Half Title
Title Page
Copyright Page
Table of contents
Acknowledgments
Introduction
	0.1 Who This Book is For
	0.2 Preservice Secondary Teachers
	0.3 Mathematics Graduate Students
	0.4 Mathematics Education Doctoral Students
	0.5 Thinking Like a Mathematician
	0.6 The Theory-Building Disposition
	0.7 Structure of the Book
	Notes
	References
1 Numbers and Number Systems
	1.1 Old and New Math
	1.2 Back to Basics
	1.3 What are Real Numbers?
	1.4 Characterizing the Reals
	1.5 Groups
	1.6 Fields and Rings
	1.7 Important Examples
	1.8 Order Properties and Ordered Fields
	1.9 Examples (and Non-Examples) of Ordered Fields
	1.10 Rational Subfields and the Completeness Property
	1.11 The Real Number Characterization Theorem, At Last
	1.12 Existence of a Complete Ordered Field
	1.13 Decimal Representations
	1.14 Recommended Reading
	Notes
2 Polynomials and Polynomial Functions
	2.1 Polynomials in the Secondary Curriculum
	2.2 Just What is a Polynomial?
	2.3 Functions
	2.4 Constant Functions and Polynomial Functions
	2.5 Formal Polynomials
	2.6 Interpreting Polynomials as Functions
	2.7 Polynomials over Finite Rings
	2.8 Recommended Reading
	Notes
3 Solving Equations
	3.1 “Equivalence” in the Secondary Curriculum
	3.2 Strings and Algebraic Strings
	3.3 Algebraic Equivalence
	3.4 Equations, Strong and Weak Equivalence, and Solutions
	3.5 A Complete (?) Algorithm for Solving Polynomial Equations in High School
	Solving Polynomial and Rational Equations
	3.6 Equations in Two Variables
	3.7 Recommended Reading
	Notes
4 Geometry, Graphs and Symmetry
	4.1 Euclidean Geometry in the Secondary Curriculum
	4.2 Compass-and-Straightedge Constructions in the Euclidean Plane
	4.3 Measuring Ratios in the Plane
	4.4 From Geometry to Algebra: Coordinatizing Lines and the Plane
	4.5 Coordinate Systems, Lines and 1st-Degree Equations
	4.6 Non-Orthonormal Coordinate Systems
	4.7 Transformations and Symmetry
	4.8 Groups of Transformations
	4.9 Operations on Functions
	4.10 Recommended Reading
	Notes
5 Exponential and Logarithmic Functions
	5.1 What We Talk About when We Talk About Logs
	5.2 Exponential Functions, Roots, and the AM–GM Inequality
	5.3 Exponential Equations and Logarithmic Functions
	5.4 Logarithm-Like and Exponential-Like Functions
	5.5 Exponentials and Logarithms in Other Fields and Rings
	5.6 Applications to Cryptography
	5.7 Recommended Reading
	Notes
6 Complex Numbers
	6.1 A World of Pure Imagination?
	6.2 Hamilton’s Construction
	6.3 Building a Multiplicative Structure from Scratch
	6.4 The Field Criterion
	6.5 The Complex Criterion
	6.6 The Case .
	6.7 Quadratic Polynomials, Factoring and Completing the Square
	6.8 Quotient Rings and Abstract Algebra
	6.9 Recommended Reading
	Notes
Index