Preface
For the Student from a Student
For the Instructor
One Semester Courses
Two Semester Course Sequence
Three Semester Courses
Focus on Middle School Content
Chapter Dependencies
Acknowledgements
I FOUNDATIONS
1
Mathematics Education Foundations
1.1
History of Mathematics Education in the United States
1.2
Mathematical Knowledge for Teaching
1.2.1
Common Content Knowledge
1.2.2
Specialized Content Knowledge
1.2.3
Horizon Content Knowledge
1.2.4
Exercises
1.3
Mathematical Practice Standards
1.3.1
Mathematical Problem Solving
1.3.2
Modeling with Mathematics
1.3.3
Communicating Mathematically
1.3.4
Understanding Mathematical Structures
1.3.5
Exercises
1.4
Mathematics Content Standards in the U.S.
1.4.1
Common Core State Standards-Math
1.4.2
NCTM-CAEP Standards
1.4.3
Exercises
2
Set Theory
2.1
Sets and Subsets
2.1.1
Venn Diagrams
2.1.2
List of Sets of Numbers
2.1.3
Exercises
2.2
Algebra of Sets
2.2.1
Set Complements
2.2.2
De Morgan’s Laws
2.2.3
Cartesian Products
2.2.4
Exercises
2.3
Collections of Sets
2.3.1
Exercises
3
Equality, Order, and Equivalence
3.1
Partitions and Equivalence Relations
3.1.1
Partitions
3.1.2
Relations
3.1.3
Equivalence Relations Induced by Partitions
3.1.4
Partitions Induced by Equivalence Relations
3.1.5
Ordered Sets and Relations
3.1.6
Exercises
3.2
Equality and Equivalence in the K-12 Curriculum
3.2.1
Exercises
3.3
Expressions, Equations, and Inequalities
3.3.1
Exercises
4
Number Systems
4.1
Natural Numbers and Integers
4.1.1
Natural Numbers
4.1.2
Integers
4.1.3
Properties of Exponents
4.1.4
Exercises
4.2
Representations of Integers
4.2.1
Numeration Systems
4.2.2
Representations of Number
4.2.3
Addition Models and Algorithms
4.2.4
Subtraction Models and Algorithms
4.2.5
Multiplication Models and Algorithms
4.2.6
Division Models and Algorithms
4.2.7
Exercises
4.3
Rational Numbers
4.3.1
Operations
4.3.2
Order
4.3.3
Algebraic Properties
4.3.4
Properties of Exponents
4.3.5
Exercises
4.4
Representations of Rational Numbers
4.4.1
Decimal Representations
4.4.2
Physical and Graphical Representations
4.4.3
Exercises
4.5
Real Numbers
4.5.1
Properties of Exponents
4.5.2
Exercises
4.6
Complex Numbers
4.6.1
Rectangular Representation
4.6.2
Complex Conjugation and Modulus
4.6.3
Euler’s Equation
4.6.4
Exercises
5
Functions
5.1
Definitions of Functions
5.1.1
Function Representations
5.1.2
Exercises
5.2
Injections, Surjections, and Bijections
5.2.1
Exercises
5.3
Composition of Functions
5.3.1
Inverse Functions
5.3.2
Exercises
5.4
Counting and Cardinality
5.4.1
Properties of Cardinality
5.4.2
Pigeonhole Principle
5.4.3
Hilbert’s Hotel
5.4.4
Countable and Uncountable
5.4.5
Exercises
6
Number Theory
6.1
Divisibility
6.1.1
Divisibility Tests
6.1.2
Exercises
6.2
Pascal’s Triangle and the Binomial Theorem
6.2.1
Binomial Theorem
6.2.2
Exercises
6.3
Counting
6.3.1
Fundamental Counting Principle
6.3.2
Factorial Notation
6.3.3
Permutations without Repetitions
6.3.4
Permutations with Repetition
6.3.5
Combinations
6.3.6
Summary
6.3.7
Exercises
II ALGEBRA
7
Groups, Rings, and Fields
7.1
Group Theory
7.1.1
Abelian Groups
7.1.2
Uniqueness of Identities and Inverses
7.1.3
Homomorphisms
7.1.4
Isomorphisms
7.1.5
Finite Groups
7.1.6
Exercises
7.2
Relationship Between Linear and Exponential
7.2.1
Exercises
7.3
Rings and Fields
7.3.1
Finite Sets with Two Operations
7.3.2
Matrices
7.3.3
Rings
7.3.4
Identities and Inverses
7.3.5
Integral Domains
7.3.6
Ring Homomorphisms
7.3.7
Exercises
8
Integral Domains and Polynomials
8.1
Properties of the Ring of the Integers
8.1.1
Composing and Decomposing Integers
8.1.2
Greatest Common Divisor and Least Common Multiple
8.1.3
Fundamental Theorem of Arithmetic
8.1.4
Exercises
8.2
Polynomial Rings
8.2.1
Exercises
8.3
Properties of Polynomial Rings
8.3.1
Polynomial Division
8.3.2
GCF and LCM for Polynomials
8.3.3
Irreducible Polynomials
8.3.4
Exercises
8.4
Rational Expressions
8.4.1
Exercises
9
Real Valued Functions
9.1
Function Properties
9.1.1
Domain and Range
9.1.2
Maxima (minima) or relative maxima (relative minima)
9.1.3
Increasing or decreasing
9.1.4
Intercepts
9.1.5
End behavior, singularities, and asymptotes
9.1.6
Special Properties
9.1.7
Exercises
9.2
Transformations of Functions
9.2.1
Horizontal Translations
9.2.2
Vertical Translations
9.2.3
Horizontal Dilations
9.2.4
Vertical Dilations
9.2.5
Reflections
9.2.6
Families of Functions
9.2.7
Exercises
9.3
Linear and Exponential Functions
9.3.1
Linear Functions
9.3.2
Exponential Functions
9.3.3
Logarithmic Functions
9.3.4
Relationships between Linear and Exponential
9.3.5
Exercises
9.4
Linear Fractional Transformations
9.4.1
Domain
9.4.2
Range
9.4.3
Increasing or decreasing
9.4.4
Intercepts
9.4.5
End behavior and asymptotes
9.4.6
Extensions
9.4.7
Exercises
9.5
Quadratic Polynomials
9.5.1
Completing the Square
9.5.2
Factoring
9.5.3
Inverse Function
9.5.4
Exercises
9.6
General Polynomial Functions
9.6.1
Short-Term Behavior
9.6.2
Long-Term Behavior
9.6.3
Polynomial Identities
9.6.4
Exercises
9.7
Rational Functions
9.7.1
Short-Term Behavior
9.7.2
Long-Term Behavior
9.7.3
Partial Fraction Decomposition
9.7.4
Exercises
9.8
Trigonometric Functions
9.8.1
Definitions
9.8.2
Additional Properties
9.8.3
Graphs of Trigonometric Functions
9.8.4
Inverse Trigonometric Functions
9.8.5
Exercises
9.9
Combinations of Functions
9.9.1
Addition, Subtraction, Multiplication, and Division of Functions
9.9.2
Composition of Functions
9.9.3
Systems of Equations
9.9.4
Exercises
III GEOMETRY
10
Axiomatic Geometry
10.1
Definitions
10.1.1
Criteria for Definitions
10.1.2
Concept Image
10.1.3
Classification of Two-Dimensional Figures
10.1.4
Definitions from Euclid’s Elements
10.1.5
Exercises
10.2
Axiomatic Systems
10.2.1
Euclid’s Common Notions and Postulates
10.2.2
Hilbert’s Axioms
10.2.3
School Mathematics Study Group Axioms
10.2.4
Discussion
10.2.5
Exercises
10.3
Euclid’s Basic Constructions
10.3.1
Initial Propositions
10.3.2
Isosceles Triangles
10.3.3
Side-Side-Side Congruence
10.3.4
Angles
10.3.5
Perpendicular Lines
10.3.6
Exercises
10.4
Angles, Parallel Lines, and Parallelograms
10.4.1
Angles and Parallel Lines
10.4.2
Parallelograms
10.4.3
Exercises
10.5
Similarity of Triangles
10.5.1
Exercises
10.6
Centers of Triangles
10.6.1
Centroid
10.6.2
Circumcenter
10.6.3
Orthocenter
10.6.4
Euler Line
10.6.5
Exercises
10.7
Circle Theorems
10.7.1
Exercises
11
Measurement
11.1
Units
11.1.1
Units of Length
11.1.2
Units of Area
11.1.3
Units of Volume
11.1.4
Units of Angle Measurements
11.1.5
Unit Conversions and Dimensional Analysis
11.1.6
Exercises
11.2
Decomposing and Composing
11.2.1
Area
11.2.2
Volume
11.2.3
Related Horizon Content Knowledge
11.2.4
Exercises
11.3
Measurements of Triangles
11.3.1
Exercises
11.4
Distance
11.4.1
Circles and Lines
11.4.2
Related Horizon Content Knowledge
11.4.3
Exercises
12
Groups and Geometry
Erlangen Program
12.1
Groups and Transformations
12.1.1
Group Review
12.1.2
Dihedral Groups
12.1.3
Subgroups
12.1.4
Exercises
12.2
Normal Subgroups and Factor Groups
12.2.1
Cosets
12.2.2
Normal Subgroups
12.2.3
Factor Groups
12.2.4
Exercises
12.3
Group Homomorphisms Revisited
12.3.1
Exercises
13
Euclidean Transformational Geometry
13.1
Introduction to Transformational Geometry
13.1.1
Neutral Geometry from Distances
13.1.2
Isometries
13.1.3
Congruence
13.1.4
Exercises
13.2
Representations of the Euclidean Plane
13.2.1
Synthetic Plane
13.2.2
Cartesian Plane
13.2.3
Vector Space
13.2.4
Complex Plane
13.2.5
Isometries
13.2.6
Exercises
13.3
Translations
13.3.1
Synthetic Plane
13.3.2
Cartesian Plane and Vector Space
13.3.3
Complex Plane
13.3.4
Compositions
13.3.5
Group of Translations
13.3.6
Exercises
13.4
Rotations
13.4.1
Synthetic Plane
13.4.2
Complex Plane
13.4.3
Cartesian Plane and Vector Space
13.4.4
Compositions and Groups of Rotations
13.4.5
Exercises
13.5
Reflections
13.5.1
Synthetic Plane
13.5.2
Complex Plane
13.5.3
Vector Space and Cartesian Plane
13.5.4
Compositions
13.5.5
Exercises
13.6
Glide-Reflections
13.6.1
Synthetic Plane
13.6.2
Complex Plane
13.6.3
Exercises
13.7
Group of Isometries
13.7.1
Fixed Points
13.7.2
Exercises
13.8
Dilations and Similarity
13.8.1
Synthetic Plane
13.8.2
Cartesian Plane
13.8.3
Vector Space
13.8.4
Complex Plane
13.8.5
Compositions
13.8.6
Exercises
IV DATA ANALYSIS
14
Data Analysis Foundations
14.1
Statistics, Data Analysis, and Mathematics
14.1.1
The Centrality of Variability
14.1.2
Exercises
14.2
Teaching and Learning of Data Analysis and Statistics
14.2.1
Statistical Habits of Mind
14.2.2
Exercises
15
Exploring Data
15.1
Types of Data
15.1.1
Categorical
15.1.2
Ordinal
15.1.3
Binary
15.1.4
Binomial
15.1.5
Count
15.1.6
Continuous
15.1.7
Exercise
15.2
Exploring Univariate Data Graphically
15.2.1
Histograms
15.2.2
Dot Plots
15.2.3
Density Plots
15.2.4
Box Plots
15.2.5
Exercises
15.3
Exploring Bivariate Data Graphically
15.3.1
Scatterplots
15.3.2
Exercises
15.4
Measures of Center
15.4.1
Exercises
15.5
Measures of Variability
15.5.1
Exercises
15.6
Exploring Bivariate Data Numerically
15.6.1
Categorical
\(\times\)
Numerical Data
15.6.2
Numerical
\(\times\)
Numerical Data
15.6.3
Categorical
\(\times\)
Categorical Data
15.6.4
Exercises
16
Samples, Simulations, and Probability
16.1
Probability Overview
16.1.1
Compound Events
16.1.2
Estimating Probabilities
16.1.3
Exercises
16.2
Probability Spaces
16.2.1
Boolean Algebra and Probability
16.2.2
Exercises
16.3
Conditional Probability
16.3.1
Exercises
16.4
Random Variables and Expected Value
16.4.1
Exercises
17
Estimating Parameters and Testing Hypotheses
17.1
Sampling Techniques and Study Types
17.1.1
Sampling Errors and Bias
17.1.2
Probability Sampling
17.1.3
Study Types
17.1.4
Exercises
17.2
Point and Interval Estimators
17.2.1
Estimating Population Means and Medians
17.2.2
Estimating Differences of Means
17.2.3
Estimating Proportions
17.2.4
Interpretations
17.2.5
Exercises
17.3
Hypothesis Testing
17.3.1
Fisher’s Approach
17.3.2
Neyman and Pearson’s Approach
17.3.3
Summary
17.3.4
Exercises
References
Mathematical Knowledge for Secondary Teachers
1.1
History of Mathematics Education in the United States