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E-raamat: Secondary Mathematics for Mathematicians and Educators: A View from Above

Michael Weiss (The University of Michigan, USA)

Formaat: 332 pages
Ilmumisaeg: 05-Oct-2020
Kirjastus: Routledge
Keel: eng
ISBN-13: 9781351587679

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Formaat: 332 pages
Ilmumisaeg: 05-Oct-2020
Kirjastus: Routledge
Keel: eng
ISBN-13: 9781351587679

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In this engaging text, Michael Weiss offers an advanced view of the secondary mathematics curriculum through the prism of theory, analysis, and history, aiming to take an intellectually and mathematically mature perspective on the content normally taught in high school mathematics courses. Rather than a secondary mathematics textbook, Weiss presents here a textbook about the secondary mathematics curriculum, written for mathematics educators and mathematicians and presenting a long-overdue modern-day integration of the disparate topics and methods of secondary mathematics into a coherent mathematical theory.

Areas covered include:

Polynomials and polynomial functions;

Geometry, graphs, and symmetry;

Abstract algebra, linear algebra, and solving equations;

Exponential and logarithmic functions;

Complex numbers;

The historical development of the secondary mathematics curriculum.

Written using precise definitions and proofs throughout on a foundation of advanced content knowledge, Weiss offers a compelling and timely investigation into the secondary mathematics curriculum, relevant for preservice secondary teachers as well as graduate students and scholars in both mathematics and mathematics education.

Acknowledgments

vii

Introduction

(11)

0.1 Who This Book is For

(1)

0.2 Preservice Secondary Teachers

(2)

0.3 Mathematics Graduate Students

(1)

0.4 Mathematics Education Doctoral Students

(1)

0.5 Thinking Like a Mathematician

(1)

0.6 The Theory-Building Disposition

(3)

0.7 Structure of the Book

(2)

1 Numbers and Number Systems

(64)

1.1 Old and New Math

(3)

1.2 Back to Basics

(3)

1.3 What are Real Numbers?

(3)

1.4 Characterizing the Reals

(2)

1.5 Groups

(5)

1.6 Fields and Rings

(6)

1.7 Important Examples

(4)

1.8 Order Properties and Ordered Fields

(5)

1.9 Examples (and Non-Examples) of Ordered Fields

(2)

1.10 Rational Subfields and the Completeness Property

(7)

1.11 The Real Number Characterization Theorem, At Last

(5)

1.12 Existence of a Complete Ordered Field

(6)

1.13 Decimal Representations

(7)

1.14 Recommended Reading

(6)

2 Polynomials and Polynomial Functions

(41)

2.1 Polynomials in the Secondary Curriculum

(1)

2.2 Just What is a Polynomial?

(2)

2.3 Functions

(5)

2.4 Constant Functions and Polynomial Functions

(5)

2.5 Formal Polynomials

(6)

2.6 Interpreting Polynomials as Functions

(11)

2.7 Polynomials over Finite Rings

106

(8)

2.8 Recommended Reading

114

(3)

3 Solving Equations

117

(33)

3.1 "Equivalence" in the Secondary Curriculum

117

(3)

3.2 Strings and Algebraic Strings

120

(4)

3.3 Algebraic Equivalence

124

(2)

3.4 Equations, Strong and Weak Equivalence, and Solutions

126

(9)

3.5 A Complete (?) Algorithm for Solving Polynomial Equations in High School

135

(2)

3.6 Equations in Two Variables

137

(9)

3.7 Recommended Reading

146

(4)

4 Geometry, Graphs and Symmetry

150

(74)

4.1 Euclidean Geometry in the Secondary Curriculum

150

(3)

4.2 Compass-and-Straightedge Constructions in the Euclidean Plane

153

(6)

4.3 Measuring Ratios in the Plane

159

(6)

4.4 From Geometry to Algebra: Coordinatizing Lines and the Plane

165

(10)

4.5 Coordinate Systems, Lines and 1st-Degree Equations

175

(10)

4.6 Non-Orthonormal Coordinate Systems

185

(6)

4.7 Transformations and Symmetry

191

(13)

4.8 Groups of Transformations

204

(9)

4.9 Operations on Functions

213

(6)

4.10 Recommended Reading

219

(5)

5 Exponential and Logarithmic Functions

224

(65)

5.7 What We Talk About when We Talk About Logs

224

(6)

5.2 Exponential Functions, Roots, and the AM-GM Inequality

230

(16)

5.3 Exponential Equations and Logarithmic Functions

246

(12)

5.4 Logarithm-Like and Exponential-Like Functions

258

(12)

5.5 Exponentials and Logarithms in Other Fields and Rings

270

(9)

5.6 Applications to Cryptography

279

(5)

5.7 Recommended Reading

284

(5)

6 Complex Numbers

289

(30)

6.1 A World of Pure Imagination?

289

(3)

6.2 Hamilton's Construction

292

(4)

6.3 Building a Multiplicative Structure from Scratch

296

(5)

6.4 The Field Criterion

301

(2)

6.5 The Complex Criterion

303

(2)

6.6 The Case μ2 + 4λ ≥ 0

305

(4)

6.7 Quadratic Polynomials, Factoring and Completing the Square

309

(3)

6.8 Quotient Rings and Abstract Algebra

312

(4)

6.9 Recommended Reading

316

(3)

Index

319

Michael Weiss is currently a member of the faculty at the Department of Mathematics at the University of Michigan, USA. His background is in mathematics education and pure mathematics, and he was formerly a high school mathematics teacher in the United States.

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