This is a textbook for pre-service elementary
school teachers and for current teachers who are taking professional
development courses. By emphasizing the precision of mathematics, the
exposition achieves a logical and coherent account of school
mathematics at the appropriate level for the readership. Wu provides a
comprehensive treatment of all the standard topics about numbers in the
school mathematics curriculum: whole numbers, fractions, and rational
numbers. Assuming no previous knowledge of mathematics, the
presentation develops the basic facts about numbers from the beginning
and thoroughly covers the subject matter for grades K through 7.
Every
single assertion is established in the context of elementary school
mathematics in a manner that is completely consistent with the basic
requirements of mathematics. While it is a textbook for pre-service
elementary teachers, it is also a reference book that school teachers
can refer to for explanations of well-known but hitherto unexplained
facts. For example, the sometimes-puzzling concepts of percent, ratio,
and rate are each given a treatment that is down to earth and devoid
of mysticism. The fact that a negative times a negative is a positive
is explained in a leisurely and comprehensible fashion.
Readership
Pre-service
elementary school teachers and current teachers interested in a
logical and coherent account of school mathematics.
Reviews
The
author has a published article related to this book entitled, "Phoenix
Rising: Bringing the Common Core State Mathematics Standards to Life."
To read the article, please go to the Fall 2011 issue of American Educator.
Contents
Acknowledgments
To the Reader
Some Conventions in this Book
Part 1. Whole Numbers
Part Preview
Chapter 1. Place Value
1.1. How to Count
1.2. Place Value
1.3. The Use of Symbolic
Notation
1.4. The Number Line
1.5. Comparing Numbers (Beginning)
1.6. Multiplication and the
Expanded Form of a Number
1.7. All about Zero
1.8. The Hindu-Arabic
Numeral System
Exercises
Chapter 2. The Basic Laws of Operations
2.1. The Equal Sign
2.2. The Associative and
Commutative Laws of +
2.3. The Associative and
Commutative Laws of X
2.4. The Distributive Law
2.5. Comparing Numbers
(Conclusion)
2.6. An Application of the
Associative and Commutative Laws of Addition
Exercises
Chapter 3. The Standard Algorithms
Chapter 4. The Addition Algorithm
4.1. The Basic Idea of the
Algorithm
4.2. The Addition Algorithm
and Its Explanation
4.3. Essential Remarks on
the Addition Algorithm
Exercises
Chapter 5. The Subtraction Algorithm
5.1. Definition of
Subtraction
5.2. The Subtraction
Algorithm
5.3. Explanation of the
Algorithm
5.4. How to Use the Number
Line
5.5. A Special Algorithm
5.6. A Property of
Subtraction
Exercises
Chapter 6. The Multiplication Algorithm
6.1. The Algorithm
6.2. The Explanation
Exercises
Chapter 7. The Long Division Algorithm
7.1. Multiplication as
Division
7.2.
Division-with-Remainder
7.3. The Algorithm
7.4. A Mathematical
Explanation (Preliminary)
7.5. A Mathematical
Explanation (Final)
7.6. Essential Remarks on
the Long Division Algorithm
Exercises 123
Chapter 8. The Number Line and the Four Operations Revisited
8.1. The Number Line Redux,
and Addition and Subtraction
8.2. Importance of the Unit
8.3. Multiplication
8.4. Division
8.5. A Short History of the
Concept of Multiplication
Chapter 9. What Is a Number?
Chapter 10. Some Comments on Estimation
10.1. Rounding 140
10.2. Absolute and Relative
Errors
10.3. Why Make Estimates?
10.4. A Short History of
the Meter
Exercises
Chapter 11. Numbers in Base b
11.1. Basic Definitions
11.2. The Representation
Theorem
11.3. Arithmetic in Base 7
11.4. Binary Arithmetic
Exercises
Part 2. Fractions
Part Preview
Chapter 12. Definitions of Fraction and Decimal
12.1. Prologue
12.2. The Basic Definitions
12.3. Decimals
12.4. Importance of the
Unit
12.5. The Area Model
12.6. Locating Fractions on
the Number Line
12.7. Issues to Consider
Exercises
Chapter 13. Equivalent Fractions and FFFP
13.1. Theorem on Equivalent
Fractions (Cancellation Law)
13.2. Applications to
Decimals
13.3. Proof of Theorem 13.1
13.4. FFFP
13.5. The
Cross-Multiplication Algorithm
13.6. Why FFFP?
Exercises
Chapter 14. Addition of Fractions and Decimals
14.1. Definition of
Addition and Immediate Consequences
14.2. Addition of Decimals
14.3. Mixed Numbers
14.4. Refinements of the
Addition Formula
14.5. Comments on the Use
of Calculators
14.6. A Noteworthy Example
of Adding Fractions
Exercises
Chapter 15. Equivalent Fractions: Further Applications
15.1. A Different View of a
Fraction
15.2. A New Look at Whole
Number Divisions
15.3. Comparing Fractions
15.4. The Concept of m
Exercises
Chapter 16. Subtraction of Fractions and Decimals
16.1. Subtraction of
Fractions and Decimals
16.2. Inequalities
Exercises
Chapter 17. Multiplication of Fractions and Decimals
17.1. The Definition and the
Product Formula
17.2. Immediate
Applications of the Product Formula
17.3. A Second
Interpretation of Fraction Multiplication
17.4. Inequalities
17.5. Linguistic vs.
Mathematical Issues
Exercises
Chapter 18. Division of Fractions
18.1. Informal Overview
18.2. The Definition and
Invert-and-Multiply
18.3. Applications
18.4. Comments on the
Division of Decimals
18.5. Inequalities
18.6. False Doctrines
Exercises
Chapter 19. Complex Fractions
19.1. The Basic Skills
19.2. Why Are Complex Fractions Important?
Exercises
Chapter 20. Percent
20.1. Percent
20.2. Relative Error
Exercises
Chapter 21. Fundamental Assumption of School Mathematics (FASM)
Chapter 22. Ratio and Rate
22.1. Ratio
22.2. Why Ratio?
22.3. Rate
22.4. Units
22.5. Cooperative Work
Exercises
Chapter 23. Some Interesting Word Problems
Exercises
Chapter 24. On the Teaching of Fractions in Elementary School
Part 3. Rational Numbers
Chapter 25. The (Two-Sided) Number Line
Chapter 26. A Different View of Rational Numbers
Chapter 27. Adding and Subtracting Rational Numbers
27.1. Definition of Vectors
27.2. Vector Addition for
Special Vectors
27.3. Addition of Rational
Numbers
27.4. Explicit Computations
27.5. Subtraction as
Addition
Exercises
Chapter 28. Adding and Subtracting Rational Numbers Redux
28.1. The Assumptions on
Addition
28.2. The Basic Facts
28.3. Explicit Computations
28.4. Basic Assumptions and
Facts, Revisited
Exercises
Chapter 29. Multiplying Rational Numbers
29.1. The Assumptions on
Multiplication
29.2. The Equality (−m)(−n) = mn for
Whole Numbers
29.3. Explicit Computations
29.4. Some Observations
Exercises
Chapter 30. Dividing Rational Numbers
30.1. Definition of
Division and Consequences
30.2. Rational Quotients
Exercises
Chapter 31. Ordering Rational Numbers
31.1. Basic Inequalities
31.2. Powers of Rational
Numbers
31.3. Absolute Value
Exercises
Part 4. Number Theory
Part Preview
Chapter 32. Divisibility Rules
32.1. Review of
Division-with-Remainder
32.2. Generalities about
Divisibility
32.3. Divisibility Rules
Exercises
Chapter 33. Primes and Divisors
33.1. Definitions of Primes
and Divisors
33.2. The Sieve of
Eratosthenes
33.3. Some Theorems and
Conjectures about Primes
Exercises
Chapter 34. The Fundamental Theorem of Arithmetic (FTA)
Exercises
Chapter 35. The Euclidean Algorithm
35.1. Common Divisors and
Gcd
35.2. Gcd as an Integral
Linear Combination
Exercises
Chapter 36. Applications
36.1. Gcd and Lcm
36.2. Fractions and
Decimals
36.3. Irrational Numbers
36.4. Infinity of Primes
Exercises
Chapter 37. Pythagorean Triples
Exercises
Part 5. More on Decimals
Part Preview
Chapter 38. Why Finite Decimals Are Important
Chapter 39. Review of Finite Decimals
Exercises
Chapter 40. Scientific Notation
40.1. Comparing Finite Decimals
40.2. Scientific Notation
Exercises
Chapter 41. Decimals
41.1. Review of
Division-with-Remainder
41.2. Decimals and Infinite
Decimals
41.3. Repeating Decimals
Exercises
Chapter 42. Decimal Expansions of Fractions
42.1. The Theorem
42.2. Proof of the Finite Case
42.3. Proof of the Repeating Case
Exercises
Bibliography
Hung-Hsi Wu
Voice Phone Number: 510-642-2071Email address: wu@math.berkeley.edu
Postal Address:
- Department of Mathematics #3840
- University of California
- Berkeley, CA, 94720-3840
These are papers about mathematics education. They are listed more or less in chronological order. The general heading before each title ("General", "Curriculum", "Professional Development") gives a rough classification of its content.
Curriculum The role of open-ended problems in mathematics education (J. Math. Behavior 13(1994), 115-128)
General Invited Comments on the NCTM Standards (1996)
General The mathematician and the mathematics education reform (Notices of the American Mathematical Society, 43(1996), 1531-1537)
Profesional Development On the education of mathematics teachers (For mathematicians and education researchers) (formerly entitled: On the training of mathematics teachers) (1997)
General The Mathematics Education Reform: Why you should be concerned and what you can do (Amer. Math. Monthly 104(1997), 946-954)
General The Mathematics education reform: What is it and why should you care ? (1998)
Professional Development Teaching fractions in elementary school: A manual for teachers (March 1998)
Professional Development On the education of mathematics majors (For mathematicians and education researchers) (Contemporary Issues in Mathematics Education, edited by E. Gavosto, S. G. Krantz, and W. G. McCallum, MSRI Publications, Volume 36, Cambridge University Press, 1999, 9-23)
Professional Development and General The joy of lecturing--With a critique of the romantic tradition of education writing (Appendix to How to Teach Mathematics by S.G. Krantz, 2nd edition, Amer. Math. Soc. 1999, pp. 261-271)
Professional Development Preservice professional development of mathematics teachers (For mathematicians and education researchers) (March 1999)
Curriculum The isoperimetric inequality: The algebraic viewpoint (April 1999)
General Mathematics Standards: A new direction for California (Address in the Northridge Conference, May 21, 1999)
Curriculum and Professional Development Some remarks on the teaching of fractions in elementary school (October 1999)
General and Curriculum Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education, American Educator, Fall 1999, Vol. 23, No. 3, pp. 14--19, 50-52.
General The 1997 mathematics standards war in California (What is at Stake in the K-12 Standards Wars: A Primer for Educational Policy Makers, Sandra Stotsky, ed., Peter Lang Publishers, NY, 2000. Pp. 3-31.)
Curriculum Review of the Interactive Mathematics Program (IMP) (March 2000)
Professional Development and Curriculum Chapter 1: Whole Numbers (Draft) (July 15, 2000; REVISED September 1, 2002) [A completely revised version of this document has appeared as Part 1 of the book, Understanding Numbers in Elementary School Mathematics, Amer. Math. Soc., 2011.]
Professional Development and Curriculum Chapter 2: Fractions (Draft) (June 20, 2001; REVISED September 3, 2002) [A completely revised version of this document has appeared as Part 2 of the book, Understanding Numbers in Elementary School Mathematics, Amer. Math. Soc., 2011.]
General On the learning of algebra (January, 2001)
Professional Development and Curriculum How to prepare students for algebra (For teachers of grades 5-8), American Educator, Summer 2001, Vol. 25, No. 2, pp. 10-17.
General Dialog with Rick Hess on the Common Core Mathematics Standards (For general audience) (October 29, 2011)
General and Professional Development Professional development and Textbook school mathematics (For mathematicians and education researchers) (October 29,2011)
Professional Development and Curriculum What is so difficult about the preparation of mathematics teachers? (For educators and mathematicians) (November 29, 2001; REVISED March 6, 2002)
Professional Development (with Mary Burmester) Some lessons from California (November 29, 2001; Revised, May 25, 2004 )
Curriculum Topics in Pre-Calculus: Functions, graphs, and basic mensuration formulas (June 15, 2003)
Curriculum ``Order of operations" and other oddities in school mathematics (April 7, 2004: Revised, June 1, 2004)
Professional Development Geometry: Our Cultural Heritage -- A Book Review (Notices of the American Mathematical Society, 51 (2004), 529-537)
Professional Development and General Must Content Dictate Pedagogy in Mathematics Education? (For educators and mathematicians) (February 1, 2005; second revision May 23, 2005)
Curriculum and General Key mathematical ideas in grades 5-8 (September 12, 2005; presentation at the 2005 NCTM Annual Meeting, April, 2005)
Professional Development Professional Development: The Hard Work of Learning Mathematics (For educators and mathematicians) (June 18, 2006; revised November 18, 2006)
General How mathematicians can contribute to K-12 mathematics education, Proceedings of International Congress of Mathematicians, Madrid 2006, Volume III, European Mathematical Society, Zuerich, 2006, 1676-1688.
General What is mathematics education? (Plenary address at the NCTM Annual Meeting, March 23, 2007) (June 2, 2007)
Professional Development and Curriculum Fractions, decimals, and rational numbers (For education researchers and K-8 teachers) (February 29, 2008)
Curriculum (with Wilfried Schmid) The major topics of school algebra (March 31, 2008)
Professional Development The mathematics K-12 teachers need to know (LONG VERSION; for educators and teachers) (December 17, 2008)
Curriculum The critical foundations of algebra (February 3, 2009)
Curriculum From arithmetic to algebra (February 23, 2009)
General and Professional Development What's sophisticated about elementary mathematics? American Educator, Fall 2009, Vol. 33, No. 3, pp. 4-14.
Professional Development Learning algebra (For teachers of grades 5-8) (October 22, 2009)
Curriculum and Professional Development Euclid and high school geometry (For high school teachers and mathematicians) (February 2, 2010)
Curriculum and Professional Development The mathematics school teachers should know (For a general audience) (February 2, 2010)
General What is school mathematics? (For educators and mathematicians) (February 2, 2010)
Professional Development Pre-Algebra (Draft of textbook for teachers of grades 5-8) (April 21, 2010)
General and Professional Development Teaching fractions: is it poetry or mathematics? (For educators and professional developers) (April 26, 2010)
Professional Development and Curriculum Introduction to School Algebra (Draft of textbook for teachers of grades 5-8) (August 14, 2010)
General and Professional Development The Mis-Education of Mathematics Teachers (For mathematicians and education researchers) (Notices Amer. Math. Soc. 58 (2011), 372-384.)
General and Professional Development The impact of Common Core Standards on the mathematics education of teachers (For mathematicians and education researchers) (April 29, 2011)
General and Professional Development What is different about the Common Core Mathematics Standards? (For teachers and educators) (June 20, 2011)
General and Professional Development The Common Core Mathematics Standards: Implications for Administrators (For teachers and administrators) (June 22, 2011)
Curriculum Syllabi of High School Courses According to the Common Core Standards (For high school teachers and adminstrators) (August 5, 2011)
Curriculum Teaching Fractions According to the Common Core Standards (For teachers of K-8 and educators) (August 5, 2011)
General and Curriculum Bringing the Common Core State Mathematics Standards to Life. American Educator, Fall 2011, Vol. 35, No. 3, pp. 3-13.
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