20 novembre 2011

Wu - Understanding Numbers in Elementary School Mathematics (2011)

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.
Pre-service elementary school teachers and current teachers interested in a logical and coherent account of school mathematics.

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.
"[This book] delivers the mathematical knowledge that elementary-grades teachers need."

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

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

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

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

Chapter 6. The Multiplication Algorithm

6.1. The Algorithm
6.2. The Explanation

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

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

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

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?

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

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

Chapter 16. Subtraction of Fractions and Decimals

16.1. Subtraction of Fractions and Decimals
16.2. Inequalities

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

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

Chapter 19. Complex Fractions

19.1. The Basic Skills
19.2. Why Are Complex Fractions Important?

Chapter 20. Percent

20.1. Percent
20.2. Relative Error

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

Chapter 23. Some Interesting Word Problems


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

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

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

Chapter 30. Dividing Rational Numbers

30.1. Definition of Division and Consequences
30.2. Rational Quotients

Chapter 31. Ordering Rational Numbers

31.1. Basic Inequalities
31.2. Powers of Rational Numbers
31.3. Absolute Value

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

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

Chapter 34. The Fundamental Theorem of Arithmetic (FTA)


Chapter 35. The Euclidean Algorithm

35.1. Common Divisors and Gcd
35.2. Gcd as an Integral Linear Combination

Chapter 36. Applications

36.1. Gcd and Lcm
36.2. Fractions and Decimals
36.3. Irrational Numbers
36.4. Infinity of Primes

Chapter 37. Pythagorean Triples


Part 5. More on Decimals

Part Preview

Chapter 38. Why Finite Decimals Are Important

Chapter 39. Review of Finite Decimals


Chapter 40. Scientific Notation

40.1. Comparing Finite Decimals
40.2. Scientific Notation

Chapter 41. Decimals

41.1. Review of Division-with-Remainder
41.2. Decimals and Infinite Decimals
41.3. Repeating Decimals


Chapter 42. Decimal Expansions of Fractions

42.1. The Theorem
42.2. Proof of the Finite Case
42.3. Proof of the Repeating Case



Hung-Hsi Wu

Voice Phone Number: 510-642-2071
Email 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.
  • Go to http://www.aft.org/newspubs/periodicals/ae/fall1999/index.cfm

    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.
  • Go to http://www.aft.org/newspubs/periodicals/ae/index.cfm
    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)
  • Go to http://www.aft.org/newspubs/periodicals/ae/summer2001/index.cfm

    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)
  • Also in http://www.cbmsweb.org/NationalSummit/Plenary_Speakers/wu.htm

    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.
  • Go to http://www.aft.org/newspubs/periodicals/ae/fall2009/index.cfm

    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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