# Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States (Anniversary Edition)

## Liping Ma

Language: English

Pages: 198

ISBN: 2:00044395

Format: PDF / Kindle (mobi) / ePub

Studies of teachers in the U.S. often document insufficient subject matter knowledge in mathematics. Yet, these studies give few examples of the knowledge teachers need to support teaching, particularly the kind of teaching demanded by recent reforms in mathematics education. *Knowing and Teaching Elementary Mathematics* describes the nature and development of the knowledge that elementary teachers need to become accomplished mathematics teachers, and suggests why such knowledge seems more common in China than in the United States, despite the fact that Chinese teachers have less formal education than their U.S. counterparts.

The anniversary edition of this bestselling volume includes the original studies that compare U.S and Chinese elementary school teachers’ mathematical understanding and offers a powerful framework for grasping the mathematical content necessary to understand and develop the thinking of school children. Highlighting notable changes in the field and the author’s work, this new edition includes an updated preface, introduction, and key journal articles that frame and contextualize this seminal work.

---

*Uploader Release Notes:*

HQ PDF, includes OCR & bookmarks

---

included in this procedure does not involve carrying so it is simple too. There is still another way to regroup. We may want to regroup the subtrahend 26 as: Â€ We first subtract one 3 from 53 and get 50. Then we subtract the other 3 from 50 and get 47. Finally we subtract 20 from 47 and get 27. (Tr. C) The teachers referred to three main ways of regrouping. One was the standard way: decompose a unit at a higher value place into units at a lower value place, combine them with the original units

Mathematics and thought that the potential zeros “didn’t really belong in there.” Therefore, even though he intended to promote conceptual learning, his teaching strategy was to have students “do their problems sideways on the lined paper, using the lines of the paper to make the vertical columns,” so as to make it clear “there is a column to skip”—no conceptual learning was evidenced at all. Ms. Fiona insisted that her students needed to be able to answer the question “Why do you move those

here is part of the discussion of “The meaning and properties of fractions” from the teacher’s manual for the Grade 4 textbook (Shen & Liang, 1992). It begins: First of all we should let students understand the meaning of fractions—“when a whole ‘1’ is divided evenly into shares, the number expressing one or more of these shares is called a ‘fraction.’” Here, the difficult points in students’ learning are understanding the concept of a whole “1” and understanding the fractional unit of a

FOCUS ON SUBSTANTIVE MATHEMATICS Like the use of textbooks, the kind of teaching advocated by reform documents is subject to different interpretations. For example, Putnam and his colleagues (1992) interviewed California teachers and state and district mathematics educators. Some thought the primary focus of the 1985 California Framework was what to teach—“important mathematical content”; others thought it was how to teach—“a call to use manipulatives and cooperative groups” (p. 214). During 1992

thanks and greatest appreciation. Without their support, not only this book, but also my whole life would be impossible. Introduction Chinese students typically outperform U.S. students on international comparisons of mathematics competency. Paradoxically, Chinese teachers seem far less mathematically educated than U.S. teachers. Most Chinese teachers have had 11 to 12 years of schooling—they complete ninth grade and attend normal school for two or three years. In contrast, most U.S. teachers