What instructional formats does the teacherís guide suggest for the classroom activities?

INSIDE TEACHER’S GUIDE

A Comparison Study of U.S. and Chinese Elementary

Mathematics Teacher’s Guides

Jianhua Li (Doctoral Student)

Department of Education, University of Chicago

E-mail: jli@midway.uchicago.edu

ICME-9, Tokyo/Makuhari, Japan

July 31- August 6, 2000

Introduction

Cross-national studies of mathematics education in China and the United States have been conducted somewhat regularly for two decades (Cai, 1995). These studies investigated a variety of issues from student achievement, curriculum, to cultural, social factors that affect mathematics education. However, studies on teacher preparation and delivery of mathematics instruction, textbook and teacher’s manuals used in China and the United States are still rare. More in-depth information is needed.

This study concerns how teacher’s guides serve the teachers from the cultural and functional perspective in the context of elementary mathematics, whether there are some differences between teacher’s guides of the two countries, namely American and Chinese teacher’s guides.

A Teacher’s guide (synonyms are teacher’s manual, teacher’s edition, teacher’s guidebook) is a tool for helping a teacher teach. As far as the function of textbooks is concerned, Eisner (1987) indicated that "the textbook and its partner, the workbook, provide the curricular hub around which much of what is taught revolved." He also mentioned that textbooks were most important in influencing what was taught in schools. The thirtieth yearbook of National Society for the Study of Education (1931) on textbook and a national survey conducted in 1976 both indicated the dominant use of textbooks in American schools (Suydam & Osborne, 1977). Tyson and Woodward (1989) concluded that "textbooks structure 75% to 95% of classroom instruction especially at the elementary level." Woodward’s observation (1986) on experienced and less experienced teachers shows that teachers followed their textbooks almost word for word. All these manifest the importance of teacher’s guides. It is also true in China that teachers follow textbooks very closely. On the other hand, textbooks and teacher’s guides have been criticized of their integrity, of offering questions that stress memory and direct recall of information, of lacking of explicit cues, of practice opportunities for students, and of proving known rules, definitions (Tyson-Bernstein, 1988, 1989; Mehlinger, 1989; Graves, Slater, 1986). Research and studies have been conducted in a variety of ways on textbooks. For example, Stodolsky (1988) studied the uses of textbooks in math and social studies. She found out that little evidence in the literature or their case studies to support that teachers teach strictly by the book and the variation exists because teachers own convictions and preferences. Graybeal (1988) examined several math and social studies materials of the 70s and concluded that conventional, teacher-centered instructional suggestions were still the main body of all teacher’s guides she examined. Flanders (1987) found that "there was a relatively steady decrease in the amount of new content over the years up through eighth grade, where less than one-third of the material is new to students." He indicated that the repetitive pattern was likely to turn off students’ interests in learning mathematics at a very young age, and the students consequently wasted many years of schooling. Stevenson and his associates (1992) pointed out that U.S. textbooks might form an obstacle to the learning of mathematics because they were long, wordy, and repetitive. Studies on teacher’s guides, though, are scarce. My interest in mathematics textbooks and teacher’s guides is that whether there are some differences between teacher’s guides of two countries, namely American and Chinese teacher’s guides. In particular, what kinds of commentaries and suggestions about the roles of teachers and students, the role of mathematics for a particular grade are given, how higher order thinking is portrayed in teacher’s guides.

Until around the early 1940s the teacher’s guide was "chiefly for elementary-school teachers and limited to the presentation and treatment of a particular student textbook" (McNeil, 1991). Since 1940 expanded guides giving more assistance had been developed. However, they were criticized for more assistance than teachers could use. Consequently, innovative simpler formats appeared. An annotated edition of the student text is such a case. The teacher’s guides from then on have been keeping the same format. Now, a teacher’s edition usually includes a rationale for the instructional textbook(s) along with an outline of the scope and sequence and description of other supplementary material. The main body of the teacher’s edition includes suggestions in overprint on a duplicate copy of the pupil’s page, insert pages to help the teachers in analyzing the material to be taught. Some new features, such as adding links to the WWW, a CD, have been added to the teacher’s edition. For the past several decades in China, the teacher’s guide, often called the teacher’s reference book, usually include general theory of instruction and child growth and development, a rationale for the text and suggestions for teaching the text. No duplicate copy of pupil’s text appears in the teacher’s reference books.

This study will examines two widely used current mathematics teacher's guides: one from the United States’ Scott Foresman - Addison Wesley Mathematics (Charles, et al., 1999) (SF thereafter) and one from the People's Republic of China’s 9-year Compulsory Education Five-Year Elementary Mathematics published by People’s Education Press of Ministry of Education (Mathematics Department of People’s Education Press, 1995) (PEP thereafter). Because third grade Chinese math book has more common topics as does fourth grade American math book, I decided to use those two grades in this study. Three main chapters of each book will be examined.

The specific questions of this study are:

What commentaries in nature do two guides contain?

What instructional formats does the teacher’s guide suggest for the classroom activities?

What kinds of teacher roles are expected?

What kinds of student roles are expected?

Does the guide suggest that the teacher ask students to clarify and justify their ideas orally and in writing?

Does the guide provide suggestions on encouraging students to try different strategies in computations and problem solving?

Does the guide provide suggestions for the teacher to pose questions that encourage students to generalize, summarize, abstract, and expand based on the examples and exercises in the text?

Methods

Three kinds of approaches have been used to explore the teaching guide: (1) applying particular criteria by the researcher, i.e., using the researcher’s own criteria to examine different areas of the textbook guide, (2) comparing what teachers want in their guides with what the guides actually offer, and (3) how guides are utilized and with what effect (McNeil, 1991). With respect to content analysis, because mathematics teaching is primarily organized by lessons, it is natural to use the lesson as basic analysis unit. For some questions, the commentary of every activity in each lesson was categorized and coded according to a checking list based on Stodolsky (1988) and Graybeal (1988)’s studies, but with less coding categories; for other questions, new checking lists were created.

Because the study compares two countries’ teacher guides, the common content of both country students’ materials has been given greater attention. After examining both third and fourth grade students’ materials, I chose to compare three chapters of Chinese third grade books with U.S. fourth grade books. The three chapters of Chinese third grade and U.S. fourth grade are as follows:

Chinese U.S.

Chapter 1 Multiplying by 2-Digit Chapter 6 Multiplying by 2-Digit Factors

Factors

Chapter 3 Dividing by 2-Digit Chapter 12 Dividing by 2-Digit Divisors

Divisors

Chapter 6 Areas of Rectangles and Chapter 8 Using Geometry

Squares

We can see that three chapters from each country match quite well on main topics. All the topics in Chinese material are new topics and are introduced for the first and the only time. Dividing by 2-digit divisors is introduced for the first time in U.S. fourth grade and multiplying by 2-digit factors is introduced first in third grade. Most of the geometric concepts in Chapter 8 of U.S. sample are first introduced at early grades.^*

Results and Findings

Seven questions were dealt with in the study and each will be discussed separately.

1. What commentaries in nature do two guides contain?

Coding

The layouts of the two country’s guides are quite different.

The main structure of SF guide can be outlined as the following:

Chapter X

Planning and Pacing (Scope and sequence chart)

Resources

Technology

Learning Environment

Assessment

Lesson X-1

At-A-Glance Planning

Facilitating and Assessing Lesson X-1 or Another Way to Learn Lesson
X-1

Options for Reaching All Learners

Lesson Organizer (with a duplicate student’s page aside)^*

Introduce (with a duplicate student’s page aside)

Teach (with a duplicate student’s page aside)

Close and Assess (with duplicate student’s page aside)

Lesson X-2

…

Unlike the American teacher’s edition, the PEP’s guide does not have the student text embedded in the teacher’s guide. The structure of the PEP’s guide can be outlined as the following:

Chapter objectives

Chapter illustration

Section A Illustration

Instructional Suggestions

1. Score and Sequence

2. (Main body of suggestions on the lesson(s)

Section B Illustration

…

Chapter Summary and Review

Illustration

Suggestions

A chapter is divided into several sections in which each section contains several lessons.

The following categories are used in examining what commentaries the teacher’s guides give:^**

--Organizer,

--Manipulatives,

--Resource,

--Script instructional suggestion ,

--Application,

--Structural explanation about the text,

--Options for reaching all learners, and

--Pedagogical mathematics knowledge.

An organizer is a section that includes any or all of the following: the objective of a lesson, materials the teacher and students are to use, and assignment guide. For example, the following is an organizer from chapter 12 of SF series:

Lesson Organizer

Objective Estimate quotients with 2-digit divisors.

Student Materials None

Assignment Guide

Basic 10-25, 33-42

Average 14-30, 33-42

Enriched 15-42

(SF Grade 4 Teacher’s Edition, volume 2, p.530. 1998)

The PEP series has similar kind of organizer, an example is included in Appendix A.

Manipulatives provide the concrete materials and their use in a chapter or a lesson. Resource refers to the text in which related resources of learning a chapter or a lesson are provided. Those resources include mathematics literature, internet, software, other materials in the teacher’s package. Structural knowledge of the text describe the rationale of how different sections of a chapter, a lesson are assembled, or explain the connection between the current text and other related sections. Pedagogical mathematics knowledge refers to students’ mathematical development, thinking, psychology of mathematics, and pedagogical aspects of a lesson. Scripted instructional suggestion is the main part of the teacher’s guide in both series. It gives suggestions on how to introduce a lesson, how to approach each example and what questions to ask or gives answer to questions in student’s book.

Result and Discussion

Table 1 shows the percentage of the teacher’s guide taken by each format. Each number in the table represents the number of sections of continuous paragraphs of one same format.

SF

PEP

Option

34

9.7%

Application

11

3.2%

Manipulatives

3

0.9%

Organizer

92

26.4%

23

21.9%

Pedagogical

34

9.7%

17

16.2%

Resource

44

12.6%

Script

110

31.5%

35

33.3%

Structural

21

6.0%

30

28.6%

Total

349

100.0%

105

100.0%

Table 1 Percentage of each teacher’s guide devoted to various formats.

First, table 1 shows that Script Instructional Suggestion tops the list in both series. Although the two guides differ in a lot of ways, to provide substantial amount of step-by-step guide for teachers is evident in both series.

Second, the another function of a teacher’s guide is to set the pace for the teacher who teaches the course. In this case, both guides give a very clear notes on what is expected to be accomplished in each lesson.

Perhaps the most salient feature of PEP series is that of Structural Knowledge of the Text that accounts a solid portion of the guide. An example from PEP series is as follows:

...

On the basis of 14x3, example 2 leads students to deduct 140x3. Because in Grade 2 students learned the convenient way with a zero at the end of a factor when they studied to use paper and pencil to solve one-digit multiplier multiplication, like 240 , knowing first to do

x 4

24x4, then add a zero at the end the result of 24x4. So even they have been not taught how to do 140x3, students should figure this out.

...

(PEP Grade 3 Teacher’s Reference Book, volume 5, p. 18. 1995.)

Structural Knowledge of the Text also tells you why the authors put this example here and that example there. They try to give you ample information about how the text has been compiled in the shape you see. Here is another example:

Multiplying by two-digit number (II)

In this section students will learn some more difficult multiplication on the basis of the algorithm of two-digit multiplication they already knew. Here, during the process of multiplying, students need to do carry over many times. It is likely that students would attend to one thing and lose sight of another when they do these computations. They may use the multiplication rule wrongly or may add wrongly of the carry numbers. Thus, the text first review multiplication-addition practice like 3x8+6, 4x9+7. Problem 5 of Exercise 3 is for the same purpose.

(Grade 3 Teacher’s Reference Book, volume 5, p. 30. 1995.)

In contrast, the SF series does not give much illustration about the structural knowledge (6% overall). Most of them are stated in tables such as when the topic is first introduced and the locations of a practice, an application, and reviews of the topic.

However, the SF series has some unique features which the PEP series does not have. Resource is one of those features. In some cases the Resource provides the Problem of the Day, Math Routines, Mental Math; in other cases the Resource provides where to find resources for further study.

The unique feature of Application in the SF series also shows input in connecting mathematics with the real world. For instance, Chapter 12’s opener theme is entertainment, as an extension, the teacher’s guide suggests students be grouped by interest to do research on the history of various forms of entertainment. The guide also suggests that students use division and probability and other math skills when reporting on their clubs (predictions about scores, seating arrangements, tickets, etc.).

In sum, the SF guide and the PEP guide differ in structure as well as in content. However, both guides provide scope and sequence and script steps for most lessons in the texts. The PEP series has more commentaries on the illustrations of the text than the SF series does. However, the SF series has some unique features that the PEP series does not have. Those unique features of the SF series include Option, Application, Manipulatives, and Resources. An example from PEP series is as follows:

...

x 4

24x4, then add a zero at the end the result of 24x4. So even they have been not taught how to do 140x3, students should figure this out.

...

(PEP Grade 3 Teacher’s Reference Book, volume 5, p. 18. 1995.)

Multiplying by two-digit number (II)

(Grade 3 Teacher’s Reference Book, volume 5, p. 30. 1995.)

In contrast, the SF series does not give much illustrations about the structural knowledge (6% overall). Most of them are stated in tables such as when the topic is first introduced and the locations of a practice, an application, and reviews of the topic.

However, the SF series has some unique features which the PEP series doesn’t have. Resource is one of those features. In some cases the Resource provides the Problem of the Day, Math Routines, Mental Math; in other cases the Resource provides where to find resources for further study. An example is included in Appendix A.

Option is another feature solely belongs to the SF series. An example is also included in Appendix A.

The unique feature of Application in the SF series also shows input in connecting mathematics with the real world. For instance, Chapter 12 opener’s theme is entertainment, as an extension, the teacher’s guide suggests students be grouped by interest to do research on the history of various forms of entertainment. The guide also suggests that students use division and probability and other math skills when reporting on their clubs (predictions about scores, seating arrangements, tickets, etc.).

In sum, the SF’s guide and the PEP’s guide differ in structure as well as in content. However, both guides provide scope and sequence and script steps for most lessons in the texts. The PEP series has more commentaries on the illustrations of the text than SF series does. However, the SF series has some unique features that the PEP series does not have. Those unique features of the SF series include Option, Application, Manipulatives, and Resources.

2. What instructional formats does the teacher’s guide suggest for the classroom activities?

Coding

Lesson Commentaries and Student Text Pages

The lesson commentaries and student text pages were examined. Activity segment in each lesson were identified and then coded for a number of instructional variables. An activity segment is a unique part of a lesson, marked by a particular format of instruction, or instructional shift from one sub-goal to another. Figure1 presents a clear case that applies to both guides. The section "Learn" is an activity segment followed by another segment "Check".

Insert Figure 1 here

An instructional format provides a global description of a norm in a classroom activity. The format classifications used in this study are:

--Recitation,

--Discussion,

--Lecture,

--Demonstration,

--Seatwork,

--Check work,

--Games,

--Q-A-C-E (Question-Answer-Clarify-Emphasize),

--Global teacher presentation,

--Work together, and

--Unknown.

Definitions for each category but not Q-A-C-E are attached in Appendix B. Every category is mutually exclusive except for the category Q-A-C-E, which is independent from the other formats. Q-A-C-E means a teacher communicates with students to clarify or emphasize something. If there is such a suggestion anywhere in the teacher’s guide, Q-A-C-E will be tallied once. This category is added as a supplement to help the analysis of the question. After all activities were coded, a reexamining process was done later in order to increase the reliability of the coding result.

Among all the categories, recitation, global teacher presentation, and work together are worth mentioning. Recitation is solely designated as the introduction part of each lesson, which is common in both countries’ guides. In this activity, the teacher either will use a projector or the blackboard to ask students answer some problems to serve as the basis of the new lesson, or ask students to explain previously learned concept(s) or procedure(s). A global teacher presentation means that when teaching the central example of a lesson, multiple teaching methods are to be applied. In most cases, this means that a teacher will apply a combination of demonstration, lecture, and discussion. As to the "work together", students are to work in peer groups on central examples of the lesson on same or different tasks.

Result and Discussion

The three chapters in U.S. and Chinese materials have a total of 38 and 54 lessons respectively.

Scott Foresman &emdash; Addison Wesley(SF)

China-

PEP(PEP)

Number of segments

%

Number of segments

%

Recitation

49

20.7%

23

12.5%

Discussion

1

0.4%

2

1.1%

Lecture

0

0.0%

3

1.6%

Demonstration

0

0.0%

0

0.0%

Seatwork

80(163)

33.8%

61

33.2%

Check Work

55

23.2%

41

22.3%

Games

1

0.4%

3

1.6%

Global Teacher

Presentation

13

5.5%

49

26.6%

Work Together

26

11.0%

0

0.0%

Unknown

12

5.1%

2

1.1%

Total

237

100.0%

184

100.0%

Q-A-C-E^*

79

71

Table 2. Instructional formats the teacher’s guides suggest.

Table 2 shows the result of each count and percentage of activity segments coded by instructional formats. Overall, instructional formats of the two countries are similar and comparable. The percentages of Seatwork and Check Work of the two guides are strikingly similar. Student discussions, teacher lectures and teacher demonstrations are none or very few in the guides. This seems to suggest that Recitation, Check work, and Q-A-C-E are common in both countries’ materials. Although the percentage of PEP’s Recitation is lower than that of SF’s, because almost in each lesson of SF series, there is a recitation at the beginning of the lesson, whereas in some of PEP’s lessons, mostly review and practice lessons, no recitations are provided, in practice, teachers will almost certain to do some Recitation in those review and practice lessons. Thus, the percentage of Recitation of the PEP series is expected to be higher than that shown in Table 2.

The big difference between the two guides is the format of presenting developmental examples. In the SF’s guide, a substantial amount of examples are students’ group work whereas all examples are global teacher presentations in the PEP’s guide. It is unclear whether this difference has an influence on students’ mathematical development and performance.

The difference between the developmental example commentaries also lies on the length of commentaries. For example, in the SF the commentary usually provides several questions for the teacher to ask students while teaching the example; a developmental example’s commentary in the PEP’s guide, on the other hand, will usually provide lengthy paragraph, detailing all the steps about what and how the teacher should proceed. The following is part of a lesson in SF’s student book:

Example

Find 60x22.

Step 1 Step 2

Multiply by the digit in the ones Multiply by the digit in the tens

place. place.

22 22

x 60 x 60

0 1,320

Estimate to check.

60x20=1,200

Since 1,320 is close to 1,200, the answer is reasonable.

There are 1,320 crew members in the race.

(SF Grade 4 Student Edition, p.254. 19998)

In the teacher’s guide, the corresponding comments are these words:

When focusing on the example, you may wish to ask these questions.

. In Step 1, why is a zero written in the product? It tells that 22 multiplied by 0 equals 0.

. In Step 2, why is the small 1 written at the top of the tens column? Because 6x2 is 12 ones. You must regroup 12 ones as 1 ten and 2 ones.

(SF Grade 4 Teacher’s Edition, volume 1, p. 254. 1998.)

In contrast, lengthy commentaries in PEP teacher’s guide are given for the following example:

In the teacher’s guide, the corresponding comments are the following paragraphs:

(1) Use a piece of paper to cover "1". First to do 3 multiplies 24, have a student tell how to multiply, what the product is, where to write the product down, and what it represents. Emphasize that 3 should multiply every digit of the multiplicand, the last digit of the product should be aligned with ones’ place, and the product 72 represents three 24 is 72. Write out "the product of 3x24" at the right of the algorithm on the blackboard.

(2) Uncover the tens’ digit "1". Ask the class: "This 1 multiplies the multiplicand, means what times what, what’s the product? How to write the answer?" Teacher may encourage any students to write the remaining process on the blackboard of the second step following the model of first sub-product. If the child is not right, ask class to discuss it and correct the mistake. Then the teacher will use red chalk to write down the second sub-product 240 and write down the explain note of "the product of 10x24" at the right of the algorithm.

(3) Ask the following question: What do we do next? The two sub-products should be added together.

After write down the whole algorithm, the teacher may lead students compare the separate steps and let them make sure that the three partial computation can be combined into one, a simple one.

At the end, summarize the whole computation process again, pay more attention to asking students when "1", which at the tens’ place of the multiplier, multiplies "4", which at the ones’ place of the multiplicand, the result is "4", What is "4" representing? Where should it be written down? Also indicate that once write down the "4" on the tens’ column, "0" on the ones’ column can be omitted.

(PEP Teacher’s Reference Book, volume 5, pp. 26-7. 1995.)

The data suggest that the PEP series places a heavier weight on developmental examples. Adding the two categories of Global Teacher Presentation and Work Together of SF series, which account for all the developmental examples, the total percentage (16.6%) is still much less than that of PEP’s Global Teacher Presentation (26.6%). Often more than one developmental example in one lesson in the PEP series contributes to this fact.

One thing needs to be pointed out, though, is that the two country’s percentages of seatwork among all activities matched very well for the two guides (33.8% vs. 33.2%) is calculated by excluding those addtionaladditional sheets in the SF series. This is one area that differs the two guides, that is, in general, SF series is more practical, it tries to provide everything teachers need whereas it is not so for the PEP’s guide. The difference between the two guides on Seatwork lies on the fact that SF series provides additional practice sheets for almost every lesson. For a typical lesson, there will be a practice sheet, a reteaching sheet, an enrichment sheet, and a problem solving sheet. This does not mean that teachers teaching SF materials will use all the sheets in a lesson. In fact, the teacher’s guide only provides all the answers to the questions in those sheets, and no instructional commentaries are given anywhere. By classroom observation and talking with teachers we know that different teachers will treat these sheets differently. The number in the blanket of table 2 shows the number of those additional instructional sheets. On the other hand, the PEP guide provides no additional seatwork exercises. In practice, a school will purchase addtionaladditional practice material on the market under the regulation of the school itself or the school district.

There is a certain amount of student cooperative work in the SF series. By examining those activities we learned that, by and large, the activities are problem solving oriented, multi-question type problems. Following is an example from the SF series.

Neighbors want to put a merry-go-round in the park. They buy one that holds 14 children.

The neighbors decide to make a sign that tells how many pounds the merry-go-round can support. The typical weight for a child who rides on merry-go-rounds is 75 pounds. Should the sign overestimate or underestimate the weight allowed?

Work Together

Understand What do you know?

What do you need to find out?

................................................................................................................................................…………….........

Plan Decide if you should To make use the merry-go-round can

overestimate or support the weight of 14 children, you

underestimate should underestimate the weight allowed.

................................................................................................................................................……….……….........

Solve To underestimate, Round 14 to 10.

round one or both 10 x 75 = 750

factors down. Then The sign should list 750 pounds as the

multiply maximum weight allowed.

................................................................................................................................................…………….........

Look Back How can you check to see if you answer is reasonable?

Talk about it

Why does it make sense to underestimate in this situation?

(SF Grade 4 Student Edition, p. 278. 19998.)

There are considerably more recitation activities suggested in the SF guide than in the PEP guide. This is because the PEP series has more exercise lessons than the SF has. Unless the new PEP lesson (new developmental example) is an easy one, one or two more exercise lessons will be followed by in order to have students practice what they’ve learned in the new lesson. In those lessons, either the students’ book or the teacher’s guide provides no recitations. On the other hand, every SF lesson is accompanied by a recitation to start with the lesson. In general, not many commentaries can be found in PEP’s exercise lessons, where teachers will have considerable flexibility on how to review and how to do the exercises.

Twelve SF activities are categorized as "unknown" in this study. Some of them are labeled by "connect" on which no commentaries are found in teacher’s guide. In addition, there are a few activities in SF series that are unique, like "home-school connection", "social studies connection" or "school-community connection". Those activities are simply ommitedomitted due to lack of information on those segments and the less relavency of this paper. How those activities take place and what their effect is are behind the scope of this study.

Summary

In sum, several forms of mathematics classroom activity are common to both series. Recitation, Check Work, Q-A-C-E, and Seatwork are what we most often see in the two series teacher’s guides. As for how to proceed the developmental examples, PEP’s guide will almost always suggest to use a Global Teacher Presentation to present the example whereas in less than half of the cases, SF will suggest teachers to use this form of approach. In other cases, the SF guide will suggest that students work together although the teacher will often pose some questions to assist students think while working on the problem at hand.

3. What kinds of teacher roles are expected?

Coding

In examining what the teacher roles are in each teacher’s guide, the following categories are used:

--Watcher-Helper,

--Recitation Leader,

--Instructor,

--Leader or Instructor,

--Action Director,

--Tester, and

--Unknown.

These categories are mutually exclusive. A description of each category is attached in Appendix C. After all activities were coded a reexamining process was done later. This time all segments were coded again according to the same classification. DescripaciesDiscrepancies were examined and final codes were given. This process was to increase the reliability of the coding result.

to increase the reliability of the coding result. Some minor change was made in this process such as one segment may belong to both categories and so definitions for each category was reevaluated and changes have been made.

Result and Discussion

Table 3 shows the result of the above classification.

SF

PEP

Watcher-Helper

83

35.0%

65

35.3%

Recitation Leader

99

41.8%

83

45.1%

Instructor

16

6.8%

4

2.2%

Leader or Instructor

10

4.2%

13

7.1%

Action Director

1

0.4%

3

1.6%

Unknown

13

5.5%

16

8.7%

Tester

15

6.3%

0

0.0%

Total

237

100.0%

184

100.0%

Table 3 Expected teacher’s roles

If we view the second to fifth categories as teacher-guided roles, then they constitute for more than 50% of teacher’s roles in both guides. The second to fifth category combine for 53.2% and 56.0% of all activities for SF and PEP series respectively. TabelTable 3 shows that in slightly more than one-third of all activities the teacher’s role is Watcher-Helper. The Watcher-Helper and Recitation Leader constitute the majority of the teacher roles. They combine for 76.8% and 80.4% of all activities of SF and PEP series samples. In both guides only a small percentage of activities are those in which teacher role is Instructor. In other words, a teacher is expected to have students’ input and contributions during most parts of math instruction. Although the two guides have almost equal percentages of Watcher-Helper, the specific teacher roles somewhat differ in the two guides. The SF guide usually suggests the teacher to present a data sheet for a problem or to give some reading assistance on a story problem. The PEP guide, on the other hand, usually provides suggestions on asking some extended questions for a problem or giving students some opportunity to talk about their thinking process..

Table 4 is a break down of teacher’s roles by instructional formats.

Teacher Roles

SF

PEP

Recitation

Recitation Leader

44

18.6%

Watcher-Helper

2

1.1%

Instructor

5

2.1%

Recitation Leader

15

8.2%

Leader or Instructor

5

2.7%

Unknown

1

0.5%

Discussion

Recitation Leader

1

0.4%

Recitation Leader

2

1.1%

Lecture

Instructor

3

1.6%

Seatwork

Watcher-Helper

50

21.1%

Watcher-Helper

42

22.8%

Recitation Leader

9

3.8%

Recitation Leader

18

9.8%

Leader or Instructor

1

0.4%

Unknown

1

0.5%

Unknown

5

2.1%

Tester

15

6.3%

Check Work

Watcher-Helper

11

4.6%

Watcher-Helper

18

9.8%

Recitation Leader

39

16.5%

Recitation Leader

12

6.5%

Unknown

5

2.1%

Unknown

11

6.0%

Games

Action Director

1

0.4%

Action Director

3

1.6%

Global Teacher Presentation

Recitation Leader

4

1.7%

Watcher-Helper

4

2.2%

Instructor

4

1.7%

Recitation Leader

36

19.6%

Leader or Instructor

5

2.1%

Leader or Instructor

8

4.3%

Unknown

1

0.5%

Work Together

Watcher-Helper

23

9.7%

Leader or Instructor

3

1.3%

Unknown

Watcher-Helper

1

0.4%

Unknown

2

1.1%

Recitation Leader

1

0.4%

Instructor

6

2.5%

Leader or Instructor

1

0.4%

Unknown

3

1.3%

Total

237

100.0%

184

100.0%

Table 4 A break down of teacher’s roles by instructional formats.

From this table, we can see some similarities and differences between the teacher roles of the two guides.

Work Together is unique for the SF series. In 23 out of 26 activities of group work, the teacher's role is that of Watcher-Helper. Although the teacher’s role is Watcher-Helper in most Work- Together activities, it doesn’t mean that the students will be "roaming" in the problem. In fact, much of the directions are in students’ book. Questions at various stages of the activity are provided to assist students working through the problem. Suggestions in teacher's guide provide information on how to prepare for the activity, how to divide students into work groups and so on. Following is an example of this role.

Lesson 1 Exploring Multiplication Patterns

Get Started You may wish to assign the roles of reader, calculator operator, and recorder, as well as the group skill Check for Understanding that students can practice. Share Level 4 of the Assessment Rubric with Students before they begin their work.

(SF Grade 4 Teacher’s Edition, volume 1. p. 250B. 19998.)

In the PEP's guide, there is no place in which the text suggests students work together or the teacher being an Action Director. However, the teacher’s role is tallied as Action Director for three games in PEP sample. The first two ask every three children to form a group to play the game. The third one asks the teacher to present nine division estimation problems and each student will use a 9-squared table to play the game. Those instructions are all stated in students’ books although no words about these games are mentioned in teacher’s guide.

The above table also shows that for Global Teacher Presentation (all activities are developmental examples), the teacher can take one of three roles (recitation leader, instructor, or leader or instructor in the SF series) which role was judged by the main part of the activity. Additionally, combining both Global Teacher Presentation and Work Together sections of Table 3 (this equals to all the developmental examples), if we view Watcher-Helper as more student-centered activity and Recitation Leader, Instructor, and Leader or Instructor as more teacher-centered activities, about 41% of those activities are more teacher-centered (Recitation Leader, Instructor, and Leader or Instructor), and 59% are more student-centered (Watcher-Helper) in the SF series. Global Teacher Presentation accounts for about 90% of all developmental activities in the PEP series, with long step-by-step suggestions in the teacher’s guide in PEP series. In contrast, in the SF series, developmental examples are often treated with some key questions for the teacher to ask students while they are working on the example. This seems to suggest that there are two different approaches in the two teacher’s guides on developmental examples.

There are some differences between two countries’ guides on the teacher’s roles as Recitation Leader. In most Recitation Leader activities of the SF series, the teacher’s role is to lead students to review previously learned skills or knowledge and get students prepared for the upcoming new examples. They account for 90% of Recitation activity. In the PEP series, although Recitation Leader is the most frequent teacher role in a Recitation activity, there are some cases in which Watcher-Helper or Leader or Instructor is a teacher’s role. Unlike developmental examples, recitation activities all are whole-class activities in both series. Interestingly enough, all recitation activities in the SF series are provided only in the teacher’s guides, whereas almost all those activities are presented in the students’ books of the PEP series. Still, whether this difference shows any effect on the degree of emphasis on recitation activities is not clear. Studies on the uses of teacher’s guides can be helpful in answering this question. From this author’s personal experiences working at PEP, we know that the PEP series authors put an effort on building bridges between the new content and students’ previously learned knowledge. Those recitation activities are new added content of the 90s series.

Another difference between the two series’ recitation activities is that in the SF series, an icon of the overhead projector is always presented in teacher’s guides, whereas in the PEP series, blackboard use by the teacher is often mentioned in the teachers’ guides. This certainly says that in China, perhaps using the blackboard is still the backbone of instruction in the classrooms.

Table 4 also showsThere are some differences between teacher’s roles in the "Check" activities. In most of the "Check" activities of the SF series, the teacher’s role is Recitation Leader, whereas the teacher’s role in 41% of PEP’s "Check" activities is Watcher-Helper. Following is an example from the SF series:

Check

Reading Assist Find Main Idea with Supporting Details

Have students read the Check and Practice exercises aloud. Ask volunteers to paraphrase the main ideas and list details.

(Grade 4 Teacher’s Edition Volume 1. pp. 279. 19998)

In the SF series, the teachers’ guide sets up a section for suggestions of each Check. This is not the case in the PEP series, where Checks are not always mentioned in teacher’s guides.

The teacher’s roles also differ in Seatwork. For instance, it is unique that SF series provides a number of tests and quizzes for which the teacher is the test supervisor. In the PEP series, in seatwork activities the teacher is usually Recitation Leader.

Something the PEP series is likely to do in recitation activity is to provide some questions or problems for students to try to establish a connection between the review questions and the new topic. For example, before learning how to multiply by a two-digit number, two review questions are presented in the student’s book:

(1) 14x2 31x30 214x3 (Paper & Pencil)

(2) One box of crayon has 24 crayons, how many crayons are there in three

identical boxes?

The teacher’s guide suggests to "ask students to talk about the procedure of (1) and appoint a student to come to the blackboard to show his/her work of (2)".

The next developmental example is:

One box of crayon has 24 crayons. How many crayons are there in 13 same boxes?

(PEP Grade 3 MathematicsGrade 3 Mathematics, volume 5, p.6. 1995)

In sum, there are some similarities and differences between the teacher’s roles in the two series. The biggest difference seems to be the approaches towards developmental examples. In the SF series, the teacher’s roles seem to be less teacher-centered. In the PEP series, teacher’s roles are more teacher-centered. However, teacher-student interactions are emphasized in both series. Even in those activities in which a teacher’s role is to be the director of an action, students can follow the questions in their book to assist them in working through the problem. Plus, key questions are provided in teacher’s guides.

4. What kinds of student roles are expected?

Coding Process

In examining what the student roles are in each teacher’s guide the following categories are used:

--Question-Answer,

--Discuss,

--Listen-Watch,

--Question-Answer-Discuss-Listen,

--Solve-Question-Answer,

--Solve-Discuss,

--Solve-Explain-Paper & Pencil,

--Play Games, and

--Unknown.

These categories are mutually exclusive. A description of each category is attached in Appendix D. After all activities were coded a reexamining process was done later. This time all segments were coded again according to the same classification. Diescreipancies were examined and final codes were given. This process was to increase the reliability of the coding result.

Result and Discussion

Table 45 shows the result of the above classification. The numbers in the second and forth column are numbers of activity segments for each category of corresponding student’s role in the teacher’s guide of the two series respectively. The percentage next to the number is the percentage of this type of student’s role of all the activities in that series.

Student Roles

SF

PEP

Question-Answer(QA)

53

22.4%

37

20.1%

Discussion(DIS)

8

3.4%

3

1.6%

Listen-Watch(LW)

5

2.1%

10

5.4%

Question-Answer-Discussion-

Listen(QADL)

10

4.2%

29

15.8%

Solve-Question-Answer(SQA)

28

11.8%

32

17.4%

Solve-Question-Answer-

Discussion(SQAD)

7

3.0%

2

1.1%

Solve-Explain-Paper &

Pencil(SEPP)

95

40.1%

60

32.6%

Games(GA)

2

0.8%

3

1.6%

Work Together(WT)

17

7.2%

0

0.0%

Unknown(UN)

12

5.1%

8

4.3%

Total

237

100.0%

184

100.0%

Table 45 Expected Students’ Behavior

Table 45 shows that only three activities constitute are more than 10% of all the activity segments in the SF series and four activities are more than 10% of all segments in the PEP series and the rest are all below 10%. Among those categories over 10%, Solve-Explain-Paper & Pencil, Question-Answer, and Solve-Question-Answer are more than 10% in both series. They combine for 74.3% and 70.1% for the SF series and the PEP series respectively. The percentages of the above three categories combined in two series are close and constitute the majority of all activities. Of the above three categories, Solve-Explain-Paper & PencilSEPP activities in both series are from practices and exercises in the text following the developmental examples. Except for problems with asterisks in the PEP series, which do not require all students to finish them, those practices and exercises are expected to be finished in class by all students. It is not clear, though, that how those problems in practices and exercises are to be grouped and finished, i.e., there are no suggestions about where to make a stop or whether to check students’ work in a teacher-led group. Most suggestions are for specific problems. So, it is totally up to the teacher to decide whether to let the whole class to do all the problems without any interruption or whether the teacher to control the pace of those activities.

Two types of classroom activity are worth mentioning. One is when a teacher asks a students to tell the class how to solve a problem and then let the whole class solve the problem(QA). The other is after students finish the problem the teacher then asks students to tell how they solved the problem (SQA). For the first type, the percentages of the two series are very close, 22.2% for the SF and 20.1% for the PEP. The PEP series has a higher percentage of second type than that of the SF series. Generally speaking, in the PEP series, there are more activities in which teacher’s guide suggests that after students solve a problem or do some practice that the teacher let students tell how they solved the problem and than solidify the procedure students have just been studying.

There are more activities in the PEP series (15.9% vs. 3.8%) in which the students’ role is "Question-Answer-Discussion-Listen"(A combination of the Question-Answer, Discussion, and Listen-Watch) first three). An Following is an example from the PEP series follows:.

Teach 48x72=

x 72

You may ask students to tell to the class which number to multiply first, which number to multiply after that, and what to do next. Emphasize on how to carry in each step. For instance, using tens-digit "7" to multiply 48, 7 times 8 is 56, write 6 on tens’ place, write a small "5" lightly on hundreds’ place. 7 times 4 is 28, 28 plus 5 is 33. Then write "3" on hundreds’ place and cover the small "5" so to avoid adding 5 twice.

As an alternative, you may let students to do the computation by themselves and ask a student come to the blackboard to show his/her work at the same time. Then correct the errors with students after observing students work and collecting the errors they made.

(PEP Teacher’s Reference Book, volume 5, p.31. 1995.)

Thus, for developmental examples, the PEP series tends to give a combination of suggestions of the first three categories (Question-Answer, Discussion, and Listen-Watch) whereas few cases in the SF series do so.

The SF series has a unique type of activity, which is Work Together. As mentioned earlier, students of one class will be divided into several groups. In each group, individual students will be assigned to different roles, such as a reader, a recorder, a dice roller, and a reporter. This type of activity provides a good opportunity for students to develop their ability to work with others as well as to take individual responsibility. However, this type of activity is not designed to let students work independently from their teachers. The Tteacher’s’ assists are provided in teacher’s guide. Following is such an example:

You may wish to ask questions such as the following as you observe students at work.

· How did you determine the total time the batteries would last?

· How did you determine how long each event lasts?

Make sure students understand what they are to do. Invite a volunteer to read aloud the questions in the Understand section. Then ask students to tell in their own words what they think their task is. ...

(SF Grade 4 Teacher’s Edition, p. 537. 19998.)

If we treat students’ role of Work Together as QADL and add it to QADL, we got an 11.4%. This seems to match PEP series’ 15.8% for that type of student’s role.

Insert Table 6 here

Table 6 includes a breakdown of the students’ behavior by types of activities. The first column contains the types of activity defined in Question 2 of this study. Numbers in the third column are numbers of activity for every student behavior. The percentage next to the number is the percent of this behavior of all the activities. Table 6 is a break down of the students’ behavior by types of activities.

Table 6. A break down of Expected students’ behavior by instructional formats.

From this table, we knowfind out that for the instructional format Recitation, QA is the dominant role of students in the SF series. This means most often the teacher will pose questions and ask students to answer them orally. For the PEP series, the student’s’ role in Recitation is SQA or QA for the most part. As mentioned earlier, SQA is one way for the teacher to lay a good basis for students’ ensuing learning of a new lesson. This is what the PEP series tried to accomplish for the whole series.

Seatwork is a major part of student’s activity. From the Seatwork section in Table 6, we learn that student’s independantindependent work, i.e., Solve-Explain-Paper & Pencil, is the most frequent type of work: 65 out of 80 activity segments for the SF series and 42 out of 61 activity segments for the PEP series were counted for this type of student’s role. A Another difference can be found through examining the teacher roles in Seatwork.minor difference exists between the two series on those independent work, though. Although using paper & pencil to solve problems is frequent in Seatwork activities, Tthe PEP teacher’s guide provides some general suggestions from time to time that the teacher may talk to students and ask them questions in order to let students understand the problem better either before asking students to solve the problem the teacher may talk to students and ask them questions in order to let students understand the problem better, or after students finish solving the problem and then ask them some follow up questions. In addition, the PEP guide sometimes suggests that the teacher to ask students how they solved the problem and help students clarify their thinking.

The data also show that there are more interactions between the students and their teacher and among students in the PEP series’ Seatwork than in the SF series’ Seatwork. The QA(10) and SQA(9) take 31.1% of all Seatwork activity in the PEP series whereas QA(1),QADL(1), and SQA(11) in SF series take only 16.3% of all Seatwork activity.

The section breakdown of Check Work break down shows that the two series have comparable percentages of teacher-students and, student-student interaction. In the PEP series QA(11), DIS(1), QADL(1), and SQA(8) take 51.2% of all Check Work activities. In the SF series QA(11), DIS(2), SQA(11), and SQAD(1) take 45.5%.

It is better tonot vieweasy to interpret the data of Global Teacher Presentation and Work Together (as the type of activity) together since these two kinds of activity are primarily developmental examples. From this angle we see from the table 6However, it is obvious that each series stresses one kind of student role, namely QADL in the PEP series and Work Together (as the student’ss’ role) in the SF series. From this author’s personal experience, one consideration of the PEP series’ authors is that they would like to see developmental examples be taught and learned effectively. and Iif the teacher let students work more freely, they would be worrying about that students couldn’t get what they should have learned by the end of the period.

In shortsum, about one-third or 40% of all activities are students’ paper- and- pencil exercises for the PEP and the SF series respectively. The sSecond highest percentage activity for both series is QA in which a teacher will lead the activity and students will be asked to answer questions (in most cases need oral answers are needed). Those questions could be a computation, a definition, a concept, or a learned procedure. Besides the above two types of activities, most activities need students to interact with their teacher or whole class discussions and short paper- and- pencil work. Roughly speaking, the student roles in both seriess seems to followhave a similar pattern. However, students’ roles are more active and less active in different type of activities. There is mMore group works for the SF students and more students-teacher interactions for certain types of activities in the PEP series.

5. Does the guide suggest that the teacher ask students to clarify and justify their ideas orally and in writing?

The Learning principle of Principles and Standards for School Mathematics: Discussion Draft (NCTM, 1998) states that "mathematics instructional programs should enable students to understand and use mathematics." Suggestingon that teachers askof asking students to explain their thinking is one way of achieving the above goal. In order to find out how the commentaries and the text of each guide ask students to explain their thinking, to clarify the concept and procedure they learn, any commentary which falls into the above categories wais tallied.

Result and Discussion

SF(38 Lessons)

PEP(54 Lessons)

Chapter 6(12)

29

Chapter 1(19)

26

Chapter 8(15)

14

Chapter 3(25)

28

Chapter 12(11)

24

Chapter 6(10)

4

Total

67

58

Table 57 Number of Commentaries about students’ thinking. Numbers in parentheseis are number of lessons in the chapter.

As stated earlier in this paper, three chapters of each series were chosen as the sample of this study. This is also the case in this particular question. As stated in the Principles and Standards for School Mathematics: Discussion Draft that "number and operation continue to be conerstonescornerstones of the curriculum in grdesgrades 3-5," (NCTM, 1998) and the same importance of number and operations in Chinese mathematics curriculum of the same grade band, two chapters, i.e., chapters 6 and 12 of the SF series and chapters 1 and 3 of the PEP series, belong to the core curriculum of the grade band. Chapter 8 of the SF series and Chapter 6 of the PEP sereisseries are geometry chapters, which are main strands in this grade band as well.

Table 57 shows the number of commentaries asking students either to explain their thinking or to talk about a concept or a procedure. From the Ttable 57, we find that the SF series has more commentaries about students’ thinking than PEP series has. In most SF lessons, there is a section near the end of the developmental example called "Talk about it" in which most often questions asked most often about the developmental example are identifiedasked for further explanations by the students. The teacher’s guide provides suggestions for each "Talk about it". For instance, Lesson 4 of Chapter 6 is Exploring Multiplying with 2-Digit Factors., Aafter students work together to show a multiplication with place-value blocks, "Talk About It" poses this question: "Can you show 12 x 15 using 1 hundred block and 8 tens? Explain". The commentary in the teacher’s guide states: "Listen for understanding of the partial products". A sample answer follows this question. In other cases, the SF guide provides suggestions about students’ thinking in the closing and assessing section. Here is an example:

Interview

Ask students to explain how they solved one of the Skills and Reasoning exercises.

(SF Grade 4 Teacher’s Edition, volume 1, p. 255. 19998.)

The PEP series provides a somewhat different pattern of commentaries about explaining students’ thinking. Because the teacher’s guide often suggests that the teacher use the dialectic method in developmental examples, commentaries about explaining students’ thinking are often inside suggestions of Check Work and lesson introduction. For instance, after teaching how to do the mental computation of 14 x 3, the commentary in the ensuing Check Work suggests that the teacher ask students to explain how they foundsolved 16 x 2, 26 x 3, and 25 x 2 after student have finished the computations.

It is interesting to find out from Ttable 67 that Cchapter 8 of the SF series and Cchapter 6 of the PEP series, both geometry chapters, have the least number of the commentaries above students’ thinking. How can this be happening? It seems that both series pay less attention to asking students to explain their thinking about geometric figures and their properties. Apparently, Tthis shouldn’t be the case for geometry as it is important for students to understand better by asking them to explain their thinking.

6. Does the guide provide suggestions on encouraging students to try different strategies in computations and problem solving?

Over the past two decades, a growing consensus among educators favors a shift in mathematics instruction from a curriculum in which children learn and practice the standard school algorithms to one in which reasoning, problem solving, and conceptual understanding play a major role. Encouraging students to try different strategies in solving computation problems and in problem solving is one way to achieve the above shift. In order to find out how the commentaries and the text of each guide ask students to try different strategies in computations and problem solving, any commentary that falls into the above description wasis tallied.

Result and Discussion

SF

PEP

Chapter 6

p.260

1

Chapter 1

p.20

1

p.266

1

p.34

1

p.267

1

Chapter 3

p.84

1

p.280

1

p.85

1

p.281

1

Chapter 6

p.169

1

Chapter 8

p.370B

1

Chapter 12

p.534

1

Total

7

5

Table 68 Number and pages of commentaries about using different strategies in computation and problem solving.

Table 68 shows that there are a few times in which teacher’s guides ask students to try different methods to compute or solve problems.ways about a computation or a problem solving. Some examples are given below.

In the SF series Chapter 6 Lesson 5, Multiplying with 2-Digit Factors, two different ways of computingsolving 12 x 26 are given.

Bryan’s Way Arianna’s Way

2 6 2 6

x 1 2 x 1 2

1 2 5 2

4 0 2 6 0

6 0 3 1 2

2 0 0

3 1 2

The teacher’s guide suggests to have groups compare Bryan and Arianna’s wWay and share their comparisons.

Following is aAnother example:

Portfolio

Have students write 4-digit by 2-digit multiplication examples that could be solved by each of these methods: calculator, paper and pencil, and mental math. Have them explain their choice and solve the examples.

(SF Grade 4 Teacher’s Edition, volume 1, p. 267. 19998.)

An example from the PEP series isshows a similar case to of the first example above.

When teaching 70¸ 14, you may direct students to discuss what number to start dividing? Then ask students to compare different ways to start dividing and decide which one is the best. Then direct student to read the text, ask them: can you explain how Fang, Ming, and Yong’s thinking. What’s your opinion about those different ways? Also, the following way is worth trying: round 14 to 15, five 15 is 75, so start 5 first.

(PEP Teacher’s Reference Book, volume 5, p.84. 1995.)

There are other PEP cases in which after students’ discussion of different approaches to a computation, the teacher’s guide suggests that the teacher to tell students to use the most efficient one. Additionally, since using calculators in grade school is not yet allowed in China and whether to use mental math or paper-and- & pencil is clearly stated, there are no cases in which students can choose one computational algorithm over the other. In the students’ text, however, there are two problem solving problems in which students are asked to use two different ways to solve them. In one problem, students are asked to answer how many ways to count the number of squares in a rectangular grid. The other one is a three-step story problem with an asterisk.

7. Did the guide provide suggestions for the teacher to pose questions that encourage students to generalize, summarize, abstract, and expand based on the examples and exercises in the text?

In order to find out how the commentaries and the text of each guide ask students to compare, to reason, to generalize and abstract, to summarize, and to extend, any commentary which falls into the above classification wasis tallied. The definitions for the above classification are attached in Appendix E.

Result and Discussion

SF

PEP

Compare

12

28.6%

Compare

17

37.8%

Generalize & Abstract

8

19.0%

Generalize & Abstract

4

8.9%

Reasoning

22

52.4%

Reasoning

13

28.9%

Summary

6

13.3%

Expand

5

11.1%

Total

42

100.0%

45

100.0%

2. What instructional formats does the teacher’s guide suggest for the classroom activities?

An instructional format provides a global description of a norm in a classroom activity. The format classifications used in this study are: Recitation, discussion, lecture,

Demonstration, seatwork, check work, games, Q-A-C-E* (Question-Answer-Clarify-Emphasize), global teacher presentation, work together, and unknown.

Every category is mutually exclusive except for the category Q-A-C-E, which is independent from the other formats. Q-A-C-E means a teacher communicates with students to clarify or emphasize something. If there is such a suggestion anywhere in the teacher’s guide, Q-A-C-E will be tallied once. This category is added as a supplement to help the analysis of the question. After all activities were coded, a reexamining process was done later in order to increase the reliability of the coding result.

Among all the categories, Recitation, Global teacher presentation, and Work together are worth mentioning. Recitation is solely designated as the introduction part of each lesson, which is common in both countries’ guides. In this activity, the teacher either will use a projector or the blackboard to ask students answer some problems to serve as the basis of the new lesson, or ask students to explain previously learned concept(s) or procedure(s). Global teacher presentation means that when teaching the central example of a lesson, multiple teaching methods are to be applied. In most cases, this means that a teacher will apply a combination of demonstration, lecture, and discussion. As for the Work together, students are to work in peer groups on central examples of the lesson on same or different tasks.

Result and Discussion

The three chapters in U.S. and Chinese materials have a total of 38 and 54 lessons respectively.

Scott Foresman &emdash; Addison Wesley (SF)

China-

PEP (PEP)

Number of segments

%

Number of segments

%

Recitation

49

20.7%

23

12.5%

Discussion

1

0.4%

2

1.1%

Lecture

0

0.0%

3

1.6%

Demonstration

0

0.0%

0

0.0%

Seatwork

80(163)

33.8%

61

33.2%

Check Work

55

23.2%

41

22.3%

Games

1

0.4%

3

1.6%

Global Teacher

Presentation

13

5.5%

49

26.6%

Work Together

26

11.0%

0

0.0%

Unknown

12

5.1%

2

1.1%

Total

237

100.0%

184

100.0%

Q-A-C-E

79

71

Table 2. Instructional formats the teacher’s guides suggest.

Table 2 shows the result of each count and percentage of activity segments coded by instructional formats. Overall, instructional formats of the two countries are similar and comparable. The percentages of Seatwork and Check Work of the two guides are strikingly similar. Student discussions, teacher lectures and teacher demonstrations are none or very few in the guides. This seems to suggest that Recitation, Check work, and Q-A-C-E are common in both countries’ materials. Although the percentage of the PEP’s Recitation is lower than that of the SF’s, because almost in each lesson of the SF series, there is a recitation at the beginning of the lesson, whereas in some of the PEP’s lessons, mostly review and practice lessons, no recitations are provided, in practice, teachers will almost certain to do some Recitation in those review and practice lessons. Thus, the percentage of Recitation of the PEP series is expected to be higher than that shown in Table 2.

The big difference between the two guides is the format of presenting developmental examples. In the SF guide, a substantial amount of examples are students’ group work whereas all examples are global teacher presentations in the PEP guide. It is unclear whether this difference has an influence on students’ mathematical development and performance.

The difference between the developmental example commentaries also lies on the length of commentaries. For example, in the SF the commentary usually provides several questions for the teacher to ask students while teaching the example; a developmental example’s commentary in the PEP guide, on the other hand, will usually provide lengthy paragraph, detailing all the steps about what and how the teacher should proceed. The following is part of a lesson in SF’s student book:

Example

Find 60x22.

Step 1 Step 2

Multiply by the digit in the ones Multiply by the digit in the tens

place. place.

22 22

x 60 x 60

0 1,320

Estimate to check.

60x20=1,200

Since 1,320 is close to 1,200, the answer is reasonable.

There are 1,320 crew members in the race.

(SF Grade 4 Student Edition, p.254. 1998)

In the teacher’s guide, the corresponding comments are these words:

When focusing on the example, you may wish to ask these questions.

. In Step 1, why is a zero written in the product? It tells that 22 multiplied by 0 equals 0.

. In Step 2, why is the small 1 written at the top of the tens column? Because 6x2 is 12 ones. You must regroup 12 ones as 1 ten and 2 ones.

(SF Grade 4 Teacher’s Edition, volume 1, p. 254. 1998.)

In contrast, lengthy commentaries in the PEP teacher’s guide are given for the following example:

In the teacher’s guide, the corresponding comments are the following paragraphs:

(3) Ask the following question: What do we do next? The two sub-products should be added together.

After write down the whole algorithm, the teacher may lead students compare the separate steps and let them make sure that the three partial computation can be combined into one, a simple one.

(PEP Teacher’s Reference Book, volume 5, pp. 26-7. 1995.)

The data suggest that the PEP series places a heavier weight on developmental examples. Adding the two categories of Global Teacher Presentation and Work Together of the SF series, which account for all the developmental examples, the total percentage (16.6%) is still much less than that of PEP’s Global Teacher Presentation (26.6%). Often more than one developmental example in one lesson in the PEP series contributes to this fact.

One thing needs to be pointed out, though, is that the two country’s percentages of seatwork among all activities matched very well for the two guides (33.8% vs. 33.2%) is calculated by excluding those additional sheets in the SF series. This is one area that differs the two guides, that is, in general, the SF series is more practical, it tries to provide everything teachers need whereas it is not so for the PEP guide. The difference between the two guides on Seatwork lies on the fact that SF series provides additional practice sheets for almost every lesson. For a typical lesson, there will be a practice sheet, a reteaching sheet, an enrichment sheet, and a problem solving sheet. This does not mean that teachers teaching SF materials will use all the sheets in a lesson. In fact, the teacher’s guide only provides all the answers to the questions in those sheets, and no instructional commentaries are given anywhere. By classroom observation and talking with teachers we know that different teachers will treat these sheets differently. The number in the blanket of table 2 shows the number of those additional instructional sheets. On the other hand, the PEP guide provides no additional seatwork exercises. In practice, a school will purchase additional practice material on the market under the regulation of the school itself or the school district.

Neighbors want to put a merry-go-round in the park. They buy one that holds 14 children.

Work Together

Understand What do you know?

What do you need to find out?

................................................................................................................................................…………….......

Plan Decide if you should To make use the merry-go-round can

overestimate or support the weight of 14 children, you

underestimate should underestimate the weight allowed.

................................................................................................................................................…………….......

Solve To underestimate, Round 14 to 10.

round one or both 10 x 75 = 750

factors down. Then The sign should list 750 pounds as the

multiply maximum weight allowed.

................................................................................................................................................…………….......

Look Back How can you check to see if you answer is reasonable?

Talk about it

Why does it make sense to underestimate in this situation?

(SF Grade 4 Student Edition, p.278. 1998.)

There are considerably more recitation activities suggested in SF guide than in the PEP guide. This is because the PEP series has more exercise lessons than the SF has. Unless the new PEP lesson (new developmental example) is an easy one, one or two more exercise lessons will be followed by in order to have students practice what they’ve learned in the new lesson. In those lessons, either the students’ book or the teacher’s guide provides no recitations. On the other hand, every SF lesson is accompanied by a recitation to start with the lesson. In general, not many commentaries can be found in the PEP’s exercise lessons, where teachers will have considerable flexibility on how to review and how to do the exercises.

Twelve SF activities are categorized as "unknown" in this study. Some of them are labeled by "connect" on which no commentaries are found in teacher’s guide. In addition, there are a few activities in the SF series that are unique, like "home-school connection", "social studies connection" or "school-community connection". Those activities are simply omitted due to lack of information on those segments. How those activities take place and what their effect is are behind the scope of this study.

Summary

In sum, several forms of mathematics classroom activity are common to both series. Recitation, Check Work, Q-A-C-E, and Seatwork are what we most often see in the two series teacher’s guides. As for how to proceed the developmental examples, the PEP guide will almost always suggest to use a Global Teacher Presentation to present the example whereas in less than half of the cases, SF will suggest teachers to use this form of approach. In other cases, the SF will suggest that students work together although the teacher will often pose some questions to assist students think while working on the problem at hand.

DiscussionTable 79. Commentaries about compare, reason, generalize and abstract, summarize, and extend.

Table 79 shows that Reasoning and Compare are the primary, if not dominant, forms of students’ activities of "doing" mathematics in the SF series^* . For "Reasoning", it is more often that the teacher’s guide poses some good questions for student to think about, like the following one asking students to reason several questions.

You may wish to ask questions such as the following as students begin work on this activity. ( based on a given roller coaster table)

· Will you select the roller coasters you think you want to ride first, or will you use to find the waiting times?

· What calculations will you use to find the waiting times?

· What method(s) will you use to find the waiting time?

(SF Grade 4 Teacher’s Edition, volume 1, p. 270. 19998.)

A number of "Reasoning" suggestions come from students’ text, like the one below:

Journal

Suppose you need to find the products for 2,400 x 10 and 2,420 x 18. If you could use a calculator for only one problem, which would it be? Explain.

(SF Grade 4 Student’s Edition, volume, 1, p.272. 19998)

In PEP series, Reasoning and Compare are strong too. Most of PEP series’ Reasoning come from students’ text (11 out of 13). OneAn example of such reasoning exercise follows:

Are the following computations correctright? What’s your opinion?

(PEP Grade 3 MathematicsGrade 3 Mathematics, volume 5, p. 15. 1995.)

Differences exist between the two guides in "Compare". First, 16 out of 17 PEP Compares are from the students’ text. It is quite evident that PEP series tries to instill some questions in the teacher’s guide and try to create opportunities for students to really understand the concepts or procedures they learn by comparing the common ground and differences of related concepts and procedures. For instance, when learning the new example 24 x 13, the teacher’s guide suggests:

Just before students are going to learn 24x13, ask students: Can you do this computation? Can you change this one into something you’ve learned? Students may have difficulty answering these questions. At this time, you can ask students to compare this example 24x13 and review problem 2, 24 x 3, ask students to tell the similarity and difference between the two computations.

(PEP Grade 3 Teacher’s Reference Book, volume 5, p.25. 1995.)

Besides Ggeneralize and Aabbstract, which is comparable to that of the SF series, the PEP series has two unique categories thatwhich the SF series doesn’t have: Ssummary and Eextend. For instance, after two developmental examples of multiplying with 2-digit factors, the teacher’s guide suggests that:

Guide students to compare the above two examples and summarize the procedure. You may wish to use one of the following ways to do this: a. ask students to supplement the key words while you summarize, ‘first to use the _____ place number of the multiplicantmultiplicand to multiply the multiplier, the last digit of the product should align with ____, ... b. ask students to use their own language to summarize what to do first, second, and third to finish the calculation.

(PEP Grade 3 Teacher’s Reference Book, volume 5, p.27-8. 1995.)

In sum, both series seem to pay more attention to reasoning as well as comparing. Other thinking skills, such as to generalize and abstract, are less stressed in the two series. However, the PEP series also provides examples of summarizing and extending, which are not cases are found in the SF series.

Discussion and Conclusion

Regardless theof differences of teachers’ background of the two countries and the differences of their curricula, the current teacher’s guides provide much more detailed step-by-step suggestions for classroom instruction. By and large, not much room has left for teachers to choose other alternatives for most of the lessons. The SF teacher’s guide is a more user-friendlier book than the PEP is. It is much easier for the SF guide users to open up the guide and begin teaching to studentsthe young kids just by following the steps provided in the guide. On the other hand, most PEP guide users, especially new teachers or new users, will find it very difficult to engage instruction with the teacher’s guide open. In fact, almost all elementary schools ask their teacher to prepare a lesson plan (with exception for some veteran teachers) based on the teacher’s guide for each lesson they teach. This may also explain in part why the PEP teacher’s guide has much more systematic illustrations for every lesson. The philosophy is simple: because the size of students’ book is only 5.5x8 (half of a standard letter sheet), you can’t includeadd everything in the book. Also, that the heavy teaching loads of American teachers documented by cross-cultural studies may explain the make up of the teacher’s guide (For example, see Adelman, 1998)Trying to beat the clock: Uses of teacher professional time in three countries, Department of Education, 1998). American teachers’ short period of time at school without teaching or being with students may not allow teachers to study those illustrations and rationales about the content they teach. Nevertheless, more detailed scripted commentaries are evident in both guides.

This study finds out that as far as the instructional formats of teachers’ guides are concerned, regardless what shapes or forms they may be, one big difference between the two guides is in the developmental examples. This difference seems to reflect the different visions on classroom instruction. In the U.S., cooperation learning, group activity and student-centered learning are the focal point of classroom instruction.

In Chinese classrooms, however, teachers always want to make sure that students can solve similar problems to the example after they teach an example. Part of the reason is that the examples are always new to the students and teachers will almost have no chance to re-teach those examples and to allow students work independently wouldn’t always guarantee they got the idea of what they should have known.

It is not surprising that the PEP guide provides lengthy commentaries on developmental examples. From the experience of this author, Chinese textbook s authors expect mathematics teachers to follow the teacher’s guides very closely, they have been trying hard to revised teacher’s guides to add more script- like questions and suggestions for teachers to follow. Additionally, without the restriction of only the marginal areas being used for commentary, commentaries can go freelancing.

The data on teacher’s roles show an interesting result. Even though teachers they may have different things to do, theirteacher’s roles in the two guides are similar. The two categories, Watcher-Helper and Recitation Leader, are the basic roles for a teacher. TeachersThey will lead a conversation or ask student to answer some questions orally, or they will be a Wwatcher-Hhelper assisting students work. The two categories, Watcher-Helper and Recitation Leader, are the basic roles for a teacher. However, comments and suggestions on which teacher’s role being discussed as Watcher-Helper are not well written in both guides, but at least they are concise. To be a skillful teacher as a Watcher-Helper, more comments seem to be needed.

The table of break down of teacher’s roles in different mathematical activities seems to indicate that the form of activity itself doesn’t mean much. Even in Seatwork, aswhere one might think that the only thing a teacher can do is to be a Watcher-Helper, a teacher can be more active. This also leads us to think about the degree of convictioncredence in of cross-cultural studies. It seems that how the teachers and students engage in classroom activities is more important than what forms of activities they take.

The data about students’ roles show that the common roles for students are: to use paper and pencil to solve or to explain math problems: , to answer their teacher’s questions orally:, and to solve a question(s) and answer questions about the problem(s) they solved. The data also show that the SF series has more student-student interaction than the PEP series has. The SF series has a combined 10.6% of Discussion and Work Together whereas the PEP series only has 1.6% of the two categories. Thise result is consistent with the teacher’s roles result in that the PEP series has more teacher-centered roles. The data do not support the conclusion that teacher-centered instruction isare still the main body of teacher’s guides as we see only small percentages of activities categorized as Lecture or, Demonstration., Rrather we see balanced programs with student-student and, students-teacher interactions throughout the texts.

The implication of this study is twofold. By comparing two guides we know that we can learn something from both guides. For one thing, the SF guide is like a resource book. It is so much convenient for the teachers, especially novices. It’s Journal section provides a good opportunity for students to write down their thinking processes. The PEP guide provides such a detailed analysis to students’ text that new teachers as well as new users will find it very helpful.

This study also raises questions such as the relationship between the teacher’s guide and effective instruction and how to support teachers based on research findings. According to Leinhardt and Greeno’s study (1986), experts can provide exemplary performances. Unfortunately, we have no extensive case literature in education, such as we have in medicine, law, or business. So, by studying videotapes or transcripts of exemplary performances, teacher’s guides can provide effective suggestions for novices in such things as lesson openings, lesson closings, or teaching tough concepts. From this perspective, both guides can be more effective by providing the above materials.

This study only examined a sample of chapters in each guide. This will limit the conclusions to apply to broader situations. For instance, more word story problems can be found in other chapters (such as the chapter about word story and mixed computation) of the PEP series which different ways of solving a word story problem is stated in the text. Also, certain types of thinking skills may be emphasized in different grades. For instance, we can expect that more examples of generalizatione and abstraction will cases to be found from upper grades of elementary textbooks. Additionally, this study is a content study, the questions this study dealt with maycan have different answers or interpretations if actual use of the teacher’s guides have been studied.conducted.

This study also raises the question of what kind of guide teachers really need or want. The results of this study seems to show that the SF guide is a more practical one and the PEP guide is somewhat a textbook for pre-service teacher. From the standpoint of teachers themselves, a convenient, practical guide would be a better one. On the other hand, this kind of guides might hinder teachers’ professional development. We often hear that "good teachers don’t follow textbooks." So what kind of teacher’s guides we should create is a dilemadilemma for different teachers.also relates to Oother issues are involved in this matter. Ball (1988) identified three goals for preservice teacher education, namely learning to justify choices, acquiring subject matter, and knowledge and developing curricular independence. These goals, too, should be sought when writing teachers’ guides.

The intention of this study is also not to compare which teacher’s guide is best, rather to shed light on some of the issues like connections of teacher’s guides and effective teaching and learning. As Usiskin (1999) pointed out, that there are so few studies and greatthe ignorance and difficulties to conduct such studies about mathematics textbooks in recent time. For instance, "Only 5 of 627 studies in 1995 and only 3 of 529 published studies in 1996 are textbook comparisons at any level" from two most recent compilations, more studies need to be done.

Acknowledgment

I deeply thank Mrs. Karen Usiskin, of Scott Foresman - Addison Wesley, for her kind help. I also much appreciate Professors John Craig, especially Susan Stodosky and Zalman Usiskin’s valuable comments and suggestions on the early stage of this paper.

References

Adelman, N. E., Trying to beat the clock: Uses of teacher professional time in three countries. Department of Education, Washington, DC. 1998.

Armbruster, B. B., The Problem of "Inconsiderate Text." In: G. Duffy, L. Roehler, & Mason (EdsEds.) Comprehension Instruction, 1990. .

Armbruster, B.B., and Anderson, T. H., Textbook Analysis. In: Lewy, Arieh (Eed) 1991, The International Encyclopedia of Curriculum., Pergamon Press,. 1991.

Ball, D. L., Feiman-Nemser, S., Using Textbooks and Teacher's Guides: A Dilemma for Beginning Teachers and Teacher Educators. Curriculum Inquiry, 1988, 18, 4(1988):401-423.

Baller, E., The Impact of Textbooks. In A. Lewy(EEd) The International Encyclopedia of Curriculum., 1991, pp.138-139, Pergamon Press,. 1991, pp.138-139.

Barr, R., Content Coverage. In: M. Dunkin, (Ed) The International Encyclopedia of Teaching and Teacher Education. 1989, pp. 364-368. Sidney, Australia,. 1989, pp. 364-368.

Ben-Peretz, M., Tamir, P., What Teachers Want to Know About Curriculum Materials. Journal of Curriculum Studies, 1981. 13(1): 45-54.

Cahen, L. S., et al., Class Size and Instruction. 1983. Longman Inc. New York and London,. 1983.

Cai, J. (1995). A cognitive analysis of U.S. and Chinese students’ mathematical performance on tasks involving computation, simple problem solving, and complex problem solving. Unpublished dissertation, University of Pittsburgh.

Charles, R. I., et al. (1999). Scott Foresman-Addison Wesley Math: Grade 4, Teacher’s Edition, Vol. 1. Menlo Park, CA. and Glenview, IL.

Charles, R. I., et al. (1999). Scott Foresman-Addison Wesley Math: Grade 4, Teacher’s Edition, Vol. 2. Menlo Park, CA. and Glenview, IL.

Cole, J. Y., and Sticht, T. G., (EdsEds.) The Textbook in American Society. 1981. Library of Congress, Washington, DC., 1981.

Cronbach, L. J., (Ed) Text Materials in Modern Education. 1955. University of Illinois Press, Urbana, IL., 1955.

Cui, X., Current Chinese national mathematics textbooks for grades 1-9 and their implications for Chinese students’ mathematics performance. An Uunpublished Master’s Paper, Department of Education, Univerisity of Chicago, 1996.

Department of Education, Trying to beat the clock: Uses of teacher professional time in three countries. Prepared by Nancy E. Adelman, Washington, DC.

Durkin, D., Is there a match between what elementary teachers do and what basal reader manuals recommend? The Reading Teacher, 1984: 734-744.

Elliott, D. L., and Woodward, A.(EdsEds.), Textbooks and Schooling in the United States, Eighty-ninth Yearbook of the National Society for the Study of Education, Part 1. 1990. The University of Chicago Press, Chicago, IL., 1990.

Flanders, J. (1987). "How much of the content on mathematics textbooks is new?" Arithmetic Teacher, 35(1), 18-23.

Graves, M. F., and Slater, W. H., Could Textbooks be Better Written and Would it Make a Difference? American Educator, Spring 1986, pp. 36-42.

Graybeal, S. S., A study of instructional suggestions in fifth-grade math and social studies teacher'ss’s guides and textbooks. Unpublished 1988. Ph. D. Dissertation. University of Chicago, Chicago, Illinois,. 1988.

Grouws, D. A., Cooney, T. J., and Jones, D., (EdsEds.) Effective Mmathematics Teaching. 1988. Lawrence Erlbaum Associates, National Council of Teachers of Mathematics. Reston, Va., 1988.

Holsti, O.R., Content Analysis. In G. Lindzey, E. Aronson(EdsEds.), 1968: 596-692, The Handbook of Social Psychology. Second Edition, Volume Two., Addison-Wesley Publishing Co., 1968: 596-692,

Jackson, P. W., (Ed) Contributing to Educational Change: Pperspectives on Research and Practice. 1988. McCutchan Publishing Co. Berkekey, Ca., 1988.

Leinhardt, G., & Greeno, J.G., The cognitive skill of teaching. Journal of Educational Psychology, 78, p. 75-95.

Mathematics Department of People’s Education Press (1995). Mathematics: Compulsory Education, Five Year Elementary School Textbook, Vol. 5. Beijing, China: People’s Education Press.

Mathematics Department of People’s Education Press (1995). Mathematics: Compulsory Education, Five Year Elementary School Textbook, Vol. 6. Beijing, China: People’s Education Press.

Mathematics Department of People’s Education Press (1995). Mathematics: Compulsory Education, Five Year Elementary School Textbook, Teacher’s Teaching Manual, Vol. 5. Beijing, China: People’s Education Press.

Mathematics Department of People’s Education Press (1995). Mathematics: Compulsory Education, Five Year Elementary School Textbook, Teacher’s Teaching Manual, Vol. 6. Beijing, China: People’s Education Press.

McCutcheon, G., How Do elementary School Teachers Plan? The Nature of Planning and Influences on It. The Elementary School Journal, 1980, 81(1): 4-23

McNeil, J., Teacher's Guide. In A. Lewy (Ed), The International Encyclopedia of Curriculum. , 1991, pp. 81-84, Pergamon Press,. 1991, pp. 81-84.

Mehlinger, H. D., American Textbook Reform: What Can We Learn from the Soviet Experience? Phi Delta Kappan, 71, 1(1989): 29-35.

National Society for the Study of Education (NSSE), Textbooks and Schooling in the United States. Eighty-ninth Yearbook of NSSE, Part 1, D. Elliott, and A. Woodward (EdsEds.), 1990.

NCTM, Principles and S standards for Sschool Mmathematics: Ddiscussion Ddraft, National Council of Teachers of Mathematics. Reston, Virginia, 1998, p. 33.

Porter, A., et al., Content Determinants in Elementary School Mathematics. In D. A. Grouws, T. J. Cooney, and D. Jones (EdsEds.), Effective Mathematics Teaching, 1988, pp. 96-113. The National Council of Teachers of Mathematics, Reston, Va. 1988, pp. 96-113.

Romberg, T. A., Carpenter, T. P., Research on Teaching and Learning Mathematics: Two Disciplines of Scientific Inquiry. In Wittrock, M. C., Handbook of Research on Teaching, 3rd edition., 1986. pp. 850-873. Macmillan, New York,. 1986, pp. 850-873.

Romberg, T. A., Can Teachers be Professionals. In : Effective Mathematics Teaching. D. A. Grouws, T. J. Cooney, and D. Jones. (EdsEds.). 1988, pp. 224-244. Lawrence Erlbaum Associates, and National Council of Teachers of Mathematics. Reston, Va., 1988, pp. 224-244.

Stodolsky, S., Is Teaching Really by the Book? In P. W. Jackson, and Harooutunian-Gordon, S., From Socrates to Software: The Teacher as Text and the Text as Teacher, Eighty-ninth Yearbook of the National Society for the Study of Education, part 1., 1989. The University of Chicago Press, Chicago, IL,. 1989.

Stodolsky, S., The Subject Matters: Classroom Activity in Math and Social Studies. 1988. The University of Chicago Press. Chicago and London,. 1988.

Suydam, M. N., and Osborne, A., The status of pre-college science, mathematics, and social science education: 1955-1975. Volume II. Mathematics Education., p.114. National Science Foundation, 1977. p.114.

Tanner, D., The Textbook Controversies. In: Critical Issues in Curriculum, 1988, pp. 122-147. Eighty-seventh Yearbook of the National society for the Study of Education, Part 1,. 1988, pp. 122-147.

Tyson-Bernstein, H., A Conspiracy of Good Intentions: America's Textbook Fiasco. The Council for Basic Education, Washington, DC, 1988..

Tyson-Bernstein, H., Textbook Development in the United States: How Good Ideas Become Bad Textbooks, In : J.Farrell and S.P. Heyneman (EdsEds.) 1989, Textbooks in the Developing World, pp.72-87. Washington, DC., 1989, pp.72-87.

Tyson-Bernstein, H,. A Conspiracy of Good Intentions: America’s Textbook Fiasco. The Council for Basic Education. Washington, DC, 1988.

Venezky, R. L., Textbooks in School and Society. In P. W. Jackson(Ed), Handbook of Research on Curriculum. , 1992, pp. 436-461. Macmillan Publishing Company, New York,. 1992, pp. 436-461.

Whiting, J. W., Methods and Problems in Cross-Cultural Research. In G. Lindzey, E. Aronson(EdsEds.), 1968: 693-728, The Handbook of Social Psychology. Second Edition, Volume Two, Addison-Wesley Publishing Co., 1968: pp.693-728.

Woodward, A., Elliott, D. L., and Nagel, K. C., Textbooks in School and Society: An Annotated Bibliography and Guide to Research. , 1988. Garland Publishing Inc., New York and London,. 1988.

Woodward, A., Taking Teaching Out of Teaching and Reading Out of Learning to Read: A Historical Study of Reading Textbook Teachers’ Guides, 1920-1980. Book Research Quarterly, Spring 1986,. pPp. 53-73.

Woodward, A., Over-Programmed Materials: Taking the Teacher out of Teaching. American Education. Spring 1986, pp. 26-31.

Usiskin, Z., Which curriculum is best? University of Chicago School Mathematics Project Newsletter, No. 24, Winter 1998-1999, p.8.

References:

Adelman, N. E. (1998). Trying to beat the clock: Uses of teacher professional time in three countries. Washington, DC, U.S. Department of Education.

Ball, D. L. (1988). Unlearning to teach mathematics (Issue Paper 88-1). East Lansing: Michigan State University, National Center for Research on Teacher Education.

Cai, J. (1995). A cognitive analysis of U.S. and Chinese students’ mathematical performance on tasks involving computation, simple problem solving, and complex problem solving. Unpublished dissertation, University of Pittsburgh.

Charles, R. I., et al. (1999). Scott Foresman-Addison Wesley Math: Grade 4, Teacher’s Edition, Vol. 1. Menlo Park, CA. and Glenview, IL.

Cui, X. (1996). Current Chinese national mathematics textbooks for grades 1-9 and their implications for Chinese students’ mathematics performance. Unpublished master’s paper, Department of Education, University of Chicago.

Flanders, J. (1987). "How much of the content on mathematics textbooks is new?" Arithmetic Teacher, 35(1), 18-23.

Leinhardt, G., and Greeno, J.G. (1986). "The cognitive skill of teaching." Journal of Educational Psychology, 78: 75-95.

Mathematics Department of People’s Education Press (1995). Mathematics: Compulsory Education, Five Year Elementary School Textbook, Teacher’s Teaching Manual, Vol. 6. Beijing, China: People’s Education Press.