The Pedagogical Content Knowledge of Middle School, Mathematics Teachers in China and the U.S

SHUHUA AN, GERALD KULM and ZHONGHE WU

THE PEDAGOGICAL CONTENT KNOWLEDGE OF MIDDLESCHOOL, MATHEMATICS TEACHERS IN CHINA AND THE U.S.

ABSTRACT. This study compared the pedagogical content knowledge of mathematicsin U.S. and Chinese middle schools. The results of this comparative study indicatedthat mathematics teachers’ pedagogical content knowledge in the two countries differsmarkedly, which has a deep impact on teaching practice. The Chinese teachers empha-sized developing procedural and conceptual knowledge through reliance on traditional,more rigid practices, which have proven their value for teaching mathematics content. TheUnited States teachers emphasized a variety of activities designed to promote creativityand inquiry in attempting to develop students’ understanding of mathematical concepts.Both approaches have benefits and limitations. The practices of teachers in each countrymay be partially adapted to help overcome deficiencies in the other.

KEY WORDS: pedagogical content knowledge, mathematics teaching, student’scognition, teacher’s knowledge, unit fraction

During the past several decades, there has been increased attention tocomparative studies in mathematics education, especially with respect tothe movement of reforming mathematics education in the beginning ofthe 21st Century. According to Robitaille and Travers (1992), compar-ative study provides opportunities for sharing, discussing, and debatingimportant issues in an international context. Stigler and Perry (1988)observe:

Cross-cultural comparison also leads researchers and educators to a more explicit under-standing of their own implicit theories about how children learn mathematics. Withoutcomparison, teachers tend not to question their own traditional teaching practices and arenot aware of the better choices in constructing the teaching process (p. 199).

In 1996, U.S. eighth and twelfth graders scored below average in math-ematics when compared with other countries in the Third InternationalMathematics and Science Study (TIMSS) assessment (Silver, 1998). In1999, U.S. eighth-grade students scored slightly above the internationalaverage in mathematics and science performance according to the ThirdInternational Mathematics and Science Study-Repeat (TIMSS-R) whencompared with students in 37 participating nations. This report indicatedthat there has been improvement in the U.S. in mathematics education.However, to compete globally and achieve a top rank internationally,

Journal of Mathematics Teacher Education 7: 145–172, 2004.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

146 SHUHUA AN ET AL.

mathematics education in the U.S. would benefit from continuing toexamine international mathematics education practices and research.

Teachers and teaching are found to be one of the major factors relatedto students’ achievement in TIMSS and other studies. According to theNational Council of Teachers of Mathematics (2000), “Effective teachingrequires knowing and understanding mathematics, students as learners,and pedagogical strategies” (p. 17). In an era of globalization and infor-mation, teachers’ knowledge in mathematics is becoming more complexand dynamic (Fennema & Franke, 1992). New aspects of teaching, suchas knowledge of technology, must be mastered. However, the balance andintegration of pedagogy and content knowledge, referred to as pedagogicalcontent knowledge (Shulman, 1987; Pinar, Reynolds, Slattery & Taubman,1995), should be the most important element in the domain of mathematicsteachers’ knowledge. Pedagogical content knowledge addresses how toteach mathematics content and how to understand students’ thinking. Thisincludes, taking into consideration both the cultural background of thestudents as well as their preferences for various teaching and learningstyles. The purpose of this study was to examine the differences betweenChina and the United States of teachers’ pedagogical content knowledgein mathematics at the middle school level.

CONCEPTUAL FRAMEWORK

Shulman’s Model of Pedagogical Content Knowledge

According to a Chinese saying, if you want to give the students one cupof water, you (the teacher) should have one bucket of water of your own.Shulman (1985) believes that “to be a teacher requires extensive and highlyorganized bodies of knowledge” (p. 47). Elbaz (1983) has the same view,“the single factor which seems to have the greatest power to carry forwardour understanding of the teacher’s role is the phenomenon of teachers’knowledge” (p. 45).

Shulman (1987) has stated further that pedagogical content knowledgeincludes knowledge of learners and their characteristics, knowledge ofeducational contexts, knowledge of educational ends, purposes and values,and their philosophical and historical bases. Pedagogical content knowl-edge refers to the ability of the teacher to transform content into forms thatare “pedagogically powerful and yet adaptive to the variations in abilityand background presented by the students” (Shulman, 1987, p. 15). TheNetwork of Pedagogical Content Knowledge In the current study, pedago-gical content knowledge is defined as the knowledge of effective teaching

MATHEMATICS TEACHERS’ PEDAGOGICAL CONTENT KNOWLEDGE 147

which includes three components, knowledge of content, knowledge ofcurriculum and knowledge of teaching. This is broader than Shulman’soriginal designation. Knowledge of content consists of broad mathematicsknowledge as well as specific mathematics content knowledge at thegrade level being taught. Knowledge of curriculum includes selecting andusing suitable curriculum materials, fully understanding the goals and keyideas of textbooks and curricula (NCTM, 2000). Knowledge of teachingconsists of knowing students’ thinking, preparing instruction, and masteryof modes of delivering instruction.

Although all three parts of pedagogical content knowledge are veryimportant to effective teaching, the core component of pedagogical contentknowledge is knowledge of teaching. Figure 1 suggests the interactiverelationship among the three components and shows that knowledge ofteaching can be enhanced by content and curriculum knowledge.

Figure 1. The network of pedagogical content knowledge.


Ma’s study (1999) calls for teachers to have “profound understandingof fundamental mathematics”. However, as indicated in Figure 1, profoundcontent knowledge alone is not sufficient for effective teaching. Aneffective teacher must also possess a deep and broad knowledge ofteaching and curriculum or profound pedagogical content knowledge. Withthis knowledge, teachers are able to connect their knowledge of content,curriculum, and teaching in a supportive network. In this network, threetypes of knowledge interact with each other and are able to make trans-formations from one form to another around the central task of teaching.Ultimately, these components together address the goal of enhancingstudents’ learning. As shown in Figure 1, this network of knowledge isimpacted by teachers’ beliefs. Ernest (1989) and Fennema and Franke(1992) also reveal the importance and impact of teachers’ beliefs on theirknowledge. Different educational belief systems produce different attrib-utes of pedagogical content knowledge. In turn, profound pedagogicalcontent knowledge plays an important role in shaping teachers’ beliefs andin determining the effectiveness of their mathematics teaching (An, Kulm,Wu, Ma & Wang, 2002).

Teaching can be seen as either a divergent or a convergent process. Adivergent process of teaching is one that is based on content and curriculumknowledge but is without focus and ignores students’ mathematicalthinking. A convergent process of teaching focuses on knowing students’thinking, which consists of four aspects: building on students’ mathe-matical ideas, addressing students’ misconceptions, engaging students inmathematics learning, and promoting students’ thinking mathematically.Together, these four aspects of convergent teaching comprise the notion ofteaching with understanding, which is an essential to effective teaching(Carpenter & Lehrer, 1999). Under a convergent process, students, nottextbooks and curriculum, are the center of teaching. Throughout theconvergent teaching process, an effective teacher attends to students’mathematical thinking: preparing instruction according to students’ needs,delivering instruction consistent with students’ levels of understanding,addressing students’ misconceptions with specific strategies, engagingstudents in activities and problems that focus on important mathematicalideas, and providing opportunities for students to revise and extend theirmathematical ideas (Kulm, Capraro, Capraro, Burghardt & Ford, 2001).

There are two kinds of teaching beliefs regarding students’ learning:learning as knowing and learning as understanding. A teacher who holdsthe belief of learning as knowing often assumes that mathematics is learnedand understood if a concept or skill is taught. This type of learning usuallyis achieved at a surface level. Teachers are often satisfied with students’


knowing or remembering facts and skills but are not aware of students’thinking or misconceptions about mathematics. This divergent teachingprocess often results in fragmented and disconnected knowledge.

A teacher who holds the belief of learning as understanding realizesthat knowing is not sufficient and that understanding is achieved at thelevel of internalizing knowledge by connecting prior knowledge througha convergent process. In this process, the teacher does not only focuson conceptual understanding and procedural development, making surestudents that comprehend and are able to apply the concepts and skills, butalso consistently inquires about students’ thinking. Teachers who use thisconvergent process develop systematic and effective ways to identify anddevelop their students’ thinking. These ideas are summarized in Figure 2,showing that with profound knowledge of students’ thinking, teachers canenhance students’ learning substantially, leading to content mastery.

Figure 2. Two types of learning.

Although there has been some research comparing Chinese and U.S mathematics teachers’ content knowledge (e.g., Ma, 1999), there hasbeen very little research comparing their profound pedagogical contentknowledge. Ma’s work (1999) focuses on comparing elementary teachers’content knowledge, without fully accounting for cultural contexts, goalsand teacher beliefs. Her study documented that Chinese teachers withmore mathematical training seemed to know more than U.S. mathematicsteachers. Many of Ma’s examples did provide indications about howteachers apply their mathematics knowledge in teaching, but stopped shortof a systematic study of how mathematical and pedagogical knowledgewas integrated with knowledge of students’ thinking.

The current study focuses on the middle-school level, addresses mathe-matics teachers’ pedagogical content knowledge within a cultural context,and explores how this knowledge is used by teachers to understand anddevelop students’ mathematical thinking. The question that provided the


focus for this study was: What are the differences in teachers’ profoundpedagogical content knowledge between middle school mathematicsteachers in China and the United States?

METHODOLOGY

Subjects

The subjects were 28 mathematics teachers in fifth- to eighth-grade levelsfrom 12 schools in four school districts in a large metropolitan area inTexas and 33 mathematics teachers in fifth- and sixth-grade levels from 22schools in four school districts in a large city in Jiangsu province in easternChina. In order to examine the teachers’ profound pedagogical contentknowledge at the middle school level (particularly in the area of fraction,ratio, and proportion), this study included U.S. teachers from fifth to eighthgrade, because U.S. mathematics curricula in these grades are similar tothose in fifth and sixth in China.1

Criteria for inclusion of teacher volunteers in the study were: (1) currentteaching of mathematics in fifth to eighth grades; (2) teaching in schooldistricts that have characteristics typical of each nation’s public schoolswith respect to the students’ ethnic, economic, and cultural diversity; (3)having at least three years of teaching experience at the fifth to eighthgrade levels; and (4) willing to provide the data relevant to the reliabilityand validity of this study, including classroom observations and interviews.

The U.S. teachers all had bachelor’s degrees; three had master’sdegrees. They had an average of 24 hours of mathematics course workand an average of 13 years teaching experience. Only one participanttaught fifth-grade mathematics, 14 of them taught sixth grade, 7 wereseventh-grade teachers, and 6 were eighth-grade teachers. It should benoted that the U.S. teachers who participated in this study only teachmathematics. All of the Chinese teachers had three-year education degreesat Normal schools after ninth grade; 23 also had three-year universitydegrees, including 10 who majored in fields other than mathematics. Theaverage number of hours in mathematics courses for the Chinese teacherswas 15. For example, with a three-year degree at university, a teacherhad calculus, modern algebra, elementary mathematics methods, history ofelementary mathematics education, and Olympic elementary mathematics.Their average length of teaching experience was 9 years. Six of them werefifth-grade teachers, and 28 taught sixth grade. As in the case of the U.S.teachers, all Chinese teachers in this study only taught mathematics.


The U.S. schools were located in both urban and suburban areas andthe school populations ranged from 800 to 1300 students. The ethniccomposition of the schools was similar and, on average, consisted of 30%African American, 27% Caucasian, 27% Hispanic and 16% Asian. Schoolsin China were located in urban areas and the number of students in eachschool ranged from 1000 to 1200. All students were from the same ethnicgroup.

Procedures

Data were collected with an author-constructed Mathematics TeachingQuestionnaire, an author-constructed Teachers’ Beliefs about MathematicsTeaching and Learning Questionnaire, and interviews and observationswith selected teachers. Both questionnaires were prepared first in English,and then translated into Chinese. This article will focus on an analysis ofthe Mathematics Teaching Questionnaire.

Mathematics Teaching QuestionnaireThe questionnaire consisted of four problems that were designed toexamine teachers’ profound pedagogical content knowledge in topicsof fractions, ratios, and proportion (see Figure 3). Each of the fourproblems focused on one aspect of teachers’ knowledge of students’cognition, with attention to assessing teachers’ knowledge and strategiesfor, namely, building on students’ mathematics ideas, identifying andcorrecting students’ misconceptions, engaging students in learning, andpromoting student thinking.

Classroom ObservationsAfter reviewing and analyzing the responses to the questionnaires, fiveteachers from each country were selected for observation to confirmthat their teaching matched their responses on the questionnaire. Theywere chosen so as to represent a range of education background, lengthof teaching experience, and level of responses to the questionnairesfor classroom observations. The observations were conducted at a pre-arranged date and time. Field notes and audiotape recordings were madeduring the classroom observations using an Instructional Criteria Observa-tion Checklist that was constructed as a guide. The Checklist was adaptedfrom criteria used for analyzing the instructional quality of mathematicstextbooks (AAAS, 2000). The observation criteria included specific activ-ities in the categories: building on student ideas in mathematics, beingalert to students’ ideas, identifying student ideas, addressing misconcep-tions, engaging students in mathematics, providing first-hand experiences,


Mathematics Teaching Questionnaire

Problem 1Adam is a 10-year-old student in 5th gradewho has average ability. His grade on the lasttest was an 82 percent. Look at Adam’s writtenwork for these problems:

3

4+ 4

5= 7

92

1

2+ 1

1

2= 3

2

5

a. What prerequisite knowledge might Adam not under-stand or be forgetting?

b. What questions or tasks would you ask Adam, in orderto determine what he understands about the meaning offraction addition?

c. What real world example of fractions is Adam likely tobe familiar with that you could use to help him?

Problem 2A fifth-grade teacher asked her students towrite the following three numbers in orderfrom smallest to largest:

3

8,

1

4,

2

3

Latoya, Robert, and Sandra placed themin order as the follows.

Latoya:1

4,

2

3,

3

8Robert:

2

3,

1

4,

3

8Sandra:

1

4,

3

8,

2

3

a. What might each of the students be thinking?b. What question would you ask Latoya to find out if your

opinion of her thinking is correct?c. How would you correct Robert’s misconception about

comparing the size of fractions?

Problem 3You are planning to teach procedures for doingthe following type of fraction multiplication.

a. Describe an introductory activity that would engage andmotivate your students to learn this procedure.

b. Multiplication can be represented by repeated addition,by area, or by combinations.Which one of these representations would you use toillustrate fraction multiplication to your students? Why?

c. Describe an activity that would help your students under-stand the procedure of multiplying fractions.

Problem 4Your students are trying to solve the followingproportion problem:The ratio of girls to boys in Math club is 3:5.If there are 40 students in the Math club, howmany are boys?

a. Describe an activity that you would use to determinethe types of solution strategies your students have usedto solve the problem. Here are two students’ solutionsto the problem:

June’s solution:3

5= x

40

There are 24 girls, so there 16 boys.Kathy’s solution:

3

8= x

40

There are 15 boys.b. What question would you ask Kathy to determine if she

could justify her answer and reasoning?c. What suggestion would you provide to June that might

help her correct her approach?d. What strategy would you use to encourage your students

to reflect on their answers and solutions?

Figure 3. Mathematics teaching questionnaire.


promoting student thinking about mathematics, guiding interpretation andreasoning, and encouraging students to think about what they had learned.

InterviewsAfter each observation, an interview was conducted using a set of interviewquestions. The objective of the interviews was to examine teachers’ beliefsabout the goals of mathematics education and the impact of their beliefs ontheir teaching practices, to investigate the teaching approaches they use inthe classrooms, to learn how the teachers prepare for instruction and howthey determine their students’ thinking. The interview questions exploredfurther the teachers’ pedagogical content knowledge and its importance intheir teaching.

Data Analysis

QuestionnaireA constant comparative data analysis method was used in the analysis ofthe Mathematics Teaching Questionnaire. In all, 18 different categorieswere identified which included the responses to the four problems. Theresponses were categorized into groups and assigned a descriptive code.Two researchers used the resulting codes to analyze the responses inde-pendently. Both sets of codes were compared, and then, through discussionwith the third researcher, the disparities were reconciled to reach valu-able agreements on the responses. Table I lists the categories and theirdefinitions. In Table II, the 18 categories are grouped according to thefour components of pedagogical content knowledge in the conceptualframework.

Interviews and ObservationsTranscriptions were made of the interviews. The transcriptions were codedusing the 18 response categories and the four components of pedagogicalcontent knowledge. The responses to interview questions and the fieldnotes and checklists from the observations were also analyzed through theuse of concept mapping to clarify key teacher ideas and beliefs. Thesedata confirmed that the teachers’ responses to the Mathematics TeachingQuestionnaire were consistent with their actual classroom practices andtheir knowledge and beliefs about effective teaching.

RESULTS

The responses of the U.S. and Chinese teachers to the MathematicsTeaching Questionnaire are presented in Table III, based on the 18


TABLE I

Categories for Describing Teachers’ Responses to Pedagogical Content KnowledgeQuestions

Category Brief definition

1. Prior knowledge: Know students’ prior knowledge and connect it to newknowledge.

2. Concept or definition: Use concept or definition to promote under-standing.

3. Rule and procedure: Focus on rule and procedure to reinforce the knowl-edge.

4. Draw picture or table: Use picture or table to show a mathematical idea.

5. Give example: Address a mathematical idea through examples.

6. Estimation: Solve problems using estimation.

7. Connect to concrete model: Use concrete model to demonstrate mathema-tical idea.

8. Students who do not understand prior knowledge: Students lack inunderstanding of prior knowledge.

9. Provide students opportunity to think and respond: Promote students tothink problems and give them chances to answer questions.

10. Manipulative activity: Provide hands-on activities for students to learnmathematics.

11. Attempts to address students’ misconceptions: Identify students’ miscon-ceptions.

12. Use questions or tasks to correct misconceptions: Pose questions orprovide activities to correct misconceptions.

13. Use questions or tasks to help students’ progress in their ideas: Posequestions or provide activities to increase the level of understanding forstudents.

14. Provide activities and examples that focus on student thinking: Createactivities and examples that encourage students to ponder questions.

15. Use one representation to illustrate concepts: Apply repeated addition toaddress the meaning of fraction multiplication, or use area to address thegeometrical meaning of fraction multiplication.

16. Use both representations to illustrate fraction multiplication: Apply bothrepeated addition and area to address the meaning of fraction multiplica-tion.

17. Unintelligible response: Provide response that is not relevant to thequestion.

18. Incorrect: Provide a wrong answer.


TABLE II

Categories for Describing Four Aspects of Teaching to Pedagogical Content KnowledgeQuestions

Pedagogical content Essential components Categoryknowledge Number

Building on students’ 1. Connect to prior knowledge 1,8

math ideas 2. Use concept or definition 2

3. Connect to concrete model 7

4. Use rule and procedure 3

Addressing students’ 1. Address students’ misconceptions 11

misconceptions 2. Use questions or tasks to correct misconceptions 12

3. Use rule and procedure 3

4. Draw picture or table 4

5. Connect to concrete model 7

Engaging students 1. Manipulative activity 10

in math learning 2. Connect to concrete model 7

3. Use one representation (area) 15

4. Use both representations (area & repeated 16

addition)

5. Give example 5

6. Connection to prior knowledge 1

Promoting students’ 1. Provide activities to focus on students’ thinking 14

thinking about 2. Use questions or tasks to help students’ progress 13

in their ideas

mathematics 3. Use Estimation 6

4. Draw picture or table 4

5. Provide opportunity to think and respond 9

categories and four components of pedagogical content knowledge. Sinceeach teacher’s responses could be coded into more than one of the 18categories, the resulting total percentage is greater than 100 for eachproblem.

On the one hand, the results show that both groups of teachers haveextensive and broad pedagogical content knowledge and are able to applyvarious methods to help students learn mathematics. However, there aresome important differences in each of the four components of teaching


TABLE III

Percentage of U.S. and Chinese Teachers for Each Category of Response to Problem 1to 4

Pedagogical content Essential components Problem Category US Chinaknowledge number number % %

Building on students’ 1. Connect to prior knowledge

math ideas Forget prior knowledge 1.a 1 46 27

Does not understand prior 1.a 8 11 55

knowledge

2. Use concept or definition 1.b 2 29 51

Concept 29 21

Unit fraction 0 30

3. Connect to concrete model 1.c 7 93 42

4. Rule and procedure 1.b 3 25 76

5. Unintelligible response 1.b 17 12 3

Addressing students’ 1. Address students’ 2.a 11 86 97

misconceptions misconceptions

2. Use questions or tasks to 2.c 12 61 100

correct misconceptions

3. Use rule and procedure 2.c 3 11 42

4. Draw picture or table 2.c 4 29 30

5. Connect to concrete model 2.c 7 26 12


Engaging students in 1. Manipulative activity 3.a 10 39 18

math learning 2. Connect to concrete model 3.a 7 36 64

3. Use one representation (area) 3.b 14 64 28

4. Use both representations 3.b 15 11 67

(area & repeated addition)

5. Give example 3.a 5 4 91

6. Connect to prior knowledge 3.a 1 7 45

7. Unintelligible response 3.c 17 25 24

Promoting students’ 1. Provide activities to focus on 4.d 14 68 94

students’ thinking students’ thinking

about mathematics 2. Use questions or tasks to 4.b 13 57 100

help students’ progress in

their ideas

3. Use estimation 4.c 6 4 6

4. Draw picture or table 4.a 4 14 15

5. Provide opportunity to think 1.b 9 61 79

and respond



for understanding between the U.S. and Chinese teachers. A discussion ofeach difference between the two groups of teachers follows.

Building on Students’ Ideas about Fractions

Understanding and ForgettingAs shown in Table III, 46% of the responses of the U.S. teachers toProblem 1(a) indicated that, in their view, Adam forgot the prerequisiteknowledge of finding common denominators, while 55% of the responsesof the Chinese teachers had an opposite view, namely, that Adam did notunderstand the prerequisite knowledge of finding common denominators.For example, Mrs. Ross mentioned that Adam forgot prior knowledge:“Adam does not remember that to add, you must add like denominators,and fourths and fifths are not alike. He does not remember to makeequivalent fractions using the lowest common denominator”.

Mrs. Wang pointed out that Adam did not understand the fractionbecause he seemed to separate numerator and denominator into inde-pendent parts. Furthermore, Mrs. Sheng, Mrs. Wang, and Mr. He indicatedthat only with like units can two numbers be added, such as 3 books + 5books = 8 books. The Chinese teachers realized that three books cannot beadded to 4 desks because “book” and “desk” are different units. A fractionis a number, so only with like unit fractions can two fractions be added;therefore with unlike unit fractions, two fractions cannot be added. Adam’smistake indicated that he did not understand the concept of like units andcould not see the connection between whole number and fraction, whichmay misdirect him to think fraction as something else but not a number.

Here “forgot” and “did not understand” have two distinct meaningsfor the teachers. The teachers who said the student “forgot” did not knowtheir students’ thinking about fraction addition and did not understand thechallenges students are likely to encounter in learning fraction addition.They appeared to believe that learning simply consists of knowing or notknowing; that is, remembering or forgetting. In contrast, teachers whosaid that the student “did not understand” showed evidence of knowingstudents’ thinking about fraction addition.

A large percentage of Chinese teachers connected prior knowledge ofwhole number addition to fraction addition, that is, that numbers with likeunits can be added. This means that no matter what numbers they are(such as whole numbers, decimals and fractions) as long as the numbershave the same unit, they can be added together. This connection not onlyhelps students to see a fraction as a number, but also helps students tounderstand and to use the rule of fraction addition easily. Understandingalso means that students are able to internalize a concept and use it in


different situations, such as understanding like units in whole numberaddition and applying like units in fraction addition. Internalizing andconnecting knowledge about like units into a coherent whole provides aclose link that makes learning easier and leads to mastery.

Use of Models to Develop Concepts and ProceduresThe results in Table III show that the 93% of the U.S. teachers tended tobuild on students’ ideas of addition of fractions with various approaches,by focusing on the connection with concrete or pictorial models. Incontrast, only 42% of Chinese teachers used concrete models. Most ofthem used definitions or the unit fraction concept to develop students’knowledge of addition of fractions by emphasizing procedural develop-ment and the following of rules.

The U.S. teachers used a wide variety of concrete and pictorial models,including pizzas, pies and cakes, a Hershey bar, an egg carton, crayonboxes, measuring cups, sports, money, time, and fraction pieces. The useof concrete models and pictures helps students to visualize and explorea mathematics concept and to connect learning with their experiences.This learning will be meaningful and will make sense to students. Thisapproach is supported by research showing that mathematics should servestudents’ needs to make sense of experience arising outside of mathematicsinstruction (Fennema & Romberg, 1999).

For Chinese teachers, conceptual understanding is very important inlearning. As shown in Table III, only 29% of the U.S. teachers empha-sized concepts in developing students’ fraction ideas, while 51% Chineseteachers focused on concepts to build understanding. Of these, 30% usedthe unit fraction concept. The Chinese teachers believed that the unit frac-tion is a critical concept in learning fractions and is more easily understoodby children. For example, Mrs. Li, a sixth grade teacher, suggested thefollowing four questions when responding to part (b) of Problem 1:

1. What is the unit of the fraction?2. What fractions can be added directly? Give the problems 1/5 + 2/5, 1/3

+ 2/3, and 5/6–1/6 to Adam. Can you give a picture problem for this?3. What does it mean to find the common denominator? Why do you need

to find the common denominator? And how do you find the commondenominator?

4. How do you add fractions with unlike denominators?

Mrs. Li provided an example and explained it with Figure 4 andFigure 5: “Look at the example 1/3 + 1/2”. 1/3 + 1/2 does not have acommon denominator; in other words, the unit fractions are unlike, so


numerators cannot be added directly. Figure 4 shows that 1/3 and 1/2 donot have equal sized parts.

Figure 4. Circles for comparing unit fractions.

To help the students find the like unit fractions, Mrs. Li explainedfurther: “To divide the unit ‘1’, a circle, into 6 equal-sized parts, the unitfraction becomes 1/6 for both circles”. The unit fraction is now the same(see Figure 5).

Figure 5. Circles for like unit fractions.

The questions that Mrs. Li asked, involving the unit fraction concept,were intended to help Adam understand the meaning of fraction addition.

The example by Mrs. Li shows that applying the unit fraction conceptin teaching fractions makes the concept of fraction addition more rigorousand meaningful than the part-whole relationship. Using unit fractionsto build conceptual understanding connects fractions to students’ priorknowledge of the concept of whole number and helps students to constructfractions in a continuous and systemic way. In addition, it places numeratorand denominator in the context of a number and it also links numer-ator and denominator by multiplication and repeated addition, which arecomponents of the prior knowledge of fractions.

In contrast, in answering part (b) of Problem 1, U.S. teachers tended touse a concrete model to build the concept of fraction addition. For example,Mrs. William used the fraction pieces to help Adam. She said:

Have Adam use fraction pieces to show 3/4 and 4/5 and ask: “What must we do in order tocombine 3/4 and 4/5? Let’s use the fraction pieces to work together to find what size piecesfourths and fifths share, then find equivalent fractions and add”.


This approach helps students to visualize the size of fractions andlook for equivalent fractions, but it emphasizes less the “why” of findingequivalent fractions, and it puts fractions in a separate and non-continuousdomain. With this approach, questions such as “Why can equivalent frac-tions be added?” “What are the connections between addition of fractionand the concept of ‘parts of whole?’ ” could not be answered clearly.

Sowder et al. (1998), agreed that the notion of a fraction as a quantity,as a number, is important for understanding. In contrast, Kerslake (1986)argued that some teachers and children have difficulty conceiving of afraction as a number and considered it either as two numbers or not asa number. Some Chinese teachers in this study also believed that a studentmight not think of a fraction as a number at all. But this misconceptionof fraction can be corrected by understanding the unit fraction concept.One of the key reasons for students having misconceptions and confu-sion about fractions is the way fraction concepts are taught. Teachers ortextbooks in the U.S. typically introduce fractions as parts of a whole.This concept separates a fraction into two parts, and increases confusionbetween numerator and denominator. This difference between the U.S. andChinese teachers in developing fraction ideas may produce the disparity inthe knowledge of students’ thinking, which is illustrated in the teachers’responses to part (a) of problem 1.

Table III indicates that in their responses to part (b) of Problem 1, 76%of Chinese teachers focused on procedures and rules to build students’ideas about fractions. In contrast, far fewer U.S. teachers (25%) believedthat using procedures and rules were effective in building fraction ideas.This result is not surprising, since Chinese teachers focus on developingskills as an integral part of learning. The history of mathematics develop-ment in China has had a great impact on mathematics teaching in China.Under the influence of the classic work Arithmetic in Nine Chapters, themain characteristic of Chinese mathematics is the development and prac-tice of accurate and efficient means of computation and to apply these inreal life. In general, Arithmetic in Nine Chapters has defined the tradi-tional mathematics style as useful in applications and calculation (Li &Chen, 1995). In addition, under the influence of the Chinese examinationsystem, in order to help most students to pass the examinations, teachersnot only pay attention to students’ conceptual understanding, but also workextremely hard to build students’ proficiency in computation and solvingnon-routine problems. Chinese teachers believe that developing proficientprocedural skills helps to reinforce what students have learned and allowsthem to transfer skills easily to new knowledge. Most importantly, it aidsstudents’ confidence in their ability to understand mathematics.


Addressing Students’ Misconceptions

Models and ConnectionsTable III shows that, in their responses to Problem 2, 86% the U.S. teachershad understood each student’s thinking on the comparison of fractions. Inresponse to part (c), they used various activities, graphs, manipulatives,and procedures to help students to correct misconceptions, focusing onthe use of concrete models. For example, Mrs. Nelson would show Robertphysical examples of a candy bar. By seeing the sections of 2/3, 1/4, and3/8, Robert would be able to conceptualize concretely each fraction.

The U.S. teachers helped students correct a misconception aboutcomparing fractions by using experience with a variety of models. Theuse of various models helps students build ideas about abstract mathema-tical concepts. This approach is supported by NCTM (2000) that says thatconcrete models provide students with concrete representations of abstractideas and support students in using representations meaningfully.

In contrast, Chinese teachers dealt with students’ misconceptions by avariety of activities, but they focused on developing the explicit connectionbetween the various models and abstract thinking. For example, to correctRobert’s misconception about ordering fractions, Mrs. Jian presented thefollowing approach:

Have Robert cut two equal-sized ropes, one in 7 pieces, and one in 2 pieces. Then have himcompare a section from each rope to find out which one is longer. Help him summarize therules: take one part from each rope, the one with the short part has a larger denomin-ator, and the fraction is smaller. Therefore, compare fractions by not only looking at thedenominator; a large denominator does not mean the fraction is larger.

In this example, Mrs. Jian not only used a concrete model to help Robertbuild understanding, but also connected the model to abstract ideas and therules for ordering fractions.

Use of QuestionsAsking questions is one of the effective ways to engage students’thinking and learning. Probing questions involving misconceptions canguide students in identifying errors by themselves and develop a deepconceptual understanding. Carroll (1999) found that probing questionsare effective in identifying student errors through engaging students inreasoning and thinking processes. In addition, questions assess learning,promote discussion, and provide direction for teachers in planning.

Posing questions in mathematics teaching is another feature of Chineseeducation, which is a reflection of Arithmetic in Nine Chapters. In thebook, teachers are urged to develop and use sequences of questions,answers, and principles during planning and instruction. In this study,


the questioning strategies were displayed extensively in the responses ofChinese teachers. As shown in Table III, 100% of Chinese teachers wereable to use questions or tasks to correct the misconceptions posed inProblem 2, compared with 61% of the U.S. teachers. We take an examplefrom Mrs. Wu: “Can you directly order fractions by comparing numeratorsonly while the numerators and denominators are all different? Have Latoyause the same-sized paper to fold 1/4, 2/3, 3/8, and then compare these threepieces”. Furthermore, Chinese teachers not only asked focusing questionsto identify each student’ thinking, but also understood student’s thinkingin different ways.

As indicated in Table III, 79% of the U.S. teachers did not poseappropriate questions in order to identify students’ thinking and helpstudents progress their mathematics ideas on comparing fractions, whileonly 39% of Chinese teachers had similar problems. Questions asked byU.S. teachers consisted of “How did you order these?” and “Explain whyyou put the fraction in that order” and provide a chance for Latoya to think,but do not directly lead her to recognize her misconception.

Engaging Students in Mathematics Learning

Use of RepresentationsThe results show that there are differences in the way the U.S. and Chineseteachers engage students in mathematics learning. Most U.S. teacherssuggested engaging and motivating the students to learn the procedureof multiplication through various activities, such as manipulatives, andpictorial representations. In their responses to Problem 3, as shown inTable III, 64% of the U.S. teachers would prefer to use one representation– area to illustrate fraction multiplication – while 67% of Chinese teachersuse two representations – area and repeated addition.

By applying manipulatives, such as cutting a paper circle, singing afraction song, playing with money, using base ten blocks, or drawing andcoloring areas, the U.S. teachers sparked their students’ interest in frac-tion multiplication and engaged students in a meaningful and concretelearning process. This “learning by doing” approach encourages studentsto acquire knowledge through inquiry and creative processes and fostersstudents’ creativity and critical thinking. The use of manipulatives in frac-tion multiplication is supported by Sowder et al. (1998), who reported theeffectiveness of the use of paper folding to learn fraction multiplication.This study reported that most U.S. teachers used area representations toillustrate fraction multiplication.

In contrast, most Chinese teachers used both area and repeated addi-tion to illustrate fraction multiplication, and they understood when and


how to use each representation. For example, Mrs. Yian provided a clearexplanation of how to use different representations for multiplication: (a)If a fraction multiplies a whole number, using repeated addition is easierfor students to understand, (b) If a fraction multiplies a fraction, usingthe area graph is better for visualization, (c) For fractions that are mixednumbers use both methods. This example shows that Mrs. Yian knowswhich representations can be helpful for students in solving multiplicationproblems in particular situations.

Connections in Introductory ActivitiesChinese teachers connected concrete models and stories related tostudents’ life more frequently than did the US teachers in their responsesto part (a) of Problem 3. As shown in Table III, 91% of Chinese teachers inthis study would engage and motivate their students to learn the procedureof multiplication by giving examples, which connect to concrete models(64%) and students’ life experiences, such as examples of stories, in addi-tion. By providing examples related to students’ real life, manipulatives,and concrete models, Chinese teachers were able to make a connectionbetween manipulatives and the strategy of solving problems, and buildunderstanding for students through developing rigorous procedures. Forexample, Mr. Wang designed an introductory activity:

In the class, 56 students were divided into four groups with 14 students in each group.The teacher has a student divide a colored paper into 4 pieces, so each group gets 1/4 ofthe paper. To illustrate the procedure of fraction multiplication, let’s use group one as anexample first: In order to share 1/4 of the paper among 14 students in group one, 1/4 of thispaper will be divided equally into 14 pieces. How much paper will each student in groupone get? How do you write this expression? It should be expressed as

1

4× 1

14Figure 6. 1/14 of 1/4 paper.

Mr. Wang continued to explain the procedure of multiplication fractions:

If all 4 groups do the same, every student in class will get one part of the paper. How muchpaper will each student get? To find 1/14 of 1/4, a student can divide 1/4 into 14 parts,taking one part of it, which means the student divide one paper into 4 × 14 parts and haveone of 4 × 14, the result can be expressed as:

1

4 × 14× 1 = 1 × 1

4 × 14Figure 7. One part of 1/14 of 1/4 paper.


Therefore,

1

4× 1

14= 1 × 1

4 × 14= 1

56Figure 8. The equivalent of the whole number 1 multiplying a fraction.

For two students, they will get 2/14 of 1/4 paper, i.e.

1

4× 1

14× 2 = 1 × 4

2 × 14= 2

56Figure 9. Two parts of 1/14 of 1/4 paper.

So two students get 2/56 of paper.

Mr. Wang concluded:

Now we can arrive at a conclusion: when multiplying fractions, the numerator will be theproduct of numerators, the denominator will be the product of denominators.

1

4× 2

14= 1 × 2

4 × 14= 2

56Figure 10. The rule of multiplying fractions.

At last, Mr. Wang applied the above conclusion to direct students to solvethe part (c) of Problem 3:

3

4× 2

3= 3 × 2

4 × 3= 6

12Figure 11. The solution of part (c) of Problem 3.

This activity addressed visually the connection between a concretemodel and procedure of fraction multiplication, which provides clear stepsfor fraction multiplication and also promotes students’ engagement inlearning. Notice that the notation from the above is different from thatof U.S. notation. In Chinese textbooks, the multiplication of fractions isrepresented differently: dividing1/4 of the paper into 14 pieces is writtenthe same as “to find 1/4’s 1/14” (Chinese language expression) and itsexpression is 1/4 × 1/14, while in U.S. it would be “to find 1/14 of 1/4”and its expression is 1/14 × 1/4. The Chinese way of defining fractionmultiplication seems to have the same order between the meaning andexpression (i.e., first having 1/4, and then having 1/4’s 1/14) which comple-ments student thinking more effectively, while the U.S. way of definingfraction multiplication tends to produce confusions for students.


Although 36% of responses from U.S. teachers focused on connectingto the concrete model, and 39% of them related to manipulatives activities,the U.S. teachers again often ignored developing the connection betweenmanipulative activities and abstract thinking. For example, Mrs. Parkerwould direct students to, “Cut a circle (paper) into 3 pieces. Take 2/3.Cut each one of these 2 pieces into 4 and take 3 of each one. You have 6out of 12 pieces”. She failed to connect the manipulative explanation andprocedure of multiplying fractions. The lack of such connection would failto build a bridge for students to understand why they use the manipulativeand how the activity would help them to use the procedure in doing themultiplication. Only 4% of responses of U.S. teachers showed exampleswith connecting concrete models or manipulatives to the procedure ofmultiplying fractions.

Use of Prior KnowledgeIn the introduction of new concepts, using prior knowledge not only helpsstudents to review and reinforce the knowledge being taught, but also helpsthem to picture mathematics as an integrated whole rather than as separateknowledge. It develops generalizations and helps students to solidify whatthey have learned and allow them to transfer the knowledge to new situ-ations (Suydam, 1984). Linking the new and prior knowledge in contextwill also help students know why and how to learn the new topic and graspnew knowledge with better understanding. This is supported by NCTM(1989), “Connection among topics will instill in students an expectationthat the ideas they learn are useful in solving other problems and exploringother mathematical concepts” (p. 84). Furthermore, NCTM (2000) pointsout, “Because students learn by connecting new ideas to prior knowledge,teachers must understand what their students already know” (p. 18).

In this study, 45% of the Chinese teachers focused on the importance ofdetermining students’ prior knowledge; in contrast, very few U.S. teachers(7%) gave examples for prior work to help students learn fraction multi-plication. For example, Mrs. Zhong presented the following introductorystory, which connects addition:

The monkey’s mother bought a watermelon and cut it into 9 pieces; every monkey ate 2/9of the pieces. How many pieces did four monkeys eat? How do you express this problemin addition? How do you express this problem in multiplication? Which method is easier?

This example helps students transfer prior knowledge of repeated addi-tion to multiplication with better conceptual understanding of fractionmultiplication.

In reviewing prior knowledge, one can, according to Confucius, alwaysfind new knowledge (Cai & Lai, 1994). Mathematics education in China


has been following this idea for classroom teaching for centuries. Obser-vation of classrooms in this study showed that Chinese teachers spent atleast one-third of the time reviewing prior knowledge at the beginning orduring class. One of the teachers said that she only teaches a new lesson 5to 10 minutes every day; the rest of the time is spent reviewing and rein-forcing the knowledge. The review process not only promotes continuityand attains a more comprehensive view of topics previously covered, but isalso a diagnostic tool that helps teachers to identify student strengths andweaknesses and provides valuable insight for future instruction (Suydam,1984).

Promoting Students’ Thinking about Mathematics

Visual Activities and Abstract Thinking

The results in Table III show different emphases on approaches to promotestudents’ thinking: U. S. teachers tend to use charts and tables, concrete orpictorial models, and manipulatives, while Chinese teachers tend to buildstudents’ abstract thinking using procedures. For example, in response toProblem 4, Mrs. Flores displayed this chart for a proportion activity:

Girls: 3 6 9 12 15 18 21 24

Boys: 5 10 15 20 25 30 35 40

Using a chart to write equivalent ratios shows a pattern and makesproblems easier for students to understand.

In Chinese texts, simple equation and direct/indirect variation are intro-duced at the sixth grade in 11th & 12th elementary math textbooks(Jiangsu, 1998). The similar content areas in U.S. 6th grade textbook(Glencoe, 2000) used by teachers in this study was also introduced to6th grade students. However, teachers’ responses in this study showed thatChinese teachers tend to use the algebraic approach in solving proportionproblems more than do U.S. teachers. For example, Mrs. Wang used thefollowing approach:

The ratio of girls to boys is 3:5, which means the girls are 3/5 of boys and boys are 5/3 ofgirls. So if the ratio of boys to girls is more than one, it will be a direct variation. Let thenumber of boys be x; the number of girls will be (40-x), so the ratio of girls to boys = x:(40-x). Therefore, from the following proportion:

3

5= x

40 − x

We can find the number of boys.


Mrs. Zhen explained the procedure without using proportion:

Since girls are 3 parts and boys are 5 parts, the total is 8 parts with 40 students. Every parthas 40/8 = 5 students. So boys will be 5 parts time 5 students, i.e., 25 students.

Furthermore, Table III shows that 43% of U.S. teachers in this studyprovided general questions that probably would not provide insight intostudents’ thinking. A question such as “Do you have the problem set upcorrectly?” could prompt a student to look at problem again, but placesless focus on students’ misconception. In contrast, all Chinese teachers inthis study used probing questions at various levels, which help teachersto explore students’ thinking directly in different ways and encouragestudents’ thinking deeply and critically. Mrs. Wang would ask her students:

What measurements are being compared in the ratio of 3:8? Girls are being compared tothe whole.What measurements are being compared in the ratio of x: 40? Boys are being compared tothe whole.How can we use unequal ratios to make a proportion? How can we make changes in orderto get two equivalent ratios?

As a basis for understanding questions she asked, Mrs. Wang would directher students to solve the problem using two different ways:

Method 1. Let x be the number of boys, so 5/8 = x/40Method 2. Let x be the number of girls, so 3/8 = x/40

Effective teachers “know how to ask questions” (NCTM, 2000, p. 18) andhow to use these questions to enhance the students’ thinking.

Self-reflection and CommunicationNCTM (2000) views communication as an essential part of mathematicsteaching and learning: “Reflection and communication are intertwinedprocesses in mathematics learning” (NCTM, 2000, p. 61). In this study,the U.S. teachers tended to provide students with opportunities to discussand share their ideas. Mrs. Baker “always asks students to explaintheir answers”. Mrs. Larson would have students engage in “discussion,reworking, discourse, and remembering and learning from their mistakes”.Chinese teachers in this study also encouraged students’ thinking andcommunication, but focused on developing reflection. Chinese teachersencouraged their students to think about problem solving, to substi-tute their answers into the original equations, and to check to see if itmakes sense. Reflection occurs when students check and analyze theirwork and thinking and helps students to reorganize knowledge and find


their errors by themselves. Importantly, reflection develops a deep under-standing and fosters good learning habits and has been viewed as a criticallearning strategy constantly taught in mathematics classrooms in China.The importance of reflection was noted by Fennema and Romberg (1999)who stated that reflection plays an important role in solving problems,and a critical factor of reflection is that teachers recognize and valuereflection. In this study, Chinese teachers Mrs. Wang and Mrs. Lu encour-aged students to be a “mathematics doctor”, which means to reflect andto examine the errors in problem solving. They also valued reflectionby giving praise to students who do well on checking procedures andanswers.

In this study, neither group of teachers used estimation very often. Forexample, in their response to Problem 4, only 4% of the U.S. teachersand 6% of Chinese teachers mentioned using estimation to help students’thinking. Estimation is an important skill in the thinking and reasoningprocess. It fosters a good number sense, helps students to think and reasonlogically, and develops proficient skills in computations and problemsolving.

DISCUSSION AND CONCLUSIONS

Four Aspects of Analysis of Pedagogical Content Knowledge

Comparative study can increase our understanding of how to produceeducational effectiveness and enhance our understanding of our owneducation and society (Kaiser, 1999). However, it is difficult to conduct avalid comparative study between different cultures without setting essen-tial components as a norm in analysis of data. This study includedcomparisons and contrasts of teachers’ pedagogical content knowledgebetween the U.S. and China by using four aspects of pedagogical contentknowledge of students’ thinking: building on student ideas in mathematics,addressing students’ misconceptions, engaging students in mathematicslearning, promoting student thinking about mathematics. Each of theseaspects consists of several essential components (see Table II). By analysisof these essential components, this study provided suggestions as to howto set up dimension and scope for further cross-cultural comparativestudies. Although the four aspects in this study are only a part of pedago-gical content knowledge, namely, knowledge of students’ thinking, thisstudy addressed a critical way of assessing teachers’ knowledge regardingstudents’ thinking.


Importance of Pedagogical Content KnowledgeTeacher knowledge of mathematics is not isolated from its effects onteaching in the classroom and student learning (Fennema & Franke, 1992).Teachers’ pedagogical content knowledge combines knowledge of content,teaching, and curriculum, focusing the knowledge of students’ thinking.It is closely connected with the content knowledge, connected with theway of transformation of content knowledge in the learning process and inthe way in which teachers know about the students’ thinking. This studyindicated that deep and broad pedagogical content knowledge is importantand necessary for effective teaching. Teaching for understanding includesa convergent process in which teachers build students mathematics ideasby connecting prior knowledge and concrete models to new knowledge,focusing on conceptual understanding and procedure development. Inaddition, teachers should be able to identify students’ misconceptionsand be able to correct misconceptions by probing questions or usingvarious tasks. Teachers should engage students in learning by providingexamples, representations and manipulatives. Finally, effective teachingrequires the effort of promoting students’ thinking by a variety of focusingquestions and activities. The results of this study supported the idea thata balance is needed between the use of manipulatives and developingprocedures. Although manipulatives develop conceptual understandingof mathematics, procedural learning is an essential learning process forreinforcing understanding and achieving mathematical proficiency and isa necessary step for problem solving. Without developing firm under-standing and skill with procedures, students will not be able to solveproblems efficiently and confidently. Last and most importantly, atten-tion to learners’ cognitions is a key component in teachers’ pedagogicalcontent knowledge and effective teaching. Knowledge of students’ mathe-matical thinking helps teachers to enhance their own knowledge of contentand curriculum, prepare lessons thoroughly, and teach mathematics effec-tively. Without knowledge of students’ thinking, teaching cannot producelearning; it may instead be like “playing piano to cows” (a Chineseidiom).

Conclusion

The results of this study indicated that mathematics teachers’ pedagogicalcontent knowledge in the two countries differed markedly and this has adeep impact on teaching practice. The Chinese system emphasizes gainingcorrect conceptual knowledge by reliance on traditional, more rigid devel-opment of procedures, which has been the practice of teaching and learning


mathematics content for many years. The United States system emphasizesa variety of activities designed to promote creativity and inquiry to developconcept mastery, but often has a lack of connection between manipulativesand abstract thinking, and between understanding and procedural develop-ment. Both approaches have shown benefits and limitations in teachingand learning mathematics, and also illustrate the different demands onteachers’ pedagogical content knowledge.

This study cannot necessarily be generalized to all mathematicsteachers in the United States and China because the samples included onlyone city from each country, with 23 schools from China and 12 schoolsfrom the U.S. However, this is an internal comparative study and, with acentralized education system in China, one city may represent the wholesystem of education in China. With a locally controlled education systemas in the U.S., one city may not reflect the whole United States. There-fore, the results cannot necessarily be applied to teachers in the UnitedStates. Nevertheless, these results do point to the importance, from aninternational perspective, of pedagogical content knowledge and to theessential components that can promote further understanding of effectivemathematics teaching.

NOTE

1 7th and 8th graders in China have already learned Algebra I.

REFERENCES

American Association for the Advancement of Science (2000). Middle grades mathematicstextbooks: A benchmarks based evaluation. Washington, DC: Author.

An, S., Kulm, G., Wu, Z., Ma, F. & Wang, L. (October, 2002). A comparative study ofmathematics teachers’ beliefs and their impact on the teaching practice between theU.S. and China. Invited paper presented at the International Conference on MathematicsInstruction, Hong Kong.

Cai, X.Q. & Lai, B. (1994). Analects of confucius. Beijing: Sinolingua.Carpenter, T.P. & Lehrer, R. (1999). Teaching and learning with understanding. In E.

Fennema & T. Romberg (Eds.), Mathematics classrooms that promote understanding(pp. 19–32). Mahwah, NJ: Erlbaum.

Carroll, W.M. (1999). Using short questions to develop and assess reasoning. In L.Stiff (Ed.), Developing mathematical reasoning in grades K-12: 1999 NCTM yearbook(pp. 247–253). Reston, VA: National Council of Teachers of Mathematics.

Education Department of Jiangsu Province (1998). Mathematics: 11th textbook forelementary school. Nanjing, JS: Jiangsu Educational Publisher.

Elbaz, F. (1983). Teacher thinking: A study of practical knowledge. London: Croom Helm.


Ernest, P. (1989). The impact of beliefs on the teaching of mathematics. In P. Ernest (Ed.),Mathematics teaching: The state of the art (pp. 249–254). New York: The Falmer Press.

Fennema, E. & Franke, M.L. (1992). Teachers knowledge and its impact. In D.A. Grouws(Ed.), Handbook of mathematics teaching and learning (pp. 147–164). New York:Macmillan Publishing Company.

Fennema, E. & Romberg, T.A. (1999). Mathematics classrooms that promote under-standing. Mahwah, NJ: Lawrence Erlbaum Associates.

Glencoe. (2000). Mathematics: Applications and connections, Course 1. Glencoe:McGraw-Hill.

Kaiser, G. (1999). International comparisons in mathematics education under theperspective of comparative education. In G. Kaiser, E. Luna & I. Huntley (Eds.), Inter-national comparisons in mathematics education (pp. 1–15). Philadelphia, PA: FalmerPress.

Kerslake, D. (1986). Fractions: Children’s strategies and errors. Windsor, England:NFER-Nelson.

Kulm, G., Capraro, R.M., Capraro, M.M., Burghardt, R. & Ford, K. (April, 2001). Teachingand learning mathematics with understanding in an era of accountability and high-stakes testing. Paper presented at the research pre-session of the 79th annual meetingof the National Council of Teachers of Mathematics. Orlando, FL.

Li, J. & Chen, C. (1995). Observations on China’s mathematics education as influenced byits traditional culture. Paper presented at the meeting of the China-Japan-U.S. Seminaron Mathematical Education. Hongzhou, China.

Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ: LawrenceErlbaum Associates.

National Council of Teachers of Mathematics (NCTM). (1989). Curriculum and evaluationstandards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (NCTM). (2000). Principles and standardsfor school mathematics. Reston, VA: Author.

Pinar, W.F., Reynolds, W.M., Slattery, P. & Taubman, P.M. (1995). Understandingcurriculum. New York: Peter Lang.

Robitaille, D.F. & Travers, K.J. (1992). International studies of achievement in math-ematics. In D.A. Grouws (Ed.), Handbook of mathematics teaching and learning(pp. 687–709). New York: Macmillan Publishing Company.

Shulman, L. (1985). On teaching problem solving and solving the problems of teaching. InE. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple researchperspectives (pp. 439–450). Hillsdale, NJ: Lawrence Erlbaum Associates.

Shulman, L. (1987). Knowledge and teaching: Foundations of the new reform. HarvardEducational Review, 57(1), 1–22.

Silver, E.A. (1998). Improving mathematics in middle school: Lessons from TIMSS andrelated research. Washington, DC: U.S. Department of Education.

Sowder, J. & Philipp, R. (1999). Promoting learning in middle-grades mathematics. In E.Fennema. & T.A Romberg (Eds.), Mathematics classrooms that promote understanding(pp. 89–108). Mahwah, NJ: Lawrence Erlbaum Associates.

Sowder, J.T., Philipp, R.A., Armstrong, B.E. & Schappelle, B.P. (1998). Middle-gradeteachers’ mathematical knowledge and its relationship to instruction. Albany, NY: StateUniversity of New York Press.

Stigler, J.W. & Perry, M. (1988). Cross-cultural studies of mathematics teaching andlearning: Recent finding and new directions. In D. Grouws & T. Cooney (Eds.),


Effective mathematics teaching directions (pp. 194–223). Reston, VA: National Councilof Teachers of Mathematics.

Suydam, M.N. (1984). The role of review in mathematics instruction. ERIC/SMEACMathematics Education Digest, 2 (ERIC Document Reproduction Service No.ED260891).

Department of Teacher Education Shuhua AnCalifornia State University1250 Bellflower BoulevardLong Beach, CA 90840-2210USAE-mail: [email protected]

Department of Teaching Gerald KulmTexas A&M University Zhonghe WuLearning and CultureCollege Station, TX 77843USAE-mail: [email protected]

[email protected]

Documents

The Pedagogical Content Knowledge of Middle School, Mathematics Teachers in China and the U.S