A COMPARISON OF PROBLEMS IN SELECTED CONTENT SECTIONS IN AMERICAN AND SINGAPORE MATHEMATICS TEXTBOOKS

Yeping Li

University of New Hampshire

yeping@math.unh.edu

This paper reports a comparative study on mathematical problems that follow content presentation of graphing linear equations/functions in several U.S. and Singapore textbooks. The results show the differences in problems' three dimensions: mathematics, context, and performance requirements. In particular, the U.S. textbooks tended to include various problems for developing students' problem-solving ability in general, whereas the Singapore textbook tended to include purely mathematical problems that have high mathematics requirement to facilitate students' acquisition of newly taught concepts and procedures. The results provide a basis for understanding students' mathematics performance documented previously and further support the use of problem-analysis approach for understanding curricular expectations of developing students' mathematics competence.

Efforts to identify contributing factors for cross-system differences in students' mathematics achievement have led to the contention that curriculum is one of the key factors (e.g., McKnight et al., 1987; Schmidt, McKnight, & Raizen, 1997). In particular, researchers have analyzed textbooks to understand their potential effect on students' mathematical achievement in the United States and other countries (e.g., Li, 2000; Mayer, Sims, & Tajika, 1995; Schmidt, McKnight, & Raizen, 1997). The results from previous textbook studies have shown cross-system similarities and differences in textbook content presentation (e.g., Mayer et al., 1995; Schmidt, McKnight, & Raizen, 1997) and exercise problems (e.g., Li, 2000). Unlike textbook content analyses that often focus on content exposure to students, an examination of textbook problems can illuminate the cross-system similarities and differences in expectations for developing students' mathematics competence. Li (2000) exemplified the feasibility and importance of textbook problem analysis in the case of China and the United States. To extend the previous study, this study was undertaken to compare the mathematical problems in several American and Singapore textbooks. Due to variations of topic sequence and content, it was necessary to designate a common content topic in the textbooks selected for comparison. Thus, in this study all mathematical problems presented in selected content sections were compared for several American and Singapore textbooks.

Theoretical Framework

Mathematical problems can be analyzed from various perspectives (e.g., Goldin & McClintock, 1984). An examination of mathematical problems can reveal their characteristics in mathematics and context (e.g., Stigler et al., 1986). Moreover, differences in problems' performance requirements can also dramatically influence students' mathematics performance (e.g., Zhang, 1992). Therefore, a three-dimensional framework was developed for examining textbook problems (see the framework below).

Framework for Coding Mathematical Problems in Textbooks*

1. Mathematics^# • same content (S)

• different content (D)

• mixed content (M)

2. Context • purely mathematical context (PM)

• illustrative context (IC)

3. Performance Requirements⁺⁺

(1) Response type: • no explanation or solution process required (NES)

• explanation or solution process required (ES)

(2) Cognitive requirement: • conceptual understanding (CU)

• performing routine procedures (RP)

• using complex procedures (CP)

• problem solving (PS)

• special requirement (SR)

(*: An elaborate description of the framework can be found in Li, 1999;

^#: A problem’s requirement in mathematics is specified in terms of whether the mathematics content of the problem is the same as, different from, or mixed other contents with the content that is introduced in the immediate content sections;

⁺⁺: Problems’ performance requirements include two aspects: response type and cognitive requirement.)

Method

Materials

Five U.S. textbooks and one Singapore textbook were selected for comparison. These textbooks are the ones that were analyzed in the TIMSS curriculum study (Schmidt, McKnight, Valverde, Houang, & Wiley, 1997). All textbooks were developed and intended for use in the eighth grade. The American textbooks were commonly used across the country in various settings and with diverse populations. The Singapore textbook was commonly used and bore the approval of the Ministry of Education of Singapore. An examination of these textbooks showed that there are striking differences in their content topic inclusions (see Li, 1999). In particular, only one common content topic on algebra, graphing linear equations/functions, can be found across the five U.S. textbooks and the Singapore textbook. Therefore, mathematical problems that were included in sections on graphing linear equations/functions were examined and compared.

Problem Analysis

Mathematical problems selected from the textbooks were those exercises or questions that did not have accompanying solutions and/or answers. In both countries' textbooks mathematical problems appeared under the headings: 'check for understanding', 'exercises', 'problems', 'practice', 'application', or 'problem solving' within or immediately following the selected content sections. Each mathematical problem was coded using the four categories listed in the above framework. A total of 308 problems were examined. The data used in this report were from a larger textbook study (Li, 1999).

A second rater independently coded problems randomly selected from each textbook. The inter-rater agreement of all corresponding codes was 97%.

Results and Discussion

Table 1 shows the percentages of mathematical problems classified in terms of problem requirements in mathematics (same content, different content, or mixed content), context (purely mathematical context or illustrative context), and performance for the American and Singapore textbooks. Problems’ performance requirements include two aspects: response type (explanation or solution process required, no explanation or solution process required), and cognitive requirement (conceptual understanding, performing routine procedure, using complex procedures, problem solving, or special requirement).

Table 1

Percentages of Textbook Problems Classified According to the Categories of Mathematics, Context, Response Type, and Cognitive Requirement

	Singapore Text	U.S. Texts
Mathematics
S	100	92
D	0	7
M	0	1
Context
PM	100	89
IC	0	11
Response Type
NES	99	93
ES	1	7
Cognitive Requirement
CU	13	32
RP	72	56
CP	15	8
PS	0	4
SR	0	0

The table shows a similar distribution pattern for the classifications in each of the three categories: mathematics, context, and response type. It was found that in both systems, the mathematical problems in sections on graphing linear equations/functions overwhelmingly required the same mathematics content as introduced, contained a purely mathematical context, and required no explanation or solution process. Although a similar pattern is generally presented across these two systems’ textbooks, some substantial variations exist. Specifically, in the dimension of mathematics, the U.S. textbooks contained a small percentage of exercise problems that required the use of mathematics content which was different from or mixed with the newly introduced content, whereas none of those included in the corresponding section in the Singapore text. The results indicate that the U.S. texts tended to provide students with problems that served the purposes of content review or connections and the Singapore textbook emphasized on students’ practice in using newly introduced concepts and procedures. In the dimension of context, the U.S. texts contained a high percentage of exercise problems given in illustrative contexts but none in the Singapore text. The results suggest that the U.S. texts tended to include fewer purely mathematical problems but more real-world-like problems for students’ practice. In the first aspect under the dimension of problems’ performance requirements, response type, the results show that the U.S. texts put more emphasis on explanation and solution process in exercise problems than did the Singapore text. There are very limited requirements for explanation in the exercise problems in the selected section from the Singapore text. In contrast, the U.S. texts contained 7% of the problems that required an explanation or solution process.

For cognitive requirement the percentage of problems classified also shows a similar pattern, but to a less degree. 56% of the U.S. textbook problems in the selected sections and 72% of those in the Singapore textbook were found to require performing routine procedures. Such big difference was also evident between the percentages of problems classified as requiring conceptual understanding for solution in the textbooks from the Singapore (13%) and the United States (32%). However, the Singapore text included a higher percentage of problems that required using complex procedures, whereas the U.S. texts tended to put less emphasis on using complex procedure but more on problem solving.

These results show that the U.S. texts tended to include problems that vary in mathematics content, context, response type, and cognitive requirement. The Singapore text tended to include purely mathematical problems that had high mathematics requirements. The cross-system differences in textbooks' problem inclusion suggest that the U.S. texts emphasized problems' variations that may have advantages for developing students' problem-solving competence in general, whereas the Singapore text emphasized the use of exercise problems to facilitate students' acquisition of newly taught concepts and procedures. Evidence exists that Singapore students outperformed their U.S. counterparts in solving school mathematics problems (Beaton et al., 1996). This study illustrated that the Singapore text did expect students higher and more on solving mathematically difficult problems. In contrast, the U.S. texts tended to provide students with opportunities of solving various problems that is less mathematically demanding. The cross-system variations in curricular expectations of developing students' mathematics competence, as illustrated from this study, show a pattern of curricular differences that is similar to what were found in a previous comparison of problems in Chinese and American mathematics textbooks (Li, 2000). Taken together, both this study and the previous study (Li, 2000) suggest not only the relevance between curricular expectations and students' mathematics performance, but also the feasibility of using problem-analysis approach for understanding curricular expectations of developing students' mathematics competence.

References

Beaton, A., Mullis, I., Martin, M., Gonzalez, E., Kelly, D., and Smith, T. (1996). Mathematics achievement in the middle school years: IEA’s Third International Mathematics and Science Study (TIMSS). Chestnut Hill, MA: TIMSS International Study Center, Boston College.

Goldin, G. A. and McClintock, C. E. (Eds.) (1984). Task variables in mathematical problem solving. Philadelphia, Pennsylvania: The Franklin Institute Press.

Li, Y. (1999). An analysis of algebra content, content organization and presentation, and to-be-solved problems in eighth-grade mathematics textbooks from Hong Kong, Mainland China, Singapore, and the United States. Unpublished doctoral dissertation, University of Pittsburgh, Pittsburgh, PA.

Li, Y. (2000). A comparison of problems that follow selected content presentations in American and Chinese mathematics textbooks. Journal for Research in Mathematics Education, 31(2), 234-241.

Mayer, R. E., Sims, V., and Tajika, H. (1995). A comparison of how textbooks teach mathematical problem solving in Japan and the United States. American Educational Research Journal, 32(2), 443-460.

McKnight, C. C., Crosswhite, F. J., Dossey, J. A., Kifer, E., Swafford, J. O., Travers, K. J., and Cooney, T. J. (1987). The underachieving curriculum: Assessing U. S. school mathematics from an international perspective. Champaign, IL: Stipes.

Schmidt, W. H., McKnight, C. E., and Raizen, S. A. (1997). A splintered vision: An investigation of U.S. science and mathematics education. Dordrecht, Netherlands: Kluwer Academic Press.

Schmidt, W. H., McKnight, C. E., Valverde, G. A., Houang, R. T., and Wiley, D. E. (1997). Many visions, many aims (Vol. 1): A cross-national investigation of curricular intentions in school mathematics. Dordrecht, Netherlands: Kluwer Academic Press.

Stigler, J. W., Fuson, K. C., Ham, M., and Kim, M. S. (1986). An analysis of addition and subtraction word problems in American and Soviet elementary mathematics textbooks. Cognition and Instruction, 3(3), 153-171.

Zhang, D. (1992). Some puzzling questions arising from mathematics education in China. In I. Wirszup & R. Streit (Eds.), Developments in school mathematics education around the world, Vol. 3, the Proceeding of the UCSMP International Conference on Mathematics Education. Reston, VA: NCTM.

Note: Preparation of this report was supported in part by a summer faculty fellowship from the University of New Hampshire.