Curriculum & cognition: a study on math problems.
Analyzing textbook problem provides a unique opportunity for exploring students' problem-solving experience, as textbook problems embody cognitive expectations for students in a specific system. In this study, we examined mathematical problems in algebra chapters of selected textbooks from Mainland China, Singapore, and the United States. The results revealed some substantial variations between the U.S. textbooks and the textbooks from Asia, which extends what we can possibly learn from textbook analysis in large-scale international studies.
Educational knowledge is a major regulator of the structure of students' experience (Bernstein, 1975). The selection, classification and transmission of educational knowledge are generally outlined in the form of curriculum. Curriculum thus envisions a body of organized knowledge that can contribute to students' cognitive development. With a focus on identifying curricular influence on students' academic achievement, researchers have come to the understanding that cross-system similarities and differences in curriculum can provide partial explanations to cross-system discrepancy in students' academic performance, especially in mathematics (Fuson, Stigler, & Bartsch, 1988; McKnight et al., 1987; Schmidt et al., 1997; Schmidt et al., 2001). In particular, when compared to the curriculum materials from some high achieving education systems in East Asia, researchers have revealed that the U.S. curriculum materials failed to provide challenging mathematics content (e.g., Schmidt, McKnight, & Raizen, 1997) and devoted more page space to student practice than to content instruction (e.g., Carter, Li, & Ferrueci, 1997; Mayer, Sims, & Tajika, 1995).
Because textbooks are often the curricular materials that are the most influential to what happens in classrooms (e.g., McKnight et al., 1987), textbooks have attracted more and more research attention from the international mathematics education community in the past two decades. In particular, the Third International Mathematics and Science Study (TIMSS) was the first time, in the history of large-scale international studies conducted by the International Association for the Evaluation of Educational Achievement (lEA), to include the analysis of textbooks and other curriculum materials from about 50 participating education systems as a major part of the study (Schmidt et al., 1997). However, large-scale international comparisons with a focus on students' mathematical performance often included broad measures of the differences and similarities in mathematics curricula but not an analysis of problems in mathematics textbooks. As students develop their mathematical skills and abilities through solving mathematics problems, problems in the textbooks should be taken as the embodiments of content and performance expectations for students in a specific system. Several cross-national studies on mathematical textbooks have conducted fruitful comparisons of mathematical problems presented in textbooks (e.g., Li, 2000; Stigler et al., 1986; Sugiyama, 1987; Zhu & Fan, 2006). The results from these studies have confirmed the feasibility of textbook problem analysis for exploring students' learning experience through problem solving. To extend what we learn from TIMSS study on textbooks, it thus becomes feasible to reveal cross-system variations in expected students' cognitive development through analyzing textbook problems.
Differences in mathematical problems have been analyzed by different approaches. Some researchers analyzed problems through an analytical approach that examined problems in multiple features (e.g., Goldin & McClintock, 1985; Li, 2000; Stigler et al., 1986). Others analyzed problems through a holistic approach that classified problems into different categories in terms of a specific feature (e.g., Stein & Smith, 1998; Zhu & Fan, 2006). As these approaches have their own merits in revealing problem characteristics that afford different opportunities for students' problem-solving experiences, this current study adopts both approaches in analyzing textbook problems. In particular, textbook problems are analyzed first from an analytical approach with several specified features, then re-examined from a holistic perspective through putting these specified features back together on problems.
Stigler et al. (1986) studied addition and subtraction word problems in U.S. and Soviet textbooks using a classification scheme based on "problem's semantic structure" and "location of unknown quantities". In a cognitive analysis of algebraic problem difficulty, Tabachneck et al. (1995) considered mathematical and contextual factors presented in algebraic problems. In both investigations, the choice and emphasis in different problem factors were a function of the characteristics of the problems being analyzed and the purpose of the study. Similarly, in the present study a comprehensive framework was developed to capture the basic features of the problems from the chapters on algebra in different textbooks. In addition to the mathematical and contextual aspects identified by Tabachneck et al., Zhang (1992) found that the requirements levied by mathematical problems from different nations could dramatically affect students' problem solving performance. Consequently, a three-dimensional framework was used in this study to analyze mathematical problems in different textbooks. The dimensions were problem requirements in mathematics, context, and performance. Several categories under each dimension were also identified (see below) and used in coding the problems for this investigation.
Framework for Coding Mathematical Problems in Textbooks
* same content (S)
* different content (D)
* mixed content (M)
2. Context * purely mathematical context (PC)
* 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)
* other (O)
(*: A problem's requirement in mathematics is specified in terms of whether the mathematics content of the problem is the same as the algebraic content that is introduced in the immediate textbook chapter. Specifically, the mathematical characteristics of each problem in the chapter is grouped into three categories: S--same as the content introduced in the chapter, D--different from the content introduced in the chapter, M--mixed contents from this chapter and outside of this chapter such as geometry and trigonometry.)
This study[l] compared all mathematical problems presented in chapters devoted to algebra in the selected textbooks from Mainland China, Singapore, and the United States. Based on the framework specified above, this study analyzed textbook problems with both analytical and holistic approaches, which aim to reveal the extent and nature, respectively, of cross-system similarities and differences in textbook problem features. The following two research questions were specified for the textbook problem analysis.
(1) What is the extent of similarities and differences in students' envisioned mathematics competence as represented by the problems provided in eighth-grade mathematics textbooks from China, Singapore, and the United States?
(2) What is the nature of similarities and differences in students' envisioned mathematics competence as represented by the problems provided in eighth-grade mathematics textbooks from China, Singapore, and the United States?
Mathematical problems in chapters on algebra in five U.S. textbooks, one textbook from Mainland China, and one Singapore textbook were selected for comparison. These textbooks are the same ones that were analyzed in the TIMSS curriculum study (see Schmidt et al., 1997). All textbooks were developed and intended for use at the eighth grade level. The American textbooks were commonly being used across the country in various settings and with diverse populations. The textbooks from China and Singapore were commonly used and bore the approval of the Ministry of Education in each education system. The use of the same textbooks analyzed in the TIMSS study made it advantageous in selecting representative textbooks from the three education systems. Such an analysis also made it possible to supplement TIMSS curriculum studies in revealing cross-system variations in expectations for developing students' mathematics competence through problem solving. At the same time, as textbook is a moving target that keeps changing in different education systems, we would like to remind the readers that the study is, essentially, an exploratory case study. It is not intended to generalize the results from these selected textbooks to any newly published ones, which would require further analysis beyond the scope of the current study.
Mathematical problems selected from the textbooks were those exercises or questions that did not have accompanying solutions and/or answers. In these textbooks, mathematical problems appeared under the headings: 'check for understanding', 'exercises', 'problems', 'practice', 'application', or 'problem solving' within or immediately following chapters that devoted to algebraic content. Mathematical problems also included relevant exercises that were given in review sections. A total of 24,991 problems were examined. Mathematical problems in the relevant chapters of each text were coded using the above framework. A second rater independently coded problems given in one randomly selected chapter from each text. The inter-rater agreement of all corresponding codes was 97%. The following are two examples of the problems and coding.
Example 1: Solve the following pair of simultaneous equations.
3x+4y = 8, 6x + 8y = 16.
Coding: same content (S), purely mathematical context (PC), no explanation or solution process required (NES), performing routine procedures (RP).
Example 2: A belt and a wallet cost $42, while 7 belts and 4 wallets cost $213. Find the cost of each item, and show your solution process. Coding: same content (S), illustrative context (IC), explanation or solution process required (ES), problem solving (PS).
Results and Discussion
The results are reported below in terms of research questions with accompanying description of the analysis.
1. The extent of textbooks' similarities and differences in their inclusion of problems in terms of problem requirements in mathematics, context, and performance.
On average, the five U.S. textbooks have about 89% problems requiring the use of same mathematics content. In contrast, Singapore textbook has about 99% and the Chinese textbook 93.7% in this category. Although there are relatively small percentages of problems that require the use of different or mixed mathematics content, the results show that the five US textbooks contain a higher percentage of the problems that require different mathematics content than the other textbooks. The results indicate that the five US textbooks are more likely to provide students with opportunities to solve problems with different content requirements, whereas the textbooks from Singapore and China emphasize more on students' practices in using newly learned concepts and procedures. The results show that all these textbooks are dominated with the problems given in a purely mathematical context. However, substantial differences in the percentage distribution of problem context exist between the five US textbooks and the ones from Singapore and China. Specifically, the five US textbooks have a range of 81.8% and 89.4% and an average of 85.3% problems given in a purely mathematical context. Comparatively, the textbooks from China and Singapore have similarly higher percentages (98.5% and 94.3%, respectively) of problems with such a feature. On the contrary, the five US textbooks contain higher percentages of problems (an average of 14.74%) that are given in illustrative contexts, in contrast to only 1.5% and 5.7% in China and Singapore textbooks. These results indicate that the five US textbooks tend to include less pure mathematics problems but more contextualized problems for students' practices.
In terms of problems' response type, the results show the U.S. textbooks put more emphasis on explanation and solution process in exercise problems (an average of 6.9%) than did the selected textbooks from Singapore (0.1%) and China (2.9%). Notably, these differences in the U.S. problem sets are in the direction of the emphasis given to enhance communication skills, applications, and connections to other subject matter in American standards-based reform documents. For cognitive requirement, the percentage of problems classified also shows a similar pattern, but to a much less degree. 62% of the U.S. textbook problems in the chapters on algebra and around 90% of those in the Asia textbooks were found to require performing routine procedures. Such dramatic difference was also evident between the percentages of problems classified as requiring conceptual understanding for solution in the textbooks from the U.S. (23.5%) as compared to the selected Asia texts (less than 7.5%). This evidence of reduced attention to procedural practice and increased emphasis on conceptual understandings is also in line with the focus of current U.S. mathematics education reform efforts.
2. The nature of problems included in different textbooks.
By specifying the types of problems in terms of different combinations of problems' requirements in mathematics, context, and performance, a total of 49 different types of problems were identified. In general, the five US textbooks include many more types of problems than other textbooks, with a range of 23 to 36 types and an average of 29 types problems included in algebraic chapters. In contrast, there are only 12 and 13 types of problems included in the Singapore and China textbooks, respectively. Although all the textbooks are dominated with the type of problems given as Example 1 above (coded as S-PC-NES-RP), the textbooks from Mainland China (86%) and Singapore (89%) include much higher percentage of this type of problems than the five US textbooks (a range of 46% to 57% with an average of 52%). The comparatively smaller percentages of the type of (S-PC-NES-RP) problems in the five US textbooks related to the distribution of their problems in two trends. The first trend is that the problems in the five US textbooks are spread to many more types of problems than the textbooks from China and Singapore. The second trend is that the five US textbooks have relatively higher percentages of problems that check students' conceptual understanding or knowledge application. In contrast to the problems in China and Singapore textbooks that show higher requirements in mathematics, the problems in the US textbooks often pose more variations and higher cognitive challenges than the ones in these two Asian textbooks.
The results provide some indications of the types of differences and similarities that may be gleaned from a detailed analysis of the mathematical problems in American and selected Asian textbooks. This study revealed that Asian textbooks expected more from students on solving traditional types of problems. Evidence exists that students from Mainland China and Singapore score higher than their U.S. counterparts on traditional algebra problems (e.g., Lapointe, Mead & Askew, 1992; Mullis et al., 2004). In contrast, the U.S. textbooks expected students to develop problem-solving competence through solving various problems. The findings of this study also rectify that analyzing textbook problems is feasible and valuable for understanding cross-system variations in curricular expectations of developing students' mathematics competence (Li, 2000; Zhu & Fan, 2006).
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Zhu, Y. & Fan, L. (2006). Focus on the representation of problem types in intended curriculum: A comparison of selected mathematics textbooks from mainland China and the United States. International Journal of Science and Mathematics Education, 4, 609-626.
 This study is part of a larger research effort that aims to investigate how, as the intended curriculum, mathematics textbooks in Hong Kong, Mainland China, Singapore, and the U.S. present and organize algebra content and problems for classroom teaching and learning.
Yeping Li, Texas A&M University
Yeping Li, Ph.D., is an Associate Professor of Mathematics Education. His interests include cross-national studies in mathematics curriculum and teacher education.
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|Publication:||Academic Exchange Quarterly|
|Date:||Jun 22, 2007|
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