East Asian students outperform US students on several measures of performance. This special issue is focused on curricular analyses that can increase our understanding of this phenomenon and also increase our understanding of social-cultural practices that help develop teacher and student understanding of mathematics in China (and other East Asian countries) and in the United States. Analyses of the mathematics topic content of the intended, implemented, and achieved curriculum are vital as a basis for understanding national differences (e.g., Cai, 2008; Li, 2007a). But there are also language differences and differences in textbook support that affect the clarity of the mathematical concepts and procedures to be taught and learned, both for teachers and for students. These classes of differences are the focus of the present paper.
Previous analyses of East Asian textbooks have emphasized their coherence in content development and the power of their meaning-making supports (Li, 2008; Murata, 2004, 2008; Murata & Fuson, 2001, 2006; Watanabe, 2006). Fuson and Murata (2007) extended these points by describing teaching principles drawn from US National Research Council reports, the NCTM process standards, and from teaching in Japanese classrooms. In addition to the importance of the above features of language and textbook illustrations as meaning-making supports, they identify the importance of a coherent learning path that supports student movement from primitive to more advanced methods that are mathematically desirable. Mathematically-desirable methods show important mathematical features, generalize across numbers and situations, and are efficient enough. Fuson and Murata also discuss how it is possible to teach mathematically-desirable methods but also make them accessible to students so that students can understand them. Therefore our analysis in this paper will also focus on the extent to which the whole range of student methods from primitive to advanced are shown in the book because this helps the teacher recognize these methods but also move students along the learning path to general mathematically-desirable methods. These focal research issues are summarized in Table 1 as a framework that future cross-cultural analyses of textbooks can use and modify. This framework is a result of the study and was clarified as we worked on the analysis. Table 1 may better be placed at the end rather than at the beginning of the paper, but its placement here allows it to act as an advance organizer for the issues to be addressed and has enabled us to introduce and explain here the terminology that will be used later in the paper.
Partners were shown initially in the CMW program with a static part-part-whole drawing (Fuson & Willis 1989) like those commonly used in the US (a rectangle with a horizontal line splitting it in half, and one of these halves further split by a vertical line). But children viewed these as static, and many did not understand why there were twice as many entities as were actually in the problem situation (because the total and both addends were there at the same time). The visual equal split for the addends in this representation was also problematic for some children because the addends were usually not the same. We began using a drawing like the addend drawing in the Chinese books and only later saw it in East Asian books. We called this a math mountain, with a story about tiny tumblers who lived at the top of a mountain and some went to play each day on one side of the mountain. In kindergarten children drew circles to show how many played on each side and then wrote these partners of the total at the bottom. These math mountains were introduced in Grade 1 in the third unit to represent unknown addends (partners), which were then related to subtraction situations. So partners/totals, addition, and subtraction situations and representations became related in problem solving as children used equations, or math drawings with circles, or math mountains to show their situations and solutions. The math mountains had a sensory-motor component that allowed children to compose/decompose the number from the total at the top to addends at the bottom, and children found this to be a powerful representation.
East Asian books all show the meanings of place-value notation using quantities such as bundled sticks or base-ten blocks, and they use such pictures to show such quantities as they relate to the steps of multidigit addition and subtraction. Many US texts now also show such pictures, especially base-ten blocks in which the ones are 1 cm2, the tens are 10 cm long, and the hundreds are a flat that is 10 by 10 by 1 cm. Many US texts now also show these blocks or other quantities with multidigit addition and subtraction, but usually not in separate steps. In the research leading to Math Expressions, we found that schools seldom had enough blocks or other multiunit quantities for children to use, such manipulatives created management issues, and they were hard to show to the whole class when explaining multidigit addition and subtraction. Therefore we moved to using math drawings like those shown in Fig. 5. Students first made the 10-sticks and 100-squares on centimeter dot grids and then moved to making sketches (the math drawings) using the quick-tens and quick-hundreds using 5-groups so that a viewer could see at a glance how many there were. This also reduced drawing errors considerably and increased the use of the separate multiunit conception that did not require counting by hundreds, tens, and ones. Each step in a written multidigit addition and subtraction method could be related to a step in the drawing during explanations of methods. When explaining their method, children used the quantity words that named the multiunits (three hundreds seven tens nine ones) that were like Chinese number words (except for the plurals and the word ones at the end), and they also became able to use English number words. They initially made numbers using secret-code-cards, which had drawings of hundreds, tens, and ones on the back, and sometimes used these when solving problems.
The two Write All Totals methods and the Expanded Notation Form shown at the top of Fig. 6 are variations of the second mathematically-desirable and accessible method used in Math Expressions. This method can be shown by making both numbers with the secret-code cards and then taking apart the cards to show the numerical values using zeroes (see especially the top right example in Fig. 6). After using the secret-code cards, students become able to see the zeroes hiding under the tens and ones digits (e.g., they can see 80 and 50 in the vertical problems). Then they do not need to see or write out the expanded notation as in the top right but can do either one of the vertical methods. Many students prefer to work left to right (they read from left to right) and so use the left-to-right version, but some students use the right-to-left method. To keep these numerical methods meaningful, students connect each step in the numeric method to steps in the math drawing as they solve and explain their method to their classmates.
The three levels identified by Ma (1999) for Chinese teachers seemed to be verified in other East Asian books. The make-a-ten methods for teen addition and subtraction were developed in separate units as Level 1. In Level 2, multiple methods were given for 2-digit problems. In Level 3 for 3-digit and larger problems, the books focused on one generalizable mathematically-desirable method. However, not all East Asian students may be able to explain these larger problems fully. Fuson and Kwon (1992b) found that all students could explain the ungrouping in two-digit problems, but that some viewed three-digit problems only as columns of single digits and did not, for example, say they were dealing with tens when they explained their adding or subtracting in the tens column. Drawings and modeling by the teacher and other students do enable Math Expressions students to make such quantity explanations for larger numbers. However, when one is actually carrying out multidigit addition and subtraction, especially with larger numbers, it may be useful to ignore the names of the places and solve as if each column is the same kind of multiunit. The consistent ten-for-one trades between all adjacent columns including decimal places enables one to do this, and a student could add the name of the place if an explanation is desired.
This in-depth analysis of the major early numerical aspects in a representative Chinese textbook series and a US textbook series with major East Asian components illustrates how linguistic issues create different teaching and learning tasks for the same mathematical topic. Previous analyses of East Asian textbooks have emphasized their coherence and the power of their meaning-making supports. This article indicates that a program in the US can have a similar coherence and power, but that additional visual-quantitative and linguistic supports are needed to compensate for the linguistic complexities that are not present in China.
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