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Can you teach in a normal way? Examining Chinese and US curricula's approach to teaching fraction divisions

Dissertation
Author: YingYing Crystal Feil
Abstract:
This dissertation presents two studies designed to examine the topic of fraction division in selected Chinese and US curricula. By comparing the structure and content of the Chinese and Everyday Mathematics textbooks and teacher's guides, Study 1 revealed many different features presented in the selected curricula. Major differences include the number of lessons on this topic, the algorithms introduced, the type of examples and exercises provided in the textbooks, and the teaching strategy suggested by the teacher's guides. Study 2 examined whether a set of lessons that represented Chinese textbook features or a set of Everyday Mathematics -style lessons were more effective in promoting U.S. sixth-grade students' understanding of fraction division algorithms and their ability to apply the algorithms to solve word problems. Results indicated that the participating U.S. students lacked understanding of mathematics concepts that are relevant to fraction division, and the participants did not effectively learn about fraction division from either the Chinese-style or Everyday Mathematics -style lessons. These studies suggest that to apply features of Chinese mathematic curricula to teach US students, it is important to take into account US students' prior knowledge, learning experiences, and learning styles.

Table of Contents

Chapter 1 Introduction and Background.......................................................................1 Introduction...................................................................................................................1 Background...................................................................................................................3

Chapter 2 Study 1: Content Analysis of Chinese and U.S. Curricula.......................20 Introduction.................................................................................................................20 Method.........................................................................................................................20 Results..........................................................................................................................24 Discussion.....................................................................................................................45

Chapter 3 Study 2: Experimental Investigation of Chinese- and Everyday Mathematics-Style Textbooks....................................................................48 Introduction.................................................................................................................48 Method.........................................................................................................................48 Results..........................................................................................................................58 Discussion.....................................................................................................................71

Chapter 4 General Discussion........................................................................................83 Concept of Division.....................................................................................................83 Examine Curriculum Design With Cognitive Load Theory...................................88 Textbook Structure.....................................................................................................91 Improve Teachers’ Knowledge Through Mathematics Curricula.........................94 Future Studies.............................................................................................................95 Conclusion...................................................................................................................99

References……………………………………………………………………………...102

Appendix A Study 2: Lesson Content.........................................................................112 Appendix B Study 2: Pretest Questions......................................................................115 Appendix C Study 2: Posttest Questions....................................................................116

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Chapter 1 Introduction and Background

Introduction

Over the past 20 years, a series of large-scale studies has shown that U.S. students underperformed in various international mathematics tests when compared to their East Asian counterparts (Hiebert, Gallimore, Garnier, Givvin, Hollingwsworth, & Jacobs, 2003; Stevenson & Stigler, 1992; Stigler, Gonzales, Kwanaka, Knoll, & Serrano, 1999; Stigler & Hiebert, 1999). To examine possible causes for such achievement differences, many researchers (Perry, 2000; Stevenson, Chen, & Lee, 1993; Stigler & Hiebert, 1998) have studied cross-cultural differences in mathematics teaching and learning; these differences include both schooling and non-schooling factors, for example, the structures of school system and policies (Cohen & Spillane, 1992), teachers’ knowledge and classroom practice (Ma, 1999; Stigler & Stevenson, 1991) , and parental expectations of children’s achievement (Stevenson, Lee, Chen, & Lummis, 1990; Stigler & Hiebert, 1998). These studies reveal some possible causes of U.S. students’ relatively low achievement and, through these efforts, they also introduce other countries’ successful mathematics education experiences that may help improve mathematics education in the United States. The purpose of this study is to partially trace such achievement differences to mathematics curricula that are used in China and the United States, and to suggest ways to improve mathematics curricula in the United States. Mathematics curricula and their impact on teaching and learning have received more and more research attention both in the United States and in international context

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(Clements, 2007; Li, 2007; Remillard, 2005; Schmidt, Wang, & McKnight, 2005; Stevenson, 1985). In addition to mathematics curriculum research, many reform curricula have been published and implemented as an attempt to improve the quality of mathematics education in the United States (Carpenter, Fennema, Franke, Levi, & Empson, 2000). Some of these reform-based mathematics curricula are widely used in the United States. However, very few studies (Fuson, Carroll, & Drueck, 2000; Riordan & Noyce, 2001) have examined the effectiveness of these curricula through experimental studies after the curriculum’s development stage. And even fewer studies (Putnam, 2003) have examined how many of these curricula actually meet the reform mathematics standards and goals. Even though Sims and her colleagues (Sims, Perry, Schleppenbach, McConney, & Wilson, 2008) have found that U.S. classes that used a reform curriculum demonstrated a teacher-student discourse pattern that was similar to Chinese classrooms, there has not been much study on the structure and content of these U.S. reform curricula, especially in terms how these curricula differ from or resemble the curricula used in countries where students have shown higher mathematics achievement than U.S. students. Among many cross-cultural studies, Chinese students are consistently rated among the top mathematics performers in international comparisons; and Chinese teachers and students have often been chosen as a baseline for U.S.-East Asia comparisons (Fan & Zhu, 2007; Fuson, 1988; Jiang & Eggleton, 1995; Ma, 1999; Perry, 2000; Stevenson, Hofer, & Randel, 2000; Zhou, Peverly, & Xin, 2006). However, because the Chinese government has only recently begun to allow scholars to study its curricula, comparison of elementary mathematics curricula used in China and the United States is a relatively new research focus.

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In this study, I choose two popular Chinese textbooks and one widely used U.S. reform-based textbook, Everyday Mathematics, as my sample curricula. First, I compared how these textbooks present the topic of fraction division to students by examining their lesson structures, examples, and exercises. Then I analyzed their accompanying teacher’s guides, in terms of what assistance they provide to help teachers understand and teach this topic. To examine whether a set of Chinese-style or a set of Everyday Mathematics- style lessons are more effective in helping U.S. children understand this topic, I conducted an experimental investigation in Study 2. Results indicated that Chinese textbooks introduce fraction division through word problems and encourage students to discover the algorithm on their own, but that Everyday Mathematics presents the algorithms to students directly. Word problems provided in Chinese textbooks exceeded the ones in Everyday Mathematics both in terms of quantity and variety. Analysis of the teacher’s guides indicated that Chinese teacher’s guides provide more detailed instructions for teaching the topic than the EM teacher’s guide. However, results of the teaching experiment indicate that a set of Chinese style lessons are not more effective than a set of Everyday Mathematics-style lessons in helping U.S. students understand and apply fraction division algorithms. Implications of these studies and suggestions for future research are discussed.

Background

Fraction division and teachers’ knowledge. Fraction division is often considered one of the most difficult and least understood topics in elementary and middle-school mathematics (Fendel, 1987; Payne, 1976). Children’s achievement on

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tasks related to this topic is usually very low (Carpenter, Lindquist, Brown, Kouba, & Silver, 1988; Hart, 1981). Therefore, it is urgent to develop a more effective curriculum for teaching this topic to children. Not only is this topic challenging for students, it is also difficult for teachers to understand fully. Studies have shown that many teachers hold misconceptions about this topic and need help to provide effective teaching (Ball, 1990; Rule & Hallagan, 2006; Tirosh, 2000). Therefore, it is also necessary to find ways to improve teachers’ content knowledge and pedagogical content knowledge (Shulman, 1986) about this topic. Another major reason that fraction division was chosen as the mathematical concept for curriculum comparison is that Ma (1999) demonstrated significant differences between Chinese and U.S. teacher’s content knowledge and pedagogical content knowledge on this topic. Ma surveyed 72 Chinese teachers and 23 U.S. teachers on their understanding of fraction division. She asked participating teachers to solve one calculation problem ( 3 1 4 2 1 ÷ ) and create one real-word problem from this equation. Her results showed that all Chinese teachers completed the computation correctly, and 65 (90%) of them created a story problem representing the meaning of the problem presented to them in symbolic form. In addition, many of them clearly pointed out the inverse relationship between fraction multiplication and division to explain the rationale for their word problems. In contrast, only 9 (43%) of U.S. teachers completed the computation correctly, and none created a story problem that was both conceptually and pedagogically correct. Through further interviews, Ma found that while Chinese teachers demonstrated a profound understanding of fundamental mathematics on this topic, U.S. teachers demonstrated relatively little understanding. Such dramatic differences between

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Chinese and U.S. teachers’ understanding of the topic motivated me to investigate possible causes for such differences and to discover ways to help U.S. teachers to improve their understanding. One reason that fraction division is so challenging is because many teachers and students find it difficult to understand the meanings of the algorithms. (Bray, 1963; Capps, 1962; Elashhab, 1978; Johnson, 1965; McMeen, 1962) There are two arithmetic algorithms to calculate a fraction division problem. One is called the common- denominator algorithm, in which both the dividend and the divisor should first be transformed into their equivalent fractions with a common denominator. In this way, the problem is transformed into a whole number division problem when the numerator of the dividend is divided by the numerator of the divisor. For example, 3 6 1 1 4 8 8 8 6 1 6÷ = ÷ = ÷ =. The other algorithm is called the invert-and-multiply algorithm, or “multiply the reciprocal of the divisor” in Chinese textbooks. For example, 3 4 3 6 4 2 4 12 1 5 3 5 2 5 2 10 5 5 1 × × ÷ = × = = = =. Studies have found that the common denominator algorithm is more meaningful to students (Brownell, 1938; Capps, 1962; Miller, 1957) because of its close connection with whole number division (Johnson, 1965). Some students can even construct it themselves (Sharp & Adams, 2002). Therefore, some researchers (Gregg & Gregg, 2007b) suggest that students should first be introduced to the common denominator algorithm so that they can establish a meaningful understanding of fraction division. However, in terms of calculation, when division results in a remainder or the divisor is greater than the dividend, for example, 5 20 3 1 3 4 12 12 20 3÷ = ÷ = ÷, the common denominator algorithm becomes difficult for many students to complete accurately (Bray, 1963; Johnson, 1965).

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In such cases, the calculation can be done easily and quickly with the invert-and-multiply algorithm ( 5 5 20 1 3 4 3 3 3 4÷ = × = = 2 6 ), although students often find little sense in it and have to learn it by rote memorization (Capps, 1962; Elashhab, 1978; McMeen, 1962). Between these two algorithms, current studies suggest that the common denominator algorithm is easy to understand but inefficient in calculation, and that the invert-and-multiply algorithm is efficient in calculation but difficult to understand. However, most mathematics textbooks in the world adopt invert-and-multiply because of its efficiency and connectedness to algebraic thinking (Bergen, 1966; Capps, 1962; Krich, 1964). Therefore, most students are taught to use the more mechanical algorithm. Although it is difficult to find meaning in the invert-and-multiply algorithm, teachers and students from China demonstrate deep understanding of this algorithm, especially in terms of using contextualized representations to explain and model the algorithm (Ma, 1999). Therefore, it is possible that teaching the invert-and-multiply algorithm through real-life models might be an effective method to help students understand this concept. Thus it is important to examine the real-life models of fraction division used in Chinese classrooms. Many researchers have proposed different ways to categorize mathematical word problems in terms of the type of numbers involved, different operations, and context (e.g., Barlow & Darke, 2008; Baroody 1998; Greer, 1994; Marshall, Barthuli, Brewer, & Rose, 1989; Peck & Wood, 2008; Perlwitz, 2005; Vergnaud, 1988). Because this study is focused on division of fractions, I argue that it is important to consider children’s understanding of different real-life situations in terms of how these situations represent the meaning of fraction division. Therefore, I chose to base my word problem

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categorization on Baroody’s (1998) summary of meanings of multiplication word problems. This categorization includes four real-life models. Because division is the inverse operation of multiplication, these four multiplication models can be mirrored to represent meanings of division word problems. I summarize the four types of models as follows: • Fair-Sharing (partitive and measure-out/quotative). This type involves equally dividing a total amount into a number of equal collections. The unknown number can either be the amount of each share (partitive), for example; or the number of shares (measure-out/quotative). For example, Scott has 5½ pounds of rice, he wants to divide it into ½ pound bags, how many bags of rice does he end up with? 1 1 2 2 5 ÷. In the case of partitive problems, because the number of shares are always given as an integer (although the amount of whole collection can be represented by a fraction). Partitive problems are not provided as an example in this topic very often. Most fair-sharing fraction division word problems are measure-out problems.

• Rate. This type involves finding a rate or speed given the total distance/productivity and usage of time. For example, it took Josh 2/5 hour to walk 1 ½ miles, how many miles does he walk for an hour? 1 2 2 5 1 ÷. A special sub- type of this meaning is area meaning, which involves calculating the area of a rectangle or the volume of a cube.

• Comparison. This type involves determining the size of a set, given the size of another set and how many times larger or smaller this set is compared to the unknown set. For example, a bag of brown sugar weights 3½ pounds, the brown sugar is 2/3 as heavy as a bag of white sugar, how much does this bag of white sugar weigh? 1 2 2 3 3 ÷

• Combinations. This type involves all the ways of combing more than one context. For example, Josh can run ¾ miles within 1/5 hour, Josh runs 1 8 faster than Mike, how many miles can Mike run within an hour? Josh: 3 3 1 4 5 4 3÷ = (m/h), Mike: ( ) 3 1 1 3 3= (m/h) 4 8 3 1÷ +

Many studies have found that the measure-out/quotative model of fair-sharing is the easiest meaning of fraction division for children to grasp. Because of children’s understanding of the fair-sharing situation, they not only can easily understand the

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measure-out model, many of them can even spontaneously solve fraction division problems in measure-out situations (Sharp & Adams, 2002). Therefore, many researchers (e.g., Gregg & Gregg, 2007b) suggested that this model should be introduced to children first so that they can start with an easy understanding of fraction division. However, no studies have examined teaching fraction division through the other models. We do not know whether we should teach these other models or how we should teach them. Interestingly, many Chinese teachers in Ma’s study (1999) created word problems that represented the other models. 62 out of a total of 80 stories created by Chinese teachers represented either the comparison or the rate model, but no US teachers created any comparison or rate model story. Such differences motivated me to study what models are included in the Chinese textbooks and teacher’s guides. Therefore, an in-depth examination of the content of Chinese and U.S. curricula is necessary. Studies on content of Chinese and U.S. textbooks. There are two types of studies that focus on specific content in mathematics textbooks. One type focuses on specific learning activities or problem types; the other type focuses on the specific mathematical topics in the curricula. For example, with the increasing effort to promote students’ problem-solving ability and the NCTM Standard’s requirement of problem- solving as an important learning goal (National Council of Teachers of Mathematics, 2000), problem-solving has become a research focus. Some studies show that although Chinese students outperform their U.S. counterparts in tests of general mathematics abilities, they tend to make similar mistakes as U.S. students do when they solve realistic word problems (Cai, 1998; Cai & Silver, 1995; Xin, Lin, Zhang, & Yan, 2007); and that when solving problems that require flexible and non-routine strategies, U.S. students

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even outperformed Chinese students in some questions because U.S. students tend to use visual representations and drawings when they do not know the routine algorithms and symbolic representations needed, but Chinese students tend to give up if they do not know the mathematical solutions (Cai, 2000). These studies show that there is discrepancy between Chinese students’ low problem-solving ability and Chinese curriculum’s focus on in-depth understanding of knowledge and flexible application ability. Many researchers (Cai, Lo, & Watanabe, 2002; Fan & Zhu, 2007) have started to look at how problem-solving is presented in Chinese and U.S. textbooks. Another example of studies on specific problem types is Zhu and Fan’s studies (Fan & Zhu, 2007; Zhu & Fan, 2006). After examining various problem types represented in one Chinese and one U.S. textbook, they found that an absolute majority of problems provided in both of the textbooks were routine and traditional; however, among the small amount of non-traditional problems, U.S. textbooks provided a larger variety of problems, especially problems with visual information. As for the difficulty level of the problems, they found that problems in Chinese textbooks were more challenging than those in the U.S. books. Compared with the large number of studies on problems types, only a very limited number of studies have focused on how specific mathematical topics are taught in China and in the United States. Most of these studies have focused on very basic and elementary concepts. After examining some Chinese and U.S. textbooks, some researchers found that Chinese textbooks focus more on the essential meaning of mathematics concepts than U.S. textbooks. For example, Capraro, Ding, Matteson, and Li (2007) studied how the equal sign is presented in U.S. and Chinese textbooks. A common misconception of

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young students is that the equal sign is a signal for “doing something” rather than a relational symbol of equivalent or quantity sameness. They found that this misconception is manifest in the U.S. and that U.S. textbooks include few activities that address this misconception; but Chinese students are able to interpret the equal sign as a relational symbol of equivalence because Chinese textbooks introduce the equal sign together with the more-than and less-than signs as relational symbols. A further analysis of this topic was done by this research team (Li, Ding, Capraro, & Robert, 2008). To find out how this difference may be traced back to other teaching materials, they examined selected Chinese and U.S. teacher preparation materials, students’ textbooks, and teacher’s guides. They found that although U.S. teacher preparation textbooks rarely interpret the equal sign as equivalence, Chinese teachers’ textbooks typically introduce the equal sign in a context of relationships and interpret the sign as “balance,” “sameness,” or “equivalence.” Another example of examining basic concepts in textbooks is Zhou and Peverly’s (2005) research on the teaching of addition and subtraction. They found that Chinese textbooks, using only small number combinations, introduce the concepts of addition and subtraction together with a focus on the inverse relationships between the two operations; but U.S. textbooks usually introduce subtraction as a very different operation than addition after students have learned addition with sums up to 20. Other studies (Lo, Cai, & Watanabe, 2001; Zhou, Peverly & Xin, 2006) have looked at how Chinese and U.S. textbooks introduce higher levels of mathematics concepts as new knowledge. They found that Chinese textbooks are very similar in terms of always using short word problems to contextualize a new concept; but U.S. textbooks vary a great deal in their ways of introducing new knowledge: some U.S. textbooks

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expect students to discover a new concept on their own through problem-solving; for others, they introduce the new concept as a very abstract concept without much context. For example, Lo and Cai (Lo, et al., 2001; Lo, Watanabe, & Cai, 2004) found that although all of their selected Chinese and U.S. textbooks used contextual problems to introduce a new concept, the types of word problems provided in the Chinese and U.S. textbooks were very different. Chinese textbooks only included word problems that were short and specific to the new concept. But U.S. textbooks varied in the types of word problems provided. They sometimes used very elaborated contextual problems with various sub-problems that were not always relevant to the new concept; but in other cases, for example, when introducing the concept of arithmetic average, some U.S. textbooks did not use any contextualized problems at all (Cai, Lo, & Watanabe, 2002). Another example is that while Chinese textbooks used “equal-sharing” and “per-unit-quantity” word problems to introduce the concept of average; U.S. textbooks introduced the concept as a measure of central tendency without any word problems, which may be abstract and difficult for elementary school students to understand. In addition to studies that compared U.S. and Chinese textbooks, there are also some articles that focus on describing how a certain topic is introduced in a Chinese textbook. For example, Li (2008b) described what Chinese students are expected to learn about fraction division and how the Chinese textbook is structured so that students can be guided to achieve the expected understanding of the topic. He pointed out that students are expected to learn more than just the algorithm (Invert-and-Multiply) and that the Chinese textbook makes good use of a problem solving approach to help students construct the algorithm. After Li’s description of the Chinese textbook, he encouraged

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educators in the United States to learn from the perspective of the Chinese textbook. However, he did not provide much analysis on how any why the structure and content of Chinese textbook might benefit students more than U.S. curricula that have been used currently, and there is also a lack of experimental evidence that the U.S. students would learn effectively if they were taught with the Chinese perspective. No matter how much difference we can find between U.S. and Chinese textbooks, one reason that we should not rush into copying each other’s textbook is that the successfulness of a textbook not only depends on the textbook’s structure and content, but also depends on policies and structures of the whole mathematics curricula of a country. The reason for many existing teaching strategies in U.S. textbooks and those in Chinese textbooks may be traced to the policies and structures of polices and structures of Chinese and U.S. curriculum. Policies and structures of Chinese and U.S. curricula. Chinese and U.S. mathematics curricula differ dramatically in terms of whether or not they follow a centralized structure. In the United States, curricula are usually developed by mathematics educators, researchers, and publishers (Remillard, 2005). Although many curriculum materials are developed to support the curriculum standards published by the National Council of Teachers of Mathematics (NCTM), there is no mandated requirement for publishing a curriculum. Curriculum decisions are made at state and local levels (Moy & Peverly, 2005), and school districts can mandate the use of a single curriculum or allow each school to choose its own curriculum. In China, a national mathematics curriculum that specifies goals, content topics and requirements at each grade level is published by China’s Ministry of Education (Li &

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Fuson, 2002; Moy & Peverly, 2005; Wang & Paine, 2003). A series of textbooks that closely follows this curriculum was commissioned by the Ministry of Education to be used in all Chinese schools (Madell & Becker, 1984). Recently, the education ministry started publishing several different textbooks that interpreted this national math curriculum in ways that were more relevant to the needs of different regions in China (Ma, 1999). Because most Chinese students need to compete in city-, province-, and nation-wide admission exams to enter high schools and colleges, and because these exams closely follow the content topics and requirements in the national curriculum, all mathematics instruction in Chinese school closely follows this national curriculum (An, 2000). Organizations of content topics in Chinese and U.S. textbooks also differ greatly. Jiang (Jiang & Eggleton, 1995) and Askey (1999) found that U.S. schools typically use a “spiral curriculum,” which means that topics are briefly introduced one year and then reviewed in successive years to build on previous learning. In this structure, U.S. textbooks usually cover many topics in each semester (Fuson, 1988; Hook, Bishop, & Hook, 2007; Stigler & Hiebert, 1999) with much review and repetition of previous knowledge (Schmidt, Wang, & McKnight, 2005) but not much new knowledge (Flanders, 1987). In contrast, Chinese textbooks introduce topics with a much more sequential and nonrepetitive approach (Askey, 1999; Jiang & Eggleton, 1995). Each topic is taught with elaborated details, and a thorough understanding of the topic is required as preparation for the future learning of other closely connected topics (Ma, 1999). Students learn a

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great deal about a small number of topics included in each semester (Fuson & et al., 1988; Zhou & Peverly, 2005). Many researchers (Schmidt, Houang, & Cogan, 2002, 2004; Schmidt, et al., 2005), after looking at U.S. textbooks from an international perspective, have argued that because U.S. textbooks cover many topics that are not necessarily related in a short period of time, and ignore the sequential connections among mathematics content topics, the textbooks do not present content topics in a coherent way that promotes students’ in- depth understanding of mathematics knowledge. Not only are the structures of curricula very different, but also teachers in the two countries approach their curricula in very different ways. Chinese and U.S. teachers’ approach to their curricula. With increased attention on the implementation of reform-oriented curriculum in the United States, more and more researchers are realizing that teachers play a very important role as they apply the curriculum in their classroom teaching; and many researchers have called for studies on the interaction between teachers and their curricula (Remillard, 2005; Remillard & Bryans, 2004; Smith & Star, 2007). Many cross-national studies (e.g. Ma, Lam, & Wong, 2006; Nicol & Crespo, 2006; Remillard & Bryans, 2004) have been done on teachers’ approaches to understanding and implementing their curricula. Existing studies that compare Chinese and U.S. teachers’ approaches to their curricula indicate that large differences exist in how teachers in the two countries study, understand, and implement their curriculum (Li & Fuson, 2002; Ma, Lam, & Wong, 2006; Moy & Peverly, 2005). In the United States, both teachers’ understanding of their textbooks and their implementation of their curricula vary greatly. Among teachers who use the same

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curriculum, their selection of topic, content emphasis, and sequences of instruction may be very different (Freeman & Porter, 1989). Such different approaches may result from teachers’ different beliefs about teaching, their content knowledge, and the professional development and support they receive (Remillard, 2005; Remillard & Bryans, 2004). When introduced to teachers, some curricula do not include much support for teachers to understand the rationale of the curriculum design; in those cases, teachers find it very difficult to understand and implement the new curriculum (Remillard, 2005; Remillard & Bryans, 2004). U.S. teachers’ different approaches to their curriculum may also be a result of teacher education programs. During their teacher education program, preservice teachers do not know the curricula they will use in the future because the curricula will be determined by their future school districts; therefore, preservice teachers do no have a chance to study their future curricula. Also, U.S. teacher education programs usually do not provide any training on any specific curriculum. In general, because there is no systematic guidance on how to use any specific curriculum, even when preservice teachers have the chance to practice teaching a curriculum in their student-teaching, their understanding and implementation of the curriculum may vary dramatically (Collopy, 2003; Lloyd & Behm, 2005; Nicol & Crespo, 2006; Pehkonen, 2004; Remillard, 2005). In contrast, Chinese teachers’ approaches to their mathematics curriculum are relatively uniform. Along with the national curriculum and the commissioned textbooks, there are also teacher’s guides accompanying the textbooks. These teacher’s guides explain in detail the rationale of the organization of the topics, the purpose of the inclusion of specific examples and exercises, and common mistakes and misconceptions

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of students. All teachers study the national curriculum, the textbooks, and the teacher’s guides very closely. Teachers’ design of their lesson plans and their implementation of their lessons are very similar; they also use the national curriculum and teacher’s guide as the most important resource for learning the mathematics as well as students’ misconceptions (Askey, 1999; Li & Fuson, 2002; Ma, 1999; Wang & Paine, 2003). Other than classroom teaching, teachers also attend professional development workshops, group discussions, and observation of other teachers’ teaching; all these activities are intended to help teachers improve their understanding and implementation of their curriculum (Ma, 1999). In addition to the teaching practice, teacher education programs in China also play an important role in helping preservice teachers become familiar with the national curriculum (Moy & Peverly, 2005). In China, each elementary school teacher teaches only one subject; preservice teachers are enrolled in programs of a specific subject and take classes that focus on the teaching of that subject. Preservice teachers of elementary school mathematics need to complete a series of courses that focus on the teaching of different topics in elementary school and middle schools. One major objective of these courses is to help preservice teachers have a systematic and in-depth understanding of the national curriculum, textbooks, and teacher’s guides, which are all required readings and learning materials in these courses (Li, 2002; Ma et al., 2006; Sun, 2000; Wang & Paine, 2003). Because the instructors of those teacher education courses are all experts in the curriculum materials, they provide a great opportunity for preservice teachers to develop their understanding of the curriculum that they will be using in the future.

Full document contains 122 pages
Abstract: This dissertation presents two studies designed to examine the topic of fraction division in selected Chinese and US curricula. By comparing the structure and content of the Chinese and Everyday Mathematics textbooks and teacher's guides, Study 1 revealed many different features presented in the selected curricula. Major differences include the number of lessons on this topic, the algorithms introduced, the type of examples and exercises provided in the textbooks, and the teaching strategy suggested by the teacher's guides. Study 2 examined whether a set of lessons that represented Chinese textbook features or a set of Everyday Mathematics -style lessons were more effective in promoting U.S. sixth-grade students' understanding of fraction division algorithms and their ability to apply the algorithms to solve word problems. Results indicated that the participating U.S. students lacked understanding of mathematics concepts that are relevant to fraction division, and the participants did not effectively learn about fraction division from either the Chinese-style or Everyday Mathematics -style lessons. These studies suggest that to apply features of Chinese mathematic curricula to teach US students, it is important to take into account US students' prior knowledge, learning experiences, and learning styles.