International comparative studies offer unique opportunities to interrogate and challenge embedded practices within education systems. In this paper we explore the textbook presentation of fractions from a Chinese text. The fraction tasks reviewed in the Chinese text involve practice on learnt knowledge as well as exercises designed to extend the learnt knowledge to generate integrated knowledge structures and to develop flexible problem-solving abilities. Designed using the theory of “teaching with variation”, these tasks reflect cultural expectations about what content is important, how mathematics can be taught and what competence is valued for students. In this paper we consider how these task activities compare with those from a recently published New Zealand text.

Teaching and learning fractions: Lessons from alternative example spaces

Abstract

International comparative studies offer unique opportunities to interrogate and challenge embedded practices within education systems. In this paper we explore the textbook presentation of fractions from a Chinese text. The fraction tasks reviewed in the Chinese text involve practice on learnt knowledge as well as exercises designed to extend the learnt knowledge to generate integrated knowledge structures and to develop flexible problem-solving abilities. Designed using the theory of “teaching with variation”, these tasks reflect cultural expectations about what content is important, how mathematics can be taught and what competence is valued for students. In this paper we consider how these task activities compare with those from a recently published New Zealand text.

Introduction

Large-scale international studies (see, for example, “2009 Program for International Student Assessment Scores”, 2010) involving Asian and Western countries consistently identify Chinese students, mainly from mainland China, Hong Kong and Taiwan, as top performers in mathematics (Fan & Zhu, 2004; Wang & Lin, 2009). Cross-cultural studies propose a range of factors that contribute to the “learning gap” in mathematics achievement (Brenner, Herman, Ho, & Zimmer, 1999; Cai & Hwang, 2002; Stevenson & Stigler, 1992). Significant factors relate to culture (Education Review Office, 2000), teacher recruitment and status (Pang, 2009), teacher expertise (Ding & Jones, 2009; Ma, 1999) and organisation of school and curricula (Zhou, Peverly, & Xin, 2006). While acknowledging this complex network of influences, we have chosen in this paper to focus our attention on the opportunities for learning that are designed into the curriculum by way of textbooks. Through an examination of the tasks presented in a mainstay Chinese textbook—the analysis of which is framed against the approach presented in a recently published New Zealand textbook—we look at task potential to engage learners with specific mathematics content and cognitive processes.

Our domain of inquiry concerns the teaching and learning of fractions. We have selected fractions for two reasons. Firstly, worldwide, fraction teaching and learning presents a major challenge. The situation is no different in New Zealand (Young-Loveridge, 2009) and China (Zhou et al., 2006). In New Zealand, findings from National Education Monitoring Project (Crooks, Smith, & Flockton, 2010) and the Numeracy Development Project (Young-Loveridge, 2009; Young-Loveridge, Taylor, Hāwera, & Sharma, 2007) all provide evidence that student performance in fractions is weak. While Wang and Lin’s (2009) meta-analysis of large-scale comparative studies indicates that Grade 8 Chinese students performed better than their United States peers in fractions and proportionality, a recent study by Wu (2008), testing 491 Grade 6 Chinese students’ mathematical proficiency in fractions, raised concerns about the procedural nature of students’ knowledge. The results showed that the students were proficient at computations, much more so than at creating appropriate visual models and solving word problems. However, the researcher takes care to stress that students in this study were using textbooks that were influenced by more traditional teaching approaches than would be found in reform-type Chinese classes (see Li, Zhang, & Ma, 2009, for discussion of textbook changes). A second reason for our focus on fractions arose through our involvement in the Learner Perspective Study (LPS) (Clarke, Keitel, & Shimizu, 2006). As members of an international project we studied three sets of lessons from Year 9 New Zealand classes, one of which was fractions. Expressions of surprise from our LPS colleagues in Asia with regard to the fact that in New Zealand we teach fractions at the secondary level piqued our interest in comparative curricula.

In looking to better understand how those practices that are embedded in educational ways of being are implicated in students’ mathematical learning and achievement, we claim that textbook tasks provide a useful site of inquiry. As Silver (2009) notes, “the tasks in which students engage, constitute, to a great extent, the domain of students’ opportunities to learn mathematics” (p. 829). They influence learners by “directing their attention to particular aspects of content and by specifying ways of processing information” (Doyle, 1983, p. 161). Informed by recent studies that focus on the role of example/task design and sequencing (for example, Runesson & Mok, 2004; Sun, 2009; Watson & Mason, 2005), we provide a comparative sample of task activities from a Chinese mathematics textbook, *The Nine Year Compulsory Education Textbook, Mathematics, First Term, Grade Six* (Shanghai Primary and Secondary School (Kindergarten) Curriculum Reform Committee, 2005), and a New Zealand text, *Mathematics & Statistics for the New Zealand Curriculum Year 9* (Brookie, Halford, Lawrence, Tiffen, & Wallace, 2008). Our analysis of the textbook tasks is informed by the theory of variation, which we briefly introduce in the following section.

Theoretical framework: Theory of variation

Variation theory seeks to account for differences in learning and the conditions that are necessary for learning. According to Marton, Runesson and Tsui (2004) the “pattern of variation inherent in the learning situation is fundamental to the development of certain capabilities” (p. 15). Within this theoretical framework, learning is defined as a change in the way something is seen, experienced or understood. In order to see something in a certain way, the learner must discern certain features of the object of learning, paying attention to what varies and what is invariant in a learning situation. Runesson (2005) illustrates the significance of variation for young children solving simple arithmetic problems (such as 2 + 3= , 3 + 6= , 3 + 5 = and so on) as follows:

A child, usually starting with the first addend, may suddenly change strategy and add 3 + 2 = , 6 + 3 = (cf. Carpenter & Moser, 1984). In this case, variation lies in the order of the addends. Through opening up variation of this feature of addition, that is, that the sum is independent of the order of the addends, it becomes possible to discern. (p. 72)

When related to tasks in mathematics, different strategies for solving problems, different numbers in problems, different problem formats (e.g., a + b = c versus d = e + f), different representations, and so on, all make up dimensions of variations.

Comparative analyses of mathematics classroom lessons (for example, Häggström, 2008; Huang, Mok, & Leung, 2006; Runesson & Mok, 2004) provide evidence that some ways of experiencing—perceiving, understanding or seeing—are more powerful than others. For instance, Huang et al.’s study of practice problems used in classrooms in Hong Kong and Shanghai concluded that “if variation is not used appropriately, such as undue varying in technical aspects for specific skills, students may be limited in a narrow space of learning and will not be challenged enough” (p. 272). Runesson and Mok’s (2004) analysis of two lessons in which the object of learning was fractional numbers as operator (for example, how to find of 12) demonstrated differences in what was possible to learn. Teacher A provided a series of tasks involving sharing a piece of string, focused on a single strategy to find of 90, of 90, of 40, of 40, of 60. In contrast, Teacher B pressed the students to come up with multiple strategies to solve a single problem ( of 56). Strategies included: counting seven squares in each column and then marking three of them; dividing the 56 squares into seven parts (and multiplying 8 by 3); and trying different numbers which when multiplied by 3 equal 56. In terms of variation, the researchers note that:

Teacher A provided the learners with the possibility to learn to solve different problems with the same strategy, whereas Teacher B provided the learners with the possibility to learn to solve the same problem with different strategies. In Teachers A’s lesson the changing of the parameter in the operation opened a variation of different examples, whereas in Teacher B’s lesson, the parameters were invariant and the strategy varied. (p. 86)

In this paper, the focus is on the object of learning, as represented by examples and exercises in the textbook—“an important, stable and visible mediator between teaching and learning” (Sun, 2009, p. 194). However, we do not wish to imply that classroom arrangements and student relationships with mathematics are unimportant. Indeed, while arguing that “the possibility to learn is provided by what it is possible to discern” (p. 87), Runesson and Mok (2004) are careful to note that arrangements such as group work can facilitate the opening of variation.

Teaching and learning fractions

In New Zealand, the teaching of fractions is informed by the Numeracy Development Project teaching model (see Ministry of Education, 2008a). The teaching model promotes the explicit teaching of problem-solving strategies alongside a knowledge-base framework. The model advocates a dynamic process of working—from using materials, to visualisation to using number properties—for introducing new ideas. The development of students’ mental strategies through a “dialogical approach in which students’ explanations are promoted prior to the introduction of the written algorithm” (Higgins & Parsons, 2009, p. 235) is a core feature of the teaching model.

In contrast to New Zealand reforms that have involved systemic professional development at a national level, the sheer size of China and its ongoing reforms in education makes it difficult to profile a specific classroom pedagogy model. Research reports note varied pedagogical practices. For example, reporting on Shanghai Grade 8 classes in the *Learners Perspective Study*, Mok and Lopez-Real (2006) noted that teachers regularly used exploratory activities in their teaching. In a study of 14 Chinese maths lessons, An (2008) noted Chinese teachers’ high expectations of their students aligned with a high level of challenge in content. Both An (2008) and Cao, Wan, Wan and Liao (2008) reported that Chinese mathematics lessons typically moved at a fast and focused pace with limited opportunity for students to collaborate or have input during the lessons. Mok (2006), however, stressed the need to appreciate the rationale behind the apparent teacher dominance:

The teacher has a clear understanding of the subject matter at a level of subtle detail and tries hard to make his students understand the same level of detail. He gives the students opportunities for discussion but he controls their activity by choosing tasks with limited options so that students will see what he expects. He welcomes students to express ideas in his own words but he corrects their language to the standardized language. (pp. 95–96)

At the same time as currently instituting educational reforms aimed “towards global perspectives” (An, 2008, p. 1), efforts to reform Chinese mathematics education remain strongly influenced by a theory of mathematics teaching/learning called *teaching with variation*. Developed by Gu (Gu, Huang, & Marton, 2004), this teaching model is based on the theory of variation as discussed above. That is, a central concern of instructional design involves attention to the process in which learners develop a “certain capability or a certain way of seeing or experiencing” (Huang et al., 2006, p. 264) by discerning certain features of the object. In the mathematics lesson, experiencing varying problems (sometimes multiple approaches to solving problems) is essential for such discernment. The purpose of the variation is to “form a hierarchical system of experiencing process through forming concepts or solving stages of problems” (Gu et al., 2004, p. 324). When accompanied with opportunities to summarise key points, practising with variation provides opportunities for students to master basic procedural knowledge and to have a deep understanding of principles and mathematical thinking. Like the Numeracy Development Project teaching model, teaching with variation acknowledges the importance of students having concrete and visual experiences in order to connect the abstract concept with its concrete embodiments. And while both models advocate that “the visual background of a concept must be disconnected in due course so that a concept is upgraded to an abstract level” (Gu et al., 2004, p. 316), the push to the abstract appears to happen at an earlier level in the Chinese school system (Zhang & Zhou, 2004), with concrete embodiments being confined to early grades.

In China, students are taught basic fraction concepts such as fractional units, part–whole relations and the concept of “dividing shares evenly” during Grade 3. In the early grades, lessons would include concrete manipulatives. At Grade 6—the final year in which fractions are studied—children complete the study of computation with fractions using visual models only. In Grade 6 the study of fractions is also integrated into the study of proportion and ratio, percentages, statistics and circle geometry. New Zealand children are also introduced to basic fraction concepts at an early age. In Years 3 and 4 students explore simple fractions in everyday contexts, and in Years 5–8 they explore computations with fractions (Ministry of Education, 2007). However, unlike students in China, students’ exploration of fractions, focusing on computations including relationship with percentages and decimals, continues in Years 9 and 10 (Ding, Anthony, & Walshaw, 2009).

Fraction tasks

Given differences in the curriculum sequencing, we turn our attention to how the curriculum is interpreted in the form of tasks within texts. Using a Chinese reform mathematics school text, *The Nine Year Compulsory Education Textbook, Mathematics, First Term, Grade Six* (Shanghai Primary and Secondary School (Kindergarten) Curriculum Reform Committee, 2005), we provide examples of the fraction tasks at Grade 6, the final year of study of fractions. Parallel content tasks are drawn from the New Zealand text, *Mathematics and Statistics for the New Zealand Curriculum Year 9* (Brookie et al., 2008).^{1} In the following discussion, the respective texts will be referred to as CT and NZT.

At first glance the CT and NZT are similar in format. Both texts contain references to real context, and have an abundance of diagrams, examples and practice exercises. However, when looking at the nature of the models used to introduce the fraction chapters (noting that in both curricula fractions are first introduced in earlier years) there are significant differences.

In the CT, the concept of a fraction is introduced using a division (quotient) model. A sharing situation is used to represent fractions—first 1 orange shared between 4 people, represented as 1 ÷ 4 = , then 2 oranges shared between 4 people, represented as 2 ÷ 4 = . This action is immediately formalised as *p ÷ q* = . What is particularly interesting is that the choice of examples focuses very quickly on the structure and number properties. Examples 3 ÷ 1 and 5 ÷ 1 are generalised to the new rule: When *q* = 1 then = *p*. In contrast to this approach, the introduction in NZT uses a part–whole model, explaining that “fractions are used in everyday life to describe parts of quantities”. The numeric example reiterates that the *3* of is the number of parts out of the whole, and the *8* is the number of equal parts altogether.

The next section in the CT presents equivalent fractions (as does the NZT)—first visually, then using the calculational rule. Of note are the explicit links made between visual, symbolic and formulaic representations of fractional numbers and computations (see Figure 1). Such algebraic generalisation is not a feature in the NZT.

In looking at the practice set of exercises within the texts, what is particularly interesting is the variation in the sample space (Watson & Mason, 2006) offered for equivalent fractions tasks. While both texts require students to generate sets of equivalent fractions or find the simplest equivalent fraction of a set, the introductory task in the CT is unique in that it requires students to explicitly complete an equality statement that includes an identity element (see Figure 2).

In extending this concept, students are asked to link to through multiplication of and then to use equivalent fractions to determine whether is greater or smaller than . These tasks presented very early in the practice exercises serve to reinforce the generalisation

Addition and subtraction of fractions is presented in a similar way in both texts—with concrete part-whole representations linked to numeric calculations. In contrast to the NZT, which introduced improper fractions in an earlier discussion of mixed numbers, the CT introduction to improper fractions is explicitly linked to the context of addition and subtraction of fractions as follows:

After completing practice exercises with addition and subtraction, the final set of exercises in the CT involve solving for *x* in the likes of Solving this form of equation has been met in earlier grades with whole numbers and decimals.

In introducing multiplication, both texts use the overlapping area model as a precursor to the formulation of the rule

Likewise, the texts use similar approaches in their introduction of division by a fraction—exploring the relationship 1 ÷ = 4 and 4 × = 1 and the equivalent algebraic statement. However, unlike the approach in the NZT that moves quickly to the “flip over the second fraction” rule, the CT continues to explore the inverse relationship to derive the “rule” for division (see Figure 3).

To reaffirm the inverse relationship, the example spaces in the CT offer problem variation not typically seen in New Zealand texts. Unlike a page of exercises in the NZT—consisting only of the form the CT introductory set for division (see Figure 4) continues to stress the inverse nature of operations of multiplication and division. The variation in the placement of the unknown across the example space requires that students are able to consider the statements relationally rather than routinely applying a “flip and multiply” algorithm.

After a set of standard division problems of the form students are also asked to solve

requiring students to use multiplicative inverses. As noted in Sun’s analysis of Chinese mathematics textbooks, this variation in problem format is explicitly intended “to link the concept of a fraction division to the concept of an equation” (2011, p. 79). The explicit linking of fractions to equations is seen to be supportive of the development of algebraic ideas. The final section in the fraction chapter in the CT links fractional notation to division to decimals. For example, and followed by an exploration of repeating decimals.

Discussion and conclusions

In reflecting on the differences in culturally specific educational practices, we are keen to stress the claim by Marton et al. (2004) that it is “highly unlikely that there is any one particular way of arranging for learning that is conducive to *all* kinds of learning” (p. 3). These researchers argue that in order to find effective ways of arranging for learning, we “need to first address *what* it is that should be learned in each case, and find the different conditions that are conducive to different kinds of learning” (p. 3). Our investigation of the textbook examples and exercises provides a window into the object of learning. However, as indicated above, this is only part of the picture. An examination of how the object of learning is co-constituted through interaction in the classroom would no doubt reveal many more culturally specific educational practices.

In China, doing exercises in the classroom occupies a large portion of teaching time and reflects the “practice makes perfect” notion that has long been an important principle in the Chinese mathematics education system. When teaching with variation, practice has a dual role of mastering skills and deepening understanding (Zhang, Li, & Tang, 2006). Indeed, the tasks reviewed in the fractions chapter of the CT involved practice on learnt knowledge as well as exercises designed to extend the learnt knowledge to generate integrated knowledge structures and to develop flexible problem-solving abilities. Rather than practising large sequences of the same format (as is the case with a sequence of 13 addition of fraction problems of the form in the NZT), the example spaces provided in the CT were explicitly structured so as to encourage learners to notice and discern variation. This use of example spaces is in accord with Watson and Mason’s (2005) view of mathematics as constructive activity:

… in which to learn is to construct objects meeting specified constraints, so that when example spaces are triggered they are complex and confidence-inspiring, with their components richly interconnected, enabling re-construction and fresh construction as appropriate. (p. 160)

Other papers looking at American versus Chinese texts report similar findings. Most notably Sun’s (2011) detailed analysis of fraction division across multiple texts from China highlights how problem variation intensifies curriculum connectedness. As we saw, for example, division was explicitly linked to multiplication and the ideas of inverse and equation.

What challenges can the CT exemplars offer New Zealand teachers and teacher educators? To date, professional development associated with the Numeracy Development Project in New Zealand has focused teachers’ attention on diagnostic interviewing, explanatory number frameworks and encouragement of pedagogical practices associated with inquiry learning. While this is admirable, we suggest that this glimpse of the types of tasks that young children in China study offers a timely reminder of the importance of task planning and sequencing. Currently much literature in Western education systems focuses on the need to ensure that tasks engage students in high-level thinking (Sullivan, Clarke, & Clarke, 2009)—typically associated with rich task formats (Anthony & Hunter, 2010). While these task formats should be to the fore and in themselves provide opportunities for practice, too often we see evidence of classroom and homework activities that involve repetitive tasks. The tasks exemplified in the Chinese text prompt us to consider alternative ways of achieving meaningful learning through structured practice involving variation. Rather than consolidate one topic or skill at a time—as is typically captured in the current WALT (What I Am Learning Today) adherence—before moving on to the next topic or skill, the issue of problem variation reflects the old Chinese proverb, “no clarification, no comparison”.

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The authors

Glenda Anthony is Professor of Mathematics Education at Massey University, Palmerston North. Glenda’s primary research interests include mathematical inquiry and effective teaching practices and teacher education. Her research contexts include early years, primary, secondary and tertiary sectors. Glenda is the co-author of the New Zealand Iterative Best Evidence Synthesis (BES) for effective mathematics teaching and *Effective Pedagogy in Mathematics*, part of the International Academy of Education’s Educational Practice Series.

Email: g.j.anthony@massey.ac.nz

Liping Ding is a mathematics teacher and mathematics education researcher at Song Ching Ling School, Shanghai, China. Liping’s research interests include classroom pedagogy and children’s cognitive process in primary and secondary mathematics.

Email: dlp_2000@hotmail.com

Notes

1This text is one of many available to be purchased by secondary schools. New Zealand teachers would also access curriculum material from the Ministry of Education Numeracy Development Project material (Ministry of Education, 2008b) and the Ministry of Education-supported website www.nzmaths.co.nz In the introduction of the NZT, the authors note that they have been guided by the strategies for calculation and problem solving as recommended by the New Zealand Secondary Numeracy Project.