Teaching and learning fractions: Lessons from alternative example spaces

Abstract

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 (Fan & Zhu, 2004; Wang & Lin, 2009). Cross-cultural studies propose a range of factors that contribute to gap in 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 opportunities for learning that are designed into curriculum by way of textbooks. Through an examination of tasks presented in a mainstay Chinese textbook--the analysis of which is framed against approach presented in a recently published New Zealand textbook--we look at task potential to engage learners with specific content and cognitive processes. Our domain of inquiry concerns 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 Numeracy Development Project (Young-Loveridge, 2009; Young-Loveridge, Taylor, Hawera, & 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 procedural nature of students' knowledge. The results showed that students were proficient at computations, much more so than at creating appropriate visual models and solving word problems. However, 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 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 fact that in New Zealand we teach fractions at 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, 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 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 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 New Zealand Curriculum Year 9 (Brookie, Halford, Lawrence, Tiffen, & Wallace, 2008). …