SO COMFORTABLE WITH THE ALGORITHM? - USE OF CONCRETE-PICTORIAL-ABSTRACT IN MATHEMATICS CLASSROOMS IN SINGAPORE
Based on Bruner’s (1966) conception of the enactive, iconic and symbolic modes of representation, this paper illustrates two teachers’ use of the Concrete-Pictorial-Abstract or CPA approach in the Primary 3 Mathematics classroom in Singapore. Drawing on data from a qualitative study of mathematics curriculum enactment, a descriptive case study highlights the teachers’ classroom enactment woven with insights into their pedagogical reasoning based on lesson videos, transcripts of classroom talk and teacher interviews, and samples of student work. A Brunerian lens is used to examine the modes of representations employed (or not) in these classrooms and the varying levels of effectiveness with which the teachers enable their students to ‘see’ conceptual connections across a range of representations such as concrete manipulatives and pictorial depictions to facilitate a gradual “decontextualizing” or “fading away” to abstract concepts (Bruner, 1966, pp. 60-62).
The official curriculum document i.e. the Primary Mathematics Teaching and Learning Syllabus (MOE, 2013) outlines learning experiences to facilitate students’ conceptual understanding through the CPA approach, and to enable students to communicate their reasoning through various mathematical tasks and activities. CPA, as an instructional heuristic has been advocated by Ministry of Education (Singapore) since the early 1980’s, embedded in school textbooks (Fan, 2012) as well as taught in pre-service courses of mathematics teachers (Chua, 2010; Edge, 2006). The paper surfaces a nuanced understanding of some important issues in relation to the effectiveness of CPA in the Singapore mathematics classroom: the need for teachers to make explicit and meaningful links across representations; opportunities for students to engage in practical, hands-on activities to clarify their own understanding; space for students to communicate and interact to develop mathematical understandings (Hiebert et al., 1997; Schoenfeld, 1998; Boaler & Greeno, 2000); and the focus on mathematical thinking through language (Marton & Tsui, 2004; Kwek, 2012).
Overall, the paper sheds light on the unique characteristics and strengths of the CPA approach in mathematics classrooms, and possible challenges of implementing the same in the Singapore context given the time pressure teachers experience in completing syllabus within the constraints of the allocated time (Assude, 2005; Leong & Chick, 2011). Broadly, problematizing the teacher’s use of CPA yields valuable insights into how the teacher’s practical reasoning (Eisner, 2002; Birmingham, 2004) and pedagogical repertoires recontextualise the curriculum into classroom practice. The issues raised have obvious implications for policy and pedagogical practice particularly so, in the realm of mathematics learning, which is conceived as an active and constructive process of sense-making, understanding, and problem solving in a community of learners (Baroody & Dowker, 2003; De Corte & Verschaffel, 2006).