Objectives and theoretical frameworks

Mathematical content requires representation in a rich and meaningful way in order for young students to be able to achieve mathematical competence, which is important for understanding and participating in public debate, and developing technology and solutions for emerging issues related to environment, poverty and health (Kilpatrick et al., 2001; UNESCO, 2012). Research has shown that teachers’ instructional practices make a difference to students’ learning (Hattie, 2012; Seidel & Shavelson, 2007) and that instruction shapes what students learn (Boaler, 2000, p. 172).

The present study examines teachers’ representation of mathematical content in Norwegian (Oslo area) and Finnish-Swedish (Helsinki area) 7^th and 8^th grade classrooms. This is the same age group of 13-year olds, since Norwegian students start school at the age of 6 and Finnish students at the age of 7. The analytical framework applied consists of parts of an observation manual (PLATO, see methods section), called Representation and use of content, which is divided into four sub-elements: Connections to prior academic knowledge, Purpose, Quality of instructional explanations, and Conceptual richness of instructional explanations. ‘Connections to prior academic knowledge’ captures to what degree teachers elicit and connect prior knowledge to new content, which is considered important for developing conceptual understanding (Greeno, 2006; Hiebert & Grouws, 2007). The element ‘Purpose’ encompasses to what degree learning goals are made explicit. Research suggests children benefit when purposes and goals of their work are clearly articulated and the relationship between what they learn and broader goals are clear (Borko & Livingston, 1989). ‘Quality of instructional explanations’ of content involves making students pay attention to similarities and differences of concepts and procedures (Zaslavsky & Sullivan, 2011) and whether teacher pays attention to normal misconceptions, focus on students’ thinking and address emerging misunderstandings (Ball & Hill, 2009). ‘Conceptual richness of instructional explanations’ captures to what degree deeper conceptual understanding is provided in explanations, in relation to procedural attention. This entails whether content is represented in a context relevant for students, facilitating later refinement of content into generalized strategies (Anthony & Walshaw, 2009) and makes mathematics meaningful for students (Cobb & Yackel, 1998). Still, researchers stress that conceptual understanding and learning procedures ought to be developed in conjunction (Baroody, Feil, & Johnson, 2007; Star, 2005). This is also in line with the standards set in the national curriculums in Norway (Kunnskapsdepartementet, 2013) and in Finland (Opetushallitus, 2014).

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