Objectives and theoretical frameworks

Mathematics proficiency is considered a prerequisite for all students to become full participants in a knowledge-based society (Organisation For Economic & Development, 2016). Research has shown that teachers’ instructional practices make a difference to students’ learning in general (Hattie, 2012; Seidel & Shavelson, 2007), and for developing students’ mathematical proficiency (Kilpatrick et al., 2001).

In order to understand and gain knowledge about what mathematical proficiencies and learning goals teachers emphasize during their instruction, the present study examines instructional practices and how new content is introduced in Norwegian and Finnish-Swedish 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. Instructional practices where teaching a concept only means pointing out definitions and rules, followed by the teacher stating procedures, provokes some students to view mathematics as an unrelated set of rules and procedures (Kaasila, 2009). While for others, this kind of procedure may be preferable. How teachers introduce content depends on what kind of content is at focus, as some content may allow a different approach than other (Hill & Grossman, 2013). Therefore this study compares the introduction of new content in both national contexts, namely algebra and geometry. Algebra related content is introduced in three different Finnish-Swedish and three Norwegian classrooms, while geometry-related content is introduced in one Finnish-Swedish and two Norwegian classrooms (N= 9).

The theoretical framework applied is Kilpatrick et al. (2001) five strands of mathematical proficiency, in order to operationalize what mathematical proficiencies teachers prioritize their students to acquire and develop when they introduce new content. The five strands are; Conceptual understanding, meaning comprehension of mathematical concepts, operations and relations; Procedural fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately; Strategic competence – ability to formulate, represent, and solve mathematical problems; Adoptive reasoning – capacity for logical thought, reflection, explanation, and justification; Productive disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. Albeit some disagreement in the field of mathematical education research on how to define mathematical proficiencies and which are of greater importance, researchers highlight the need to develop strands of proficiency in conjunction, especially conceptual understanding and procedural fluency (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).

Baroody, A. J., Feil, Y., & Johnson, A. R. (2007). An Alternative Reconceptualization of Procedural and Conceptual Knowledge. Journal for Research in Mathematics Education, 38(2), 115-131. Blikstad-Balas, M. (2016). Key challenges of using video when investigating social practices in education: contextualization, magnification, and representation. International Journal of Research & Method in Education, 1-13. doi:10.1080/1743727X.2016.1181162 Cohen, J. (2015). Challenges in Identifying High-Leverage Practices. Teach. Coll. Rec., 117(7). Grossman, P. (2015). Protocol for Language Arts Teaching Observations (PLATO 5.0). Palo Alto: Stanford University. Haapasalo, L. (2003). The conflict between conceptual and procedural knowledge: Should we need to understand in order to be able to do, or vice versa?2017. Retrieved from Hannula, M. S., Pehkonen, E., Maijala, H., & Soro, R. (2006). Levels of students' understanding on infinity. Teaching Mathematics and Computer Science, 4(2), 317-337. doi:10.5485/TMCS.2006.0129 Hattie, J. (2012). Visible learning for teachers : maximizing impact on learning. London: Routledge. Hill, H., & Grossman, P. (2013). Learning from Teacher Observations: Challenges and Opportunities Posed by New Teacher Evaluation Systems. Harvard Educational Review, 83(2), 371-384,401. Kaasila, R. P., Erkki. (2009). Effective mathematics teaching in Finland through the eyes of elementary student teachers. In J. K. Cai, Gabriel; Perry, Bob; Wong, Ngai-Ying (Ed.), Effective Mathematics Teaching from Teachers' Perspectives (pp. 203-216). Rotterdam: Sense Publishers. Kilpatrick, J., Swafford, J., Findell, B., Mathematics Learning Study, C., National Research Council Center for Education, D. o. b., & social sciences, e. (2001). Adding it up : helping children learn mathematics. Washington, DC: National Academy Press. Knoblauch, Hubert, Schnettler, & Bernt. (2012). Videography: analysing video data as a ‘focused’ ethnographic and hermeneutical exercise. Qualitative Research, 12(3), 334-356. doi:10.1177/1468794111436147 Kunnskapsdepartementet. (2013). Læreplan i matematikk fellesfag. https://www.udir.no/kl06/MAT1-04. Opetushallitus. (2014). Grunderna för läroplanen för den grundläggande utbildningen 2014. http://www.oph.fi/lp2016/grunderna_for_laroplanen. Organisation For Economic, C.-O., & Development. (2016). PISA 2015 Results (Volume I): Excellence and Equity in Education: Paris: OECD Publishing. Seidel, T., & Shavelson, R. J. (2007). Teaching Effectiveness Research in the Past Decade: The Role of Theory and Research Design in Disentangling Meta-Analysis Results. Review of Educational Research, 77(4), 454-499. doi:10.3102/0034654307310317 Simola, H. (2005). The Finnish miracle of PISA: historical and sociological remarks on teaching and teacher education. Comparative Education, 41(4), 455-470. doi:10.1080/03050060500317810 Star, J. R. (2005). Reconceptualizing Procedural Knowledge. Journal for Research in Mathematics Education, 36(5), 404-411.