The teaching of mathematical problem-solving in Swedish classrooms: a case study of one grade five teacher’s practice
Abstract
In this paper we examine the teaching of mathematical problem-solving to grade five students of one well-regarded and experienced Swedish teacher, whom we call Mary. Working within a decentralised curriculum in which problem-solving is centrally placed, Mary is offered little systemic support in her professional decision making with respect to problem-solving instruction. Drawing on Lester’s and Schroeder’s descriptions of teaching for, about and through problem-solving, we draw on multiple sources of data, derived from interviews and videotaped lessons, to examine how Mary’s problem-solving-related teaching is constituted in relation to the weakly-framed curriculum and the unregulated textbooks that on which she draws. The analyses indicate that Mary’s emphases are on teaching for and about problem-solving rather than through, although the ambiguities that can be identified throughout her practice with respect to goals, curricular aims and the means of their achievement can also be identified in the curricular documents from which she draws.
References
Andrews, P. (2015). Mathematics, PISA, and culture: an unpredictable relationship. Journal of Educational Change, 16 (3), 251-280. https://doi.org/10.1007/s10833-015-9248-2
Andrews, P. & Xenofontos, C. (2015). Analysing the relationship between problem-solving-related beliefs, competence and teaching of three Cypriot primary teachers. Journal of Mathematics Teacher Education, 18 (4), 299-325. https://doi.org/10.1007/s10857-014-9287-2
Andrews, P. (2016). Is the "telling case" a methodological myth? International Journal of Social Research Methodology. https://doi.org/10.1080/13645579.2016.1198165
Arcavi, A. & Friedlander, A. (2007). Curriculum developers and problem solving: the case of Israeli elementary school projects. ZDM, 39 (5-6), 355-364. https://doi.org/10.1007/s11858-007-0050-3
Artigue, M. & Houdement, C. (2007). Problem solving in France: didactic and curricular perspectives. ZDM, 39 (5-6), 365-382. https://doi.org/10.1007/s11858-007-0048-x
Bell, E. (1937). Men of mathematics. London: Victor Gollancz.
Bergqvist, E., Bergqvist, T., Boesen J., Helenius O., Lithner J. et al. (2010). Matematikutbildningens mål och undervisningens ändamålsenlighet: grundskolan våren 2009. National Centre for Mathematics Education, University of Gothenburg.
Bernstein, B. (2000). Pedagogy, symbolic control and identity: theory, research, critique (Rev. ed.). Lanham: Rowman & Littlefield Publishers.
Boero, P. & Dapueto, C. (2007). Problem solving in mathematics education in Italy: dreams and reality. ZDM, 39 (5), 383-393. https://doi.org/10.1007/s11858-007-0051-2
Boesen, J., Helenius, O., Bergqvist, E., Bergqvist, T., Lithner, J. et al. (2014). Developing mathematical competence: from the intended to the enacted curriculum. The Journal of Mathematical Behavior, 33, 72-87. https://doi.org/10.1016/j.jmathb.2013.10.001
Braun, V. & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3 (2), 77-101. https://doi.org/10.1191/1478088706qp063oa
Bryman, A. (2008). Social research methods. New York: Oxford University Press.
Burkhardt, H. & Bell, A. (2007). Problem solving in the United Kingdom. ZDM, 39 (5), 395-403. https://doi.org/10.1007/s11858-007-0041-4
Cai, J. & Nie, B. (2007). Problem solving in Chinese mathematics education: research and practice. ZDM, 39 (5-6), 459-473. https://doi.org/10.1007/s11858-007-0042-3
Clarke, D., Goos, M. & Morony, W. (2007). Problem solving and working mathematically: an Australian perspective. ZDM, 39 (5-6), 475-490. https://doi.org/10.1007/s11858-007-0045-0
Cockcroft, W. (1982). Mathematics counts. London: HMSO.
Donmoyer, R. (2009). Generalizability and the single-case study. In M. Hammersley & R. Gomm (Eds.), Case study method (pp. 45-69). London: SAGE. https://doi.org/10.4135/9780857024367.d7
Doorman, M., Drijvers, P., Dekker, T., Heuvel-Panhuizen, M. van den, Lange, J. de & Wijers, M. (2007). Problem solving as a challenge for mathematics education in the Netherlands. ZDM, 39 (5-6), 405-418. https://doi.org/10.1007/s11858-007-0043-2
Engle, R. A. & Conant, F. R. (2002). Guiding principles for fostering productive disciplinary engagement: explaining an emergent argument in a community of learners classroom. Cognition and Instruction, 20 (4), 399-483. https://doi.org/10.1207/S1532690XCI2004_1
Fan, L. & Zhu, Y. (2007). From convergence to divergence: the development of mathematical problem solving in research, curriculum, and classroom practice in Singapore. ZDM, 39 (5-6), 491-501. https://doi.org/10.1007/s11858-007-0044-1
Geertz, C. (1994). Thick description: toward an interpretive theory of culture. In M. Martin & L. Mcintyre (Eds.), Readings in the philosophy of social science (pp. 213-232). London: Bradford Books.
Gerring, J. (2006). Case study research: principles and practices. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511803123
Grevholm. B. (1989). Lilla utmaningen: problem och tankenötter i matematik. Malmö: Liber.
Hammersley, M. & Gomm, R. (2009). Introduction. In R. Gomm & M. Hammersley (Eds.), Case study method (pp. 1-17). London: SAGE.
Heath, S. B. & Street, B. V. (2008). On ethnography: approaches to language and literacy research. London: Routledge.
Hiebert, J. (2003). Signposts for teaching mathematics through problem- solving. In F. K. Lester (Ed.), Teaching mathematics through problem solving : prekindergarten-grade 6 (pp. 53-62). Reston: National Council of Teachers of Mathematics.
Hino, K. (2007). Toward the problem-centred classroom: trends in mathematical problem solving in japan. ZDM, 39 (5-6), 503-514. https://doi.org/10.1007/s11858-007-0052-1
Holmes, B. & McLean, M. (1989). The curriculum: a comparative perspective. London: Unwin Hyman.
Katz, M. (1976). The origins of public education: a reassessment. History of Education Quarterly, 16 (4), 381-407. https://doi.org/10.2307/367722
Kilpatrick, J., Martin, W. G. & Schifter, D. (2003). A research companion to principles and standards for school mathematics. Reston: National Council of Teachers of Mathematics.
Kilpatrick, J., Swafford, J. & Findell, B. (2001). Adding it up: helping children learn mathematics. Washington: National Academy Press.
Klein, D. (2003). A brief history of American K-12 mathematics education in the 20th century. In J. Royer (Ed.), Mathematical cognition: current perspectives on cognition, learning and instruction (pp. 175-225). Charlotte: Information Age Publishing.
Lauwerys, J. (1959). The philosophical approach to comparative education. International Review of Education, 5 (3), 281-298. https://doi.org/10.1007/BF01416896
Lester, F. (Ed.) (2003). Teaching mathematics through problem solving: prekindergarten-grade 6. Reston: National Council of Teachers of Mathematics.
Mitchell, J. (1984). Typicality and the case study. In R. Ellen (Ed.), Ethnographic research: a guide to general conduct (pp. 238-241). London: Academic Press.
NCTM. (1980). An agenda for action: recommendations for school mathematics of the l980s. Reston: National Council of Teachers of Mathematics.
NCTM. (2000). Principles and standards for school mathematics. Reston: National Council of Teachers of Mathematics.
Niss, M. & Jensen, T. (2002). Kompetencer og matematiklaering [Competencies and mathematical learning]. Copenhagen: Undervisningsministeriet.
OECD (2013). PISA 2012. Assessment and analytical framework. Paris: OECD publishing.
Ponte, J. da (2007). Investigations and explorations in the mathematics classroom. ZDM, 39 (5-6), 419-430. https://doi.org/10.1007/s11858-007-0054-z
Polya, G. (1945). Mathematical discovery: on understanding, learning, and teaching problem solving (Vol. 1). New York: Wiley.
Reiss, K. & Törner, G. (2007). Problem solving in the mathematics classroom: the German perspective. ZDM, 39(5), 431-441. https://doi.org/10.1007/s11858-007-0040-5
Roulston, K. (2001). Data analysis and "theorizing as ideology". Qualitative Research, 1 (3), 279-302. https://doi.org/10.1177/146879410100100302
Russel, S. J., Eston, R., Rook, J., Scott, M. & Sweeney, L. (2003). How to focus the mathematics curriculum on solving problems. In F. Lester (Ed.), Teaching mathematics through problem solving: prekindergarten-grade 6 (pp. 85-100). Reston: National Council of Teachers of Mathematics.
Santos-Trigo, M. (2007). Mathematical problem solving: an evolving research and practice domain. ZDM, 39 (5-6), 523-536. https://doi.org/10.1007/s11858-007-0057-9
Schoen, H. & Charles, R. (Eds.) (2003). Teaching mathematics through problem solving. Grades 6 -12. Reston: National Council of Teachers of Mathematics.
Schoenfeld, A. (2004). The math wars. Educational Policy, 18 (1), 253-286. https://doi.org/10.1177/0895904803260042
Schoenfeld, A. (2007). Problem solving in the United States, 1970-2008: research and theory, practice and politics. ZDM, 39 (5), 537-551. https://doi.org/10.1007/s11858-007-0038-z
Schroeder, T., & Lester, F. K. (1989). Developing understanding in mathematics via problem solving. In A. P. Shulte & P. R. Trafton (Eds.), New directions for elementary school mathematics (pp. 31-42). Reston: National Council of Teachers of Mathematics.
Skolverket. (1999). Curriculum for the compulsory school, the pre-school class and the after school centre: Lpo 94. Stockholm: Swedish National Agency for Education.
Skolverket. (2011a). Curriculum for the compulsory school system, the pre-school class and the leisure-time centre 2011. Stockholm: Swedish National Agency for Education.
Skolverket. (2011b). Kommentarmaterial till kursplanen i matematik. Stockholm: Skolverket.
Skott, J. (2001). The emerging practices of a novice teacher: the roles of his school mathematics images. Journal of Mathematics Teacher Education, 4 (1), 3-28. https://doi.org/10.1023/A:1009978831627
Skott, J. (2004). The forced autonomy of mathematics teachers. Educational Studies in Mathematics, 55 (1-3), 227-257. https://doi.org/10.1023/B:EDUC.0000017670.35680.88
Stacey, K. (2005). The place of problem solving in contemporary mathematics curriculum documents. The Journal of Mathematical Behavior, 24(3-4), 341-350. https://doi.org/10.1016/j.jmathb.2005.09.004
Stake, R. E. (1995). The art of case study research. London: SAGE.
Sterner, G. (2007). Lässvårigheter och räknesvårigheter. Nämnaren, 2007(2), 8-13.
Szendrei, J. (2007). When the going gets tough, the tough gets going problem solving in Hungary, 1970-2007: research and theory, practice and politics. ZDM, 39 (5), 443-458. https://doi.org/10.1007/s11858-007-0037-0
Undvall, L., Olofsson, K. & Forsberg, S. (2001). Matematikboken X. Stockholm: Almqvist & Wiksell. p. 187.
Wyndhamn, J., Riesbeck, E. & Schoultz, J. (2000). Problemlösning som metafor och praktik: studier av styrdokument och klassrumsverksamhet i matematik- och teknikundervisningen. Linköping: Institutionen för tillämpad lärarkunskap.
Xenofontos, C. & Andrews, P. (2014). Defining mathematical problems and problem solving: prospective primary teachers' beliefs in Cyprus and England. Mathematics Education Research Journal, 26 (2), 279-299. https://doi.org/10.1007/s13394-013-0098-z
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