Defining mathematical problems and problem solving: prospective primary teachers’ beliefs in Cyprus and England
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Abstract
This paper presents a comparative analysis of prospective elementary teachers’ mathematical problem solving-related beliefs in Cyprus and England. Twenty-four participants, twelve from a well-regarded university in each country, were interviewed qualitatively at the exit point of their undergraduate teacher education studies. Analyses revealed both similarities and differences in the ways in which prospective teachers in each country construe both mathematical problems and mathematical problem solving, indicating not only that their beliefs are culturally situated but also that the concepts of “mathematical problem” and “problem solving” have different meanings cross-culturally. Such findings challenge the received view in mathematics education research of definitional convergence with respect to both mathematical problems and problem solving. Some implications for policy making are discussed.
Keywords
Cyprus England Mathematical problems Problem solving Prospective teachers’ beliefsReferences
- Adamson, B., & Morris, P. (2007). Comparing curricula. In M. Bray, B. Adamson, & M. Mason (Eds.), Comparative education research: Approaches and methods (Vol. 19, pp. 263–282). Dordrecht: Springer.CrossRefGoogle Scholar
- Aguirre, J., & Speer, N. (1999). Examining the relationship between beliefs and goals in teacher practice. The journal of mathematical behavior, 18(3), 327–356.CrossRefGoogle Scholar
- Anderson, J. (2005). Implementing problem solving in mathematics classrooms: What support do teachers want? In A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce, & A. Roche (Eds.), Building connections: Theory, research and practice. Proceedings of the 28th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 89–96). Melbourne: Victoria.Google Scholar
- Anderson, J., White, P., & Sullivan, P. (2005). Using a schematic model to represent influences on, and relationships between, teachers’ problem-solving beliefs and practices. Mathematics Education Research Journal, 17(2), 9–38.CrossRefGoogle Scholar
- Andreescu, T., Gallian, J., Kane, J., & Mertz, J. (2008). Cross-cultural analysis of students with exceptional talent in mathematical problem solving. Notices of the American Mathematical Society, 55(10), 1248–1260.Google Scholar
- Andrews, P. (2003). Opportunities to learn in the Budapest mathematics classroom. International Journal of Science and Mathematics Education, 1(2), 201–225.CrossRefGoogle Scholar
- Andrews, P. (2007). The curricular importance of mathematics: A comparison of English and Hungarian teachers' espoused beliefs. Journal of Curriculum Studies, 39(2), 317–318.CrossRefGoogle Scholar
- Andrews, P. (2011). The cultural location of teachers’ mathematical knowledge: Another hidden variable in mathematics education research? In T. Rowland & K. Ruthven (Eds.), Mathematical knowledge in teaching (Vol. 50, pp. 99–118). New York: Springer.CrossRefGoogle Scholar
- Arcavi, A., & Friedlander, A. (2007). Curriculum developers and problem solving: The case of Israeli elementary school projects. ZDM, 39(5), 355–364.CrossRefGoogle Scholar
- Artigue, M., & Houdement, C. (2007). Problem solving in France: Didactic and curricular perspectives. ZDM, 39(5/6), 365–382.CrossRefGoogle Scholar
- Avcu, S., & Avcu, R. (2010). Pre-service elementary mathematics teachers' use of strategies in mathematical problem solving. Procedia - Social and Behavioral Sciences, 9, 1282–1286.CrossRefGoogle Scholar
- Bassok, M., Pedigo, S., & Oskarsson, A. (2008). Priming addition facts with semantic relations. Journal of Experimental Psychology: Learning, Memory, and Cognition, 34(2), 343–352.Google Scholar
- Blum, W., & Niss, M. (1991). Applied mathematical problem solving, modelling, applications, and links to other subjects—State, trends and issues in mathematics instruction. Educational Studies in Mathematics, 22(1), 37–68.CrossRefGoogle Scholar
- Booth, T. (1999). Viewing inclusion from a distance: Gaining perspective from comparative study. Support For Learning, 14(4), 164–168.CrossRefGoogle Scholar
- Borasi, R. (1986). On the nature of problems. Educational Studies in Mathematics, 17(2), 125–141.CrossRefGoogle Scholar
- Brown, A. (2008). Gesture viewpoint in Japanese and English: Cross-linguistic interactions between two languages in one speaker. Gesture, 8(2), 256–276.CrossRefGoogle Scholar
- Cai, J. (2004). Why do U.S. and Chinese students think differently in mathematical problem solving? Exploring the impact of early algebra learning and teachers’ beliefs. The Journal of Mathematical Behavior, 23, 135–167.Google Scholar
- Cai, J., & Nie, B. (2007). Problem solving in Chinese mathematics education: Research and practice. ZDM, 39(5), 459–473.CrossRefGoogle Scholar
- Callejo, M., & Vila, A. (2009). Approach to mathematical problem solving and students’ belief systems: Two case studies. Educational Studies in Mathematics, 72(1), 111–126.CrossRefGoogle Scholar
- Cassell, J., McNeill, D., & McCullough, K.-E. (1999). Speech-gesture mismatches: Evidence for one underlying representation of linguistic and nonlinguistic information. Pragmatics & Cognition, 7(1), 1–34.CrossRefGoogle Scholar
- Chapman, O. (1997). Metaphors in the teaching of mathematical problem solving. Educational Studies in Mathematics, 32(3), 201–228.CrossRefGoogle Scholar
- Chapman, O. (1999). Inservice teacher development in mathematical problem solving. Journal of Mathematics Teacher Education, 2(2), 121–142.CrossRefGoogle Scholar
- Chapman, O. (2002). Belief structure and in-service high school mathematics teacher growth. In G. Leder, E. Pehkonen, & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 177–193). Springer.; Mathematics Education Library 31:177-193Google Scholar
- Chapman, O. (2006). Classroom practices for context of mathematics word problems. Educational Studies in Mathematics, 62(2), 211–230.CrossRefGoogle Scholar
- Charalambous, C., Philippou, G., & Kyriakides, L. (2004). Towards a unified model on teachers’ concerns and efficacy beliefs related to a mathematics reform. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 199–206).Google Scholar
- Christou, C., & Philippou, G. (1998). The developmental nature of ability to solve one-step word problems. Journal for Research in Mathematics Education, 29(4), 436–442.CrossRefGoogle Scholar
- Cohen, D. (1990). A revolution in one classroom: The case of Mrs Oublier. Educational evaluation and policy analysis, 12(3), 311–329.CrossRefGoogle Scholar
- Cooney, T. (1985). A beginning teacher’s view of problem solving. Journal for Research in Mathematics Education, 16(5), 324–336.CrossRefGoogle Scholar
- Cooney, T., Shealy, B., & Arvold, B. (1998). Conceptualizing belief structures of preservice secondary mathematics teachers. Journal for Research in Mathematics Education, 29(3), 306–333.CrossRefGoogle Scholar
- Elia, I., Panaoura, A., Gagatsis, A., Gravvani, K., & Spyrou, P. (2008). Exploring different aspects of the understanding of function: Towards a four-facet model. Canadian Journal of Science, Mathematics, and Technology Education, 8(1), 49–69.CrossRefGoogle Scholar
- Enfield, N. J., Kita, S., & de Ruiter, J. P. (2007). Primary and secondary pragmatic functions of pointing gestures. Journal of Pragmatics, 39(10), 1722–1741.CrossRefGoogle Scholar
- Ernest, P. (1989). The knowledge, beliefs and attitudes of the mathematics teacher: A model. Journal of Education for teaching, 15(1), 13–33.CrossRefGoogle Scholar
- 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), 491–501.CrossRefGoogle Scholar
- Fuchs, L., Fuchs, D., Stuebing, K., Fletcher, J., Hamlett, C., & Lambert, W. (2008). Problem solving and computational skill: Are they shared or distinct aspects of mathematical cognition? Journal of Educational Psychology, 100(1), 30–47.CrossRefGoogle Scholar
- Givvin, K., Hiebert, J., Jacobs, J., Hollingsworth, H., & Gallimore, R. (2005). Are there national patterns of teaching? Evidence from the TIMSS 1999 video study. Comparative Education Review, 49(3), 311–343.CrossRefGoogle Scholar
- Goos, M., Galbraith, P., & Renshaw, P. (2000). A money problem: A source of insight into problem solving action. International Journal for Mathematics Teaching and Learning. Retrieved May 8, 2013 from http://www.cimt.plymouth.ac.uk/journal/pgmoney.pdf
- Grant, N. (2000). Tasks for comparative education in the new millennium. Comparative Education, 36(3), 309–317.CrossRefGoogle Scholar
- Handal, B. (2003). Teachers’ mathematical beliefs: A review. The Mathematics Educator, 13(2), 47–57.Google Scholar
- Handal, B., & Herrington, A. (2003). Mathematics teachers’ beliefs and curriculum reform. Mathematics Education Research Journal, 15(1), 59–69.CrossRefGoogle Scholar
- Haylock, D., & Cockburn, A. (2008). Understanding mathematics for young children. London: SAGE.Google Scholar
- Hensberry, K., & Jacobbe, T. (2012). The effects of Polya’s heuristics and diary writing on children’s problem solving. Mathematics Education Research Journal, 24(1), 59–85.CrossRefGoogle Scholar
- Jennings, S., & Dunne, R. (1996). A critical appraisal of the national curriculum by comparison with the French experience. Teaching Mathematics and its Applications, 15(2), 49–55.CrossRefGoogle Scholar
- Jitendra, A. (2002). Teaching students math problem-solving through graphic representations. Teaching exceptional children, 34(4), 34–38.Google Scholar
- Jitendra, A., Griffin, C., Deatline-Buchman, A., & Sczesniak, E. (2007). Mathematical problem solving in third-grade classrooms. Journal of Educational Research., 100(5), 283–302.CrossRefGoogle Scholar
- Jungheim, N. O. (2006). Learner and native speaker perspectives on a culturally-specific Japanese refusal gesture. International Review of Applied Linguistics in Language Teaching, 44(2), 125–143.CrossRefGoogle Scholar
- Kaiser, G. (2002). Educational philosophies and their influence on mathematics education—An ethnographic study in English and German mathematics classrooms. ZDM, 34(6), 241–257.Google Scholar
- Kelly, C. (2006). Using manipulatives in mathematical problem solving: A performance-based analysis. The Montana Mathematics Enthusiast, 3(2), 184–193.Google Scholar
- Kvale, S., & Brinkmann, S. (2009). InterViews: Learning the craft of qualitative research interviewing. London: SAGE Publications.Google Scholar
- Lam, T. (2006). Group problem-solving among teachers: A case study of how to improve a colleague’s teaching. Social Psychology of Education, 9, 273–299.CrossRefGoogle Scholar
- Leikin, R., & Kawass, S. (2005). Planning teaching an unfamiliar mathematics problem: The role of teachers’ experience in solving the problem and watching pupils solving it. The Journal of Mathematical Behavior, 24(3–4), 253–274.CrossRefGoogle Scholar
- Lester, F. (1994). Musings about mathematical problem-solving research: 1970–1994. Journal for Research in Mathematics Education, 25(6), 660–675.CrossRefGoogle Scholar
- Leung, F. (1995). The mathematics classroom in Beijing, Hong Kong and London. Educational Studies in Mathematics, 29(4), 297–325.CrossRefGoogle Scholar
- Marshall, S. (1995). Schemas in problem solving. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
- Mason, J. (2004). Are beliefs believable? Review of Leder, G., Pehkonen, E. & Törner, G. (Eds.) 2002, Beliefs: A hidden variable in mathematics education?, Kluwer, Dordrecht. Mathematical Thinking and Learning, 6(3), 343–352.Google Scholar
- Mason, J., Burton, L., & Stacey, K. (1982). Thinking mathematically. London: Addison-Wesley.Google Scholar
- Metallidou, P. (2009). Pre-service and in-service teachers’ metacognitive knowledge about problem-solving strategies. Teaching and Teacher Education, 25(1), 76–82.CrossRefGoogle Scholar
- NCTM. (2000). Principles and standards for school mathematics. Reston: NCTM.Google Scholar
- Nesher, P., & Hershkovitz, S. (1994). The role of schemes in two-step problems: Analysis and research findings. Educational Studies in Mathematics, 26(1), 1–23.CrossRefGoogle Scholar
- Nesher, P., Hershkovitz, S., & Novotna, J. (2003). Situation model, text base and what else? Factors affecting problem solving. Educational Studies in Mathematics, 52(2), 151–176.CrossRefGoogle Scholar
- Nunokawa, K. (2005). Mathematical problem solving and learning mathematics: What we expect students to obtain. The Journal of Mathematical Behavior, 24(3–4), 325–340.CrossRefGoogle Scholar
- Nyström, H. (2000). The postmodern challenge—From economic to creative management. Creativity & Innovation Management, 9(2), 109–114.CrossRefGoogle Scholar
- Pajares, M. F. (1992). Teachers beliefs and educational research: Cleaning up a messy construct. Review of Educational Research, 62(3), 307–332.CrossRefGoogle Scholar
- Palm, T. (2008). Impact of authenticity on sense making in word problem solving. Educational Studies in Mathematics, 67(1), 37–58.CrossRefGoogle Scholar
- Philippou, G., & Christou, C. (1999). A schema-based model for teaching problem solving. In O. Zaslavsky (Ed.), Proceedings of the 23rd International Conference for the Psychology of Mathematics Education (Vol. 4, pp. 57–64). Haifa: Israel Institute of Technology.Google Scholar
- Pólya, G. (1945). How to solve it: A new aspect of mathematics. New Jersey: Princeton University Press.Google Scholar
- Rowland, T. (2003). Mathematics as human activity: A different handshakes problem. The Mathematics Educator, 7(2), 55–70.Google Scholar
- Schoenfeld, A. (1985). Mathematical problem solving. New York: Academic Press.Google Scholar
- Schoenfeld, A. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). New York: MacMillan.Google Scholar
- Schroeder, T., & Lester, F. (1989). Developing understanding in mathematics via problem solving. In A. Shulte & P. Trafton (Eds.), New directions for elementary school mathematics (pp. 31–56). Reston: National Council of Teachers of Mathematics.Google Scholar
- Smidt, S., & Weiser, W. (1995). Semantic structures of one-step word problems involving multiplication or division. Educational Studies in Mathematics, 28(1), 55–72.CrossRefGoogle Scholar
- Stein, M., & Kaufman, J. (2010). Selecting and supporting the use of mathematics curricula at scale. American Educational Research Journal, 47(3), 663–693.CrossRefGoogle Scholar
- Strauss, A., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory. London: SAGE Publications.Google Scholar
- Sweller, J., Clark, R., & Kirschner, P. (2010). Teaching general problem-solving skills is not a substitute for, or a viable addition to, teaching mathematics. Notices of the American Mathematical Society, 57(10), 1303–1304.Google Scholar
- Taplin, M., & Chan, C. (2001). Developing problem-solving practitioners. Journal of Mathematics Teacher Education, 4(4), 285–304.CrossRefGoogle Scholar
- Thompson, A. (1984). The relationship of teachers’ conceptions of mathematics and mathematics teaching to instructional practice. Educational Studies in Mathematics, 15(2), 105–127.CrossRefGoogle Scholar
- Thompson, A. (1985). Teachers’ conceptions of mathematics and the teaching of problem solving. In E. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 281–294). Hillsdale: Lawrence Erlbaum.Google Scholar
- Tresch, J. (2001). On going native: Thomas Kuhn and anthropological method. Philosophy of the Social Sciences, 31(3), 302–322.CrossRefGoogle Scholar
- Van den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. Educational Studies in Mathematics, 54(1), 9–35.CrossRefGoogle Scholar
- Verschaffel, L., De Corte, E., & Vierstraete, H. (1999a). Upper elementary school pupils’ difficulties in modeling and solving nonstandard additive word problems involving ordinal numbers. Journal for Research in Mathematics Education, 30(3), 265–285.CrossRefGoogle Scholar
- Verschaffel, L., De Corte, E., Lasure, S., Van Vaerenbergh, G., Bogaerts, H., & Ratinckx, E. (1999b). Learning to solve mathematical application problems: A design experiment with fifth graders. Mathematical Thinking and Learning, 1(3), 195–229.CrossRefGoogle Scholar
- Wong, N.-Y., Lam, C.–. C., Sun, X., & Chan, A. (2009). From "exploring the middle zone" to "constructing a bridge": Experimenting in the spiral bianshi mathematics curriculum. International Journal of Science and Mathematics Education, 7(2), 363–382.CrossRefGoogle Scholar
- Xenofontos, C., & Andrews, P. (2012). Prospective teachers’ beliefs about problem-solving: Cypriot and English cultural constructions. Research in Mathematics Education, 14(1), 49–65.CrossRefGoogle Scholar
- Yeap, B.–H., Ferrucci, B., & Carter, J. (2006). Comparative study of arithmetic problems in Singaporean and American mathematics textbooks. In F. Leung, K. Graf & Lopez-Real, F. (eds.), Mathematics education in different cultural traditions - A comparative study of east Asia and the West. (pp. 213–226). New York: Springer.Google Scholar