On Teaching Problem Solving in School Mathematics
Keywords:
Mathematics teaching, Open problems, Problem solving
Abstract
The article begins with a brief overview of the situation throughout the world regarding problem solving. The activities of the ProMath group are then described, as the purpose of this international research group is to improve mathematics teaching in school. One mathematics teaching method that seems to be functioning in school is the use of open problems
(i.e., problem fields). Next we discuss the objectives of the Finnish curriculum that are connected with problem solving. Some examples and research results are taken from a Finnish–Chilean research project that monitors the development of problem-solving skills in third grade pupils. Finally, some ideas on “teacher change†are put forward. It is not possible to change teachers, but only to provide hints for possible change routes: the teachers themselves should work out the ideas and their implementation.
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References
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Becker, J. P., & Shimada, S. (1997). The Open-Ended Approach. Reston (VA): NCTM.
Bereiter, C. (1990). Aspects of an Educational Learning Theory. Review of Educational Research, 60(4), 603–624.
Bereiter, C., & Scardamalia, M. (1996). Rethinking learning. In D. R. Olson & N. Torrance (Eds.), The handbook of education and learning. New models of learning, teaching and schooling. Cambridge (MA): Blackwell.
Bergqvist, T. (Ed.) (2012). Learning Problem Solving And Learning Through Problem Solving. University of Umeå.
Blanc, P., & Sutherland, R. (1996). Student teachers’ approaches to investigative mathematics: iterative engagement or disjointed mechanisms? In L. Puig & A. Gutierrez (Eds.), Proceedings of the PME-20 conference, Vol. 2 (pp. 97–104). Valencia: University of Valencia.
Blumenfeld, P. C., Soloway, E., Marx, R. W., Krajcik, J. S., Guzdial, M., & Palincsar, A. (1991). Motivating project-based learning: Substaining the doing, supporting the learning. Educational
Psychologist, 26(3&4), 369–398.
Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for research in mathematics education, 29(1), 41–62.
Brown, S. I. (1997). Thinking Like a Mathematician: A Problematic Perspective. For the Learning of Mathematics, 17(2), 36–38.
Brown, J. S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher, 18(1), 32–42.
Clarke, D. J., & Sullivan, P. A. (1992). Responses to open-ended tasks in mathematics: characteristics and implications. In W. Geeslin & K. Graham (Eds.), Proceedings of the PME 16, Vol I (pp. 137–144). Durham (NH): University of New Hampshire.
Cockcroft, W. (Chair) (1982). Mathematics Counts, Report of the Committee of Enquiry into the teaching of Mathematics in Schools. London: HMSO.
Collins, A., Brown, J. S., & Newman, S. (1989). Cognitive apprenticeship: Teaching the crafts of reading, writing and mathematics. In L. B. Resnick (Ed.), Knowing, Learning and Instruction. Essays in Honor of Robert Glaser (pp. 453–494). Hilldale, N. J.: Lawrence Erlbaum Associates.
Evens, H., & Houssart, J. (2004). Categorizing pupils’ written answers to a mathematics test question: “I know but I can’t explainâ€. Educational Research, 46(3), 269–282.
Kantowski, M. G. (1980). Some Thoughts on Teaching for Problem Solving. In S. Krulik & R. E. Reys (Eds.), Problem Solving in School Mathematics. NCTM Yearbook 1980. (pp. 195–203). Reston (VA): Council.
Kwon, O. N., Park, J. H., & Park, J. S. (2006). Cultivating divergent thinking in mathematics through an open-ended approach. Asia Pacific Education Review, 7(1), 51–61.
Laine, A., Näveri, L., Pehkonen, E., Ahtee, M., & Hannula, M. S. (2012). Third-graders’ problem solving performance and teachers’ actions. In T. Bergqvist (Ed.), Learning Problem Solving And
Learning Through Problem Solving (pp. 69–81). University of Umeå.
Mason, J. (1991). Mathematical problem solving: open, closed and exploratory in the UK. International Reviews on Mathematical Education (= ZDM), 23(1), 14–19.
Näveri, L., Ahtee, M., Laine, A., Pehkonen, E., & Hannula, M. S. (2012). Erilaisia tapoja johdatella ongelmanratkaisutehtävään - esimerkkinä aritmagonin ratkaiseminen alakoulun kolmannella
luokalla [Different ways to introduce a problem task – as an example the solving of aritmagon in the third grade]. In H. Krzywacki, K. Juuti, & J. Lampiselkä (Eds.), Matematiikan ja luonnontieteiden opetuksen ajankohtaista tutkimusta (pp. 81–98). Helsinki: Suomen ainedidaktisen tutkimusseuran julkaisuja. Ainedidaktisia tutkimuksia 2.
NBE. (2004). Perusopetuksen opetussuunnitelman perusteet 2004 [The basics of the curriculum for the basic instruction]. Helsinki: Opetushallitus.
NBE. (2010). Esiopetuksen opetussuunnitelman perusteet 2010 [Basics of the curriculum for pre-school instruction 2010]. Retrieved from www.oph.fi/download/131115_Esiopetuksen_
opetussuunnitelman_perusteet_2010
NCTM. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
Nohda, N. (1988). Problem solving using “open-ended problems†in mathematics teaching. In H. Burkhardt, S. Groves, A. Schoenfeld, & K. Stacey (Eds.), Problem Solving – A World View. Proceedings of problem solving theme group at ICME-5 (Adelaide) (pp. 225–234). Nottingham: Shell Centre.
Nohda, N. (1991). Paradigm of the “open-approach†method in mathematics teaching: Focus on mathematical problem solving. International Reviews on Mathematical Education (= ZDM), 23(2), 32–37.
Pehkonen, E. (1989). Verwenden der geometrischen Problemfelder. In E. Pehkonen (Ed.), Geometry Teaching – Geometrieunterricht. Research Report 74 (pp. 221–230). University of Helsinki. Department of Teacher Education.
Pehkonen, E. (1995). Introduction: Use of Open-Ended Problems. International Reviews on Mathematical Education (= ZDM), 27(2), 55–57.
Pehkonen, E. (Ed.) (1997). Use of open-ended problems in mathematics classroom. Research Report 176. University of Helsinki. Department of Teacher Education.
Pehkonen, E. (Ed.) (2001). Problem Solving Around the World. Report Series C:14. University of Turku. Faculty of Education.
Pehkonen, E. (2004). State-of-the-Art in Problem Solving: Focus on Open Problems. In H. Rehlich & B. Zimmermann (Eds.), ProMath Jena 2003. Problem Solving in Mathematics Education (pp. 93–111). Hildesheim: Verlag Franzbecker.
Pehkonen, E. (2007). Über “teacher change†(Lehrerwandel) in der Mathematik. In A. Peter-Koop & A. Bikner-Ahsbahs (Eds.), Mathematische Bildung - mathematische Leistung: Festschrift für Michael Neubrand zum 60. Geburtstag (pp. 349–360). Hildesheim: Franzbecker.
Pehkonen, L. (2000). Written arguments in a conflicting mathematical situation. Nordic Studies in Mathematics Education, 8(1), 23–33.
Schroeder, T. L., & Lester, F. K. (1989). Developing understanding in mathematics via problem solving. In P. R. Trafton (Ed.), New Directions for Elementary School Mathematics. NCTM 1989
Yearbook. (pp. 31–42). Reston, Va: NCTM.
Schupp, H. (2002). Thema mit Variationen. Aufgabenvariation im Mathematikunterricht. Hildesheim: Verlag Franzbecker.
Shaw, K. L., Davis, N. T., & McCarty, J. (1991). A cognitive framework for teacher change. In R. G. Underhill (Ed.), Proceedings of PME-NA 13, Vol 2 (pp. 161–167). Blacksburg (VA): Virginia Tech.
Shimada, S. (Ed.) (1977). Open-end approach in arithmetic and mathematics – A new proposal toward teaching improvement. Tokyo: Mizuumishobo. [in Japanese]
Sierpinska, A. (1994). Understanding in mathematics. Studies in mathematics education series: 2. London: Falmer.
Silver, E. (1995). The Nature and Use of Open Problems in Mathematics Education: Mathematical and Pedagogical Perspectives. International Reviews on Mathematical Education (= ZDM), 27(2), 67–72.
Stacey, K. (1995). The Challenges of Keeping Open Problem-Solving Open in School Mathematics. International Reviews on Mathematical Education (= ZDM), 27(2), 62–67.
Törner, G., Schoenfeld, A. H., & Reiss, K. M. (Eds.) (2007). Problem solving around the world: summing up the state of the art. ZDM Mathematics Education, 39(5/6), 353–551.
Wiliam, D. (1994). Assessing authentic tasks: alternatives to mark-schemes. Nordic Studies in Mathematics Education, 2(1), 48–68.
Williams, D. (1989). Assessment of open-ended work in the secondary school. In D. F. Robitaille (Ed.), Evaluation and Assessment in Mathematics Education. Science and Technology Education. Document Series 32. (pp. 135–140). Paris: Unesco.
Wu, H. (1994). The Role of Open-Ended Problems in Mathematics Education. Journal of Mathematical Behavior, 13(1), 115–128.
Zaslavsky, O. (1995). Open-ended tasks as a trigger for mathematics teachers’ professional development. For the Learning of Mathematics, 15(3), 15–20.
Zimmermann, B. (1991). Offene Probleme für den Mathematikunterricht und ein Ausblick auf Forschungsfragen. International Reviews on Mathematical Education (= ZDM), 23(2), 38–46.
Zimmermann, B. (2010). “Open ended problem solving in mathematics instruction and some perspectives on research question†revisited – new bricks from the wall? In A. Ambrus & E.
Vasarhelyi (Eds.), Problem Solving in Mathematics Education. Proceedings of the 11th ProMath conference in Budapest (pp. 143–157). Eötvös Lorand University.
Becker, J. P., & Shimada, S. (1997). The Open-Ended Approach. Reston (VA): NCTM.
Bereiter, C. (1990). Aspects of an Educational Learning Theory. Review of Educational Research, 60(4), 603–624.
Bereiter, C., & Scardamalia, M. (1996). Rethinking learning. In D. R. Olson & N. Torrance (Eds.), The handbook of education and learning. New models of learning, teaching and schooling. Cambridge (MA): Blackwell.
Bergqvist, T. (Ed.) (2012). Learning Problem Solving And Learning Through Problem Solving. University of Umeå.
Blanc, P., & Sutherland, R. (1996). Student teachers’ approaches to investigative mathematics: iterative engagement or disjointed mechanisms? In L. Puig & A. Gutierrez (Eds.), Proceedings of the PME-20 conference, Vol. 2 (pp. 97–104). Valencia: University of Valencia.
Blumenfeld, P. C., Soloway, E., Marx, R. W., Krajcik, J. S., Guzdial, M., & Palincsar, A. (1991). Motivating project-based learning: Substaining the doing, supporting the learning. Educational
Psychologist, 26(3&4), 369–398.
Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for research in mathematics education, 29(1), 41–62.
Brown, S. I. (1997). Thinking Like a Mathematician: A Problematic Perspective. For the Learning of Mathematics, 17(2), 36–38.
Brown, J. S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher, 18(1), 32–42.
Clarke, D. J., & Sullivan, P. A. (1992). Responses to open-ended tasks in mathematics: characteristics and implications. In W. Geeslin & K. Graham (Eds.), Proceedings of the PME 16, Vol I (pp. 137–144). Durham (NH): University of New Hampshire.
Cockcroft, W. (Chair) (1982). Mathematics Counts, Report of the Committee of Enquiry into the teaching of Mathematics in Schools. London: HMSO.
Collins, A., Brown, J. S., & Newman, S. (1989). Cognitive apprenticeship: Teaching the crafts of reading, writing and mathematics. In L. B. Resnick (Ed.), Knowing, Learning and Instruction. Essays in Honor of Robert Glaser (pp. 453–494). Hilldale, N. J.: Lawrence Erlbaum Associates.
Evens, H., & Houssart, J. (2004). Categorizing pupils’ written answers to a mathematics test question: “I know but I can’t explainâ€. Educational Research, 46(3), 269–282.
Kantowski, M. G. (1980). Some Thoughts on Teaching for Problem Solving. In S. Krulik & R. E. Reys (Eds.), Problem Solving in School Mathematics. NCTM Yearbook 1980. (pp. 195–203). Reston (VA): Council.
Kwon, O. N., Park, J. H., & Park, J. S. (2006). Cultivating divergent thinking in mathematics through an open-ended approach. Asia Pacific Education Review, 7(1), 51–61.
Laine, A., Näveri, L., Pehkonen, E., Ahtee, M., & Hannula, M. S. (2012). Third-graders’ problem solving performance and teachers’ actions. In T. Bergqvist (Ed.), Learning Problem Solving And
Learning Through Problem Solving (pp. 69–81). University of Umeå.
Mason, J. (1991). Mathematical problem solving: open, closed and exploratory in the UK. International Reviews on Mathematical Education (= ZDM), 23(1), 14–19.
Näveri, L., Ahtee, M., Laine, A., Pehkonen, E., & Hannula, M. S. (2012). Erilaisia tapoja johdatella ongelmanratkaisutehtävään - esimerkkinä aritmagonin ratkaiseminen alakoulun kolmannella
luokalla [Different ways to introduce a problem task – as an example the solving of aritmagon in the third grade]. In H. Krzywacki, K. Juuti, & J. Lampiselkä (Eds.), Matematiikan ja luonnontieteiden opetuksen ajankohtaista tutkimusta (pp. 81–98). Helsinki: Suomen ainedidaktisen tutkimusseuran julkaisuja. Ainedidaktisia tutkimuksia 2.
NBE. (2004). Perusopetuksen opetussuunnitelman perusteet 2004 [The basics of the curriculum for the basic instruction]. Helsinki: Opetushallitus.
NBE. (2010). Esiopetuksen opetussuunnitelman perusteet 2010 [Basics of the curriculum for pre-school instruction 2010]. Retrieved from www.oph.fi/download/131115_Esiopetuksen_
opetussuunnitelman_perusteet_2010
NCTM. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
Nohda, N. (1988). Problem solving using “open-ended problems†in mathematics teaching. In H. Burkhardt, S. Groves, A. Schoenfeld, & K. Stacey (Eds.), Problem Solving – A World View. Proceedings of problem solving theme group at ICME-5 (Adelaide) (pp. 225–234). Nottingham: Shell Centre.
Nohda, N. (1991). Paradigm of the “open-approach†method in mathematics teaching: Focus on mathematical problem solving. International Reviews on Mathematical Education (= ZDM), 23(2), 32–37.
Pehkonen, E. (1989). Verwenden der geometrischen Problemfelder. In E. Pehkonen (Ed.), Geometry Teaching – Geometrieunterricht. Research Report 74 (pp. 221–230). University of Helsinki. Department of Teacher Education.
Pehkonen, E. (1995). Introduction: Use of Open-Ended Problems. International Reviews on Mathematical Education (= ZDM), 27(2), 55–57.
Pehkonen, E. (Ed.) (1997). Use of open-ended problems in mathematics classroom. Research Report 176. University of Helsinki. Department of Teacher Education.
Pehkonen, E. (Ed.) (2001). Problem Solving Around the World. Report Series C:14. University of Turku. Faculty of Education.
Pehkonen, E. (2004). State-of-the-Art in Problem Solving: Focus on Open Problems. In H. Rehlich & B. Zimmermann (Eds.), ProMath Jena 2003. Problem Solving in Mathematics Education (pp. 93–111). Hildesheim: Verlag Franzbecker.
Pehkonen, E. (2007). Über “teacher change†(Lehrerwandel) in der Mathematik. In A. Peter-Koop & A. Bikner-Ahsbahs (Eds.), Mathematische Bildung - mathematische Leistung: Festschrift für Michael Neubrand zum 60. Geburtstag (pp. 349–360). Hildesheim: Franzbecker.
Pehkonen, L. (2000). Written arguments in a conflicting mathematical situation. Nordic Studies in Mathematics Education, 8(1), 23–33.
Schroeder, T. L., & Lester, F. K. (1989). Developing understanding in mathematics via problem solving. In P. R. Trafton (Ed.), New Directions for Elementary School Mathematics. NCTM 1989
Yearbook. (pp. 31–42). Reston, Va: NCTM.
Schupp, H. (2002). Thema mit Variationen. Aufgabenvariation im Mathematikunterricht. Hildesheim: Verlag Franzbecker.
Shaw, K. L., Davis, N. T., & McCarty, J. (1991). A cognitive framework for teacher change. In R. G. Underhill (Ed.), Proceedings of PME-NA 13, Vol 2 (pp. 161–167). Blacksburg (VA): Virginia Tech.
Shimada, S. (Ed.) (1977). Open-end approach in arithmetic and mathematics – A new proposal toward teaching improvement. Tokyo: Mizuumishobo. [in Japanese]
Sierpinska, A. (1994). Understanding in mathematics. Studies in mathematics education series: 2. London: Falmer.
Silver, E. (1995). The Nature and Use of Open Problems in Mathematics Education: Mathematical and Pedagogical Perspectives. International Reviews on Mathematical Education (= ZDM), 27(2), 67–72.
Stacey, K. (1995). The Challenges of Keeping Open Problem-Solving Open in School Mathematics. International Reviews on Mathematical Education (= ZDM), 27(2), 62–67.
Törner, G., Schoenfeld, A. H., & Reiss, K. M. (Eds.) (2007). Problem solving around the world: summing up the state of the art. ZDM Mathematics Education, 39(5/6), 353–551.
Wiliam, D. (1994). Assessing authentic tasks: alternatives to mark-schemes. Nordic Studies in Mathematics Education, 2(1), 48–68.
Williams, D. (1989). Assessment of open-ended work in the secondary school. In D. F. Robitaille (Ed.), Evaluation and Assessment in Mathematics Education. Science and Technology Education. Document Series 32. (pp. 135–140). Paris: Unesco.
Wu, H. (1994). The Role of Open-Ended Problems in Mathematics Education. Journal of Mathematical Behavior, 13(1), 115–128.
Zaslavsky, O. (1995). Open-ended tasks as a trigger for mathematics teachers’ professional development. For the Learning of Mathematics, 15(3), 15–20.
Zimmermann, B. (1991). Offene Probleme für den Mathematikunterricht und ein Ausblick auf Forschungsfragen. International Reviews on Mathematical Education (= ZDM), 23(2), 38–46.
Zimmermann, B. (2010). “Open ended problem solving in mathematics instruction and some perspectives on research question†revisited – new bricks from the wall? In A. Ambrus & E.
Vasarhelyi (Eds.), Problem Solving in Mathematics Education. Proceedings of the 11th ProMath conference in Budapest (pp. 143–157). Eötvös Lorand University.
Published
2013-12-31
How to Cite
Pehkonen, E., NäveriL., & Laine, A. (2013). On Teaching Problem Solving in School Mathematics. Center for Educational Policy Studies Journal, 3(4), 9-23. https://doi.org/10.26529/cepsj.220
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