Development of Finnish Elementary Pupils’ Problem-Solving Skills in Mathematics
Keywords:
open problem, development of problem-solving skills, infinity, Finnish elementary school, mathematics
Abstract
The purpose of this study is to determine how Finnish pupils’ problem-solving skills develop from the 3rd to 5th grade. As research data, we use one non-standard problem from pre- and post-test material from a three-year follow-up study, in the area of Helsinki, Finland. The problems in both tests consisted of four questions related to each other. The purpose of the formulation of the problem was to help the pupils to find how many solutions for a certain answer exist. The participants in the study were 348 third-graders and 356 fifth graders. Pupils’ fluency, i.e. ability to develop different solutions, was found to correlate with their ability to solve the problem. However, the proportions of the pupils (17% of the 3rd graders and 21% of the 5th graders) who answered that there were an infinite number of solutions are of the same magnitude. Thus, the pupils’ ability to solve this kind of problem does not seem to have developed from the 3rd to the 5th grade. The lack and insufficiency of pupils’ justifications reveal the importance of the teacher carefully listening to the pupils’ ideas in order to be able to promote pupils’ understanding of the concept of infinity, as well as the basic calculations.Downloads
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References
Becker, J. P., & Shimada, S. (1997). The Open-Ended Approach: A New Proposal for Teaching Mathematics. Reston (VA): NCTM.
Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for Research in Mathematics Education, 29(1), 41–62.
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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.
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NBE. (2004). National Core Curriculum for Basic Education 2004. Retrieved from http://www.oph.fi/english/publications/2009/national_core_curricula_for basic education
Nohda, N. (2000). Teaching by Open-Approach Method in Japanese Mathematics Classroom. In T. Nakahara & M. Koyama (Eds.), Proceedings of the PME-24 Conference: Vol. 1 (pp. 39–53). Hiroshima: Hiroshima University.
OECD. (2006). Assessing Scientific, Reading and Mathematical Literacy: A Framework for PISA 2006. Paris: OECD.
Patton, M. (2002). Qualitative research and evaluation methods. Thousand Oaks: Sage.
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., & Hannula, M. S. (2006). Infinity of numbers: a complex concept to be learnt? In S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of PME-NA in Merida: Vol. 2 (pp. 152–154). Merida: Universidad Pedagógica Nacional.
Polya, G. (1945). How to Solve It? A New Aspect of Mathematical Method. Princeton, NJ: Princeton University Press.
Schoenfeld, A. H. (1985). Mathematical Problem Solving. Orlando (FL): Academic Press.
Shimada, S. (1997). The Significance of an Open-Ended Approach. In J. Becker & L. Shimada (Eds.), The open-ended approach (pp. 1–9). Reston (VA): NCTM.
Silver, E. A. (1997). Fostering Creativity through Instruction Rich in Mathematical Problem Solving and Problem Posing. Zentralblatt für Didaktik der Mathematik, 29(3), 75–80.
Steinberg, R., Empson, S., & Carpenter, T. (2004). Inquiry into children’s mathematical thinking as a means to teacher change. Journal of Mathematics Teacher Education, 7, 237–267.
Torrance, E. P. (1974). Torrance Tests of Creative Thinking. Lexington (Mass.): Personnal Press/ Ginn and Company (Xerox Corporation).
Treffinger, T. J. (1995). Creative problem solving: Overview and educational implications. Educational Psychology Review, 7(3), 301–312.
Walshaw, M., & Anthony, G. (2008). The Teacher’s Role in Classroom Discourse: A Review of Recent Research Into Mathematics Classrooms. Review of Educational Research, 78(3), 516–551.
Wheeler, M. M. (1987). Children‘s understanding of zero and infinity. Arithmetic Teacher, 15(3), 42–44.
Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for Research in Mathematics Education, 29(1), 41–62.
Ericsson, K. A. (2003). The Acquisition of Expert Performance as Problem Solving. In J. E. Davidson & R. J. Sternberg (Eds.), The Psychology of Problem Solving (pp. 31–83). Cambridge: Cambridge University Press.
Guilford, J. P. (1956). The structure of intellect. Psychological Bulletin, 53(4), 267–293.
Hannula, M. S., Laine, A., Pehkonen, E., & Kaasila, R. (2012). Learning density of numbers in elementary teacher education. In G. H. Gunnarsdottir, F. Hreinsdottir, G. Palsdottir, M. Hannula, M. Hannula-Sormunen, E. Jablonka, U. T. Jankvist, A. Ryve, P. Valero, & K. Waege (Eds.), Proceedings of Norma 11: The Sixth Conference on Mathematics Education in ReykjavÃk, May 11.-14. 2011 (pp. 289-297). Reykjavik: University of Iceland Press.
Huhtala, S., & Laine, A. (2004). Mini-theories as part of pupils’ views of mathematics – division as an example. In A. Engström (Ed.), Democracy and Participation. A Challenge for Special Needs Education in Mathematics: Reports from the Department of Education 7 (pp. 177-188). Örebro: University of Örebro.
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.
NBE. (2004). National Core Curriculum for Basic Education 2004. Retrieved from http://www.oph.fi/english/publications/2009/national_core_curricula_for basic education
Nohda, N. (2000). Teaching by Open-Approach Method in Japanese Mathematics Classroom. In T. Nakahara & M. Koyama (Eds.), Proceedings of the PME-24 Conference: Vol. 1 (pp. 39–53). Hiroshima: Hiroshima University.
OECD. (2006). Assessing Scientific, Reading and Mathematical Literacy: A Framework for PISA 2006. Paris: OECD.
Patton, M. (2002). Qualitative research and evaluation methods. Thousand Oaks: Sage.
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., & Hannula, M. S. (2006). Infinity of numbers: a complex concept to be learnt? In S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of PME-NA in Merida: Vol. 2 (pp. 152–154). Merida: Universidad Pedagógica Nacional.
Polya, G. (1945). How to Solve It? A New Aspect of Mathematical Method. Princeton, NJ: Princeton University Press.
Schoenfeld, A. H. (1985). Mathematical Problem Solving. Orlando (FL): Academic Press.
Shimada, S. (1997). The Significance of an Open-Ended Approach. In J. Becker & L. Shimada (Eds.), The open-ended approach (pp. 1–9). Reston (VA): NCTM.
Silver, E. A. (1997). Fostering Creativity through Instruction Rich in Mathematical Problem Solving and Problem Posing. Zentralblatt für Didaktik der Mathematik, 29(3), 75–80.
Steinberg, R., Empson, S., & Carpenter, T. (2004). Inquiry into children’s mathematical thinking as a means to teacher change. Journal of Mathematics Teacher Education, 7, 237–267.
Torrance, E. P. (1974). Torrance Tests of Creative Thinking. Lexington (Mass.): Personnal Press/ Ginn and Company (Xerox Corporation).
Treffinger, T. J. (1995). Creative problem solving: Overview and educational implications. Educational Psychology Review, 7(3), 301–312.
Walshaw, M., & Anthony, G. (2008). The Teacher’s Role in Classroom Discourse: A Review of Recent Research Into Mathematics Classrooms. Review of Educational Research, 78(3), 516–551.
Wheeler, M. M. (1987). Children‘s understanding of zero and infinity. Arithmetic Teacher, 15(3), 42–44.
Published
2014-09-30
How to Cite
Laine, A., Näveri, L., Ahtee, M., & Pehkonen, E. (2014). Development of Finnish Elementary Pupils’ Problem-Solving Skills in Mathematics. Center for Educational Policy Studies Journal, 4(3), 111–129. https://doi.org/10.26529/cepsj.198
Section
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