Skip to main content
SpringerLink
Log in

Modelling problem-solving situations into number theory tasks: The route towards generalisation

  • Articles
  • Published:
Mathematics Education Research Journal Aims and scope Submit manuscript

Abstract

This paper examines the way two 10th graders cope with a non-standard generalisation problem that involves elementary concepts of number theory (more specifically linear Diophantine equations) in the geometrical context of a rectangle’s area. Emphasis is given on how the students’ past experience of problem solving (expressed through interplay among different modes of thinking and actions that show executive control and decision-making skills) supported them in their route towards generalisation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Amit, M., & Neria, D. (2008). “Rising to challenge”: Using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students.ZDM, 40(1), 111–129.

    Article  Google Scholar 

  • Arcavi, A. (1994). Symbol sense: Informal sense-making in formal mathematics.For the Learning of Mathematics, 14, 24–35.

    Google Scholar 

  • Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.),Mathematics, teachers and children (pp. 216–235). Milton Keynes: Open University.

    Google Scholar 

  • Ball, D. L. (1988). Unlearning to teach mathematics.For the Learning of Mathematics, 8(1), 40–48.

    Google Scholar 

  • Becker, J. R., & Rivera, F. (2004). An investigation of beginning algebra students’ ability to generalize linear patterns. In M. J. Hoins & A. B. Fuglestad (Eds.),Proceedings of the 28th International Conference for the Psychology of Mathematics Education, (vol. 1, p. 286). Bergen, Norway: Bergen University College.

    Google Scholar 

  • Brown, A., Thomas, K., & Tolias, G. (2002). Conceptions of divisibility: Success and understanding. In S. R. Campbell & R. Zazkis (Eds.),Learning and teaching number theory: Research in cognition and instruction (pp. 41–82). Westport, CT: Ablex.

    Google Scholar 

  • Carraher, D., Martinez, M., & Schliemann, A. (2008). Early algebra and mathematics generalization.ZDM, 40(1), 3–22.

    Article  Google Scholar 

  • Cooper, M., & Sakane, H. (1986). Comparative experimental study of children’s strategies with deriving a mathematical law. InProceedings of the 10th International Conference for the Psychology of Mathematics Education (pp. 410–414). London: University of London, Institute of Education.

    Google Scholar 

  • Cooper, T. & Warren, E. (2008). The effect of different representations on Year 3 to 5 students’ ability to generalise.ZDM, 40(1), 23–37.

    Article  Google Scholar 

  • Douady, R., & Parzysz, B. (1998). Geometry in the classroom. In C. Mammana & V. Villani (Eds.),Perspectives on the teaching of geometry for the 21 st century (pp. 159–192). Kluwer: Netherlands.

    Chapter  Google Scholar 

  • Ellis, A. (2007). Connections between generalizing and justifying: Students’ reasoning with linear relationships.Journal for Research in Mathematics Education, 38(3), 194–229.

    Google Scholar 

  • Ginat, D. (2006). Overlooking number patterns in algorithmic problem solving. In R. Zazkis & S. R. Campbell (Eds.),Number theory in mathematics education: Perspectives and prospects (pp. 223–248). Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Harel, G. & Tall, D. (1991). The general, the abstract, and the generic in advanced mathematics.For the Learning of Mathematics, 11(1), 38–42.

    Google Scholar 

  • Iatridou, M., & Papadopoulos, I. (2010). From area to number theory: A case study. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.),Proceedings of the 6 th Conference of European Research in Mathematics Education (pp. 599–608). Institut National De Recherche Pedagogique, Lyon.

    Google Scholar 

  • Ishida, J. (1997). The teaching of general solution methods to pattern finding problems through focusing on an evaluation and improvement process.School Science and Mathematics, 97, 155–162.

    Article  Google Scholar 

  • Kaput, J. J. (1992). Teaching and learning a new algebra. In E. Fennena & T. Romberg (Eds.),Mathematics classroom that promote understanding (pp.133–155). Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Kieran, C. & Guzman, J. (2006). Number theoretic experience of 12 to 15 year olds in a calculator environment: The intertwining coemergence of technique and theory. In R. Zazkis & S. R. Campbell (Eds.),Number theory in mathematics education: Perspectives and prospects (pp. 173–200). Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Krutetskii, V. A. (1976).The psychology of mathematical abilities in schoolchildren. Chicago: University of Chicago Press.

    Google Scholar 

  • Lee, L., & Wheeler, D. (1987).Algebraic thinking in high school students: Their conceptions of generalisation and justification. Research Report. Montreal: Concordia University.

    Google Scholar 

  • Lester, F. K. (1985). Methodological considerations in research on mathematical problem solving. In E. A. Silver (Ed.),Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 41–70). Hillsdale, NJ: Erlbaum.

    Google Scholar 

  • Mamona-Downs, J. & Papadopoulos, I. (in press). Problem-solving activity ancillary to the concept of area.Mediterranean Journal for Research in Mathematics Education.

  • Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.),Approaches to algebra: Perspectives for research and teaching (pp. 65–86). Dordrecht, Netherlands: Kluwer.

    Google Scholar 

  • Mason, J., Burton, L., & Stacey, K. (1982).Thinking mathematically. New York: Addison Wesley.

    Google Scholar 

  • Orton, J., Orton, A., & Roper, T. (1999). Pictorial and practical contexts and the perception of pattern. In A. Orton (Ed.),Patterns in the teaching and learning of mathematics (pp. 121–136). London: Cassell Publishers.

    Google Scholar 

  • Polya, G. (1957).How to solve it. Princeton: Princeton University Press.

    Google Scholar 

  • Polya, G. (1968).Mathematics and plausible reasoning: Vol. 2. Patterns of plausible inference. Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Rico, L. (1996). The role of representation systems in the learning of numerical structures. In L. Puig & A. Gutierrez (Eds.),Proceedings of the 20th International Conference for the Psychology of Mathematics Education (Vol. 1, pp. 87–102). Valencia, Spain.

    Google Scholar 

  • Schoenfeld, H. A. (1985).Mathematical problem solving. Academic Press, Inc.

  • Schoenfeld, H. A. (1988). Mathematics, technology and higher order thinking. In R. Nickerson & P. Zodhiates (Eds.),Technology in education: Looking toward 2020 (pp. 67-96). Lawrence Erlbaum Associates.

  • Sinclair, N., Zazkis, R., & Liljedahl, P. (2004). Number worlds: Visual and experimental access to elementary number theory concepts.International Journal of Computers for Mathematical Learning, 8, 235–263.

    Article  Google Scholar 

  • Sriraman, B. (2003). Mathematical giftedness, problem solving, and the ability to formulate generalizations: The problem-solving experiences of four gifted students.Journal of Secondary Gifted Education, 14, 151–165.

    Google Scholar 

  • Sriraman, B. (2004). Reflective abstraction, uniframes and the formulation of generalizations.Journal of Mathematical behavior, 23(2), 205–222.

    Article  Google Scholar 

  • Stacey, K. (1989). Finding and using patterns in linear generalising problems.Educational Studies of Mathematics, 20(2), 147–164.

    Article  Google Scholar 

  • Steele, D. (2008). Seventh-grade students’ representations for pictorial growth and change problems.ZDM, 40(1), 97–110.

    Article  Google Scholar 

  • Steele, D. F., & Johanning, D. I. (2004). A schematic-theoretic view of problem solving and the development of algebraic thinking.Educational Studies in Mathematics, 57, 65–90.

    Article  Google Scholar 

  • Zazkis, R., & Campbell, S. R. (1996). Divisibility and multiplicative structure of natural numbers: Preservice teachers’ understanding.Journal for Research in Mathematics Education, 27, 540–563.

    Article  Google Scholar 

  • Zazkis, R., & Liljedahl, P. (2004). Understanding primes: The role of representation.Journal for Research in Mathematics Education, 35, 164–186.

    Article  Google Scholar 

  • Zazkis, R., Liljedahl, P., & Chernoff, E. (2008). The role of examples in forming and refuting generalizations.ZDM, 40(1), 131–141.

    Article  Google Scholar 

  • Zazkis, R. (1998). Odds and ends of odds and evens: An inquiry into students’ understanding of even and odd numbers.Educational Studies in Mathematics, 36(1), 73–89.

    Article  Google Scholar 

  • Zazkis, R. (2000). Factors, divisors and multiples: Exploring the web of students’ connections. In A. Schoenfels, J. Kaput, & E. Dubinsky (Eds.),Research in collegiate mathematics education (Vol. 4, pp. 210–238). Providence, RI: American Mathematical Society.

    Google Scholar 

  • Zazkis, R. (2007). Number Theory in mathematics education: Queen and servant. In J. Novotna & H. Moraova (Eds.),Proceedings of the International Symposium Elementary Maths Teaching SEMT ’07 (pp. 46–59). The Czech Republic: Charles University, Prague.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Primary Education, Ptolemeon 21 Str., 55535, Pylea, Thessaloniki, Greece

    Ioannis Papadopoulos

  2. Secondary Education, Pedini, 264, 45500, Ioannina, Greece

    Maria Iatridou

Authors
  1. Ioannis Papadopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maria Iatridou
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Papadopoulos, I., Iatridou, M. Modelling problem-solving situations into number theory tasks: The route towards generalisation. Math Ed Res J 22, 85–110 (2010). https://doi.org/10.1007/BF03219779

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03219779

Keywords

Search

Navigation