Preservice Middle and High School Mathematics Teachers’ Strategies when Solving Proportion Problems

Abstract

The purpose of this study was to investigate eight preservice middle and high school mathematics teachers’ solution strategies when solving single and multiple proportion problems. Real-world missing-value word problems were used in an interview setting to collect information about preservice teachers’ (PSTs) reasoning about proportional relationships. An explanatory case study methodology with multiple cases was used to make comparisons within and across cases. Analysis of the semi-structured interviews with each PST revealed that using practical problems, in which plastic gears and a mini balance system were provided, and multiple proportion problems facilitated the PSTs’ recognition of the proportional relationships in their solutions. Therefore, they avoided using cross-multiplication and erroneous strategies in those problems. Among the strategies that the PSTs used in solving single and multiple proportion problems, the ratio table strategy was the most frequent and effective strategy. The ratio table strategy enabled the PSTs to recognize the constant ratio and product relationships more than the other strategies. The results of this study illuminate how PSTs reason about proportional relationships when they cannot rely on computation methods like cross-multiplication.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Change history

  • 28 November 2018

    This correction stands to correct Acknowledgments in the original article.

References

  1. Avcu, R. & Dogan, M. (2014). What are the strategies used by seventh grade students while solving proportional reasoning problems? International Journal of Educational Studies in Mathematics, 1(2), 34–55.

    Article  Google Scholar 

  2. Beckmann, S. (2011). Mathematics for elementary teachers (3rd ed.). Boston, MA: Pearson.

  3. Ben-Chaim, D., Keret, Y. & Ilany, B. (2012). Ratio and proportion. Research and teaching in mathematics teachers’ education. Rotterdam, The Netherlands: Sense Publishers.

  4. Bernard, H. (1994). Research methods in anthropology (2nd ed.). Thousand Oaks, CA: Sage.

  5. Boyatzis, R. E. (1998). Transforming qualitative information: Thematic analysis and code development. Thousand Oaks, CA: Sage.

  6. Canada, D., Gilbert, M. & Adolphson, K. (2008). Conceptions and misconceptions of elementary preservice teachers in proportional reasoning. In O. Figueras, J. Cortina, S. Alatorre, T. Rojano & A. Sepulveda (Eds.), Proceedings of the 32th conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 249–256). Michoacán, Mexico: The University of Saint Nicholas of Hidalgo.

  7. Clark, H. J. (2008). Investigating students’ proportional reasoning strategies (Master’s thesis). Available from ProQuest Dissertations and Theses database. (UMI No. 1453188).

  8. Common Core State Standards Initiative (2010). The common core state standards for mathematics. Washington, DC: Author.

    Google Scholar 

  9. Cox, D. C. (2013). Similarity in middle school mathematics: At the crossroads of geometry and number. Mathematical Thinking and Learning, 15(1), 3–23.

    Article  Google Scholar 

  10. Cramer, K. & Post, T. (1993). Making connections: A case for proportionality. Arithmetic Teacher, 60(6), 342–346.

    Google Scholar 

  11. Cramer, K., Post, T. & Currier, S. (1993). Learning and teaching ratio and proportion: Research implications. In D. Owens (Ed.), Research ideas for the classroom: Middle grades mathematics (pp. 159–178). New York, NY: Macmillan.

  12. De Bock, D., Verschaffel, L. & Janssens, D. (1998). The predominance of the linear model in secondary school students solutions of word problems involving length and area of similar plane figures. Educational Studies in Mathematics, 35(1), 65–83.

    Article  Google Scholar 

  13. Fisher, L. C. (1988). Strategies used by secondary mathematics teachers to solve proportion problems. Journal for Research in Mathematics Education, 19(2), 157–168.

    Article  Google Scholar 

  14. Harel, G. & Behr, M. (1995). Teachers’ solutions for multiplicative problems. Hiroshima Journal of Mathematics Education, 3, 31–51.

    Google Scholar 

  15. Hart, K. (1984). Ratio: Children’s strategies and errors. Windsor, UK: NFER-Nelson.

  16. Inhelder, B. & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence. New York, NY: Basic Books.

  17. Izsák, A., & Jacobson, E. (2013). Understanding teachers’ inferences of proportionality between quantities that form a constant difference or constant product. Paper presented at the National Council of Teachers of Mathematics Research Presession, Denver, CO.

  18. Karplus, R., Pulos, S. & Stage, E. (1983a). Early adolescents proportional reasoning on ‘rate’ problems. Educational Studies in Mathematics, 14(3), 219–233.

    Article  Google Scholar 

  19. Karplus, R., Pulos, S. & Stage, E. (1983b). Proportional reasoning of early adolescents. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 45–90). New York, NY: Academic.

  20. Lamon, S. J. (1993). Ratio and proportion: Connecting content and childrens thinking. Journal for Research in Mathematics Education, 24(1), 41–61.

    Article  Google Scholar 

  21. Lamon, S. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 1, pp. 629–667). Charlotte, NC: Information Age.

  22. Lesh, R. & Harel, G. (2003). Problem solving, modeling, and local conceptual development. Mathematical Thinking and Learning, 5(2-3), 157–189.

    Article  Google Scholar 

  23. Lesh, R., Post, T. & Behr, M. (1988). Proportional reasoning. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 93–118). Reston, VA: National Council of Teachers of Mathematics.

  24. Lim, K. (2009). Burning the candle at just one end: Using nonproportional examples helps students determine when proportional strategies apply. Mathematics Teaching in the Middle School, 14(8), 492–500.

    Google Scholar 

  25. Lobato, J. & Ellis, A. (2010). Developing essential understanding of ratios, proportions, and proportional reasoning for teaching mathematics: Grades 6-8. Reston, VA: National Council of Teachers of Mathematics.

  26. Misailadou, C. & Williams, J. (2003). Measuring children’s proportional reasoning, the “tendency” for an additive strategy and the effect of models. In N. A. Pateman, B. J. Dougherty & J. T. Zilliox (Eds.), Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 293–300). Honolulu, HI: University of Hawaii.

  27. Modestou, M. & Gagatsis, A. (2007). Students’ improper proportional reasoning: A result of the epistemological obstacle of “linearity”. Educational Psychology, 27(1), 75–92.

    Article  Google Scholar 

  28. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author.

  29. Riley, K. R. (2010). Teachers’ understanding of proportional reasoning. In P. Brosnan, D. B. Erchick & L. Flevares (Eds.), Proceedings of the 32nd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1055–1061). Columbus, OH: The Ohio State University.

  30. Simon, M. & Blume, G. (1994). Mathematical modeling as a component of understanding ratio-as-measure: A study of prospective elementary teachers. Journal of Mathematical Behavior, 13(2), 183–197.

    Article  Google Scholar 

  31. Stemn, B. S. (2008). Building middle school students’ understanding of proportional reasoning through mathematical investigation. Education 3–13, 36(4), 383–392.

    Google Scholar 

  32. Tourniaire, F. & Pulos, S. (1985). Proportional reasoning: A review of the literature. Educational Studies in Mathematics, 16(2), 181–204.

    Article  Google Scholar 

  33. Van Dooren, W., De Bock, D., Janssens, D. & Verschaffel, L. (2007). Pupils’ overreliance on linearity: A scholastic effect? British Journal of Educational Psychology, 77(2), 307–321.

    Article  Google Scholar 

  34. Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 127–174). New York, NY: Academic.

  35. Vergnaud, G. (1988). Multiplicative structures. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in middle grades (pp. 141–161). Reston, VA: National Council of Teachers of Mathematics.

  36. Vergnaud, G. (2009). The theory of conceptual fields. Human Development, 52(2), 83–94.

    Article  Google Scholar 

  37. Yin, R. K. (1993). Applications of case study research. Newbury Park, CA: Sage.

  38. Yin, R. K. (2009). Case study research: Design and methods (Vol. 5). Thousand Oaks, CA: Sage.

Download references

Acknowledgments

Parts of this study were presented at the 2016 annual meeting of the American Educational Research Association Conference, Washington, DC, USA.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Muhammet Arican.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Arican, M. Preservice Middle and High School Mathematics Teachers’ Strategies when Solving Proportion Problems. Int J of Sci and Math Educ 16, 315–335 (2018). https://doi.org/10.1007/s10763-016-9775-1

Download citation

Keywords

  • Proportions
  • Proportional reasoning
  • Proportional relationships
  • Ratios