Preservice Mathematics Teachers’ Understanding of and Abilities to Differentiate Proportional Relationships from Nonproportional Relationships

  • Muhammet AricanEmail author


This study investigated preservice middle school mathematics teachers’ (PSTs) understanding of proportional and nonproportional relationships and their abilities to differentiate these relationships from each other. The PSTs’ abilities to interpret and represent proportional and nonproportional situations and solution strategies were also investigated. Forty PSTs who attended a mathematics education course on fractions, ratios, and proportions participated in the study. The data included the PSTs’ written responses to four open-ended problems and semi-structured interviews conducted with six of the PSTs. The analysis of the data showed that the PSTs mostly attended to the simultaneous increases or decreases and constancy of the rate of change when determining relationships. The PSTs’ over attention to these specific features constrained their understanding of the proportional and nonproportional relationships. Therefore, they had difficulty differentiating proportional relationships from nonproportional relationships even after instruction on these relationships. Moreover, some PSTs still identified nonproportional relationships as proportional after determining correct additive relationships. In addition, the PSTs had difficulty representing and interpreting proportional and nonproportional relationships and relied on cross-multiplication and across-multiplication strategies when solving the given problems.


Preservice teachers Proportional reasoning Proportional and nonproportional relationships Ratios and proportions 



Parts of this paper were presented at the 2018 annual meeting of the 27th International Conference on Educational Sciences, Antalya, Turkey.

Compliance with Ethical Standards

Conflict of Interest

The author declares there is no conflict of interest.


  1. Arican, M. (2015). Exploring preservice middle and high school mathematics teachers’ understanding of directly and inversely proportional relationships (Unpublished doctoral dissertation). Athens, GA: University of GeorgiaGoogle Scholar
  2. Arican, M. (2018). Preservice middle and high school mathematics teachers’ strategies when solving proportion problems. International Journal of Science and Mathematics Education,  16(2), 315–335.Google Scholar
  3. Atabas, S. & Oner, D. (2017). An examination of Turkish middle school students’ proportional reasoning. Boğaziçi University Journal of Education, 33(1), 63–85.Google Scholar
  4. Beckmann, S. (2013). Mathematics for elementary teachers. Boston: Pearson.Google Scholar
  5. Beckmann, S. & Izsák, A. (2015). Two perspectives on proportional relationships: Extending complementary origins of multiplication in terms of quantities. Journal for Research in Mathematics Education, 46(1), 17–38.CrossRefGoogle Scholar
  6. Ben-Chaim, D., Keret, Y. & Ilany, B. (2007). Designing and implementing authentic investigative proportional reasoning tasks: The impact on preservice mathematics teachers’ content and pedagogical knowledge and attitudes. Journal of Mathematics Teacher Education, 10, 333–340.Google Scholar
  7. Boyer, T. W., Levine, S. C. & Huttenlocher, J. (2008). Development of proportional reasoning: Where young children go wrong. Developmental Psychology, 44, 1478–1490.Google Scholar
  8. Clark, H. J. (2008). Investigating students’ proportional reasoning strategies (Master’s thesis). Available from ProQuest Dissertations and Theses database. (UMI No. 1453188).Google Scholar
  9. Common Core State Standards Initiative (2010). The common core state standards for mathematics. Washington, D.C.: Author.Google Scholar
  10. Cramer, K. & Post, T. (1993). Making connections: A case for proportionality. Arithmetic Teacher, 60(6), 342–346.Google Scholar
  11. 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.Google Scholar
  12. Degrande, T., Van Hoof, J., Verschaffel, L. & Van Dooren, W. (2017). Open word problems: Taking the additive or the multiplicative road? ZDM, 50(1–2), 91–102.Google Scholar
  13. Hsieh, H. F. & Shannon, S. E. (2005). Three approaches to qualitative content analysis. Qualitative Health Research, 15(9), 1277–1288.Google Scholar
  14. Hull, L. S. H. (2000). Teachers' mathematical understanding of proportionality: Links to curriculum, professional development, and support (Unpublished doctoral dissertation). Austin: The University of Texas at AustinGoogle Scholar
  15. Izsák, A. & Jacobson, E. (2017). Preservice teachers’ reasoning about relationships that are and are not proportional: A knowledge-in-pieces account. Journal for Research in Mathematics Education, 48(3), 300–339.Google Scholar
  16. Jeong, Y., Levine, S. & Huttenlocher, J. (2007). The development of proportional reasoning: Effect of continuous vs. discrete quantities. Journal of Cognition and Development, 8, 237–256.Google Scholar
  17. Kilpatrick, J., Swafford, J. & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.Google Scholar
  18. 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, France: Information Age Publishing.Google Scholar
  19. 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.Google Scholar
  20. 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
  21. Livy, S. & Herbert, S. (2013). Second-year pre-service teachers’ responses to proportional reasoning test items. Australian Journal of Teacher Education, 38(11), 17–32.Google Scholar
  22. Lobato, J. & Ellis, A. (2010). Developing essential understanding of ratios, proportions, and proportional reasoning for teaching mathematics: Grades 6–8 (pp. 20191–21502). Reston: National Council of Teachers of Mathematics.Google Scholar
  23. 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.Google Scholar
  24. Modestou, M. & Gagatsis, A. (2007). Students’ improper proportional reasoning: A result of the epistemological obstacle of “linearity”. Educational Psychology, 27(1), 75–92.Google Scholar
  25. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author.Google Scholar
  26. Patton, M. Q. (2005). Qualitative research. Hoboken, NJ: Wiley.Google Scholar
  27. 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.Google Scholar
  28. Siemon, D., Breed, M. & Virgona, J. (2005). From additive to multiplicative thinking—The big challenge of the middle years. In J. Mousley, L. Bragg & C. Campbell (Eds.), Proceedings of the 42nd Conference of the Mathematical Association of Victoria. Bundoora, Australia.Google Scholar
  29. 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, 183–197.Google Scholar
  30. Van Dooren, W., De Bock, D., Janssens, D. & Verschaffel, L. (2008). The linear imperative: An inventory and conceptual analysis of students’ overuse of linearity. Journal for Research in Mathematics Education, 39(3), 311–342.Google Scholar
  31. Van Dooren, W., De Bock, D., Hessels, A., Janssens, D. & Verschaffel, L. (2005). Not everything is proportional: Effects of age and problem type on propensities for overgeneralization. Cognition and Instruction, 23(1), 57–86.Google Scholar
  32. 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.Google Scholar
  33. Van Dooren, W., De Bock, D. & Verschaffel, L. (2010). From addition to multiplication… and back: The development of students’ additive and multiplicative reasoning skills. Cognition and Instruction, 28, 360–381.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.Google Scholar
  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.Google Scholar
  36. Vergnaud, G. (2009). The theory of conceptual fields. Human Development, 52(2), 83–94.Google Scholar
  37. Wagner, J. (2006). Transfer in pieces. Cognition and Instruction, 24(1), 1–71.CrossRefGoogle Scholar

Copyright information

© Ministry of Science and Technology, Taiwan 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Science EducationAhi Evran UniversityKirşehirTurkey

Personalised recommendations