# Influence of proportional number relationships on item accessibility and students’ strategies

## Abstract

Proportional reasoning is important to students’ future success in mathematics and science endeavors. More specifically, students’ fluent and flexible use of scalar and functional relationships to solve problems is critical to their ability to reason proportionally. The purpose of this study is to investigate the influence of systematically manipulating the location of an integer multiplier—to press the scalar or functional relationship—on item difficulty and student solution strategies. We administered short-answer assessment forms to 473 students in grades 6–8 (approximate ages 11–14) and analyzed the data quantitatively with the Rasch model to examine item accessibility and qualitatively to examine student solution strategies. We found that manipulating the location of the integer multiplier encouraged students to make use of different aspects of proportional relationships without decreasing item accessibility. Implications for proportional reasoning curricular materials, instruction, and assessment are addressed.

## Keywords

Proportional reasoning Assessment Task characteristics Student strategies## Notes

### Acknowledgments

This project was supported by the Idaho State Department of Education (ISDE) through the Mathematical Thinking for Instruction grant. The views expressed in the report are those of the authors and do not necessarily reflect the ISDE.

## References

- Ahl, V. A., Moore, C. F., & Dixon, J. A. (1992). Development of intuitive and numerical proportional reasoning.
*Cognitive Development, 7*(1), 81–108.CrossRefGoogle Scholar - Andrich, D., De Jong, J., & Sheridan, B. E. (1997). Diagnostic opportunities with the Rasch model for ordered response categories.
*Applications of latent trait and latent class models in the social sciences*, 59–70.Google Scholar - Behr, M. J., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 296–333). NY: McMillan.Google Scholar - Ben-Chaim, D., Fey, J. T., Fitzgerald, W. M., Benedetto, C., & Miller, J. (1998). Proportional reasoning among 7th grade students with different curricular experiences.
*Educational Studies in Mathematics, 36*(3), 247–273.CrossRefGoogle Scholar - Bond, T. G., & Fox, C. M. (2013). Applying the Rasch model: fundamental measurement in the human sciences: Psychology PressGoogle Scholar
- Boyer, T. W., Levine, S. C., & Huttenlocher, J. (2008). Development of proportional reasoning: where young children go wrong.
*Developmental Psychology, 44*(5), 1478.Google Scholar - Brahmia, S., Boudreaux, A., & Kanim, S. E. (2016). Obstacles to Mathematization in Introductory Physics.
*arXiv preprint arXiv:1601.01235*.Google Scholar - Callingham, R., & Bond, T. (2006). Research in mathematics education and Rasch measurement.
*Mathematics Education Research Journal, 18*(2), 1–10.CrossRefGoogle Scholar - Capon, N., & Kuhn, D. (1979). Logical reasoning in the supermarket: adult females’ use of a proportional reasoning strategy in an everyday context.
*Developmental Psychology, 15*(4), 450.CrossRefGoogle Scholar - Carney, M. B., & Crawford, A. (2016).
*Students’ Reasoning Around The Functional Relationship In Proportional Situations*. Paper presented at the Psychology of Mathematics Education: North America (PME-NA) chapter, Tuscon, AZGoogle Scholar - Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. (2004).
*Children’s mathematics: Cognitively Guided Instruction*: HeinemannGoogle Scholar - Cohen, A., Anat Ben, M., & Chayoth, R. (1999). Hands-on method for teaching the concept of the ratio between surface area & volume.
*The American Biology Teacher, 61*(9), 691–695. doi: 10.2307/4450805.CrossRefGoogle Scholar - Dole, S., & Shield, M. (2008). The capacity of two Australian eighth-grade textbooks for promoting proportional reasoning.
*Research in Mathematics Education, 10*(1), 19–35.Google Scholar - Fernández, C., Llinares, S., Van Dooren, W., De Bock, D., & Verschaffel, L. (2011). Effect of number structure and nature of quantities on secondary school students’ proportional reasoning.
*Studia Psychologica, 53*(1), 69–81.Google Scholar - Freudenthal, H. (1973).
*Mathematics as an educational task*. Dordrecht: Reidel.Google Scholar - Freudenthal, H. (1991).
*Revisiting mathematics education*. Dordrecht: Kluwer.Google Scholar - Gabel, D. (1984). Problem-solving skills of high school chemistry students.
*Journal of Research in Science Teaching, 21*(2), 221–233.CrossRefGoogle Scholar - Heller, P. M., Ahlgren, A., Post, T., Behr, M., & Lesh, R. (1989). Proportional reasoning: the effect of two context variables, rate type, and problem setting.
*Journal of Research in Science Teaching, 26*(3), 205–220.CrossRefGoogle Scholar - Hoyles, C., Noss, R., & Pozzi, S. (2001). Proportional reasoning in nursing practice.
*Journal for Research in Mathematics Education, 32*, 4–27.CrossRefGoogle Scholar - Jackson, K., Garrison, A., Wilson, J., Gibbons, L., & Shahan, E. (2013). Exploring relationships between setting up complex tasks and opportunities to learn in concluding whole-class discussions in middle-grades mathematics instruction.
*Journal for Research in Mathematics Education, 44*(4), 646–682.CrossRefGoogle Scholar - Johnson, H. L. (2015). Secondary students’ quantification of ratio and rate: a framework for reasoning about change in covarying quantities.
*Mathematical Thinking and Learning: An International Journal, 17*(1), 64–90.CrossRefGoogle Scholar - Karplus, R., Pulos, S., & Stage, E. K. (1983a). Early adolescents’ proportional reasoning on ‘rate’problems.
*Educational Studies in Mathematics, 14*(3), 219–233.CrossRefGoogle Scholar - Karplus, R., Pulos, S., & Stage, E. K. (1983b). Proportional reasoning of early adolescents. In R. A. Lesh & M. Landau (Eds.),
*Acquisition of mathematics concepts and processes*(pp. 45–90). London: Academic Press.Google Scholar - Lamon, S. J. (1993). Ratio and proportion: connecting content and children’s thinking.
*Journal for Research in Mathematics Education, 24*(1), 41–61.CrossRefGoogle Scholar - Lamon, S. J. (2007). Rational numbers and proportional reasoning: toward a theoretical framework for research. In F. K. Lester (Ed.),
*Second handbook of research on mathematics teaching and learning*(Vol. 1, pp. 629–667).Google Scholar - Lawton, C. A. (1993). Contextual factors affecting errors in proportional reasoning.
*Journal for Research in Mathematics Education, 24*(5), 460–466.CrossRefGoogle Scholar - Lesh, R., Post, T., & Behr, M. (1988). Proportional reasoning. In J. E. Hiebert & M. Behr (Eds.),
*Number concepts and operations in the middle grades*(Vol. 2, pp. 93–118).Google Scholar - Linacre, J. M. (2010). Winsteps Rasch Measurement (Version 3.70.0.5). Retrieved from www.winsteps.com
- Lobato, J., Ellis, A. B., & Charles, R. I. (2010).
*Developing essential understanding of ratios, proportions, and proportional reasoning for teaching mathematics in grades 6–8*. Reston: National Council of Teachers of Mathematics.Google Scholar - Long, C., Wendt, H., & Dunne, T. (2011). Applying Rasch measurement in mathematics education research: steps towards a triangulated investigation into proficiency in the multiplicative conceptual field.
*Educational Research and Evaluation: An International Journal of Theory and Practice, 17*(5), 387–407.CrossRefGoogle Scholar - Merbitz, C., Morris, J., & Grip, J. C. (1989). Ordinal scales and foundations of misinference.
*Archives of Physical Medicine and Rehabilitation, 70*(4), 308–312.Google Scholar - Modestou, M., & Gagatsis, A. (2010). Cognitive and metacognitive aspects of proportional reasoning.
*Mathematical Thinking and Learning, 12*(1), 36–53.CrossRefGoogle Scholar - Moore, K. C., & Carlson, M. P. (2012). Students images of problem contexts when solving applied problems.
*Journal of Mathematical Behavior, 31*(1), 48–59.CrossRefGoogle Scholar - Ramful, A., & Narod, F. (2014). Proportional reasoning in the learning of chemistry: levels of complexity.
*Mathematics Education Research Journal, 26*(1), 25–46. doi: 10.1007/s13394-013-0110-7.CrossRefGoogle Scholar - Saunders, W. L., & Jesunathadas, J. (1988). The effect of task content upon proportional reasoning.
*Journal of Research in Science Teaching, 25*(1), 59–67.CrossRefGoogle Scholar - Schwartz, D. L., & Moore, J. L. (1998). On the role of mathematics in explaining the material world: mental models for proportional reasoning.
*Cognitive Science, 22*(4), 471–516.CrossRefGoogle Scholar - Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective.
*Journal for Research in Mathematics Education, 26*, 114–145.CrossRefGoogle Scholar - Simon, M. (2006). Key developmental understandings in mathematics: a direction for investigating and establishing learning goals.
*Mathematical Thinking and Learning, 8*(4), 359–371.CrossRefGoogle Scholar - Simon, M. A., & Placa, N. (2012). Reasoning about intensive quantities in whole-number multiplication? A possible basis for ratio understanding.
*For the Learning of Mathematics, 32*(2), 35–41.Google Scholar - Simon, M. A., & Tzur, R. (2004). Explicating the role of mathematical tasks in conceptual learning: an elaboration of the hypothetical learning trajectory.
*Mathematical Thinking and Learning, 6*(2), 91–104.CrossRefGoogle Scholar - Smith, E. V., Jr. (2001). Evidence for the reliability of measures and validity of measure interpretation: a Rasch measurement perspective.
*Journal of Applied Measurement, 2*, 281–311.Google Scholar - Stein, M. K., Smith, M. S., Henningsen, M., & Silver, E. A. (2000).
*Implementing standards-based mathematics instruction: A casebook for professional development*: Teachers College PressGoogle Scholar - Steinthorsdottir, O. B., & Sriraman, B. (2009). Icelandic 5th-grade girls’ developmental trajectories in proportional reasoning.
*Mathematics Education Research Journal, 21*(1), 6–30.CrossRefGoogle Scholar - Tjoe, H., & de la Torre, J. (2014). On recognizing proportionality: does the ability to solve missing value proportional problems presuppose the conception of proportional reasoning?
*The Journal of Mathematical Behavior, 33*, 1–7.CrossRefGoogle Scholar - Tourniaire, F., & Pulos, S. (1985). Proportional reasoning: a review of the literature.
*Educational Studies in Mathematics, 16*(2), 181–204.CrossRefGoogle Scholar - Treffers, A. (1987).
*Three dimensions: a model of goal and theory description in mathematics instruction—the Wiskobas Project*. Dordrecht: Reidel.CrossRefGoogle Scholar - 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 - Wilson, M. (2004).
*Constructing measures: an item response modeling approach*: RoutledgeGoogle Scholar - Wolfe, E. W., & Smith, E. V., Jr. (2006a). Instrument development tools and activities for measure validation using Rasch models: part I-instrument development tools.
*Journal of Applied Measurement, 8*(1), 97–123.Google Scholar - Wolfe, E. W., & Smith, E. V., Jr. (2006b). Instrument development tools and activities for measure validation using Rasch models: part II—validation activities.
*Journal of Applied Measurement, 8*(2), 204–234.Google Scholar - Wright, B. D., & Linacre, J. M. (1989). Observations are always ordinal; measurements, however, must be interval.
*Archives of Physical Medicine and Rehabilitation, 70*(12), 857–860.Google Scholar