# Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction

- 939 Downloads
- 23 Citations

## Abstract

As part of individual interviews incorporating whole number and rational number tasks, 323 grade 6 children in Victoria, Australia were asked to nominate the larger of two fractions for eight pairs, giving reasons for their choice. All tasks were expected to be undertaken mentally. The relative difficulty of the pairs was found to be close to that predicted, with the exception of fractions with the same numerators and different denominators, which proved surprisingly difficult. Students who demonstrated the greatest success were likely to use benchmark (transitive) and residual thinking. It is hypothesised that the methods of these successful students could form the basis of instructional approaches which may yield the kind of connected understanding promoted in various curriculum documents and required for the development of proportional reasoning in later years.

### Keywords

Fractions Strategies Assessment tasks One-to-one interviews Understanding Misconceptions### References

- Behr, M. J., Lesh, R., Post, T. R., & Silver, E. A. (1983). Rational number concepts. In M. Landau (Ed.),
*Acquisition of mathematics concepts and processes*(pp. 91–126). Hillsdale, NJ: Erlbaum.Google Scholar - Behr, M. J., Post, T. R., & Wachsmuth, I. (1986). Estimation and children’s concept of rational number size. In H. Schoen & M. Zweng (Eds.),
*Estimation and mental computation*(1986 National Council of Teachers of Mathematics Yearbook pp. 101–111). Reston, VA: NCTM.Google Scholar - Behr, M. J., Wachsmuth, I., & Post, T. R. (1985). Construct a sum: A measure of children’s understanding of fraction size.
*Journal for Research in Mathematics Education*,*16*(2), 120–131.CrossRefGoogle Scholar - Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. (1984). Order and equivalence of rational numbers: A clinical teaching experiment.
*Journal for Research in Mathematics Education*,*15*(5), 323–341.CrossRefGoogle Scholar - Carpenter, T. P., Kepner, H., Corbitt, M. K., Lindquist, M. M., & Reys, R. E. (1980). Results and implications of the Second NAEP Mathematics Assessments: Elementary school.
*Arithmetic Teacher*,*2*(8), 10–13.Google Scholar - Clarke, B. A., Sullivan, P., & McDonough, A. (2002). Measuring and describing learning: The Early Numeracy Research Project. In A. Cockburn & E. Nardi (Eds.),
*Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education*(pp. 181–185). Norwich, UK: PMEGoogle Scholar - Clarke, D. M., Mitchell, A., & Roche, A. (2005). Student one-to-one assessment interviews in mathematics: A powerful tool for teachers. In J. Mousley, L. Bragg & C. Campbell (Eds.),
*Mathematics: Celebrating achievement*(Proceedings of the 100th Annual Conference of the Mathematical Association of Victoria pp. 66–80). Melbourne: MAV.Google Scholar - Kieren, T. (1976). On the mathematical, cognitive and instructional foundations of the rational numbers. In R. A. Lesh (Ed.),
*Number and measurement: Papers from a research workshop*(pp. 101–144). Athens, GA: ERIC/SMEAC.Google Scholar - Kilpatrick, J., Swafford, J., & Findell, B. (Eds.) (2001).
*Adding it up: Helping children learn mathematics*. Washington, DC: National Academy Press.Google Scholar - Lamon, S. (1999).
*Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers*. Mahwah, NJ: Lawrence Erlbaum.Google Scholar - Litwiller, B., & Bright, G. (2002).
*Making sense of fractions, ratios, and proportions*(Yearbook of the National Council of Teachers of Mathematics). Reston, VA: NCTM.Google Scholar - Ma, L. (1999).
*Knowing and teaching elementary mathematics: Teachers’ knowledge of fundamental mathematics in China and the United States*. Mahwah, NJ: Lawrence Erlbaum.Google Scholar - Mamede, E., Nunes, T., & Bryant, P. (2005). The equivalence of ordering of fractions in part-whole and quotient situations. In H. L. Chick & J. L. Vincent (Eds.),
*Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education*(pp. 281-288). Melbourne: PME.Google Scholar - McDonough, A., Clarke, B. A., & Clarke, D. M. (2002). Understanding assessing and developing young children’s mathematical thinking: The power of the one-to-one interview for preservice teachers in providing insights into appropriate pedagogical practices.
*International Journal of Education Research*,*37*, 107–112.CrossRefGoogle Scholar - Mitchell, A., & Clarke, D. M. (2004). When is three quarters not three quarters? Listening for conceptual understanding in children’s explanations in a fractions interview. In I. Putt, R. Farragher & M. McLean (Eds.),
*Mathematics education for the third millenium: Towards 2010*(Proceedings of the 27th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 367–373). Townsville, Queensland: MERGA.Google Scholar - Moss, J., & Case, R. (1999). Developing children’s understanding of the rational numbers: A new model and experimental curriculum.
*Journal for Research in Mathematics Education*,*30*(2), 122–147.CrossRefGoogle Scholar - Noelting, G. (1980). The development of proportional reasoning and the ratio concept: Part 1—Differentiation of stages.
*Educational Studies in Mathematics*,*11*, 217–253.CrossRefGoogle Scholar - Pearn, C., & Stephens, M. (2004). Why you have to probe to discover what year 8 students really think about fractions. In I. Putt, R. Faragher & M. McLean (Eds.),
*Mathematics education for the third millenium: Towards 2010*(Proceedings of the 27th annual conference of the Mathematics Education Research Group of Australasia, pp. 430–437). Sydney, Australia: MERGA.Google Scholar - Post, T., Behr, M. J., & Lesh, R. (1986). Research-based observations about children’s learning of rational number concepts.
*Focus on Learning Problems in Mathematics*,*8*(1), 39–48.Google Scholar - Post, T., & Cramer, K. (2002). Children’s strategies in ordering rational numbers. In D. Chambers (Ed.),
*Putting research into practice in the elementary grades*(Readings from Journals of the National Council of Teachers of Mathematics, pp. 141–144). Reston, VA: NCTM.Google Scholar - Post, T., Cramer, K., Behr, M., Lesh, R., & Harel, G. (1993). Curriculum implications of research on the learning, teaching and assessing of rational number concepts. In T. Carpenter, E. Fennema & T. Romberg (Eds.),
*Rational numbers: An integration of research*(pp. 327–361). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar - Post, T., Harel, G., Behr, M., & Lesh, R. (1991). Intermediate teachers’ knowledge of rational number concepts. In E. Fennema, T. P. Carpenter & S. J. Lamon (Eds.),
*Integrating research on teaching and learning mathematics*(pp. 177–198). NY: State University of New York Press.Google Scholar - Post, T., Wachsmuth, I., Lesh, R., & Behr, M. (1985). Order and equivalence of rational numbers: A cognitive analysis.
*Journal for Research in Mathematics Education*,*16*(1), 18–36.CrossRefGoogle Scholar - Riddle, M., & Rodzwell, B. (2000). Fractions: What happens between kindergarten and the army?
*Teaching Children Mathematics*,*7*(4), 202–206.Google Scholar - Schorr, R. Y. (2001). A study of the use of clinical interviewing techniques with prospective teachers. In M. van den Heuvel-Panhuizen (Ed.),
*Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education*(pp. 153–160). Utrecht, The Netherlands: PME.Google Scholar - Sowder, J. T. (1988). Mental computation and number comparisons: The role in development of number sense and computational estimation. In J. Hiebert & M. Behr (Eds.),
*Number concepts and operations in the middle grades*(pp. 182–197). Reston, VA: Lawrence Erlbaum and National Council of Teachers of Mathematics.Google Scholar - Streefland, L. (1991).
*Fractions in realistic mathematics education: A paradigm of developmental research*. Dordrecht, The Netherlands: Kluwer Academic Publications.Google Scholar