Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction
 Doug M. Clarke,
 Anne Roche
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
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.
Inside
Within this Article
 Background literature on relative size of fractions
 Method
 Results
 Discussion and implications for further research
 References
 References
Other actions
 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.
 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.
 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. CrossRef
 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. CrossRef
 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.
 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: PME
 Clarke, D. M., Mitchell, A., & Roche, A. (2005). Student onetoone 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.
 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.
 Kilpatrick, J., Swafford, J., & Findell, B. (Eds.) (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
 Lamon, S. (1999). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. Mahwah, NJ: Lawrence Erlbaum.
 Litwiller, B., & Bright, G. (2002). Making sense of fractions, ratios, and proportions (Yearbook of the National Council of Teachers of Mathematics). Reston, VA: NCTM.
 Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ knowledge of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum.
 Mamede, E., Nunes, T., & Bryant, P. (2005). The equivalence of ordering of fractions in partwhole 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. 281288). Melbourne: PME.
 McDonough, A., Clarke, B. A., & Clarke, D. M. (2002). Understanding assessing and developing young children’s mathematical thinking: The power of the onetoone interview for preservice teachers in providing insights into appropriate pedagogical practices. International Journal of Education Research, 37, 107–112. CrossRef
 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.
 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. CrossRef
 Noelting, G. (1980). The development of proportional reasoning and the ratio concept: Part 1—Differentiation of stages. Educational Studies in Mathematics, 11, 217–253. CrossRef
 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.
 Post, T., Behr, M. J., & Lesh, R. (1986). Researchbased observations about children’s learning of rational number concepts. Focus on Learning Problems in Mathematics, 8(1), 39–48.
 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.
 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.
 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.
 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. CrossRef
 Riddle, M., & Rodzwell, B. (2000). Fractions: What happens between kindergarten and the army? Teaching Children Mathematics, 7(4), 202–206.
 Schorr, R. Y. (2001). A study of the use of clinical interviewing techniques with prospective teachers. In M. van den HeuvelPanhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education (pp. 153–160). Utrecht, The Netherlands: PME.
 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.
 Streefland, L. (1991). Fractions in realistic mathematics education: A paradigm of developmental research. Dordrecht, The Netherlands: Kluwer Academic Publications.
 Title
 Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction
 Journal

Educational Studies in Mathematics
Volume 72, Issue 1 , pp 127138
 Cover Date
 20090901
 DOI
 10.1007/s1064900991989
 Print ISSN
 00131954
 Online ISSN
 15730816
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Fractions
 Strategies
 Assessment tasks
 Onetoone interviews
 Understanding
 Misconceptions
 Industry Sectors
 Authors

 Doug M. Clarke ^{(1)}
 Anne Roche ^{(1)}
 Author Affiliations

 1. Australian Catholic University, Melbourne, Australia