Profiles in understanding the density of rational numbers among primary and secondary school students

  1. Juan Manuel González-Forte
  2. Ceneida Fernández
  3. Jo Van Hoof
  4. Wim Van Dooren
Journal:
Avances de investigación en educación matemática: AIEM

ISSN: 2254-4313

Year of publication: 2022

Issue: 22

Pages: 48-70

Type: Article

DIALNET GOOGLE SCHOLAR lock_openDialnet editor

More publications in: Avances de investigación en educación matemática: AIEM

Abstract

The present cross-sectional study investigated 953 fifth to tenth grade students’ understand-ing of the dense structure of rational numbers. After an inductive analysis, coding the answers based on three types of items on density, a TwoStep Cluster Analysisrevealed different intermediate profiles in the understanding of density along grades. The analysis highlighted qualitatively different ways of thinking: i) the idea of consecutiveness, ii) the idea of a finite number of numbers, and iii) the idea that between fractions, there are only fractions, and between decimals, there are only decimals. Furthermore, our pro-files showed differences regarding rational number representation since students first recognised the dense nature of decimal numbers and then of fractions. Learners, however, were still found to have a nat-ural number-based idea of the rational number structure by the end of secondary school, especially when they had to write a number between two pseudo-consecutive rational numbers.

Bibliographic References

  • Broitman, C., Itzcovich, H., & Quaranta, M. E. (2003). La enseñanza de los números deci-males: el análisis del valor posicional y una aproximación a la densidad. Revista La-tinoamericana de Investigación en Matemática Educativa, 6(1), 5-26.
  • Carpenter, T. P., Fennema, E., & Romberg, T. A. (Eds.) (1993). Rational numbers: An inte-gration of research. Erlbaum.
  • DeWolf, M., & Vosniadou, S. (2015). The representation of fraction magnitudes and the whole number bias reconsidered. Learning and Instruction, 37, 39-49. https://doi.org/10.1016/j.learninstruc.2014.07.002
  • Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16, 3-17. https://doi.org/10.2307/748969
  • González-Forte, J. M., Fernández, C., & Llinares, S. (2018). La influencia del conocimiento de los números naturales en la comprensión de los números racionales. In L. J. Rodríguez-Muñiz, L. Muñiz-Rodríguez, A. Aguilar-González, P. Alonso, F. J. García García, & A. Bruno (Eds.), Investigación en Educación Matemática XXII (pp. 241-250). SEIEM.
  • González-Forte, J. M., Fernández, C., Van Hoof, J., & Van Dooren, W. (2020). Various ways to determine rational number size: an exploration across primary and secondary education. European Journal of Psychology of Education, 35(3), 549-565. https://doi:10.1007/s10212-019-00440-w
  • González-Forte, J. M., Fernández, C., Van Hoof, J., & Van Dooren, W. (2021). Profiles in understanding operations with rational numbers. Mathematical Thinking and Learning, 24(3), 230-247. https://doi.org/10.1080/10986065.2021.1882287
  • Khoury, H. A., & Zazkis, R. (1994). On fractions and non-standard representations: Pre-service teachers’ concepts. Educational Studies in Mathematics, 27(2), 191–204. https://doi.org/10.1007/BF01278921
  • Kieren, T. E. (1993). Rational and fractional numbers: From quotient fields to recursive understanding. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 49-84). Lawrence Erlbaum Associates, Inc.
  • Markovits, Z., & Sowder, J. T. (1991). Students' understanding of the relationship be-tween fractions and decimals. Focus on Learning Problems in Mathematics, 13(1), 3-11.
  • McMullen, J., Laakkonen, E., Hannula-Sormunen, M., & Lehtinen, E. (2015). Modeling the developmental trajectories of rational number concept(s). Learning and In-struction, 37, 14-20. https://doi.org/10.1016/j.learninstruc.2013.12.004
  • McMullen, J., & Van Hoof, J. (2020). The role of rational number density knowledge in mathematical development. Learning and Instruction, 65. https://doi.org/10.1016/j.learninstruc.2019.101228
  • Merenluoto, K., & Lehtinen, E. (2002). Conceptual change in mathematics: Understand-ing the real numbers. In M. Limon, & L. Mason (Eds.), Reconsidering conceptual change: Issues in theory and practice (pp. 233-258). Kluwer Academic Publishers.
  • Moss, J. (2005). Pipes, tubs, and beakers: New approaches to teaching the rational-number system. In M. S. Donovan, & Bransford, J. D. (Eds.), How students learn: History, Math, and Science in the classroom (pp. 309-349). National Academies Press.
  • Ni, Y., & Zhou, Y. D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27-52. https://doi.org/10.1207/s15326985ep4001_3
  • Tirosh, D., Fischbein, E., Graeber, A. O., & Wilson, J. W. (1999). Prospective elementary teachers’ conceptions of rational numbers. http://jwilson.coe.uga.edu/Texts.Folder/Tirosh/Pros.El.Tchrs.html
  • Vamvakoussi, X., Christou, K. P., Mertens, L., & Van Dooren, W. (2011). What fills the gap between discrete and dense? Greek and Flemish students’ understanding of densi-ty. Learning and Instruction, 21(5), 676-685. https://doi.org/10.1016/j.learninstruc.2011.03.005
  • Vamvakoussi, X., Van Dooren, W., & Verschaffel, L. (2012). Naturally biased? In search for reaction time evidence for a natural number bias in adults. The Journal of Mathe-matical Behavior, 31(3), 344-355. https://doi.org/10.1016/j.jmathb.2012.02.001
  • Vamvakoussi, X., & Vosniadou, S. (2004). Understanding the structure of the set of ra-tional numbers: A conceptual change approach. Learning and Instruction, 14(5), 453-467.
  • Vamvakoussi, X., & Vosniadou, S. (2007). How many numbers are there in a rational numbers interval? Constraints, synthetic models and the effect of the number line. In S. Vosniadou, A. Baltas, & X. Vamvakoussi (Eds.), Reframing the conceptual change approach in learning and instruction (pp. 265-282). Elsevier.
  • Vamvakoussi, X., & Vosniadou, S. (2010). How many decimals are there between two fractions? Aspects of secondary school students’ understanding of rational num-bers and their notation. Cognition and Instruction, 28(2), 181-209. https://doi.org/10.1080/07370001003676603
  • Van Dooren, W., Lehtinen, E., & Verschaffel, L. (2015). Unraveling the gap between natu-ral and rational numbers. Learning and Instruction, 37, 1-4. https://doi.org/10.1016/j.learninstruc.2015.01.001
  • Van Hoof, J., Verschaffel, L., & Van Dooren, W. (2015). Inappropriately applying natural number properties in rational number tasks: Characterizing the development of the natural number bias through primary and secondary education. Educational Studies in Mathematics, 90(1), 39-56. https://doi.org/10.1007/s10649-015-9613-3
Data source: Dialnet