Profiles in understanding the density of rational numbers among primary and secondary school students
DOI:
https://doi.org/10.35763/aiem22.4034Keywords:
Rational numbers, Density, Discreteness, Learner profiles, FractionsAbstract
The present cross-sectional study investigated 953 fifth to tenth grade students’ understanding 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 Analysis revealed 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 profiles 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 natural 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.
Downloads
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
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Juan Manuel González Forte, Ceneida Fern´ández, Jo Van Hoof, Wim Van Dooren
This work is licensed under a Creative Commons Attribution 4.0 International License.
The articles published in this journal are under a license Creative Commons: By 4.0 España from number 21 (2022).
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and keep the acknowledgement of authorship.
- The texts published in this journal are – unless indicated otherwise – covered by the Creative Commons Attribution 4.0 international licence. You may copy, distribute, transmit and adapt the work, provided you attribute it (authorship, journal name, publisher) in the manner specified by the author(s) or licensor(s). The full text of the licence can be consulted here: http://creativecommons.org/licenses/by-nc/4.0.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
