Learning rational numbers through an integrated approach of their different representations
DOI:
https://doi.org/10.48489/quadrante.27802Keywords:
rational numbers, basic education, representations, conversion between representationsAbstract
This study aims to understand grade 5 students’ understanding of the different rational number representations and the conversion between them, before and after a teaching experiment. The participants are four students from a class, and, for data collection, two tests were used – pre-test and post-test –, complemented with individual semi-structured interviews. The results indicate that, before the teaching experiment, students had limited knowledge of rational number symbolic representations and the conversion between them, which they did procedurally. They just showed more familiarity with the fraction representation. After the teaching experiment, these students showed knowledge about fractions, decimals and percents and started to make conversions between them with conceptual understanding. The use of models seems to have contributed to the development of these concepts.
References
Aksoy, N., & Yazlik, D. (2017). Student errors in fractions and possible causes of these errors. Journal of Educations and Training Studies, 5(11), 219-233. https://doi.org/10.11114/jets.v5i11.2679
Alkhateeb, M. (2019). Common errors in fractions and the thinking strategies that accompany them. International Journal of Instruction, 12(2), 339-416. https://doi.org/10.29333/iji.2019.12226a
Behr, M., & Post, T. (1992). Teaching rational number and decimal concepts. In T. Post (Ed.), Teaching mathematics in grades K-8: Research-based methods (2nd ed.) (pp. 201-248). Allyn and Bacon.
Bennett, A., Inglis, M., & Gilmore, C. (2019). The cost of multiple representations: Learning number symbols with abstract and concrete representations. Journal of Educational Psychology, 111(5), 847-860. https://doi.org/10.1037/edu0000318
Blomberg, J., Schukajlow, S., & Rellensmann, J. (2021). Why don’t you make a drawing? Motivations and strategy use in mathematical modelling. In M. Inprasitha, N. Changsri, & N. Boonsena (Eds.), Proceedings of the 44th Conference of the International Group for the Psychology of Mathematics Education (online) (Vol. 2, pp. 111-119). PME.
Bruner, J. (1999). Para uma teoria da educação. Relógio d’Água.
Büscher, C. (2021). Teachers’ adaptions of the percentage bar model for creating different learning opportunities. International Electronic Journal of Mathematics Education, 16(3). https://doi.org/10.29333/iejme/10942
Cadez, T., & Kolar, V. (2018). How fifth-grade pupils reason about fractions: A reliance on part-whole subconstructs. Educational Studies in Mathematics, 99, 335-357. https://doi.org/10.1007/s10649-018-9838-z
Choy, B. H. (2021). Where to put the decimal point? Noticing opportunities to learn through typical problems. In M. Inprasitha, N. Changsri, & N. Boonsena (Eds.), Proceedings of the 44th Conference of the International Group for the Psychology of Mathematics Education (online) (Vol. 2, pp. 180-188). PME.
Cobb, P., Jackson, K., & Dunlap, C. (2016). Design research: An analysis and critique. In L. D. English, &. D. Kirshner (Eds.). Handbook of international research in mathematics education (pp. 481-503).
Cramer, K., Wyberg, T., & Leavitt, S. (2009). Rational Number Project: Fraction operations & initial decimal ideas. [Companion module to RNP: Fraction Lessons for the Middle Grades]. https://documents.pub/document/rational-number-project-college-of-education-and-number-project-fraction-operations.html?page=1
Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1), 103–131. https://doi.org/10.1007/s10649-006-0400-z
Erickson, F. (1986). Qualitative methods in research on teaching. In M. C. Wittrock (Org.), Handbook of research on teaching (pp. 119-161). Macmillan.
Fosnot, C., & Dolk, M. (2002). Young mathematician at work: Constructing fractions, decimals and percents. Heinemann.
Gray, M., DeWolf, M., Bassok, M., & Holyoak, K. (2018). Dissociation between magnitude comparison and relation identification across different formats for rational numbers. Thinking & Reasoning, 24(2), 179-197. https://doi.org/10.1080/13546783.2017.1367327
Guerreiro, H., Serrazina, L., & Ponte, J. P. (2018a). Uma trajetória na aprendizagem dos números racionais através da percentagem. Educação Matemática Pesquisa, 20(1), 359-384. http://dx.doi.org/10.23925/1983-3156.2018v20i1p359-384
Guerreiro, H., Serrazina, L., & Ponte, J. P. (2018b). A percentagem na aprendizagem com compreensão dos números racionais. Zetetiké, 26(2), 354-374. https://doi.org/10.20396/zet.v26i2.8651281
Hawera, N., & Taylor, M. (2011). “Twenty percent free!” So how much does the original bar weigh? Australian Primary Mathematics Classroom, 16(4), 3-7.
Kara, F., & Incikabi, L. (2018). Sixth grade students’ skills of using multiple representations in addition and subtraction operations in fractions. International Electronic Journal of Elementary Education, 10(4), 463-474. https://doi.org/10.26822/iejee.2018438137
Lamon, S. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. Lester (Ed.), Second handbook of research on mathematics and learning (pp. 629-668). Information Age.
Lee, Y., Chen, H., & Chang, S. (2017). Learning effects of iconic representation animation teaching on the mathematics problem solving process. In Kasetsart University (Org.), Proceedings of the 10th International Conference on Ubi-media Computing and Workshops (Ubi-Media) (pp. 1-4). Curran Associates. https://doi.org/10.1109/UMEDIA.2017.8074132.
Liu, R., Ding, Y., Zong, M., & Zhang, D. (2014). Concept development of decimals in Chinese elementary students: A conceptual change approach. School Science and Mathematics, 114(7), 326-338. https://doi.org/10.1111/ssm.12085
Morais, C., Serrazina, L., & Ponte, J. P. (2018a). Mathematical reasoning fostered by (fostering) transformations of rational number representations. Acta Scientiae, 20(4), 552-570. https://doi.org/10.17648/acta.scientiae.v20iss4id3892
Morais, C., Serrazina, L., & Ponte, J. P. (2018b). Números racionais no 1.º ciclo: Compreensão de grandeza e densidade apoiada pelo uso de modelos. Quadrante, 27(1), 25-45. https://doi.org/10.48489/quadrante.22963
Moss, J., & Case, R. (1999). Developing children’s understanding of the rational numbers: a new model and an experimental curriculum. Journal for Research in Mathematics Education, 30(2), 122-147. https://doi.org/10.2307/749607
Norton, A., Wilkins, J., & Xu, C. (2018). A progression of fraction schemes common to chinese and U.S. students. Journal for Research in Mathematics Education, 49(2), 210-226. https://doi.org/10.5951/jresematheduc.49.2.0210
Payne, J. N., & Allinger, G. D. (1984). Insights into teaching percent to general mathematics students. Unpublished manuscript, Montana State University, Department of Mathematical Sciences, Bozeman.
Pérez, J. C. (1997). Números decimais: Porquê? Para quê?. Editorial Sintesis.
Ponte, J. P. (2005). Gestão curricular em Matemática. In GTI (Ed.), O professor e o desenvolvimento curricular (pp. 11-34). APM.
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 Integrations of Research (pp. 327-362). Lawrence Erlbaum Associates.
Roell, M., Viarouge, A., Houdé, O., & Borst, G. (2019). Inhibition of the whole number bias in decimal number comparison: A developmental negative priming study. Journal of Experimental Child Psychology, 117, 240-247. https://doi.org/10.1016/j.jecp.2018.08.010
Siegler, R. S., Fazio, L., Bailey, D., & Zhou, X. (2013). Fractions: The new frontier for theories of numerical development. Trends in Cognitive Sciences, 17(1), 13-19. https://doi.org/10.1016/j.tics.2012.11.004
Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114-145. https://doi.org/10.2307/749205
Tripathi, P. N. (2008). Developing mathematical understanding through multiple representations. Mathematics Teaching in the Middle School, 13(8), 438–445. https://doi.org/10.5951/MTMS.13.8.0438
Van den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. Educational Studies in Mathematics, 54(1), 9–35. https://doi.org/10.1023/B:EDUC.0000005212.03219.dc
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