Razonamientos de estudiantes en tareas de comparación, ordenación y representación de fracciones y números decimales

Juan Manuel González-Forte; Ceneida Fernández

doi:10.30827/pna.v18i2.27218

Autores/as

Juan Manuel González-Forte Universidad de Alicante https://orcid.org/0000-0003-4020-7869
Ceneida Fernández Universidad de Alicante https://orcid.org/0000-0002-4791-9247

DOI:

https://doi.org/10.30827/pna.v18i2.27218

Palabras clave:

Educación primaria y secundaria, Fracciones, Números decimales, Sesgo del número natural

Resumen

Se ha llevado a cabo un estudio transversal desde 5º de Educación Primaria hasta 4º de Educación Secundaria (ESO), en el que se analiza los niveles de éxito y razonamientos de los estudiantes en tareas de comparación de fracciones, comparación y ordenación de números decimales, y de representación en la recta numérica de fracciones y números decimales. Nuestro estudio aporta evidencias del uso de diferentes razonamientos incorrectos inferidos en estudios cuantitativos y, además, aporta información sobre su evolución. Los resultados muestran que, aunque disminuyó el razonamiento centrado en el uso del conocimiento del número natural, aparecen otros razonamientos incorrectos en este tipo de actividades.

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Biografía del autor/a

Juan Manuel González-Forte, Universidad de Alicante

Doctor en Didáctica de la Matemática por la Universidad de Alicante (2020), su investigación está vinculada al desarrollo de la comprensión de los números racionales en Educación Primaria y Secundaria.

Ceneida Fernández, Universidad de Alicante

Doctora en Didáctica de la Matemática por la Universidad de Alicante (2010), su investigación está vinculada a la formación de maestros y profesores de matemáticas y a la investigación en la comprensión de los números racionales, conceptos de razón y proporción y desarrollo del razonamiento proporcional en Educación Primaria y Secundaria.

Citas

Barraza, P., Avaria, R. y Leiva, I. (2017). The role of attentional networks in the access to the numerical magnitude of fractions in adults/El rol de las redes atencionales en el acceso a la magnitud numérica de fracciones en adultos. Estudios de Psicología, 38(2), 495-522. https://doi.org/10.1080/02109395.2017.1295575 DOI: https://doi.org/10.1080/02109395.2017.1295575

Behr, M. J., Lesh, R., Post, T. y Silver E. (1983). Rational number concepts. En R. Lesh y M. Landau (Eds.), Acquisition of mathematics concepts and processes, (pp. 91-125). Academic Press.

Behr, M., Wachsmuth, I., Post T. y Lesh R. (1984). Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 15(5), 323-341. https://doi.org/10.2307/748423 DOI: https://doi.org/10.5951/jresematheduc.15.5.0323

Braithwaite, D. W. y Siegler, R. S. (2018). Developmental changes in the whole number bias. Developmental Science, 21(2). https://doi.org/10.1111/desc.12541 DOI: https://doi.org/10.1111/desc.12541

Carpenter, T. P., Fennema, E. y Romberg, T. A. (Eds.) (1993). Studies in mathematical thinking and learning. Rational numbers: An integration of research. Erlbaum.

Clarke, D. M. y Roche, A. (2009). Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction. Educational Studies in Mathematics, 72(1), 127-138. https://doi.org/10.1007/s10649-009-9198-9 DOI: https://doi.org/10.1007/s10649-009-9198-9

Cramer, K., Post, T. y delMas, R. (2002). Initial fraction learning by fourth-and fifth-grade students: A comparison of the effects of using commercial curricula with the effects of using the rational number project curriculum. Journal for Research in Mathematics Education, 33(2), 111-144. https://doi.org/10.2307/749646 DOI: https://doi.org/10.2307/749646

DeWolf, M. y Vosniadou, S. (2011). The whole number bias in fraction magnitude comparisons with adults. En L. Carlson, C. Hoelscher y T. F. Shipley (Eds.), Proceedings of the 33rd Annual Conference of the Cognitive Science Society (pp. 1751-1756). Cognitive Science Society.

DeWolf, M. y 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 DOI: https://doi.org/10.1016/j.learninstruc.2014.07.002

Durkin, K. y Rittle-Johnson, B. (2015). Diagnosing misconceptions: Revealing changing decimal fraction knowledge. Learning and Instruction, 37, 21-29. https://doi.org/10.1016/j.learninstruc.2014.08.003 DOI: https://doi.org/10.1016/j.learninstruc.2014.08.003

Fazio, L. K., DeWolf, M. y Siegler, R. S. (2016). Strategy use and strategy choice in fraction magnitude comparison. Journal of Experimental Psychology: Learning, Memory, and Cognition, 42(1), 1-16. https://doi.org/10.1037/xlm0000153 DOI: https://doi.org/10.1037/xlm0000153

Fischbein, E., Deri, M., Nello, M. S. y 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 DOI: https://doi.org/10.5951/jresematheduc.16.1.0003

Gómez, D. M. y Dartnell, P. (2015). Is there a natural number bias when comparing fractions without common components? A meta-analysis. En K. Beswick, T. Muir y J. Wells (Eds.), Proceedings of the 39th Conference of the International Group for the Psychology of Mathematics Education (vol. 3, pp. 1-8). PME.

Gómez, D. M. y Dartnell, P. (2019). Middle schoolers’ biases and strategies in a fraction comparison task. International Journal of Science and Mathematics Education, 17(6), 1233-1250. https://doi.org/10.1007/s10763-018-9913-z DOI: https://doi.org/10.1007/s10763-018-9913-z

Gómez, D. M., Silva, E. y Dartnell, P. (2017). Mind the gap: congruency and gap effects in engineering students’ fraction comparison. En B. Kaur, W. K. Ho, T. L. Toh y B. H. Choy (Eds.), Proceedings of the 41st Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 353-360). PME.

González-Forte, J. M., Fernández, C. y Llinares, S. (2019). El fenómeno natural number bias: un estudio sobre los razonamientos de los estudiantes en la multiplicación de números racionales. Quadrante, 28(2), 32-52.

González-Forte, J. M., Fernández, C., Van Hoof, J. y 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 DOI: https://doi.org/10.1007/s10212-019-00440-w

González-Forte, J. M., Fernández, C., Van Hoof, J. y Van Dooren, W. (2022). Profiles in understanding the density of rational numbers among primary and secondary school students. AIEM - Avances de investigación en educación matemática, 22, 47-70. https://doi.org/10.35763/aiem22.4034 DOI: https://doi.org/10.35763/aiem22.4034

González-Forte, J. M., Fernández, C., Van Hoof, J. y Van Dooren, W. (2023). Incorrect ways of thinking about the size of fractions. International Journal of Science and Mathematics Education, 21, 2005-2025. https://doi.org/10.1007/s10763-022-10338-7 DOI: https://doi.org/10.1007/s10763-022-10338-7

Hiebert, J. y Wearne, D. (1986). Procedures over concepts: The acquisition of decimal number knowledge. En J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 199-223). Lawrence Erlbaum Associates, Inc.

Khoury, H. A. y Zazkis, R. (1994). On fractions and non-standard representations: Preservice teachers’ concepts. Educational Studies in Mathematics, 27(2), 191–204. https://doi.org/10.1007/BF01278921 DOI: https://doi.org/10.1007/BF01278921

Kieren, T. E. (1992). Rational and fractional numbers as mathematical and personal knowledge: Implications for curriculum and instruction. En G. Leinhardt, R. Putnam y R. Hattrup (Eds.), Analysis of arithmetic for mathematics teaching (pp. 323–372). Lawrence Erlbaum. DOI: https://doi.org/10.4324/9781315044606-6

McMullen, J. y 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 DOI: https://doi.org/10.1016/j.learninstruc.2019.101228

Merenluoto, K. y Lehtinen, E. (2002). Conceptual change in mathematics: Understanding the real numbers. En M. Limon y L. Mason (Eds.), Reconsidering conceptual change: Issues in theory and practice (pp. 233-258). Kluwer Academic Publishers.

Mitchell, A. y Horne, M. (2010). Gap thinking in fraction pair comparisons is not whole number thinking: Is this what early equivalence thinking sounds like? En L. Sparrow, B. Kissane y C. Hurst (Eds.), Shaping the future of mathematics education (Vol. 2, pp. 414–421). MERGA.

Moss, J. (2005). Pipes, tubs, and beakers: New approaches to teaching the rational-number system. En Donovan, M. S. y Bransford, J. D. (Eds.), How students learn: History, Math, and Science in the classroom (pp. 309-349). National Academies Press.

Ni, Y. 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 DOI: https://doi.org/10.1207/s15326985ep4001_3

Nunes, T. y Bryant, P. (2008). Rational numbers and intensive quantities: challenges and insights to pupils’ implicit knowledge. Anales de Psicología/Annals of Psychology, 24(2), 262-270.

Obersteiner, A., Alibali, M. W. y Marupudi, V. (2020). Complex fraction comparisons and the natural number bias: the role of benchmarks. Learning and Instruction, 67. https://doi.org/10.1016/j.learninstruc.2020.101307 DOI: https://doi.org/10.1016/j.learninstruc.2020.101307

Pearn, C. y Stephens, M. (2004). Why you have to probe to discover what year 8 students really think about fractions. En I. Putt, R. Faragher y 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). MERGA.

Resnick, I., Rinne, L., Barbieri, C. y Jordan, N. C. (2019). Children’s reasoning about decimals and its relation to fraction learning and mathematics achievement. Journal of Educational Psychology, 111(4), 604-618. https://doi.org/10.1037/edu0000309 DOI: https://doi.org/10.1037/edu0000309

Resnick, L. B., Nesher, P., Leonard, F., Magone, M., Omanson, S. y Peled, I. (1989). Conceptual bases of arithmetic errors: The case of decimal fractions. Journal for Research in Mathematics Education, 20, 8-27. https://doi.org/10.2307/749095 DOI: https://doi.org/10.5951/jresematheduc.20.1.0008

Rinne, L. F., Ye, A. y Jordan, N. C. (2017). Development of fraction comparison strategies: A latent transition analysis. Developmental Psychology, 53(4), 713-730. https://doi.org/10.1037/dev0000275 DOI: https://doi.org/10.1037/dev0000275

Sackur-Grisvard, C. y Léonard, F. (1985). Intermediate cognitive organizations in the process of learning a mathematical concept: The order of positive decimal numbers. Cognition and Instruction, 2(2), 157-174. https://doi.org/10.1207/s1532690xci0202_3 DOI: https://doi.org/10.1207/s1532690xci0202_3

Siegler, R. S. y Lortie-Forgues, H. (2015). Conceptual knowledge of fraction arithmetic. Journal of Educational Psychology, 107(3), 909-918. https://doi.org/10.1037/edu0000025 DOI: https://doi.org/10.1037/edu0000025

Siegler, R. S., Thompson, C. A. y Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273-296. https://doi.org/10.1016/j.cogpsych.2011.03.001 DOI: https://doi.org/10.1016/j.cogpsych.2011.03.001

Smith, C. L., Solomon, G. E. y Carey, S. (2005). Never getting to zero: Elementary school students’ understanding of the infinite divisibility of number and matter. Cognitive Psychology, 51(2), 101-140. https://doi.org/10.1016/j.cogpsych.2005.03.001 DOI: https://doi.org/10.1016/j.cogpsych.2005.03.001

Smith, J. P. (1995). Competent reasoning with rational numbers. Cognition and Instruction, 13(1), 3-50. https://doi.org/10.1207/s1532690xci1301_1 DOI: https://doi.org/10.1207/s1532690xci1301_1

Stafylidou, S. y Vosniadou, S. (2004). The development of students’ understanding of the numerical value of fractions. Learning and Instruction, 14(5), 503-518. https://doi.org/10.1016/j.learninstruc.2004.06.015 DOI: https://doi.org/10.1016/j.learninstruc.2004.06.015

Steinle, V. y Stacey, K. (1998). The incidence of misconceptions of decimal notation amongst students in Grades 5 to 10. En C. Kanes, M. Goos y E. Warren (Eds.), Teaching mathematics in new times (pp. 548–555). Mathematics Education Research Group of Australasia.

Torbeyns, J., Schneider, M., Xin, Z. y Siegler, R. S. (2015). Bridging the gap: Fraction understanding is central to mathematics achievement in students from three different continents. Learning and Instruction, 37, 5-13. https://doi.org/10.1016/j.learninstruc.2014.03.002 DOI: https://doi.org/10.1016/j.learninstruc.2014.03.002

Vamvakoussi, X., Christou, K. P., Mertens, L. y Van Dooren, W. (2011). What fills the gap between discrete and dense? Greek and Flemish students’ understanding of density. Learning and Instruction, 21(5), 676-685. https://doi.org/10.1016/j.learninstruc.2011.03.005 DOI: https://doi.org/10.1016/j.learninstruc.2011.03.005

Vamvakoussi, X., Christou, K. P. y Vosniadou, S. (2018). Bridging psychological and educational research on rational number knowledge. Journal of Numerical Cognition, 4(1), 84-106. https://doi.org/10.5964/jnc.v4i1.82 DOI: https://doi.org/10.5964/jnc.v4i1.82

Vamvakoussi, X., Van Dooren, W. y Verschaffel, L. (2012). Naturally biased? In search for reaction time evidence for a natural number bias in adults. The Journal of Mathematical Behavior, 31(3), 344-355. https://doi.org/10.1016/j.jmathb.2012.02.001 DOI: https://doi.org/10.1016/j.jmathb.2012.02.001

Vamvakoussi, X. y Vosniadou, S. (2007). How many numbers are there in a rational numbers interval? Constraints, synthetic models and the effect of the number line. En S. Vosniadou, A. Baltas y X. Vamvakoussi (Eds.), Reframing the Conceptual Change Approach in Learning and Instruction (pp. 265-282). Elsevier.

Van Dooren, W., Lehtinen, E. y Verschaffel, L. (2015). Unraveling the gap between natural and rational numbers. Learning and Instruction, 37, 1-4. https://doi.org/10.1016/j.learninstruc.2015.01.001 DOI: https://doi.org/10.1016/j.learninstruc.2015.01.001

Van Hoof, J., Degrande, T., Ceulemans, E., Verschaffel, L. y Van Dooren, W. (2018). Towards a mathematically more correct understanding of rational numbers: A longitudinal study with upper elementary school learners. Learning and Individual Differences, 61, 99-108. https://doi.org/10.1016/j.lindif.2017.11.010 DOI: https://doi.org/10.1016/j.lindif.2017.11.010

Van Hoof, J., Verschaffel, L. y 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 DOI: https://doi.org/10.1007/s10649-015-9613-3