Learning equidistance through variation: a study with primary school children

Authors

Abstract

We present 10 year-old children’s learning moments on equidistance, that occur when experiencing the variation of elements in geometric figures. The intervention was carried out in a public school in Mexico City in three 90-minute sessions. Children used for the first time a dynamic geometry program to construct and explore invariant properties of circumferences, isosceles and equilateral triangles. The results, based on the theory of variation, show that the experience was significant for conceptualizing equidistance, in contrast to collinearity and congruence, when solving geometry problems.

Keywords

Learning of geometry, Equidistance conceptualization, Theory of variation, primary school, Dynamic geometry

References

Aravena, M., Gutiérrez, Á. y Jaime, A. (2016). Estudio de los niveles de razonamiento de Van Hiele en alumnos de centros de enseñanza vulnerables de educación media en Chile. Enseñanza de las Ciencias, 34(1), 107-128. http://dx.doi.org/10.5565/rev/ensciencias.1664

Camargo, L. y Sandoval, I. (2017). Acceso equitativo al razonamiento científico mediante la tecnología. Revista Colombiana de Educación, 73, 179-211. https://doi.org/10.17227/01203916.73rce177.209

González, G. y Herbst, P. (2009). Students’ conceptions of congruency through the use of dynamic geometry software. International Journal of Computers for Mathematical Learning, 14(2), 153-182. https://doi.org/10.1007/s10758-009-9152-z

Gu, L., Huang, R. y Marton, F. (2004). Teaching with variation: An effective way of mathematics teaching in China. En L. Fan, N. Y. Wong, J. Cai y S. Li (Eds.), How Chinese learn mathematics: Perspectives from insiders (pp. 309-345). Singapore: World Scientific. https://doi.org/10.1142/9789812562241_0012

Healy, L. (2000). Identifying and explaining geometrical relationship: Interactions with robust and soft Cabri constructions. En T. Nakahara y M. Koyama (Eds.), Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education (vol. 1, pp. 103-117). Hiroshima: Universidad de Hiroshima.

Huang, R., Barlow, A. y Prince, K. (2016). The same tasks, different learning opportunities: An analysis of two exemplary lessons in China and the US from a perspective of variation. The Journal of Mathematical Behavior, 41, 141-158. https://doi.org/10.1016/j.jmathb.2015.12.001

Huang, R. y Li, Y. (Eds.) (2017). Teaching and learning mathematics through variation: Confucian heritage meets western theories. The Netherlands: Sense Publishers. https://doi.org/10.1007/978-94-6300-782-5

Jones, K. (2002). Issues in the Teaching and Learning of Geometry. En L. Haggarty (Ed.), Aspects of Teaching Secondary Mathematics: Perspectives on Practice (pp. 121-139). Londres: Routledge. https://doi.org/10.4324/9780203165874

Jones, K. y Herbst, P. (2012). Proof, proving, and teacher-student interaction: Theories and contexts. En G. Hanna y M. de Villiers (Eds.), Proof and Proving in Mathematics Education (vol. 15, pp. 261-277). Dordrecht: Springer. https://doi.org/10.1007/978-94-007-2129-6_11

Jones, K. y Tzekaki, M. (2016). Research on the teaching and learning of geometry. En A. Gutiérrez, G. Leder y P. Boero (Eds.), The Second Handbook of Research on the Psychology of Mathematics Education (pp. 109-149). Rotterdam: Sense Publishers. https://doi.org/10.1007/978-94-6300-561-6_4

Leung, A. (2003). Dynamic geometry and the theory of variation. En N. Pateman, B. Dougherty y J. Zilliox (Eds.), Proceeding of the 27th Conference of the International Group for the Psychology of Mathematics Education (vol. 3, pp. 197-204). Honolulu: PME.

Leung, A. (2008). Dragging in a dynamic geometry environment through the lens of variation. International Journal of Computers for Mathematical Learning, 13(2), 135-157. https://doi.org/10.1007/s10758-008-9130-x

Leung, A. (2015). Discernment and Reasoning in Dynamic Geometry Environments. En S. J. Cho (Ed.), Selected Regular Lectures from the 12th International Congress on Mathematical Education (pp. 451-469). Cham: Springer. https://doi.org/10.1007/978-3-319-17187-6_26

Lo, M. (2012). Variation theory and the improvement of teaching and learning. Göteborg: Acta Universitatis Gothoburgensis.

Mariotti, M. (2000). Introduction to proof: the mediation of a dynamic software environment. Educational Studies in Mathematics, 44(1), 25-53. https://doi.org/10.1023/A:1012733122556

Marton, F. y Pang, M. F. (2009). On some necessary conditions of learning. The Journal of the Learning Sciences, 15(2), 193-220. https://doi.org/10.1207/s15327809jls1502_2

Marton, F., Runesson, U. y Tsui, A. (2004). The Space of Learning. En F. Marton y T. Amy (Eds.), Classroom Discourse and the space of learning (pp. 3-42). Nueva York: Taylor & Francis Group. https://doi.org/10.4324/9781410609762

MEN (Ministerio de Educación Nacional) (2006). Estándares básicos de competencias en lenguaje, matemáticas, ciencias y ciudadanas. Bogotá: Ministerio de Educación Nacional.

Morera, L., Fortuny, J. y Planas, N. (2012). Momentos clave en el aprendizaje de isometrías en un entorno colaborativo y tecnológico. Enseñanza de las Ciencias, 30(1), 143-154. https://doi.org/10.5565/rev/ec/v30n1.569

Mulligan, J. y Vergnaud, G. (2006). Research on children’s early mathematical development. En A. Gutiérrez y P. Boero (Eds.), Handbook of Research on the Psychology of Mathematics Education: Past, Present and Future (pp. 117-146). UK: Sense Publishers. https://doi.org/10.1163/9789087901127_006

Owens, K. y Outhred, L. (2006). The complexity of learning geometry and measurement. En A. Gutiérrez y P. Boero (Eds.), Handbook of Research on the Psychology of Mathematics Education: Past, Present and Future (pp. 83-115). UK: Sense Publishers. https://doi.org/10.1163/9789087901127_005

Pang, M., Bao, J. y Ki, W. (2017). «Bianshi» and the variation theory of learning: illustrating two frameworks of variation and invariance in the teaching of mathematics. En R. Huang y Y. Li (Eds.), Teaching and Learning Mathematics through Variation (pp. 43-67). The Netherlands: Sense Publisher. https://doi.org/10.1007/978-94-6300-782-5_3

Samper, C., Molina, O. y Echeverry A. (2013). Elementos de Geometría. Bogotá: Fondo editorial Universidad Pedagógica Nacional.

Sandoval, I. (2009). La geometría dinámica como una herramienta de mediación entre el conocimiento perceptivo y el geométrico. Educación Matemática, 21(1), 5-27.

Sarama, J. y Clements, D. (2009). Early Childhood Mathematics Education Research: Learning Trajectories for Young Children. Nueva York: Routledge. https://doi.org/10.4324/9780203883785

SEP (Secretaría de Educación Pública) (2011). Programas de estudio 2011. Guía para el maestro. Educación Básica. Primaria. Quinto grado. México: Conaliteg.

SEP (Secretaría de Educación Pública) (2013). Desafíos. Quinto grado. Docente. México: Conaliteg.

SEP (Secretaría de Educación Pública) (2017). Aprendizajes Clave para la Educación Integral. Plan y Programas de Estudio para la Educación Básica. México: Conaliteg.

Sinclair, N., Bussi, M., de Villiers, M., Jones, K., Kortenkamp, U., Leung, A. y Owens, K. (2016). Recent research on geometry education: an ICME-13 survey team report. ZDM, 48(5), 691-719. https://doi.org/10.1007/s11858-016-0796-6

Soldano, C., Luz, Y., Arzarello, F. y Yerushamy, M. (2018). Technology-based inquiry in geometry: semantic games through the lens of variation. Educational Studies in Mathematics, 100(1), 7-23. http://doi-org-443.webvpn.fjmu.edu.cn/10.1007/s10649-018-9841-4

Steffe, L. y Thompson, P. (2000). Teaching Experiment Methodology: Underlying Principles and Essential Elements. En A. Kelly y R. Lesh (Eds.), Handbook of Research Design in Mathematics and Science Education (pp. 267-307). Nueva York: Routledge. https://doi.org/10.4324/9781410602725

Author Biographies

Ivonne Twiggy Sandoval Caceres, Universidad Pedagógica Nacional, Unidad Ajusco, México

Profesora- Investigadora de Tiempo completo, titular C.

Área Académica 4. Tecnologías de la Información y Modelos Alternativos 

Leonor Camargo Uribe, Universidad Pedagógica Nacional de Colombia

Profesora de Tiempo Completo

Departamento de Matemáticas, Facultad de Ciencia y Tecnología

Published

2021-06-03

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