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Artículos

Vol. 21 No. 1 (2018): March

ANALYSIS OF ERRORS IN GEOMETRIC TASKS OF VISUAL ARGUMENTATION BY STUDENTS WITH MATHEMATICAL TALENT

DOI
https://doi.org/10.12802/relime.18.2112
Submitted
November 3, 2022
Published
2018-03-11

Abstract

This paper analyzes the errors made by a group of twenty-five students between the ages of 13 and 16, who participate in a project to stimulate mathematical talent, when solving geometric tasks during three sessions of curricular enrichment focused on argumentation techniques. Specific errors of visual argumentation are identified (establishing false analogies between plane and space, not discussing all possible cases and reasoning from limited concrete examples) and derived from the incorrect use of the elements of reasoning, content and mathematical procedures. The correlation study shows that, for the most part, the errors are independent, both among themselves and with the scores on three tests that measure their visual and intellectual capacity. The results of the repeated measures ANOVA indicate that throughout the three sessions the frequency with which they manifest the specific errors of visual argumentation decreases significantly.

References

  1. Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52 (3), 215-241. doi: 10.1023/A:1024312321077
  2. Arteaga, P., Batanero, C., Contreras, J.M. y Cañadas, G. (2016). Evaluación de errores en la construcción de gráficos estadísticos elementales por futuros profesores. Revista Latinoamericana de Investigación en Matemática Educativa, 19 (1), 15-40. doi: 10.12802/relime.13.1911
  3. Assouline, S. G. & Lupkowski-Shoplik, A. E. (2003). Developing mathematical talent: A guide for challenging and educating gifted students. Waco, TX: Prufrock Press.
  4. Battista, M. (2007). The development of geometric and spatial thinking. In F. Lester (Ed.), 2nd Handbook of Research on Mathematics Teaching and Learning, 2 (pp. 843-908). Charlotte, NC: NCTM/Information Age Publishing.
  5. Battista, M. T. & Clements, D. H. (1996). Students’ understanding of three-dimensional rectangular arrays of cubes. Journal for Research in Mathematics Education, 27 (3), 258-292. doi:10.2307/749365
  6. Benavides, M. (2008). Caracterización de sujetos con talento en resolución de problemas de estructura multiplicativa (tesis doctoral no publicada). Universidad de Granada, Granada, España.
  7. Ben-Chaim, D. & Lappan, G. (1989). The Role of Visualization in the middle school mathematics curriculum. Focus on Learning Problems in Mathematics, 11 (1), 49-60.
  8. Ben-Chaim, D., Lappan, G. & Houang, R. T. (1989). Adolescents’ ability to communicate spatial information: Analyzing and affecting students’ performance. Educational Studies in Mathematics, 20 (2), 121-146. doi: 10.1007/BF00579459
  9. Bennett, G. K., Seashore, H. G. & Wesman, A. G. (2000). Test de Aptitudes Diferenciales (dat-5). Manual. Madrid: TEA Ediciones.
  10. Bishop, A. J. (1980). Spatial Abilities and Mathematics Education: A Review. Educational Studies in Mathematics, 11 (3), 257-269. doi: 10.1007/BF00697739
  11. Bishop, A. J. (1983). Space and geometry. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 175-203). New York: Academic Press.
  12. Bishop, A. J. (1989). Review of Research on visualization in mathematics education. Focus on Learning Problems in Mathematics, 11 (1), 7-16.
  13. Biza, I., Nardi. E. & Zachariades, T. (2009). Do images disprove but do not prove? Teachers’ beliefs about visualization. In F. L. Lin, F. J. Hsieh, G. Hanna y M. de Villiers (Eds.), Proceedings of the ICMI Study 19 Conference: Proof and Proving in Mathematics Education (Vol. 1, pp. 59-64). Taipei, Taiwan: The Department of Mathematics, National Taiwan Normal University.
  14. Cohen, N. (2003). Preference of directions in 3-D space. Proceedings of the Third Conference of the European Society for Research in Mathematics Education (CERME 3). Bellaria, Italy.
  15. David, M. M. & Tomaz, V. S. (2012). The role of visual representations for structuring classroom mathematical activity. Educational Studies in Mathematics, 80 (3), 413-431. doi:10.1007/s10649-011-9358-6
  16. De Guzmán, M. (1996). El rincón de la pizarra. Madrid: Pirámide.
  17. Del Grande, J. J. (1987). Spatial Perception and Primary Geometry. In M. M. Lindquist (Ed.), Learning and Teaching Geometry, K-12 (pp. 127-135). Reston, VA: National Council of Teachers of Mathematics.
  18. Del Grande, J. J. (1990). Spatial sense. Arithmetic teacher, 37 (6), 14-20
  19. Dickson, L., Brown, M. & Gibson, O. (1991). El aprendizaje de las matemáticas. Barcelona: Editorial Labor, S.A. Centro de Publicaciones del MEC.
  20. Eisenberg, T. & Dreyfus, T. (1991). On the reluctance to visualize in mathematics. In W. Zimmermann y S. Cunningham (Eds.), Visualization in teaching and learning mathematics (pp. 25-38). Washington, DC: Mathematical Association of America.
  21. Ferrándiz, C., Prieto, M., Fernández, M., Soto, G., Ferrando, M. y Badía, M. (2010). Modelo de identificación de alumnos con altas habilidades de Educación Secundaria. reifop, 13 (1), 63-74.
  22. Freiman, V. (2006). Problems to discover and to boost mathematical talent in early grades: A Challenging Situations Approach. The Montana Mathematics Enthusiast, 3 (1), 51-75.
  23. Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24 (2), 139–162.
  24. Gal, H. & Linchevski, L. (2010). To see or not to see: Analyzing difficulties in geometry from the perspective of visual perception. Educational Studies in Mathematics, 74 (2), 163-183. doi:10.1007/s10649-010-9232-y
  25. Greenes, C. (1981). Identifying the Gifted Student in Mathematics. Arithmetic Teacher, 28 (8), 14-17.
  26. Gruessing, M. (2011). Spatial abilities and mathematics achievement among elementary school children. In B. Ubuz (Ed.). Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, p. 306). Ankara, Turkey: pme.
  27. Guillén, G. (2010). ¿Por qué usar los sólidos como contexto en la enseñanza/aprendizaje de la geometría? ¿Y en la investigación? En M. M. Moreno, A. Estrada, J. Carrillo, y T. A. Sierra, (Eds.), Investigación en Educación Matemática XIV (pp. 21-68). Lleida: seiem.
  28. Gutiérrez, A. (1996). Visualization in 3-dimensional geometry: In search of a framework. En L. Puig y A. Gutierrez (Eds.), Proceedings of the 20th P.M.E. Conference (Vol. 1, pp. 3-19). Valencia, España: Universidad de Valencia.
  29. Gutiérrez, A. (1998). Tendencias actuales de investigación en geometría y visualización. Texto de la ponencia invitada en el Encuentro de Investigación en Educación Matemática, tiem98. Centre de Recerca Matemática, Institut d’Estudis Catalans, Barcelona, España.
  30. Gutiérrez. A. (2006). La investigación sobre enseñanza y aprendizaje de la geometría. En P. Flores, F. Ruiz y M. de la Fuente, M. (Eds.), Geometría para el siglo xxi (pp. 13-58). Badajoz: Federación Española de Profesores de Matemáticas y saem Thales.
  31. Hanna, G. & Sidoli, N. (2007). Visualisation and proof: a brief survey of philosophical Perspectives. zdm Mathematics Education, 39 (1-2), 73-78. doi: 10.1007/s11858-006-0005-0
  32. Hershkowitz, R. (1990). Psychological aspects of learning geometry. In P. Nesher y J. Kilpatrick (Eds.), Mathematics and cognition (pp. 70-95). Cambridge, G. B.: Cambridge U. P.
  33. Jaime, A. y Gutiérrez, A. (1996). El grupo de las isometrías del plano. Madrid: Síntesis.
  34. Jones, K., & Tzekaki, M. (2016). Research on the Teaching and Learning of Geometry. In Á. Gutiérrez, G. C. Leder, y P. Boero (Eds.), The Second Handbook of Research on the Psychology of Mathematics Education (pp. 109-149). Rotterdam: Sense Publishers.
  35. Kageyama, K. (2009). Justification identified in mathematics classroom. In M. Tzekaki, M. Kaldrimidou & H. Sakonidis (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, p. 401.9). Greece: pme.
  36. Kim, J., Lee, K., Ko, E., Park, M., & Park, M. (2009). Are gifted students aware of unjustified assumsions in geometric constructions? En M. Tzekaki, M. Kaldrimidou y H. Sakonidis. (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 337 344). Thessaloniki, Greece: pme.
  37. Kinach, B., & Coulson, A. (2014). Visualization as learning tool: what should prospective teachers know and teacher educators teach? In P. Liljedahl, C. Nicol, S. Oesterle & D. Allan (Eds.), Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education (Vol. 1, pp. 245). Vancouver, Canada: pme.
  38. Kliapis, P. & Tzekaki, M. (2011). Strategies in early spatial reasoning. In B. Ubuz (Ed.), Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, p.408). Ankara, Turkey: pme.
  39. Krippendorff, K. (1990). Metodología de análisis de contenido: teoría y práctica. Barcelona, España: Paidós Comunicación.
  40. Krutetskii, V. A. (1976). The psychology of Mathematical Abilities in Schoolchildren. Chicago: University of Chicago Press.
  41. Kwon, S. & Song, S. (2007). Views on mathematical proof of able students in the 3rd to 7th grades. En J. H. Woo, H.C. Lew, K. S. Park & D. Y. Seo (Eds.). Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, p. 249). Seoul: pme.
  42. Lee, K. H. (2005). Mathematically gifted student´s geometrical reasoning and informal proof. In Chick, H. L. y Vincent, J. L. (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, 3 (pp. 241-248). Melbourne: pme.
  43. Lee K., Kim M., Na, G., Han, D. & Song, S. (2007). Induction, analogy, an imagery in geometric reasoning. En J. H. Wo., H. C. Lew, K. S. Park y D. Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 145-152). Seoul: pme.
  44. Lee K., Ko, E. & Song, S. (2007). The analysis of activity that gifted students construct definition of regular polyhedra. En J. H. Wo., H. C. Lew, K. S. Park y D. Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 153-160). Seoul: pme.
  45. Lee, S. & Pang, J. (2007). A survey on the uderstanding ofspatialsense of elementary schoolstudents. En J. H. Wo., H. C. Lew, K. S. Park y D. Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 255). Seoul: pme.
  46. Mann, R. (2006). Effective Teaching Strategies for Gifted/Learning Disabled Students with Spatial Strengths. The Journal of Secondary Gifted Education, 17 (2), 112-121.
  47. Meavilla, V. (2005). Razonamiento visual y matemáticas. Sigma, 27, 109-116.
  48. Miller, R. C. (1990). Discovering Mathematical Talent. eric Digest E482. Washington, D.C.: Office of Educational Research and Improvement.
  49. Nardi, E. (2014). Reflections on visualization in mathematics and in mathematics education. In M. Fried (Ed.), Mathematics and mathematics education: Searching for common ground (pp. 193-220). New York: Springer.
  50. Neria, D. & Amit, M. (2010). Talented middle school students’ strategies and reasoning in solving analytic reasoning problems. En M. M. Pinto, F. Pinto y T. F. Kawasaki. Proceedigns of the 34th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 321-328). Belo Horizonte, Brazil: pme.
  51. Park, M., Ko, E., Lee, D. & Lee, K. (2011). Mathematical gifted students’ analogy in statistics. In B. Ubuz (Ed.), Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 345-352). Ankara, Turkey: pme.
  52. Pittalis, M. & Christou, C. (2011). Types of reasoning in 3D geometry thinking and their relation with spatial ability. Educational Studies in Mathematics, 75 (2), 191-212. doi: 10.1007/s10649-010-9251-8
  53. Presmeg, N. (1986). Visualisation and mathematical giftedness. Educational Studies in Mathematics, 17 (3), 297-311. doi: 10.1007/BF00305075
  54. Presmeg, N. (1991). Classroom aspects with influence use of visual imagery in high school mathematics. In F. Furinghetti (Ed.), Proceedings of the 15th pme International Conference (Vol. 3 pp.191-198). Assisi, Italy: pme.
  55. Presmeg, N. (1999). Las posibilidades y peligros del pensamiento basado en imágenes en la resolución de problemas matemáticos. SUMA, 32, 17-22.
  56. Presmeg, N. (2006). Research on visualization in learning and teaching mathematics. In A. Gutiérrez & P. Boero (Eds.), Handbook of Research on the Psychology of Mathematics Education (pp. 205-235). Rotterdam, Netherlands: Sense Publishers.
  57. Presmeg, N. (2014). Contemplating visualization as an epistemological learning tool in mathematics. zdm, The International Journal on Mathematics Education, 46 (1), 151-157. doi:10.1007/s11858-013-0561-z
  58. Prior, J., y Torregrosa, G. (2013). Razonamiento configural y procedimientos de verificación en contexto geométrico. Revista Latinoamericana de Investigación en Matemática Educativa, 16 (3), 339-368. doi: 10.12802/relime.13.1633
  59. Rabab’h, B., & Veloo, A. (2015). Spatial Visualization as Mediating between Mathematics Learning Strategy and Mathematics Achievement among 8th Grade Students. International Education Studies, 8 (5), 1-11.
  60. Ramírez, R. (2012). Habilidades de visualización de los alumnos con talento matemático (tesis doctoral no publicada). Universidad de Granada, Granada, España.
  61. Ramírez-Uclés, R., Flores, P. y Castro, E. (2010). Visualización y talento matemático: una experiencia docente. En M. M. Moreno, A. Estrada, J. Carrillo, y T. A. Sierra (Eds.). Investigación en Educación Matemática XIV (pp. 499-510). Lleida:seiem.
  62. Ramírez-Uclés, R., Ramírez-Uclés, I., Flores, P., y Castro, E. (2013). Análisis de las capacidades de visualización espacial e intelectual en los alumnos con talento matemático. Revista Mexicana de Psicología, 30 (1), 24-31.
  63. Raven, J. C., Court, J. H. y Raven, J. (1993). Test de Matrices Progresivas. Escalas Coloreadas, General y Avanzadas. Buenos Aires: Paidós.
  64. Rico, L. (1993). Errores y dificultades en el aprendizaje de las matemáticas. En J. Kilpatrick, P. Gómez y L. Rico (Eds.), Educación Matemática (pp. 60-108). México: Grupo Editorial Iberoamericano. Rico, L., Castro, E., Castro, E., Coriat, M., Marín, A., Puig, L. et al. (1997). La Educación Matemática en la Enseñanza Secundaria. Barcelona: Editorial Horsori e I.C.E. Universitat Barcelona.
  65. Rivera, F. (2011). Towards a Visually-Oriented School Mathematics Curriculum. New York: Springer Science+Business Media. doi: 10.1007/978-94-007-0014-7. Rivera, F., Steinbring, H. & Arcavi, A. (2014). Visualization as an epistemological learning tool: an introduction. zdm, The International Journal on Mathematics Education, 46 (1), 1-2.
  66. Ryu, H. Chong, Y. & Song, S. (2007). Mathematically gifted students’ spatial visualization ability of solid figures. En J. H. Wo., H. C. Lew, K. S. Park y D. Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 137-144). Seoul: pme.
  67. Secadas, F. (1961). El test ampe. Test de inteligencia. Manual del examinador. Madrid: CSIC, Instituto San José de Calasanz.
  68. Seo, D. (2007). Generalization by comprehension and by apprehension of the Korena 5th grade gifted students. In J. H. Wo., H. C. Lew, K. S. Park & D. Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, p. 280). Seoul: pme.
  69. Sheffet, M. & Bassan-Cincinatus, R. (2009). Hiding shapes. In M. Tzekaki, M. Kaldrimidou & H. Sakonidis (Eds.). Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (vol. 1, p. 465). Thessaloniki, Greece: pme.
  70. Sriraman, B. (2004). Gifted ninth graders’ notions of proof: Investigating parallels in approaches of mathematically gifted students and professional mathematicians. Journal for the Education of the Gifted, 27 (4), 267-292. doi:10.4219/jeg-2004-317.
  71. Stacey, K. (1989). Finding and using patterns in linear generalising problems. Educational Studies in Mathematics, 20 (2), 147-164. doi: 10.1007/BF00579460
  72. Thurstone, L. L. y Thurstone, T. G. (1941). Factorial studies of intelligence. Chicago: University of Chicago Press.
  73. Thurstone, L. L. y Thurstone, T. G. (1976). P.M.A.: Aptitudes Mentales Primarias. Madrid: tea.
  74. Torregrosa, G. y Quesada, H. (2007). Coordinación de procesos cognitivos en geometría. Revista Latinoamericana de Investigación en Matemática Educativa, 10 (2), 275-300. Torregrosa, G., Quesada, H. y Penalva, M. C. (2010). Razonamiento configural como coordinación de procesos de visualización. Enseñanza de las Ciencias, 28 (3), 327-340. doi:10.5565/rev/ec/v28n3.187
  75. Treffers, A. (1987). Three dimensions: a model of goal and theory description in mathematics instruction the Wiskobas Project. Dordrecht: D. Reidel.
  76. Tzekaki, M. & Ikonomou, A. (2009). Investigating spatial representations in early childhood. In M.
  77. Tzekaki, M. Kaldrimidou y H. Sakonidis (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 5, pp. 241-248). Thessaloniki, Greece: pme.
  78. Van Garderen, D. (2006). Spatial visualization, visual imagery, and mathematical problem solving of students with varying abilities. Journal of Learning Disabilities, 39 (6), 496-506.
  79. Van Garderen, D. & Montague, M. (2003). Visual-Spatial Representation, Mathematical Problem Solving, and Students of Varying Abilities. Learning Disabilities Research & Practice, 18 (4), 246- 254. doi: 10.1111/1540-5826.00079
  80. Wheatley, G. H. (1998). Imagery and Mathematics learning. Focus on Learning problems in Mathematics, 20 (2 y 3), 65-77.
  81. Yim, J., Song, S. & Kim, J. (2008). The mathematically gifted elementary students' revisiting of Euler's polyhedron theorem. The Montana Mathematics Enthusiast, 5 (1), 125-142.
  82. Zodik, I. & Zaslavsky, O. (2007). Is a visual example in geometry always helpful? En J. H. Woo, H. C. Lew, K. S. Park & D. Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 265-272). Seoul: pme.

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