Article Sidebar

Download
PDF
Statistic
Read Counter : 188 Download : 68

Downloads

Download data is not yet available.

Main Article Content

Abstract

This report delineates the outcomes of an intervention conducted with in-service high school educators, focusing on elucidating three distinct scenarios within geometric and arithmetic domains: the infinitely large, infinitely numerous, and infinitesimally close. Grounded in the theoretical framework of conceptual change, it is posited that when an individual exhibits entrenched conceptions, it signifies a misclassification of the pertinent concept, necessitating a categorical shift to effectuate a transformation in their cognitive schema, particularly concerning the notion of infinity. Thus, the principal objective of this investigation was to ameliorate the entrenched conceptions held by educators pertaining to infinity through a workshop-based intervention. Preceding the workshop, educators predominantly exhibited conceptions aligned with natural and potential infinities. However, after the workshop, a discernible transition was observed, with educators engendering an actual conception of infinity or an omega-epsilon position, exemplified by their acceptance of equivalences such as 0.999…=1 and the parity in the cardinality of sets comprising natural numbers, even numbers, and perfect squares. Nonetheless, notwithstanding this progress, confident educators evinced resistance to embracing the concept of actual infinity, particularly in instances such as the hypothetical scenario depicted in Hilbert's Grand Hotel. Consequently, drawing upon the framework of conceptual change theory, it can be postulated that a complete categorical shift was not universally realized among educators due to their reluctance to revise entrenched beliefs concerning natural or potential infinity.

Keywords

Conceptual Change High School Teachers Infinity Workshop

Article Details

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite
Díaz-Espinoza, I. A., Juárez-López, J. A., & Miranda, I. (2024). Promoting conceptual change regarding infinity in high school mathematics teachers through a workshop. Journal on Mathematics Education, 15(2), 473–494. https://doi.org/10.22342/jme.v15i2.pp473-494
Download Citation
Endnote/Zotero/Mendeley (RIS)
BibTeX

References

  1. Ángeles-Navarro, M., & Pérez-Carreras, P. (2010). A socratic methodological proposal for the study of the equality 0 . 999 . . . = 1. The Teaching of Mathematics, XIII(1), 17–34. http://www.teaching.math.rs/vol/tm1312.pdf
  2. Belmonte, J. L., & Sierra, M. (2011). Modelos intuitivos del infinito y patrones de evolución nivelar. Revista Latinoamericana de Investigación en Matematica Educativa, 14(2), 139–171. https://www.scielo.org.mx/pdf/relime/v14n2/v14n2a2.pdf
  3. Carey, S. (1991). Knowledge aquisition: enrichment or conceptual change? In S. Carey & R. Gelman (Eds.), The Epigenesis of Mind: Essays on biology and cognition (pp. 257–291). Psychology Press. https://doi.org/10.4324/9781315807805-18
  4. Chi, M. T. H. (2008). Three types of conceptual change: Belief revision, mental model transformation, and categorical shift. In S. Vosniadou (Ed.), International Handbook of Research on Conceptual Change (pp. 61–82). Routledge, Taylor & Francis Group. https://education.asu.edu/sites/default/files/lcl/chi_concpetualchangechapter_0.pdf
  5. Cihlář, J., Eisenmann, P., & Krátká, M. (2015). Omega Position – a specific phase of perceiving the notion of infinity. Scientia in Educatione, 6(2), 51–73. https://doi.org/10.14712/18047106.184
  6. Cihlář, J., Eisenmann, P., Krátká, M., & Vopĕnka, P. (2009). Cognitive conflict as a tool of overcoming obstacles in understanding infinity. Teaching Mathematics and Computer Science, 7(2), 279–295. https://doi.org/10.5485/TMCS.2009.0240
  7. Date-Huxtable, E., Cavanagh, M., Coady, C., & Easey, M. (2018). Conceptualisations of infinity by primary pre-service teachers. Mathematics Education Research Journal, 30(4), 545–567. https://doi.org/10.1007/s13394-018-0243-9
  8. Díaz-Espinoza, I. A., & Juárez-López, J. A. (2023). Mathematics teachers’ conceptions about infinity: A preliminary study at the secondary and high school level. Union: Jurnal Ilmiah Pendidikan Matematika, 11(3), 426–435. https://doi.org/10.30738/union.v11i3.16006
  9. Díaz-Espinoza, I. A., Juárez-López, J. A., & Juárez-Ruiz, E. (2023). Exploring a mathematics teacher’s conceptions of infinity: The case of Louise. Indonesian Journal of Mathematics Education, 6(1), 1–6. https://doi.org/10.31002/ijome.v6i1.560
  10. Dubinsky, E., Weller, K., McDonald, M. A., & Brown, A. (2005a). Some historical issues and paradoxes regarding the concept of infinity: An APOS-based analysis: Part 1. Educational Studies in Mathematics, 58(3), 335–359. https://doi.org/10.1007/s10649-005-2531-z
  11. Dubinsky, E., Weller, K., McDonald, M. A., & Brown, A. (2005b). Some historical issues and paradoxes regarding the concept of infinity: An APOS analysis: Part 2. Educational Studies in Mathematics, 60(2), 253–266. https://doi.org/10.1007/s10649-005-0473-0
  12. Eisenmann, P. (2008). Why is it not true that 0.999 … < 1? Teaching of Mathematics, 11(1), 35–40. http://teaching.math.rs/vol/tm1114.pdf
  13. Fischbein, E., Tirosh, D., & Hess, P. (1979). The intuition of infinit. Educational Studies in Mathematics, 10(1), 3–40. https://doi.org/https://doi.org/10.1007/BF00311173
  14. Hannula, M. S., Pehkonen, E., Maijala, H., & Soro, R. (2006). Levels of students’ understanding on infinity. Teaching Mathematics and Computer Science, 4(2), 317–337. https://doi.org/10.5485/tmcs.2006.0129
  15. Holton, D., & Symons, D. (2021). ‘Infinity-based thinking’ in the primary classroom: a case for its inclusion in the curriculum. Mathematics Education Research Journal, 33(3), 435–450. https://doi.org/10.1007/s13394-020-00311-4
  16. Homaeinejad, M., Barahmand, A., & Seif, A. (2021). The relationship between notions of infinity and strategies used to compare enumerable infinite sets. International Journal of Mathematical Education in Science and Technology, 53(12), 3307-3325. https://doi.org/10.1080/0020739X.2021.1941362
  17. Juter, K. (2019). University students’ general and specific beliefs about infinity, division by zero and denseness of the number line. Nordic Studies in Mathematics Education, 24(2), 69–88. https://ncm.gu.se/wp-content/uploads/2020/06/24_2_069088_juter-1.pdf
  18. Kattou, M., Thanasia, M., Katerina, K., Constantinos, C., & George, P. (2010). Teachers’ perceptions about infinity : a process or an object ? In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of the 6th Congress of the European Society for Research in Mathematics Education (pp. 1771–1780). www.inrp.fr/editions/cerme6
  19. Kidron, I., & Tall, D. (2015). The roles of visualization and symbolism in the potential and actual infinity of the limit process. Educational Studies in Mathematics, 88(2), 183–199. https://doi.org/10.1007/s10649-014-9567-x
  20. Krátká, M. (2013). Zdroje epistemologických překážek v porozumění nekonečnu. Scientia in Educatione, 1(1), 87–100. https://doi.org/10.14712/18047106.7
  21. Krátká, M., Eisenmann, P., & Cihlář, J. (2021). Four conceptions of infinity. International Journal of Mathematical Education in Science and Technology, 53(10), 2661-2685. https://doi.org/10.1080/0020739X.2021.1897894
  22. Mamolo, A., & Zazkis, R. (2008). Paradoxes as a window to infinity. Research in Mathematics Education, 10(2), 167–182. https://doi.org/10.1080/14794800802233696
  23. Manfreda Kolar, V., & Čadež, T. H. (2012). Analysis of factors influencing the understanding of the concept of infinity. Educational Studies in Mathematics, 80(3), 389–412. https://doi.org/10.1007/s10649-011-9357-7
  24. Medina Ibarra, L., Romo-Vázquez, A., & Sánchez Aguilar, M. (2019). Using the work of Jorge Luis Borges to identify and confront students’ misconceptions about infinity. Journal of Mathematics and the Arts, 13(1–2), 48–59. https://doi.org/10.1080/17513472.2018.1504270
  25. Mena-Lorca, A., Mena-Lorca, J., Montoya-Delgadillo, E., Morales, A., & Parraguez, M. (2015). El obstáculo epistemológico del infinito actual: persistencia, resistencia y categorías de análisis. Revista Latinoamericana de Investigación en Matemática Educativa, 18(3), 329–358. https://doi.org/10.12802/relime.13.1832
  26. Monaghan, J. (2001). Young peoples’ ideas of infinity. Educational Studies in Mathematics, 48(2–3), 239–257. https://doi.org/10.1023/A:1016090925967
  27. Montes, M. A., Carrillo, J., & Ribeiro, C. M. (2014). Teachers knowledge of infinity, and its role in classroom practice. In P. Liljedahl, S. Oesterle, C. Nicol, & 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. 4, pp. 233–239). North American Chapter of the International Group for the Psychology of Mathematics Education. https://eric.ed.gov/?id=ED599915
  28. Moreno Armella, L. E., & Waldegg, G. (1991). The conceptual evolution of actual mathematical infinity. Educational Studies in Mathematics, 22(3), 211–231. https://doi.org/10.1007/BF00368339
  29. Roa-Fuentes, S., & Oktaç, A. (2014). El infinito potencial y actual : descripción de caminos cognitivos para su construcción en un contexto de paradojas. Educación Matemática, 26(1), 73–102. http://www.revista-educacion-matematica.com/revista/2016/05/15/vol26-1-3/
  30. Schwarzenberger, R. L. E., & Tall, D. (1978). Conflicts in the learning of real numbers and limits. Mathematics Teaching, 82, 44–49. https://wrap.warwick.ac.uk/494/1/WRAP_Tall_dot1978c-with-rolph.pdf
  31. Singer, M., & Voica, C. (2003). Perception of infinity: does it really help in problem solving? In A. Rogerson (Ed.), Proceedings of the International Conference The Decidable and the Undecidable in Mathematics Education (pp. 1–7). http://dipmat.math.unipa.it/~grim/21_project/21_brno_03.htm
  32. Smith III, J. P., DiSessa, A. A., & Roschelle, J. (1994). Misconceptions reconceived: a constructivist analysis of knowledge in transition. Journal of the Learning Sciences, 3(2), 115–163. https://doi.org/10.1207/s15327809jls0302_1
  33. Tall, D. (1980). The notion of infinite measuring number and its relevance in the intuition of infinity. Educational Studies in Mathematics, 11(3), 271–284. https://doi.org/10.1007/BF00697740
  34. Tall, D. (2001). Natural and formal infinities. Educational Studies in Mathematics, 48(2–3), 199–238. https://doi.org/10.1023/A:1016000710038
  35. Tall, D., & Tirosh, D. (2001). The never-ending struggle. Educational Studies in Mathematics, 48(2–3), 129–136. https://doi.org/10.1023/A:1016019128773
  36. Tsamir, P. (1999). The transition from comparison of finite to the comparison of infinite sets: Teaching prospective teachers. Educational Studies in Mathematics, 38, 209–234. https://doi.org/10.1023/A:1003514208428
  37. Villabona Millán, D., Roa Fuentes, S., & Oktaç, A. (2022). Concepciones dinámicas y estáticas del infinito: procesos continuos y sus totalidades. Enseñanza de Las Ciencias. Revista de Investigación y Experiencias Didácticas, 40(1), 179–197. https://doi.org/10.5565/rev/ensciencias.3277
  38. Vinner, S., & Kidron, I. (1985). The concept of repeating and non-repeating decimals at the senior high level. In L. Streefland (Ed.), Proceedings of the 9th Annual Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 357–361). https://eric.ed.gov/?id=ED411130
  39. von Aufschnaiter, C., & Rogge, C. (2010). Misconceptions or Missing Conceptions? EURASIA Journal of Mathematics, Science and Technology Education, 6(1), 3–18. https://doi.org/10.12973/ejmste/75223
  40. Vosniadou, S., Vamvakoussi, X., & Skopeliti, I. (2008). The framework theory approach to the problem of conceptual change. In S. Vosniadou (Ed.), International Handbook of Research on Conceptual Change (pp. 3–34). Routledge, Taylor & Francis Group. https://www.researchgate.net/profile/Stella-Vosniadou/publication/284495667_The_framework_theory_approach_to_the_problem_of_conceptual_change/links/5c57fd05a6fdccd6b5e1128d/The-framework-theory-approach-to-the-problem-of-conceptual-change.pdf
  41. Waldegg, G. (2005). Bolzano’s approach to the paradoxes of infinity: Implications for teaching. Science & Education, 14(6), 559–577. https://doi.org/10.1007/s11191-004-2014-0
  42. Wijeratne, C., & Zazkis, R. (2015). On painter’s paradox: Contextual and mathematical approaches to infinity. International Journal of Research in Undergraduate Mathematics Education, 1(2), 163–186. https://doi.org/10.1007/s40753-015-0004-z
  43. Wistedt, I., & Martinsson, M. (1996). Orchestrating a mathematical theme: Eleven-year olds discuss the problem of infinity. Learning and Instruction, 6(2), 173–185. https://doi.org/10.1016/0959-4752(96)00001-1
  44. Yopp, D. A., Burroughs, E. A., & Lindaman, B. J. (2011). Why it is important for in-service elementary mathematics teachers to understand the equality .999...=1. Journal of Mathematical Behavior, 30(4), 304–318. https://doi.org/10.1016/j.jmathb.2011.07.007
  45. Zippin, L. (1962). Uses of infinity. The Mathematical Association of America. https://www.ams.org/books/nml/007/nml007-endmatter.pdf