Analysis of definitions of limit provided by University students

Authors

DOI:

https://doi.org/10.15359/ru.35-2.18

Keywords:

limit concept, calculus, mathematical content, didactics of mathematical analysis, university students, advanced mathematical thinking

Abstract

This paper serves two objectives: analyze the definitions provided by college students for the concept of limit at a point in a function as well as design and validate a category system to carry it out. The definitions of limit were provided by 38 university students in the Biology and Engineering in Industrial Chemistry program, enrolled in a Differential and Integral Calculus course during the first semester of 2018, at Universidad Nacional, Costa Rica. A priori categories were used and complemented with other categories of an inductive nature, which emerged during the analysis of the responses. These categories were validated through a reliability analysis. Approximately half of the students’ answers showed the limit category as an object and over three-quarters showed the limit category as a process. In addition, approximately half of their answers showed the categories terms of relative position and general lack of coordination of the processes. The category system created allowed for the analysis of information units in an organized, simple, and replicable way. Finally, students showed a dual conception of limit either as an object, fixed and static notion, or as a process, processual and dynamic notion. Furthermore, the content analysis described in the methodology may allow future researchers to create a similar category system or use this one for research in other contexts.

References

Artigue, M. (1995). La enseñanza de los principios del cálculo: Problemas epistemológicos, cognitivos y didácticos. En M. Artigue, R. Douady, L. Moreno y P. Gómez (Eds.), Ingeniería didáctica en educación matemática (pp. 97-140). Grupo Editorial Iberoamérica. http://funes.uniandes.edu.co/676/1/Artigueetal195.pdf#page=105

Azcárate, C. y Camacho, M. (2003). Sobre la investigación en didáctica del análisis. Boletín de la Asociación Matemática Venezolana, 10(2), 135- 149.

Bell, A. W., Costello, J. & Küchemann, D. E. (1983). A review of research in mathematical education: research on learning and teaching. The NFER-NELSON Publishing Company.

Blázquez, S. (1999). Sobre la noción del límite en las matemáticas aplicadas a las ciencias sociales. En T. Ortega (Ed.), Investigación en educación matemática III (pp. 167-184). SEIEM.

Blázquez, S. (2000). Noción de límite en matemáticas aplicadas a las ciencias sociales [Tesis doctoral no publicada]. Universidad de Valladolid.

Blázquez, S. y Ortega, T. (1998). Rupturas en la comprensión del concepto de límite en alumnos de bachillerato. Aula, 10, 119-135. https://gredos.usal.es/bitstream/handle/10366/69322/Rupturas_en_la_comprension_del_concepto_.pdf?sequence=1&isAllowed=y

Blázquez, S. y Ortega, T. (2002). Nueva definición de límite funcional. UNO, 30, 67-82.

Byerley, C. y Thompson, P. (2017). Secondary mathematics teachers’ meanings for measure, slope, and rate of change. Journal of Mathematical Behavior, 48, 168-193. https://doi.org/10.1016/j.jmathb.2017.09.003

Castro, E. y Castro, E. (1997). Representaciones y modelización. En L. Rico (Ed.), La educación matemática en la enseñanza secundaria (pp. 95-124). Horsori.

Cohen, L., Manion, L. y Morrison, K. (2007). Research Methods in Education (sixth edition). Routledge. https://doi.org/10.4324/9780203029053

Cornu, B. (2002). Limits. In D. Tall (Ed.), Advanced Mathematical Thinking (pp. 153–166). 10.1007/0-306-47203-1_10

Fernández-Plaza, J. (2016). Análisis del contenido. En L. Rico y A. Moreno (Eds.), Elementos de didáctica de la matemática para el profesor de secundaria (pp. 103–118). Ediciones Pirámide.

Fernández-Plaza, J. A., Ruiz-Hidalgo, J. F., Rico, L. y Castro, E. (2013). Definiciones personales y aspectos estructurales del concepto de límite finito de una función en un punto. PNA, 7(3), 117-130. https://digibug.ugr.es/bitstream/handle/10481/23475/PNA7%283%29-3.pdf?sequence=1&isAllowed=y

Hernández, R., Fernández, C. y Baptista, P. (2014). Metodología de la investigación (Sexta edición). McGraw Hill.

Hiebert, J. y Lefevre, P. (1986). Conceptual and Procedural Knowledge in Mathematics: An Introductory Analysis. En J. Hiebert (Ed.), Conceptual and Procedural Knowledge in Mathematics: The case of Mathematics (pp. 1-28). Lawrence Erlbaum Associates.

Juter, K. (2007a). Students’Concept Development of Limits. Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education (CERME). Working group 14, 2320-2329.

Juter, K. (2007b). Students’Conceptions of Limits: High Achievers versus Low Achievers. The Montana Mathematics Enthusiast (TMME), 4(1), 53-65. https://www.diva-portal.org/smash/get/diva2:207620/FULLTEXT01.pdf

Kaput, J. (1987). Representations Systems and Mathematics. En C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 19-26). Lawrence Erlbaum Associated.

Kidron, I. (2014). Calculus teaching and learning. Encyclopedia of mathematics education. (pp. 69-75). Springer. 10.1007/978-94-007-4978-8

Kidron, I., & Tall, D. (2014). The roles of visualization and symbolism in the potential and actual infinity of the limit process. Educational Studies in Mathematics, 88, 183–199. https://doi.org/10.1007/s10649-014-9567-x

Kilpatrick, J., Hoyles, C. y Skovsmose, O. (Eds.). (2005). Meanings of meaning of mathematics. En Meaning in Mathematics Education (pp. 9-16). Springer. https://doi.org/10.1007/0-387-24040-3_2

Krippendorff, K. (2004). Content analysis: An introduction to its methodology (Second Edition). Sage publications.

Monaghan, J. (1991). Problems with the Language of Limits. For the Learning of Mathematics, 11(3), 20-24. https://flm-journal.org/Articles/3905D8E2472320B3602153840B1E86.pdf

Neuendorf, K. (2017). The Content Analysis Guidebook (Second Edition). Sage publications. https://doi.org/10.4135/9781071802878

Rico, L. (1997). La educación matemática en la enseñanza secundaria. En L. Rico, E. Castro, M. Coriat, A. Marín, L. Puig, M. Sierra, M. M. Socas (Eds.), Consideraciones sobre el currículo de matemáticas para educación secundaria (pp. 15-38). ice - Horsori.

Rico, L. (2012). Aproximación a la investigación en didáctica de la matemática. Avances de Investigación en Educación Matemática, 1, 39-63. https://doi.org/10.35763/aiem.v1i1.4

Rico, L. (2013). El método del análisis didáctico. Revista Iberoamericana de Educación Matemática, 33, 11–27. http://funes.uniandes.edu.co/15988/1/Rico2013El.pdf

Rico, L. (2016a). Matemática y análisis didáctico. En L. Rico y A. Moreno (Eds.), Elementos de didáctica de la matemática para el profesor de secundaria (pp. 85–100). Ediciones Pirámide.

Rico, L. (2016b). Significados de los contenidos matemáticos. En L. Rico y A. Moreno (Eds.), Elementos de didáctica de la matemática para el profesor de secundaria (pp. 153–174). Ediciones Pirámide.

Rico, L. y Fernández-Cano, A. (2013). Análisis didáctico y metodología de investigación. En L. Rico., J. Lupiañez. y M. Molina (Eds.), Análisis didáctico en educación matemática: Metodología de investigación, formación de profesores e innovación curricular (pp.1-22). Comares.

Rico, L. & Ruiz-Hidalgo, J. F. (2018). Ideas to Work for the Curriculum Change in School Mathematics. En Y. Shimizu y R. Vital (Eds.), ICMI Study 24 Conference proceedings. School Mathematics Curriculum Reforms: Challenges, Changes and Opportunities (pp. 301-308). ICMI.

Romero, I. (1997). La introducción del número real en educación secundaria: Una experiencia de investigación-acción. Comares.

Ruiz-Hidalgo, J. (2016). Sentido y modos de uso de un concepto. En L. Rico y A. Moreno (Eds.), Elementos de didáctica de la matemática para el profesor de secundaria (pp. 139–151). Ediciones Pirámide.

Sierpinska, A. (1987). Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics, 18(4), 371-397. https://doi.org/10.1007/BF00240986

Sfard, A. (1991). On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. Educational Studies of Mathematics, 22, 1-36. https://doi.org/10.1007/BF00302715

Sierra, M., González, M. y López, C. (2000). Concepciones de los alumnos de bachillerato y curso de orientación universitaria sobre límite funcional y continuidad. Revista Latinoamericana de Investigación en Matemática Educativa, 3(1), 71-85.

Sim, J., & Wright, C. (2005). The Kappa Statistic in Reliability Studies: Use, Interpretation, and Sample Size Requirements. Physical Therapy, 85(3), 257-268. https://doi.org/10.1093/ptj/85.3.257

Steinbring, H. (1997). Epistemological investigation of classroom interaction in elementary mathematics teaching. Educational Studies in Mathematics, 32(1), 49-92. https://doi.org/10.1023/A:1002919830949

Steinbring, H. (2006). What makes a sign a mathematical sign? – An epistemological perspective on mathematical interaction. Educational Studies in Mathematics, 61, 133-162. https://doi.org/10.1007/s10649-006-5892-z

Swinyard, C. (2011). Reinventing the formal definition of limit: The case of Amy and Mike, The Journal of Mathematical Behavior, 7(4), 765-790. https://www.sciencedirect.com/science/article/pii/S0732312311000022

Tall D. O. (1980). Mathematical intuition, with special reference to limiting processes. In R. Karplus (Ed.), Proceedings of the Fourth International Conference for the Psychology of Mathematics Education (pp. 170-176). PME.

Tall, D & Vinner, S. (1981). Concept image and concept definition in mathematics, with special reference to limits and continuity, Educational Studies in Mathematics, 12, 151- 169. https://doi.org/10.1007/BF00305619

Tall, D., & Katz, M. (2014). A cognitive analysis of Cauchy’s conceptions of function, continuity, limit, and infinitesimal, with implications for teaching the calculus. Educational Studies in Mathematics, 86(1), 97–124. https://doi.org/10.1007/s10649-014-9531-9

Thompson, P. (2013). In the absence of meaning…. En K. Leatham (Ed.), Vital directions for research in mathematics education (pp. 57-93). Springer. https://doi.org/10.1007/978-1-4614-6977-3_4

Thompson, P. (2016). Researching mathematical meanings for teaching. En L. English y D. Kirshner (Eds.), Handbook of International Research in Mathematics Education (pp. 435-461). Taylor and Francis.

Thompson, P. y Milner, F. (2019). Teachers’ Meanings for Function and Function Notation in South Korea and the United States. En H. G. Weigand, W. McCallum, M. Menghini, M. Neubrand y G. Schubring (Eds.), The Legacy of Felix Klein (pp. 55-66). Springer. https://doi.org/10.1007/978-3-319-99386-7_4

Vergnaud, G. (2009). The Theory of Conceptual Fields. Human Development, 52, 83-94. https://doi.org/10.1159/000202727

Vergnaud, G. (2013). Conceptual development and learning. Revista Curriculum, 26, 39-59. https://qurriculum.webs.ull.es/0_materiales/articulos/Qurriculum%2026/Qurriculum%2026-2013(3).pdf

Vrancken, S., Gregorini, M. I., Engler, A., Muller, D., & Hecklein, M. (2006). Dificultades relacionadas con la enseñanza y el aprendizaje del concepto de límite. Revista PREMISA, 8(29), 9-19.

Wang, W. (2011). A Content Analysis of Reliability in Advertising Content Analysis Studies [Master's Thesis]. East Tennessee State University.

Williams, S. (1991). Models of limit held by college calculus students. Journal for research in Mathematics Education, 22(3), 219-236. https://doi.org/10.5951/jresematheduc.22.3.0219

Published

2021-07-31

How to Cite

Analysis of definitions of limit provided by University students. (2021). Uniciencia, 35(2), 1-20. https://doi.org/10.15359/ru.35-2.18

Issue

Section

Original scientific papers (evaluated by academic peers)

How to Cite

Analysis of definitions of limit provided by University students. (2021). Uniciencia, 35(2), 1-20. https://doi.org/10.15359/ru.35-2.18

Comentarios (ver términos de uso)

Most read articles by the same author(s)

1 2 3 4 5 6 7 8 9 10 > >>