RESEARCH PAPER
The Understanding of the Derivative Concept in Higher Education
 
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1
Facultad de Ciencias de la Ingeniería, Universidad Austral de Chile, Valdivia, CHILE
2
Departament de Didàctica de la Matemàtica i de les Ciències Experimentals, Universitat Autònoma de Barcelona, Barcelona, SPAIN
3
Departamento de Didáctica de las Matemáticas, Universidad de Sevilla, Sevilla, SPAIN
Publish date: 2018-12-10
 
EURASIA J. Math., Sci Tech. Ed 2019;15(2):em1662
KEYWORDS
ABSTRACT
The aim of this work was to identify and characterize the levels of development of derivative schema. In order to do so, a questionnaire to 103 university students with previous instruction in Differential Calculus was applied. The questionnaire was composed of three tasks. For the identification of the levels of development of schema and their subsequent characterization, we consider the framework proposed by the APOS theory. In particular, this framework was operationalized through the establishment of 27 variables that allowed for the breakdown of the resolution protocols from the questionnaire into discrete elements. In this way, we obtained a vector associated with each of these variables. The identification of students assigned to each level of development of schema was carried out by a cluster analysis. Subsequently, we performed a statistical analysis of frequencies and implicative, with the 27 variables, which allowed to characterize the levels of development identified.
 
REFERENCES (44)
1. Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Fuentes, S. R., Trigueros, M., & Weller, K. (2014). APOS theory: A framework for research and curriculum development in mathematics education. Berlin: Springer. https://doi.org/10.1007/978-1-....
2. Asiala, M., Brown, A., DeVries, D. J., Dubinsky, E., Mathews, D., & Thomas, K. (1996). A Framework for Research and Curriculum Development in Undergraduate Mathematics Education. Research in Collegiate Mathematics Education, 2, 1-32. https://doi.org/10.1090/cbmath....
3. Asiala, M., Cottrill, J., Dubinsky, E., & Schwingendorf, K. (1997). The Development of Students’ Graphical Understanding of the Derivate. Journal of Mathematics Behavior, 16(4), 399-430. https://doi.org/10.1016/S0732-....
4. Badillo, E., Azcárate, C., & Font, V. (2011). Análisis de los niveles de comprensión de los objetos f’(a) y f’(x) de profesores de matemáticas. Enseñanza de las Ciencias, 29(2), 191-206. https://doi.org/10.5565/rev/ec....
5. Baker, B., Cooley, L., & Trigueros, M. (2000). A calculus graphing schema. Journal for research in mathematics education, 31(5), 557–578. https://doi.org/10.2307/749887.
6. Berry, J., & Nyman, M. A. (2003). Promoting students’ graphical understanding of the calculus. Journal of Mathematical Behavior, 22(4), 479-495. https://doi.org/10.1016/j.jmat....
7. Bressoud, D.M., Mesa, V., & Rasmussen, C. (2015). Insights and recommendations from the MAA National Study of College Calculus. New York: MAA Press.
8. Cooley, L., Trigueros, M., & Baker, B. (2007). Schema thematization: A framework and a example. Journal for Research in Mathematics Education, 38(4), 370–392. https://doi.org/10.2307/300348....
9. Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 95-126). Dordrecht: Kluwer.
10. Ferrini-Mundy, J., & Graham, G. (1991). An Overview of the Calculus Curriculum Reform Effort: Issues for Learning, Teaching, and Curriculum Development. American Mathematical Monthly, 98(7), 627-635. https://doi.org/10.2307/232493... https://doi.org/10.1080/000298....
11. Ferrini-Mundy, J., & Graham, K. (1994). Research in calculus learning: understanding limits, derivates and integrals. In E. Dubinsky, & J. Kaput (Eds.), Research Issues in Undergraduate Mathematics Learning, MAA Notes 33, (pp. 31-45). Washington DC: Mathematical Association of America.
12. Ferrini-Mundy, J., & Lauten, D. (1994). Learning about calculus learning. The Mathematics Teacher, 87(2), 115-121.
13. Font, V., Trigueros, M., Badillo, E., & Rubio, N. (2016). Mathematical objects through the lens of two different theoretical perspectives: APOS and OSA. Educational Studies in Mathematics, 91(1), 107-122. https://doi.org/10.1007/s10649....
14. Fuentealba, C., Badillo, E., & Sánchez-Matamoros, G. (in press). Puntos de no-derivabilidad de una función y su importancia en la comprensión del concepto de derivada. Educação e Pesquisa.
15. Fuentealba, C., Sánchez-Matamoros, G., Badillo, E. (2015). Análisis de tareas que pueden promover el desarrollo de la comprensión de la derivada. Uno: Revista de didáctica de las matematicas, (71), 72-78.
16. Fuentealba, C., Sánchez-Matamoros, G., Badillo, E., & Trigueros, M. (2017). Thematization of derivative schema in university students: nuances in constructing relations between a function’s successive derivatives. International Journal of Mathematical Education in Science and Technology, 48(3), 374-392. https://doi.org/10.1080/002073....
17. García, M., Llinares, S., & Sánchez-Matamoros, G. (2011). Characterizing thematized derivative schema by the underlying emergent structures. International journal of science and mathematics education, 9(5), 1023-1045. https://doi.org/10.1007/s10763....
18. Hiebert, J., & Carpenter, T. (1992). Learning and teaching with understanding. Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 65-97).
19. Kleiner, I. (2001). History of the infinitely small and the infinitely large in calculus.
20. Educational Studies in Mathematics, 48(2), 134–174. https://doi.org/10.1023/A:1016....
21. Lerman, I., Gras, R., & Rostam, H. (1981). Élaboration et évaluation d’un indice d’implication pour des données binaires. 2. Mathématiques et sciences humaines, 75, 5-47.
22. Orton, A. (1983). Students’ understanding of differentiation. Educational Studies in Mathematics, 14(3), 235-250. https://doi.org/10.1007/BF0041....
23. Piaget J. (1973). Las estructuras matemáticas y las estructuras operatorias de.
24. la inteligencia. En La colección Psicología y Educación. La enseñanza.
25. de las matemáticas (pp. 3-28). Madrid: Editorial Aguilar.
26. Piaget, J., & García, R. (1983). Psicogénesis e Historia de la Ciencia. México, España, Argentina, Colombia: Siglo Veintiuno Editores, S.A.
27. Pino-Fan, L., Godino, J., & Font, V. (2011). Faceta epistémica del conocimiento didáctico-matemático sobre la derivada. Educação Matemática Pesquisa, 13(1), 141-178.
28. Pino-Fan, L., Godino, J., & Font, V. (2018). Assessing key epistemic features of didactic-mathematical knowledge of prospective teachers: the case of the derivative. Journal of Mathematics Teacher Education, 21(1), 63-94. https://doi.org/10.1007/s10857....
29. Sánchez-Matamoros, G. (2004). Análisis de la comprensión de los alumnos de.
30. bachillerato y primer curso de la universidad sobre la noción matemática de derivada. (Desarrollo del concepto) (Doctoral dissertation), Universidad de Sevilla, Spain.
31. Sánchez-Matamoros, G., García, M., & Llinares, S. (2006). El desarrollo del esquema de.
32. derivada. Enseñanza de las Ciencias, 24(1), 85–98.
33. Sánchez-Matamoros, G., García, M., & Llinares, S. (2008). La comprensión de la derivada como objeto de investigación en didáctica de la Matemática. RELIME, 11(2), 267-296.
34. Selden, A., Selden, J., Hauk, S., & Mason, A. (1999). Do calculus students eventually learn to solve non-routine problems. Tennessee Technical University Department of Mathematics. Technical Report.
35. Siegler, R. (1986). Unities across domains in childrens strategy choices. In Minnesota Symposia on Child Psychology (pp. 1-48). Mahwah: Lawrence Erlbaum Assoc Inc.
36. Sokal, R. R., & Rohlf, F. J. (1962). The comparison of dendrograms by objective methods. Taxon, 11(2), 33-40. https://doi.org/10.2307/121720....
37. Tall, D. (1992). The transition to Advanced Mathematical Thinking: Functions, Limits, Infinity, and Proof. Handbook of Research on Mathematics Teaching and Learning (pp. 495-514). New York: MacMillan Publishing Company.
38. Tall, D. (1997). Informatie technologie en Wiskunde Onderwijs, Niewe Wiskrant Functions and Calculus. In A. Bishop, M. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International Handbook of Mathematics Education (pp. 289–325). Dordrecht: Kluwer.
39. Trigueros, M. (2005). La noción de esquema en la investigación enmatemática educativa a nivel superior. Educación Matemática, 17(1), 5–31.
40. Trigueros, M., & Escandon, C. (2008). Los conceptos relevantes en el aprendizaje de la graficación. Revista Mexicana de Investigación Educativa, 13(36), 59–85.
41. Von Glasersfeld, E. (1983). On the concept of interpretation. Poetics, 12(2-3), 207-218. https://doi.org/10.1016/0304-4....
42. Vrancken, S., & Engler, A. (2014). Una Introducción a la Derivada desde la Variación y el Cambio: resultados de una investigación con estudiantes de primer año de la universidad. Bolema, 28(48), 449-468. https://doi.org/10.1590/1980-4....
43. White, P., & Mitchelmore, M. (1996). Conceptual knowledge in introductory calculus. Journal for Research in Mathematics Education, 27(1), 79-95. https://doi.org/10.2307/749199.
44. Zamora, L., Gregori, P., & Orús, P. (2009). Conceptos fundamentales del Análisis Estadístico Implicativo (ASI) y su soporte computacional CHIC. Contribuciones al ASI, 4, 65-101.
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