Exploring Students’ Understanding of Integration by Parts: A Combined Use of APOS and OSA

Vahid Borji; Vicenç Font

doi:10.29333/ejmste/106166

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Vahid Borji ¹ ² ^* , Vicenç Font ¹

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¹ Departament de Didàctica de les CCEE i la Matemàtica, Facultat de Formació del Professorat, Universitat de Barcelona, Passeig de la Vall d’Hebrón, 171, Barcelona 08035 Catalonia, SPAIN² Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, IRAN^* Corresponding Author

Abstract

Our goal in this paper is to study students’ understanding of integration by parts based on two theories, APOS and OSA. We make an epistemic configuration (EC) of primary objects that a student activate for solving tasks in relation to the integration by parts, and then we design a genetic decomposition (GD) of mental constructions that he/she might need to learn the integration by parts. We then describe the EC and GD in terms of the levels of development of Schema (i.e., intra, inter and trans). Three tasks in a semi-structured interview were used to explore twenty three first-year students’ understanding of integration by parts and classify their schemas. Results showed that students had difficulties in integration by parts, especially in using this technique to obtain a simpler integral than the one they started with. Using APOS and OSA gave us a clear insight about students’ difficulties and helped us to better describe students’ understanding of integration by parts.

Keywords

student’s understanding
integration by parts
mental constructions
primary objects
schema

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, 2019, Volume 15, Issue 7, Article No: em1721

https://doi.org/10.29333/ejmste/106166

Publication date: 03 Apr 2019

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Article Downloads: 3363

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References

Anton, H., Bivens, I., & Davis, S. (2010). Calculus: early transcendentals. Jefferson City (Missouri): Wiley Global Education.
Arnon, I., Cottrill, J., Dubinsky, E., Oktac, A., Roa, S., Trigueros, M., & Weller, K. (2014). APOS Theory: A Framework for Research and Curriculum Development in Mathematics Education. New York, Heidelberg, Dordrecht, London: Springer.
Asiala, M., Cottrill, J., Dubinsky, E., & Schwingendorf, K. E. (1997). The development of students’ graphical understanding of the derivative. Journal of Mathematical Behavior, 16(4), 399-430. https://doi.org/10.1016/S0732-3123(97)90015-8.
Badillo, E., Azcárate, C., & Font, V. (2011). Analysis of Mathematics teachers’ level of understanding of the objects f'(a) and f'(x). Enseñanza de las ciencias, 29(2), 191-206. https://doi.org/10.5565/rev/ec/v29n2.546.
Bikner-Ahsbahs, A., & Prediger, S. (eds) (2014). Networking of theories as a research practice in mathematics education. Advances in Mathematical Education. Springer. https://doi.org/10.1007/978-3-319-05389-9.
Borji, V., Alamolhodaei, H., & Radmehr, F. (2018). Application of the APOS-ACE Theory to improve Students’ Graphical Understanding of Derivative. EURASIA Journal of Mathematics, Science and Technology Education, 14(7), 2947-2967. https://doi.org/10.29333/ejmste/91451.
Borji, V., Font, V., Alamolhodaei, H., & Sánchez, A. (2018). Application of the Complementarities of Two Theories, APOS and OSA, for the Analysis of the University Students’ Understanding on the Graph of the Function and its Derivative. EURASIA Journal of Mathematics, Science and Technology Education, 14(6), 2301-2315. https://doi.org/10.29333/ejmste/89514.
Clark, J. M., Cordero, F., Cottrill, J., Czarnocha, B., DeVries, D. J., St. John, D., Tolias, G., & Vidakovic, D. (1997). Constructing a schema: The case of the chain rule? Journal of Mathematical Behavior, 16(4), 345-364. https://doi.org/10.1016/S0732-3123(97)90012-2.
Font, V., & Contreras, A. (2008). The problem of the particular and its relation to the general in mathematics education. Educational Studies in Mathematics, 69(1), 33-52. https://doi.org/10.1007/s10649-008-9123-7.
Font, V., Godino, J. D., & Gallardo, J. (2013). The emergence of objects from mathematical practices. Educational Studies in Mathematics, 82(1), 97–124. https://doi.org/10.1007/s10649-012-9411-0.
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/s10649-015-9639-6.
Godino, J. D., Batanero, C., & Font, V. (2007). The onto-semiotic approach to research in mathematics education. ZDM. The International Journal on Mathematics Education, 39(1), 127-135. https://doi.org/10.1007/s11858-006-0004-1.
Haspekian, M., Bikner-Ahsbahs, A., & Artigue, M. (2013). When the fiction of learning is kept: A case of networking two theoretical views. In A. Lindmeier, & A. Heinze (Eds.), Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education, Vol. 3, pp. 9–16. Kiel, Germany: PME.
Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1-27). Hillsdale, NJ, US: Lawrence Erlbaum Associates, Inc.
Jones, S. R. (2013). Understanding the integral: Students’ symbolic forms. The Journal of Mathematical Behavior, 32(2), 122–141. https://doi.org/10.1016/j.jmathb.2012.12.004.
Kiat, S. E. (2005). Analysis of students’ difficulties in solving integration problems. The Mathematics Educator, 9(1), 39–59.
Kouropatov, A., & Dreyfus, T. (2014). Learning the integral concept by constructing knowledge about accumulation. ZDM Mathematics Education, 46(4), 533–548. https://doi.org/10.1007/s11858-014-0571-5.
Llinares, S., Boigues, F., & Estruch, V. (2010). Desarrollo de un esquema de la integral definida en estudiantes de ingenierıas relacionadas con las ciencias de la naturaleza. Un analisis a travesdela logica Fuzzi. Revista Latinoamericana de Investigacion en Matematica Educativa, 13, 255–282.
Mahir, N. (2009). Conceptual and procedural performance of undergraduate students in integration. Int J Math Educ Sci Technol, 40(2), 201–211. https://doi.org/10.1080/00207390802213591.
Mateus, E. (2016). Análisis Didáctico a un Proceso de Instrucción del Método de Integración por Partes [Teaching Analysis to Process Integration Method Instruction by Parties]. BOLEMA, 30(55), 559–585. https://doi.org/10.1590/1980-4415v30n55a13.
Piaget, J., & García, R. (1983). Psychogenesis and the history of science. New York: Columbia University Press.
Pino-Fan, L., Font, V., Gordillo, W., Larios, V. & Breda, A. (2017). Analysis of the Meanings of the Antiderivative Used by Students of the First Engineering Courses. International Journal of Science and Mathematics Education, 16(6), 1091-1113. https://doi.org/10.1007/s10763-017-9826-2.
Pino-Fan, L., Godino, J. D., & 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/s10857-016-9349-8.
Pino-Fan, L., Guzmán, I., Duval, R., & Font, V. (2015). The theory of registers of semiotic representation and the onto-semiotic approach to mathematical cognition and instruction: linking looks for the study of mathematical understanding. In Beswick, K., Muir, T., & Wells, J. (Eds.). Proceedings of the 39th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, pp. 33-40. Hobart, Australia: PME.
Thomas, G. B., Weir, M. D., Hass, J., & Giordano, R. F. (2010). Thomas’ calculus: Early transcendentals. Boston: Pearson Addison-Wesley.
Trigueros, M. (2005). La nocion del esquema en la investigacion en matematica educativa a nivel superior. Educacion Matematica, 17(1), 5–31.
Oaks, A.B. (1990). Writing to learn mathematics: Why do we need it and how can it help us? Paper presented at Association of Mathematics Teachers of New York State Conference, November 1990, Ellenville, NY.
Orton, A. (1983). Students’ understanding of integration. Educ Stud Math, 14(1), 1-18. https://doi.org/10.1007/BF00704699.
Radmehr, F., & Drake, M. (2017). Exploring students’ mathematical performance, metacognitive experiences and skills in relation to fundamental theorem of calculus. International Journal of Mathematical Education in Science and Technology, 48(7), 1043-1071. https://doi.org/10.1080/0020739X.2017.1305129.
Stewart, J. (2010). Calculus, 7th Edition. Brooks/Cole Cengage Learning, Mason.
Thompson, P. W. (1994). Images of rate and operational understanding of the fundamental theorem of calculus. Educ Stud Math, 26(2), 229-274. https://doi.org/10.1007/BF01273664.
Weller, K., Arnon, I., & Dubinsky, E. (2009). Pre-service teachers’ understanding of the relation between a fraction or integer and its decimal expansion. Canadian Journal of Science, Mathematics, and Technology Education, 9(1), 5-28. https://doi.org/10.1080/14926150902817381.

How to cite this article

APA

Borji, V., & Font, V. (2019). Exploring Students’ Understanding of Integration by Parts: A Combined Use of APOS and OSA. Eurasia Journal of Mathematics, Science and Technology Education, 15(7), em1721. https://doi.org/10.29333/ejmste/106166

Vancouver

Borji V, Font V. Exploring Students’ Understanding of Integration by Parts: A Combined Use of APOS and OSA. EURASIA J Math Sci Tech Ed. 2019;15(7):em1721. https://doi.org/10.29333/ejmste/106166

AMA

Borji V, Font V. Exploring Students’ Understanding of Integration by Parts: A Combined Use of APOS and OSA. EURASIA J Math Sci Tech Ed. 2019;15(7), em1721. https://doi.org/10.29333/ejmste/106166

Chicago

Borji, Vahid, and Vicenç Font. "Exploring Students’ Understanding of Integration by Parts: A Combined Use of APOS and OSA". Eurasia Journal of Mathematics, Science and Technology Education 2019 15 no. 7 (2019): em1721. https://doi.org/10.29333/ejmste/106166

Harvard

Borji, V., and Font, V. (2019). Exploring Students’ Understanding of Integration by Parts: A Combined Use of APOS and OSA. Eurasia Journal of Mathematics, Science and Technology Education, 15(7), em1721. https://doi.org/10.29333/ejmste/106166

MLA

Borji, Vahid et al. "Exploring Students’ Understanding of Integration by Parts: A Combined Use of APOS and OSA". Eurasia Journal of Mathematics, Science and Technology Education, vol. 15, no. 7, 2019, em1721. https://doi.org/10.29333/ejmste/106166