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RESEARCH PAPER
Exploring Students’ Understanding of Integration by Parts: A Combined Use of APOS and OSA
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, SPAIN
 
2
Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, IRAN
 
 
Online publication date: 2019-04-03
 
 
Publication date: 2019-04-03
 
 
EURASIA J. Math., Sci Tech. Ed 2019;15(7):em1721
DOI: https://doi.org/10.29333/ejmste/106166
 
 
Article (PDF)
References (32)
 
KEYWORDS
student’s understanding
integration by parts
mental constructions
primary objects
schema
 
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.
 
REFERENCES (32)
1.
Anton, H., Bivens, I., & Davis, S. (2010). Calculus: early transcendentals. Jefferson City (Missouri): Wiley Global Education.
Google Scholar
 
 
2.
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.
Google Scholar
 
 
3.
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-....
CrossRef
 
Google Scholar
 
 
4.
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....
CrossRef
 
Google Scholar
 
 
5.
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-....
CrossRef
 
Google Scholar
 
 
6.
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/ejmst....
CrossRef
 
Google Scholar
 
 
7.
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/ejmst....
CrossRef
 
Google Scholar
 
 
8.
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-....
CrossRef
 
Google Scholar
 
 
9.
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....
CrossRef
 
Google Scholar
 
 
10.
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....
CrossRef
 
Google Scholar
 
 
11.
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....
CrossRef
 
Google Scholar
 
 
12.
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....
CrossRef
 
Google Scholar
 
 
13.
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.
Google Scholar
 
 
14.
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.
Google Scholar
 
 
15.
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.jmat....
CrossRef
 
Google Scholar
 
 
16.
Kiat, S. E. (2005). Analysis of students’ difficulties in solving integration problems. The Mathematics Educator, 9(1), 39–59.
Google Scholar
 
 
17.
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....
CrossRef
 
Google Scholar
 
 
18.
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.
Google Scholar
 
 
19.
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/10.1080/002073....
CrossRef
 
Google Scholar
 
 
20.
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/10.1590/1980-4....
CrossRef
 
Google Scholar
 
 
21.
Piaget, J., & García, R. (1983). Psychogenesis and the history of science. New York: Columbia University Press.
Google Scholar
 
 
22.
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....
CrossRef
 
Google Scholar
 
 
23.
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....
CrossRef
 
Google Scholar
 
 
24.
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.
Google Scholar
 
 
25.
Thomas, G. B., Weir, M. D., Hass, J., & Giordano, R. F. (2010). Thomas’ calculus: Early transcendentals. Boston: Pearson Addison-Wesley.
Google Scholar
 
 
26.
Trigueros, M. (2005). La nocion del esquema en la investigacion en matematica educativa a nivel superior. Educacion Matematica, 17(1), 5–31.
Google Scholar
 
 
27.
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.
Google Scholar
 
 
28.
Orton, A. (1983). Students’ understanding of integration. Educ Stud Math, 14(1), 1-18. https://doi.org/10.1007/BF0070....
CrossRef
 
Google Scholar
 
 
29.
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/10.1080/002073....
CrossRef
 
Google Scholar
 
 
30.
Stewart, J. (2010). Calculus, 7th Edition. Brooks/Cole Cengage Learning, Mason.
Google Scholar
 
 
31.
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/BF0127....
CrossRef
 
Google Scholar
 
 
32.
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/10.1080/149261....
CrossRef
 
Google Scholar
 
 
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