Application of the APOS-ACE Theory to improve Students’ Graphical Understanding of Derivative

Vahid Borji; Hassan Alamolhodaei; Farzad Radmehr

doi:10.29333/ejmste/91451

Full Text (PDF)

Vahid Borji ¹, Hassan Alamolhodaei ¹ ^* , Farzad Radmehr ¹

More Detail

¹ Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, IRAN^* Corresponding Author

Abstract

APOS-ACE (Action, Process, Object, and Schema-Activities, Classroom discussion, and Exercises) is applied in this article to explore the teaching and learning of derivative by giving emphasis on its graphical understanding. For this purpose, a Genetic Decomposition is developed based on the outcomes of previous studies and on our personal teaching experiences. An ACE cycle is designed with the help of the Maple software and implemented on a group of freshmen Iranian students (experimental group). The outcomes of this implementation are evaluated by comparing the performance of the experimental group to the performance of another equivalent student group (control group), to which the same subject was taught in the traditional, lecture-based way. Our findings demonstrated students’ who were in the experimental group shown a better understanding of the derivate compared to the control group. Therefore, such ACE cycle with Maple could be used more frequently for teaching calculus, especially derivative.

Keywords

teaching and learning
derivative
graphs of functions
tangent at a point of a graph
APOS theory
genetic decomposition
ACE cycle
Maple software

License

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, Volume 14, Issue 7, July 2018, 2947-2967

https://doi.org/10.29333/ejmste/91451

Publication date: 11 May 2018

Article Views: 5279

Article Downloads: 7450

Open Access References How to cite this article

References

Abu-Naja, M. (2008). The influence of graphic calculators on secondary school pupils’ ways of thinking about the topic “positivity and negativity of functions”. International Journal for Technology in Mathematics Education, 15(3), 103-117.
Andresen, M. (2007), Introduction of a new construct: The conceptual tool “Flexibility”, The Montana Mathematics Enthusiast, 4(2), 230-250.
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, Springer, NY, Heidelberg, Dondrecht, London. https://doi.org/10.1007/978-1-4614-7966-6.
Asiala, M., Brown, A., DeVries, D. J., Dubinsky, E., Mathews, D., & Thomas, K. (1996), A framework for research and development in undergraduate mathematics education. Research in Collegiate Mathematics Education, 6(2), 1-32. https://doi.org/10.1090/cbmath/006/01.
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), 399-431. https://doi.org/10.1016/S0732-3123(97)90015-8.
Badillo, E. (2003). La Derivada como objeto matemático y como objeto de enseñanza y aprendizaje en profesores de matemática de Colombia (Doctoral Thesis). University Autonoma of Barcelona, Espain.
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/v29n2.546.
Baker, B., Cooley, L., & Trigueros, M. (2000). A Calculus Graphing Schema, Journal for Research in Mathematics Education, 31(5), 557-576. https://doi.org/10.2307/749887.
Beth, E. W., & Piaget, J. (1974). Mathematical epistemology and psychology (W. Mays, Trans.). Dordrecht, the Netherlands: D. Reidel. https://doi.org/10.1007/978-94-017-2193-6.
Blyth, B., & Labovic, A. (2009). Using Maple to implement eLearning integrated with computer aided assessment. International Journal of Mathematical Education in Science and Technology, 40(7), 975-988. https://doi.org/10.1080/00207390903226856.
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.
Borji, V., & Voskoglou, M. Gr. (2016). Applying the APOS Theory to Study the Student Understanding of the Polar Coordinates, American Journal of Educational Research, 4(16), 1149-1156.
Borji, V., & Voskoglou, M.Gr. (2017), Designing an ACE Approach for Teaching the Polar Coordinates, American Journal of Educational Research, 5(3), 303-309.
Breda, A., Pino-Fan, L., & Font, V. (2017). Meta didactic-mathematical knowledge of teachers: criteria for the reflection and assessment on teaching practice. EURASIA Journal of Mathematics, Science & Technology Education, 13(6), 1893-1918. https://doi.org/10.12973/eurasia.2017.01207a.
Chamblee, G. E., Slough, S. W., & Wunsch, G. (2008). Measuring high school mathematics teachers’ concerns about graphing calculators and change: A yearlong study. Journal of Computers in Mathematics and Science Teaching, 27(2), 183–194.
Clark J. M., Cordero, F., Cottrill, J., Czarnocha, B., DeVries, D. J., St. John, D., Tolias T., & 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.
Cooley, L., Trigueros, M., & Baker, B. (2007). Schema Thematization: A Framework and an Example. Journal for Research in Mathematics Education, 38(4), 370-392. https://doi.org/10.2307/30034879.
Dominguez, A., Barniol, P., & Zavala, G. (2017). Test of Understanding Graphs in Calculus: Test of Students’ Interpretation of Calculus Graphs. EURASIA Journal of Mathematics, Science and Technology Education. 13(10), 6507-6531. https://doi.org/10.12973/ejmste/78085.
Dubinsky, E., & Leron, U. (1994). Learning abstract algebra with ISETL. New York: Springer. https://doi.org/10.1007/978-1-4612-2602-4.
Dubinsky, E., & McDonald, M. A. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education research, in D Holton (Ed.), The teaching and learning of mathematics at university level: An ICMI study. Kluwer, Dordrecht.
Dubinsky, E., Weller, K., & Arnon, I. (2013). Pre-service teachers’ understanding of the relation between a fraction or integer and its decimal expansion: The Case of 0.999...and 1.Canadian Journal of Science, Mathematics, and Technology Education, 13(3), 5-28. https://doi.org/10.1080/14926150902817381.
Ferrini-Mundy, J., & Graham, K. (1994). Research in calculus learning: Understanding limits, derivatives, and integrals, in E. Dubinsky & J. Kaput (Eds.), Research issues in undergraduate mathematics learning (pp. 19-26), Mathematical Association of America.
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.
García, M., Gavilán, J. M., & Llinares, S. (2012). Perspectiva de la práctica del profesor de matemáticas de secundaria sobre la enseñanza de la derivada. Relaciones entre la práctica y la perspectiva del profesor. Enseñanza de las Ciencias, 30(3), 219-235.
Gavilán, J. M. (2005). El papel del profesor en la enseñanza de la derivada. Análisis desde una perspectiva cognitiva (Doctoral Thesis). Departamento de Didáctica de las Matemáticas. Universidad de Sevilla, España.
Gavilán, J. M., García, M., & Llinares, S. (2007a). Una perspectiva para el análisis de la práctica del profesor de matemáticas. Implicaciones metodológicas. Enseñanza de las Ciencias, 25(2), 157-170.
Gavilán, J. M., García, M., & Llinares, S. (2007b). La modelación de la descomposición genética de una noción matemática. Explicando la práctica del profesor desde el punto de vista del aprendizaje de los estudiantes. Educación Matemática, 19(2), 5-39.
Giraldo, V., Carvalho, L. M., & Tall, D. (2003). Descriptions and Definitions in the Teaching of Elementary Calculus. Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education, 2, 445-452.
Hähkiöniemi, M. (2004). Perceptual and symbolic representations as a starting point of the acquisition of the derivative. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, 3, 73-80.
Lyublinskaya, I., & Zhou, G. (2008). Integrating graphing calculators and Probe ware into science methods courses: Impacts on pre-service elementary teachers’ confidence and perspectives on technology for learning and teaching. Journal of Computers in Mathematics and Science Teaching, 27(2), 163–182.
Maharaj, A. (2005). Investigating the Senior Certificate Mathematics examination in South Africa: Implications for teaching (Doctoral Thesis), Pretoria, University of South Africa.
Maharaj, A. (2010). An APOS Analysis of Students’ Understanding of the Concept of a Limit of a Function. Pythagoras, 71(6), 41-52. https://doi.org/10.4102/pythagoras.v0i71.6.
Maharaj, A. (2013). An APOS Analysis of natural science students’ understanding of the derivatives, South African Journal of Education, 33(1), 1-19. https://doi.org/10.15700/saje.v33n1a458.
Mallet, D. (2007). Multiple representations for Systems of Linear Equations via the Computer Algebra system Maple. International Electronic Journal of Mathematics Education, 2(1), 16-31.
Merriweather, M., & Tharp, M. L. (1999). The effects of instruction with graphing calculators on how general mathematics students naturalistically solve algebraic problems. Journal of Computers in Mathematics and Science Teaching, 18(1), 7–22.
National Council of Teachers and Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. National Council of Teachers and Mathematics, Reston, VA.
Ng, W. L., Tan, W. C., & Ng, M. L. (2009). Teaching and learning calculus with the TI-Nspire: A design experiment. In T. Alwis, M. Majewski, & W.C. Yang (eds.), Proceedings of Asian Technology Conference in Mathematics (pp. 345–356). Beijing, China: Beijing Normal University.
Orton, A. (1983). Students’ Understanding of Differentiation, Educational Studies in Mathematics, 14(3), 235-250. https://doi.org/10.1007/BF00410540.
Özmantar, M. F., Akkoç, H., Bingölbali, E., Demir, S., & Ergene, B. (2010). Pre-Service Mathematics Teachers’ Use of Multiple Representations in Technology-Rich Environments. EURASIA Journal of Mathematics, Science and Technology Education, 6(1), 19-36. https://doi.org/10.12973/ejmste/75224.
Park, J. (2015). Is the derivative a function? If so, how do we teach it? Educational Studies in Mathematics, 89(2), 233–250. https://doi.org/10.1007/s10649-015-9601-7.
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, 1-23. https://doi.org/10.1007/s10763-017-9826-2.
Research and Educational Planning Organization. (2013). Secretariat of designing and producing the mathematics curriculum of Islamic Republic of Iran [In Persian].
Roorda, G., Vos, P. & Goedhart, M. (2009). Derivatives and applications; development of one student’s understanding. Proceedings of CERME 6. Retrieved from www.inrp.fr/editions/cerme6.
Sahin, Z., Erbas, A. K., & Yenmez, A. A. (2015). Relational Understanding of the Derivative Concept through Mathematical Modeling: A Case Study. EURASIA Journal of Mathematics, Science & Technology Education, 11(1), 177-188. https://doi.org/10.12973/eurasia.2015.1149a.
Samkova, L. (2012). Calculus of one and more variables with Maple. International Journal of Mathematical Education in Science and Technology, 43(2), 230-244. https://doi.org/10.1080/0020739X.2011.582248.
Sánchez-Matamoros, G., Fernández, C., & Llinares, S. (2015). Developing pre-service teachers’ noticing of students’ understanding of the derivative concept. International Journal of Science and Mathematics Education. 13(6), 1305–1329. https://doi.org/10.1007/s10763-014-9544-y.
Stewart, J. (2010). Calculus (7th Ed.). Brooks/Cole Cengage Learning, Mason.
Tall, D. (1993). Students’ Difficulties in Calculus, Plenary Address, Proceedings of ICME-7, 13–28, Québec, Canada.
Tall, D. (1996). Functions and calculus, in International Handbook of Mathematics Education, A. J. Bishop, K. Clement, J. Kilpatrick, and C. Laborde (Eds.), Dordrecht, The Netherlands: Kluwer Academic. https://doi.org/10.1007/978-94-009-1465-0_10.
Tall, D. (2010). A sensible approach to the calculus (Plenary talk), National and International Meeting on the Teaching of Calculus, Puebla, Mexico. Retrieved from http://homepages.warwick.ac.uk/staff/David.Tall/downloads.html.
Thompson, P. W. (1994). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics, 26(2), 229-274. https://doi.org/10.1007/BF01273664.
Tiwari, T. K. (2007). Computers graphics as an instructional aid in an introductory differential Calculus course. International Electronic Journal of Mathematics Education, 2(1), 32-48.
Uygur, T., & Özdaş, A. (2005). Misconceptions and difficulties with the chain rule, The Mathematics Education into the 21st century Project. Malaysia: University of Technology. Retrieved from http://math.unipa.it/~grim/21 _project/21_malasya_Uygur209-213_05.pdf.
Voskoglou, M. Gr. (2013). An application of the APOS/ACE approach in teaching the irrational numbers, Journal of Mathematical Sciences and Mathematics Education, 8(1), 30-47.
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.
Weller, K., Arnon, I., & Dubinsky, E. (2011). Pre-service teachers’ understanding of the relation between a fraction or integer and its decimal expansion: Strength and stability of belief. Canadian Journal of Science, Mathematics, and Technology Education, 11(2), 129–159. https://doi.org/10.1080/14926156.2011.570612.
Zandieh M. (2000). A theoretical framework for analyzing students understanding of the concept of derivative. In E. Dubinsky, A. H. Schoenfeld & J. Kaput, Research in Collegiate Mathematics Education (Vol IV, pp.103–127). Providence, RI: American Mathematical Society.

How to cite this article

APA

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

Vancouver

Borji V, Alamolhodaei H, Radmehr F. Application of the APOS-ACE Theory to improve Students’ Graphical Understanding of Derivative. EURASIA J Math Sci Tech Ed. 2018;14(7):2947-67. https://doi.org/10.29333/ejmste/91451

AMA

Borji V, Alamolhodaei H, Radmehr F. Application of the APOS-ACE Theory to improve Students’ Graphical Understanding of Derivative. EURASIA J Math Sci Tech Ed. 2018;14(7), 2947-2967. https://doi.org/10.29333/ejmste/91451

Chicago

Borji, Vahid, Hassan Alamolhodaei, and Farzad Radmehr. "Application of the APOS-ACE Theory to improve Students’ Graphical Understanding of Derivative". Eurasia Journal of Mathematics, Science and Technology Education 2018 14 no. 7 (2018): 2947-2967. https://doi.org/10.29333/ejmste/91451

Harvard

Borji, V., Alamolhodaei, H., and 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), pp. 2947-2967. https://doi.org/10.29333/ejmste/91451

MLA

Borji, Vahid et al. "Application of the APOS-ACE Theory to improve Students’ Graphical Understanding of Derivative". Eurasia Journal of Mathematics, Science and Technology Education, vol. 14, no. 7, 2018, pp. 2947-2967. https://doi.org/10.29333/ejmste/91451