Archives
Current issue
About
About us
Aims and Scope
Abstracting and Indexing
Editorial Office
Open Access Policy
Publication Ethics
Journal History
Publisher
Follow us: Twitter
Contact
For Authors
Editorial Policy
Peer Review Policy
Manuscript Preparation Guidelines
Copyright & Licencing
Publication Fees
Fee Waiver Policy for Doctoral Students
EJMSTE Language Editing Service
More
Statistics
Special Issues
Special Issue Announcements
Published Special Issues
RESEARCH PAPER
Application of the APOS-ACE Theory to improve Students’ Graphical Understanding of Derivative
1
Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, IRAN
Online publication date: 2018-05-11
Publication date: 2018-05-11
EURASIA J. Math., Sci Tech. Ed 2018;14(7):2947-2967
KEYWORDS
teaching and learningderivativegraphs of functionstangent at a point of a graphAPOS theorygenetic decompositionACE cycleMaple software
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.
REFERENCES (57)
1.
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.
2.
Andresen, M. (2007), Introduction of a new construct: The conceptual tool “Flexibility”, The Montana Mathematics Enthusiast, 4(2), 230-250.
3.
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-....
4.
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....
5.
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-....
6.
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.
7.
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....
8.
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.
9.
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....
10.
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/10.1080/002073....
11.
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....
12.
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.
13.
Borji, V., & Voskoglou, M.Gr. (2017), Designing an ACE Approach for Teaching the Polar Coordinates, American Journal of Educational Research, 5(3), 303-309.
14.
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/euras....
15.
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.
16.
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-....
17.
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/10.2307/300348....
18.
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/ejmst....
19.
Dubinsky, E., & Leron, U. (1994). Learning abstract algebra with ISETL. New York: Springer. https://doi.org/10.1007/978-1-....
20.
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.
21.
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/10.1080/149261....
22.
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.
23.
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....
24.
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.
25.
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.
26.
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.
27.
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.
28.
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.
29.
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.
30.
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.
31.
Maharaj, A. (2005). Investigating the Senior Certificate Mathematics examination in South Africa: Implications for teaching (Doctoral Thesis), Pretoria, University of South Africa.
32.
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/pythag....
33.
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.....
34.
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.
35.
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.
36.
National Council of Teachers and Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. National Council of Teachers and Mathematics, Reston, VA.
37.
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.
38.
Orton, A. (1983). Students’ Understanding of Differentiation, Educational Studies in Mathematics, 14(3), 235-250. https://doi.org/10.1007/BF0041....
39.
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/ejmst....
40.
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....
41.
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....
42.
Research and Educational Planning Organization. (2013). Secretariat of designing and producing the mathematics curriculum of Islamic Republic of Iran [In Persian].
43.
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.
44.
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/euras....
45.
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/10.1080/002073....
46.
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....
48.
Tall, D. (1993). Students’ Difficulties in Calculus, Plenary Address, Proceedings of ICME-7, 13–28, Québec, Canada.
49.
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....
50.
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....
51.
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/BF0127....
52.
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.
53.
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.
54.
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.
55.
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....
56.
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/10.1080/149261....
57.
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.
Share
RELATED ARTICLE
| eISSN: | 1305-8223 |
| ISSN: | 1305-8215 |
We process personal data collected when visiting the website. The function of obtaining information about users and their behavior is carried out by voluntarily entered information in forms and saving cookies in end devices. Data, including cookies, are used to provide services, improve the user experience and to analyze the traffic in accordance with the Privacy policy. Data are also collected and processed by Google Analytics tool (more).
You can change cookies settings in your browser. Restricted use of cookies in the browser configuration may affect some functionalities of the website.
You can change cookies settings in your browser. Restricted use of cookies in the browser configuration may affect some functionalities of the website.
