Content Analysis of the APOS Theory Studies on Mathematics Education Conducted in Turkey and Internationally: A Meta-Synthesis Study

Yıl 2021, Cilt: 15 Sayı: 2, 404 - 428, 31.12.2021

Özgün Şefik Özge Erdem Uzun Şenol Dost

https://doi.org/10.17522/balikesirnef.1020526

Cited By: 2

Öz

Studies on the APOS Theory, which was developed within the context of conceptual understanding as one of the main aims of mathematics education, have been increasing in recent years. The content analysis of these studies can be useful both for the development of the APOS Theory and for identifying the needs for future research. This study aimed to review the APOS Theory studies in the field of mathematics education. For this purpose, the studies with national and international samples related to the APOS Theory were subjected to descriptive content analysis, and a systematic summary was presented. As a result of the analysis, three themes, namely the purpose of using the theory, the use of genetic decomposition, and the aim of the study were identified. It was determined that genetic decomposition, which is the main component of the APOS Theory, was not used in 13 percent of the studies reviewed. Although the theory was mainly used for data analysis, it can be said that not using genetic decomposition in data analysis is not compatible with the purpose of the theory. On the other hand, there are descriptive studies (39%) in the literature that determine the mental structures and mechanisms predicted by the APOS Theory rather than the structure of mathematical concepts. It can be said that such studies are weak in terms of the theory’s aim of reinforcing conceptual understanding. As a result, it has been revealed that, the APOS Theory has an important place in concept teaching.

Anahtar Kelimeler

APOS theory, conceptual understanding, concept teaching, mathematics education, meta sentez

Kaynakça

• Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Roa Fuentes, S., Trigueros, M., & Weller, K. (2014). APOS theory. New York, Heidelberg, Dordrecht, London: Springer.
• Au, W. (2007). High-stakes testing and curricular control: A qualitative metasynthesis. Educational researcher, 36(5), 258-267.
• Avcu, S., & Cetinkaya, B. (2019). An instructional unit for prospective teachers' conceptualization of geometric transformations as functions. International Journal of Mathematical Education in Science and Technology, 30. doi:10.1080/0020739x.2019.1699966
• Bansilal, S. (2011). Assessment reform in South Africa: opening up or closing spaces for teachers? Educational Studies in Mathematics, 78(1), 91-107. doi:10.1007/s10649-011-9311-8 Bansilal, S., Brijlall, D., & Trigueros, M. (2017). An APOS study on pre-service teachers' understanding of injections and surjections. Journal of Mathematical Behavior, 48, 22-37. doi:10.1016/j.jmathb.2017.08.002
• Bayraktar, F., Tutak, T., & İlhan, A. (2019). An Analysis of The Studies on The APOS Theory. Electronic Journal of Education Sciences, 8(16), 242-251.
• Borji, V., & Martinez-Planell, R. (2019). What does 'y is defined as an implicit function of x' mean?: An application of APOS-ACE. Journal of Mathematical Behavior, 56, 18. doi:10.1016/j.jmathb.2019.100739
• Brijlall, D., & Maharaj, A. (2015). Exploring Pre-service Teachers' Mental Constructions When Solving Problems Involving Infinite Sets. International Journal of Educational Sciences, 9(3), 273-281. Retrieved from <Go to ISI>://WOS:000209993200002
• Brijlall, D., & Ndlazi, N. J. (2019). Analysing engineering students' understanding of integration to propose a genetic decomposition. Journal of Mathematical Behavior, 55, 12. doi:10.1016/j.jmathb.2019.01.006
• Brijlall, D., & Ndlovu, Z. (2013). High school learners' mental construction during solving optimisation problems in Calculus: a South African case study. South African Journal of Education, 33(2), 18. Retrieved from <Go to ISI>://WOS:000327851800005
• Chimhande, T., Naidoo, A., & Stols, G. (2017). An analysis of grade 11 learners' levels of understanding of functions in terms of APOS theory. Africa Education Review, 14(3-4), 1-19. doi:10.1080/18146627.2016.1224562
• Clements, D. H., Battista, M. T., Sarama, J., & Swaminathan, S. (1997). Development of students' spatial thinking in a unit on geometric motions and area. The Elementary School Journal, 98(2), 171-186.
• Çalık, M., & Sözbilir, M. (2014). İçerik analizinin parametreleri [Parameters of Content Analysis]. Eğitim ve Bilim, 39(174). Çetin, İ. (2009). Students' understanding of limit concept: an APOS perspective. (Doctoral dissertation). Middle East Technical University, Ankara.
• Deniz, Ö. (2014). 8. sınıf öğrencilerinin gerçekçi matematik eğitimi yaklaşımı altında eğim kavramını oluşturma süreçlerinin APOS teorik çerçevesinde incelenmesi [Examination of 8th grade students' construction of the concept of slope based on realistic mathematics education in APOS framework]. (Master Thesis). Anadolu Üniversitesi, Eskişehir.
• Dubinsky, E. (1984). The cognitive effect of computer experiences on learning abstract mathematical concepts. Korkeakoulujen Atk-Vutiset, 2, 41-47.
• Dubinsky, E., & McDonald, M. A. (2001). Apos: A constructivist theory of learning in undergraduate mathematics education research. Teaching and Learning of Mathematics at University Level, 7, 275-282. Retrieved from <Go to ISI>://WOS:000224702200025
• Figueroa, A. P., Possani, E., & Trigueros, M. (2018). Matrix multiplication and transformations: an APOS approach. Journal of Mathematical Behavior, 52, 77-91. doi:10.1016/j.jmathb.2017.11.002
• Hannah, J., Stewart, S., & Thomas, M. (2016). Developing conceptual understanding and definitional clarity in linear algebra through the three worlds of mathematical thinking. Teaching Mathematics and Its Applications, 35(4), 216-235. doi:10.1093/teamat/hrw001
• Harel, G. (2017). The learning and teaching of linear algebra: Observations and generalizations. Journal of Mathematical Behavior, 46, 69-95. doi:10.1016/j.jmathb.2017.02.007
• Hollebrands, K. F. (2003). High school students’ understandings of geometric transformations in the context of a technological environment. The Journal of Mathematical Behavior, 22(1), 55-72.
• Maharaj, A. (2013). An APOS analysis of natural science students' understanding of derivatives. South African Journal of Education, 33(1), 19. Retrieved from <Go to ISI>://WOS:000327850500002
• Maharaj, A. (2015). A Framework to Gauge Mathematical Understanding: A Case Study on Linear Algebra Concepts. International Journal of Educational Sciences, 11(2), 144-153. Retrieved from <Go to ISI>://WOS:000209993700003
• Maharaj, A. (2018a). An Investigation into the Preparedness of Teachers to Teach Grade 12 Mathematics: A Case Study. International Journal of Educational Sciences, 21(1-3), 112-123. doi:10.31901/24566322.2018/21.1-3.775
• Maharaj, A. (2018b). Students' Understanding of Solving a System of Linear Equations Using Matrix Methods: A Case Study. International Journal of Educational Sciences, 21(1-3), 124-134. doi:10.31901/24566322.2018/21.1-3.774
• Makonye, J. P. (2017). Pre-service mathematics student teachers' conceptions of nominal and effective interest rates. Pythagoras, 38(1), 10. doi:10.4102/pythagoras.v38i1.307
• Martin, W., Loch, S., Cooley, L., Dexter, S., & Vidakovic, D. (2010). Integrating learning theories and application-based modules in teaching linear algebra. Linear Algebra and Its Applications, 432(8), 2089-2099. doi:10.1016/j.laa.2009.08.030
• Martinez-Planell, R., Gaisman, M. T., & McGee, D. (2017). Students' understanding of the relation between tangent plane and directional derivatives of functions of two variables. Journal of Mathematical Behavior, 46, 13-41. doi:10.1016/j.jmathb.2017.02.001
• Ministry of National Education [MoNE]. (2009). İlköğretim Matematik Dersi 1-5. Sınıflar Öğretim Programı [Elementary school mathematics curriculum (1-5th grades)]. Ankara: MEB Talim ve Terbiye Kurulu Başkanlığı
• Ministry of National Education [MoNE]. (2018a). İlköğretim matematik dersi öğretim programı (İlkokul ve Ortaokul 1, 2, 3, 4, 5, 6, 7 ve 8. Sınıflar) [Elementary school mathematics curriculum (Grades of 1,2,3,4,5,6,7 and 8]. Ankara: MEB Yayınları
• Ministry of National Education [MoNE]. (2018b). Ortaöğretim matematik dersi öğretim programı (9, 10, 11 ve 12. Sınıflar) [Secondary school mathematics curriculum (Grades of 9, 10, 11 and 12)]. Ankara: MEB Yayınları
• Moll, V. F., 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. doi:10.1007/s10649-015-9639-6
• Moon, K. (2019). Graphs of Two Variable Inequalities: Alternate Approaches to the Solution Test. Mathematics Enthusiast, 16(1-3), 107-126. Retrieved from <Go to ISI>://WOS:000484167900008
• National Council of Teachers of Mathematics [NCTM]. (1989). Curriculum and evaluation standards for school mathematics. USA: National Council of Teachers of Matematics.
• National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
• National Council of Teachers of Mathematics [NCTM]. (2014). Principles to actions: Ensuring mathematical success for all. USA: National Council of Teachers of Mathematics, Incorporated.
• Ndlovu, Z., & Brijlall, D. (2016). Pre-service Mathematics Teachers' Mental Constructions of the Determinant Concept. International Journal of Educational Sciences, 14(1-2), 145-156. Retrieved from <Go to ISI>://WOS:000402865300017
• Ndlovu, Z., & Brijlall, D. (2019). Pre-service mathematics teachers’ mental constructions when using Cramer’s rule. South African Journal of Education, 39(1), 1-13. doi:10.15700/saje.v39n1a1550
• Ponte, J. P. d. (1992). The history of the concept of function and some educational implications. The Mathematics Educator, 3-8.
• Portnoy, N., Grundmeier, T. A., & Graham, K. J. (2006). Students’ understanding of mathematical objects in the context of transformational geometry: Implications for constructing and understanding proofs. The Journal of Mathematical Behavior, 25(3), 196-207.
• Possani, E., Trigueros, M., Preciado, J. G., & Lozano, M. D. (2010). Use of models in the teaching of linear algebra. Linear Algebra and Its Applications, 432(8), 2125-2140. doi:10.1016/j.laa.2009.05.004
• Siyepu, S. W. (2015). Analysis of errors in derivatives of trigonometric functions. International Journal of Stem Education, 2, 16. doi:10.1186/s40594-015-0029-5
• Stenger, C., Weller, K., Arnon, I., Dubinsky, E., & Vidakovic, D. (2008). A search for a constructivist approach for understanding the uncountable set P(N). Revista Latinoamericana De Investigacion En Matematica Educativa-Relime, 11(1), 93-125. Retrieved from <Go to ISI>://WOS:000264753500004
• Trigueros, M., & Martinez-Planell, R. (2010). Geometrical representations in the learning of two-variable functions. Educational Studies in Mathematics, 73(1), 3-19. doi:10.1007/s10649-009-9201-5
• Trigueros, M., & Oktac, A. (2019). Task design in APOS Theory. Avances De Investigacion En Educacion Matematica(15), 43-56. doi:10.35763/aiem.v0i15.256
• Trigueros, M., & Possani, E. (2013). Using an economics model for teaching linear algebra. Linear Algebra and Its Applications, 438(4), 1779-1792. doi:10.1016/j.laa.2011.04.009
• Yanik, H. B., & Flores, A. (2009). Understanding rigid geometric transformations: Jeff's learning path for translation. The Journal of Mathematical Behavior, 28(1), 41-57.
• Yin, R. K. (2018). Case study research and applications: Design and methods (Sixth ed.). Los Angeles, London, New Delhi, Singapore, Washington DC, Melbourne: Sage Publications.