APOS analysis on cognitive process in mathematical proving activities

Syamsuri Syamsuri, Indiana Marethi

Abstract


Thinking is very necessary in learning mathematics, both at school and college level. Several studies have attempted to reveal students' thinking in learning mathematics at college. This article aims to describe the mental structure that occurs when constructing mathematical proofs in terms of APOS theory. The APOS theory has been widely used in analyzing the formation of mathematical concepts in universities. This research explores a thinking process in proof constructing. It uses a qualitative approach. The research was conducted on 26 students majored in mathematics education in public university at Banten, Indonesia. The consideration of that was because the students were able to think a formal proof in mathematics. Results show that there are two types of thinking process in mathematical proving activities, namely:  the deductive-holistic and the inductive-partial type of thinking process. Based on the results, some suitable learning activities should be designed to support the construction of these mental categories.

Keywords


cognitive process; proving; mental structure; mental mechanisms; APOS Theory

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References


Arnon, I., Cottrill, J., Dubinsky, E., Oktac, A., Fuentes, S. R., Trigueros, M., & Weller, K. (2013). APOS theory : A framework for research and curriculum development in mathematics education. New York: Springer.

Asiala, M., Cottrill, J., Dubinsky, E., & Schwingendorf, K. E. (1997). The development of students’ graphical understanding of the derivative. The Journal of Mathematical Behavior, 16(4), 399–431.

Baker, D., & Campbell, C. (2004). Fostering the development of mathematical thinking : Observations from a proofs course. PRIMUS : Problems, Resources, and Issues in Mathematics Undergraduate Studies, 14(4), 345–353.

Balacheff, N. (1988). Aspects of proof in pupils’ practice on school mathematics. In D. Pimm (Ed.), Mathematics, Teacher and Children (pp. 216–235). London: Holdder & Stoughton.

Cresswell, W. J. (2012). Educational research : planning, conducting, and evaluating quantitative and qualitative research (4th edition). Boston: Pearson Education, Inc.

Dreyfus, T. (2002). Advanced mathematical thinking processes. In D. O. Tall (Ed.), Advanced mathematical thinking (pp. 25–41). Dordrecht: Springer.

Dubinsky, E. D., & McDonald, M. A. (2001). APOS : A constructivist theory of learning in undergraduate mathematics education research. In D. Holton, M. Artigue, U. Kirchgräber, J. Hillel, M. Niss, & A. Schoenfeld (Eds.), The teaching and learning of mathematics at university level (Vol. 7, pp. 275–282). Netherlands: Kluwer Academic Publishers.

Galbraith, P. L. (1981). Aspects of proving : A clinical investigation of process. Educational Studies in Mathematics, 12(1), 1–28.

García-Martínez, I., & Parraguez, M. (2017). The basis step in the construction of the principle of mathematical induction based on APOS theory. Journal of Mathematical Behavior, 46, 128–143.

Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington DC: National Academies Press.

Lee, K. S. (2016). Students’ proof schemes for mathematical proving and disproving of propositions. Journal of Mathematical Behavior, 41, 26–44.

Mason, J., Burton, L., & Stacey, K. (2010). Thinking mathematically. Dorset: Pearson.

Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27(3), 249–266.

NCTM. (2000). Principles and standards for school mathematics. Boston: The National Council of Teachers of Mathematics.

Recio, A. M., & Godino, J. D. (2001). Institutional and personal meanings of mathematical proof. Educational Studies in Mathematics, 48(1), 83–99.

Salgado, H., & Trigueros, M. (2015). Teaching eigenvalues and eigenvectors using models and APOS theory. Journal of Mathematical Behavior, 39, 100–120.

Sowder, L., & Harel, G. (2003). Case studies of mathematics majors’ proof understanding, production, and appreciation. Canadian Journal of Science, Mathematics and Technology Education, 3(2), 251–267.

Stylianides, A. J., & Stylianides, G. J. (2009). Proof constructions and evaluations. Educational Studies in Mathematics, 72(2), 237–253.

Syamsuri, S., Marethi, I., & Mutaqin, A. (2018). Understanding on strategies of teaching mathematical proof for undergraduate students. Cakrawala Pendidikan, XXXVII(2), 282-293.

Syamsuri, Purwanto, Subanji, & Irawati, S. (2016). Characterization of students formal-proof construction in mathematics learning. Communications in Science and Technology, 1(2), 42–50.

Syamsuri, Purwanto, Subanji, & Irawati, S. (2017). Using APOS theory framework : Why did students unable to construct a formal proof ? International Journal on Emerging Mathematics Education, 1(2), 135–146.

Syamsuri, & Santosa, C. (2017). Karakteristik pemahaman mahasiswa dalam mengonstruksi bukti matematis. Jurnal Review Pembelajaran Matematika, 2(2), 131–143.

Tall, D. O. (2010). Perception, operations and proof in undergraduate mathematics. Community for Undergraduale Learning in the Mathematical Sciences (CULMS) Newsletter, 2, 21–28.

Tall, D. O. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5–24.

Van Dormolen, J. (1977). Learning to understand what giving a proof really means. Educational Studies in Mathematics, 8(1), 27–34.

Varghese, T. (2011). Balacheff ’ s 1988 taxonomy of mathematical proofs. Eurasia Journal of Mathematics, Science & Technology Education, 7(3), 181–192.

Weber, K. (2010). Mathematics majors’ perceptions of conviction, validity, and proof. Mathematical Thinking and Learning, 12, 306–336.

Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56, 209–234.

Weller, K., Arnon, I., & Dubinsky, E. (2011). Preservice teachers’ understandings 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.

Weller, K., Clark, J., & Dubinsky, E. (2003). Student performance and attitudes in courses based on APOS theory and the ACE teaching cycle. Research in Collegiate Mathematics Education, 12, 97–131.




DOI: https://doi.org/10.18860/ijtlm.v1i1.5613

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