DEDUCTIVE OR INDUCTIVE? PROSPECTIVE TEACHERS’ PREFERENCE OF PROOF METHOD ON AN INTERMEDIATE PROOF TASK

Tatag Yuli Eko Siswono, Sugi Hartono, Ahmad Wachidul Kohar

Abstract


The emerging of formal mathematical proof is an essential component in advanced undergraduate mathematics courses. Several colleges have transformed mathematics courses by facilitating undergraduate students to understand formal mathematical language and axiomatic structure. Nevertheless, college students face difficulties when they transition to proof construction in mathematics courses. Therefore, this descriptive-explorative study explores prospective teachers' mathematical proof in the second semester of their studies. There were 240 pre-service mathematics teachers at a state university in Surabaya, Indonesia, determined using the conventional method. Their responses were analyzed using a combination of Miyazaki and Moore methods. This method classified reasoning types (i.e., deductive and inductive) and types of difficulties experienced during the proving. The results conveyed that 62.5% of prospective teachers tended to prefer deductive reasoning, while the rest used inductive reasoning. Only 15.83% of the responses were identified as correct answers, while the other answers included errors on a proof construction. Another result portrayed that most prospective teachers (27.5%) experienced difficulties in using definitions for constructing proofs. This study suggested that the analytical framework of the Miyazaki-Moore method can be employed as a tool to help teachers identify students' proof reasoning types and difficulties in constructing the mathematical proof.


Keywords


deductive-inductive reasoning; proving difficulties; mathematical proof; prospective teachers

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References


Almeida, D. (2000). A survey of mathematics undergraduates' interaction with proof: some implications for mathematics education. International Journal of Mathematical Education in Science and Technology, 31(6), 869-890. https://doi.org/10.1080/00207390050203360

Ayalon, M., & Even, R. (2008). Deductive reasoning: in the eye of the beholder. Educational Studies in Mathematics, 69(3), 235-247. https://doi.org/10.1007/s10649-008-9136-2

Baki, A. (2008). Mathematics education from theory to practice. Ankara: Harf Educational Publications

Balacheff, N. (2010). Bridging knowing and proving in mathematics: A didactical perspective. In G. Hanna, H. N. Jahnke & H. Pulte (Eds.), Explanation and Proof in Mathematics––Philosophical and Educational Perspectives (vol. 45, pp. 115-135). New York: Springer. https://doi.org/10.1007/978-1-4419-0576-5_9

Blanton, M. L., Stylianou, D. A., & David, M. M. (2003). The nature of scaffolding in undergraduate students’ transition to mathematical proof. In N. Pateman, B. J. Dougherty, & J. Zilliox (Eds.), Proceedings of the 27th Annual Meeting of the International Group for Psychology in Mathematics Education (vol. II, pp. 113-120). Honolulu: University of Hawaii.

Buchbinder, O., & McCrone, S. (2020). Preservice teachers learning to teach proof through classroom implementation: Successes and challenges. The Journal of Mathematical Behavior, 58, 100779. https://doi.org/10.1016/j.jmathb.2020.100779

Carrillo-yañez, J., Climent, N., Montes, M., Contreras, L. C., Flores-medrano, E., Escudero-ávila, D., ... & ribeiro, M. (2018). The mathematics teacher’s specialised knowledge (MTSK) model. Research in Mathematics Education, 20(3), 236-253. https://doi.org/10.1080/14794802.2018.1479981

Chin, E., & Lin, F. (2009). A comparative study on junior high school students’ proof conceptions in algebra between Taiwan and the UK. Journal of Mathematics Education, 2(2), 52-67.

Clark, M., & Lovric, M. (2008). Suggestion for a theoretical model for secondary–tertiary transition in mathematics. Mathematics Education Research Journal, 20(2), 25–37. https://doi.org/10.1007/BF03217475

Christou, C., & Papageorgiou, E. (2007). A framework of mathematics inductive reasoning. Learning and Instruction, 17(1), 55-66. https://doi.org/10.1016/j.learninstruc.2006.11.009

De Villiers, M. D. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17-24.

Demiray, E., & Bostan, M. I. (2017). An investigation of pre-service middle school mathematics teachers’ ability to conduct valid proofs, methods used, and reasons for invalid arguments. International Journal of Science and Mathematics Education, 15(1), 109-130. https://doi.org/10.1007/s10763-015-9664-z

Dickerson, D. S., & Pitman, D. J. (2016). An examination of college mathematics majors’ understandings of their own written definitions. The Journal of Mathematical Behavior, 41, 1-9. https://doi.org/10.1016/j.jmathb.2015.11.001

Edwards, B. S., & Ward, M. B. (2004). Surprises from mathematics education research: Student (mis) use of mathematical definitions. The American Mathematical Monthly, 111(5), 411- 424. https://doi.org/10.1080/00029890.2004.11920092

Epp, S. S. (2003). The role of logic in teaching proof. The American Mathematical Monthly, 110(10), 886-899. https://doi.org/10.2307/3647960

Hanna, G., De villiers, M., Arzarello, F., Dreyfus, T., Durand- Guerrier, V., Jahnke, N.H., (...) Yevdokimov, O. (2009). ICMI Study 19: Proof and Proving in Mathematics Education (Discussion Document). In Lin, Fou-Lai (Eds.). Proceeding of The ICMI Study 19 Conference: Proof and Proving in Mathematics Education (vol.1). Taipei: The Department of Mathematics, National Taiwan Normal University

Jones, K. (1997). Student-teachers’ conceptions of mathematical proof. Mathematics Education Review, 9, 21–32.

Kögce, D., Aydin, M. & Yildiz, C. (2010). The views of high school student about proof and their levels of proof (the case of Trabzon). Procedia Social and Behavioral Sciences, 2, 2544-2549. https://doi.org/10.1016/j.sbspro.2010.03.370

Knuth, E. J. (2002a). Teachers’ conceptions of proof in the context of secondary school mathematics. Journal of Mathematics Teacher Education, 5(1), 61-88. https://doi.org/10.1023/A:1013838713648

Knuth, E. J. (2002b). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379–405. https://doi.org/10.2307/4149959

Landis, J. R., & Koch, G. G. (1977). The measurement of observer agreement for categorical data. Biometrics, 33(1), 159-174. https://doi.org/10.2307/2529310

Lee, H. S. (2002). Optimal consensus of fuzzy opinions under group decision making environment. Fuzzy sets and systems, 132(3), 303-315. https://doi.org/10.1016/S0165-0114(02)00056-8

Martin, W. G., & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 20(1), 41–51. https://doi.org/10.2307/749097

Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis: An expanded sourcebook. Thousand Oaks, CA: Sage

Miyazaki, M. (2000). Levels of proof in lower secondary school mathematics. Educational Studies in Mathematics, 41(1), 47-68. https://doi.org/10.1023/A:1003956532587

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

Moral?, S., U?urel, I., Türnüklü, E. B. ve Ye?ildere, S. (2006). Matematik ö?retmen adaylar?n?n ispat yapmaya yönelik görü?leri. Kastamonu E?itim Dergisi, 14(1), 147-160.

Morris, A. K. (2002). Mathematical reasoning: Adults' ability to make the inductive-deductive distinction. Cognition and Instruction, 20(1), 79-118. https://doi.org/10.1207/S1532690XCI2001_4

Morselli, F. (2006). Use of examples in conjecturing and proving: An exploratory study. International Group for the Psychology of Mathematics Education, 4, 185.

National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.

Ndemo, Z. (2019). Flaws in Proof Constructions of Postgraduate Mathematics Education Student Teachers. Journal on Mathematics Education, 10(3), 379-396. https://doi.org/10.22342/jme.10.3.7864.379-396

Ozdemir, E., & Ovez, F. T. D. (2012). A research on proof perceptions and attitudes towards proof and proving: Some implications for elementary mathematics prospective teachers. Procedia-Social and Behavioral Sciences, 46, 2121-2125. https://doi.org/10.1016/j.sbspro.2012.05.439

Özer, Ö., & Arõkan, A. (2002). Students’ levels of doing proof in high school mathematics classes. Paper presented at the meeting of 5th National Science and Mathematics Education Congress (pp. 1083-1089). Ankara: Middle East Technical University.

Selden, A., & Selden, J. (2007). Overcoming students’ difficulties in learning to understand and construct proofs (Report No. 2007-1). Cookeville: Mathematics Department, Tennesse Technological University.

Selden, J., Benkhalti, A., & Selden, A. (2014). An analysis of transition-to-proof course students’ proof constructions with a view towards course redesign. In Proceedings of the 17th Annual Conference on Research in Undergraduate Mathematics Education (pp. 246-259).

Shaker, H., & Berger, M. (2016). Students’ difficulties with definitions in the context of proofs in elementary set theory. African Journal of Research in Mathematics, Science and Technology Education, 20(1), 80-90. https://doi.org/10.1080/10288457.2016.1145449

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14. https://doi.org/10.3102%2F0013189X015002004

Simon, M. A., & Blume, G. W. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. The Journal of Mathematical Behavior, 15(1), 3-31. https://doi.org/10.1016/S0732-3123(96)90036-X

Smith, J. C. (2006). A sense-making approach to proof: Strategies of students in traditional and problem-based number theory courses. The Journal of Mathematical Behavior, 25(1), 73-90. https://doi.org/10.1016/j.jmathb.2005.11.005

Stavrou, S. G. (2014). Common Errors and Misconceptions in Mathematical Proving by Education Undergraduates. In the Undergraduate Mathematics Preparation of School Teachers: The Journal, 1, 1-8

Steele, M. D., & Rogers, K. C. (2012). Relationships between mathematical knowledge for teaching and teaching practice: The case of proof. Journal of Mathematics Teacher Education, 15(2), 159-180. https://doi.org/10.1007/s10857-012-9204-5

Stylianides, A. J. (2007). The notion of proof in the context of elementary school mathematics. Educational Studies in Mathematics, 65, 1-20. https://doi.org/10.1007/s10649-006-9038-0




DOI: https://doi.org/10.22342/jme.11.3.11846.417-438

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Journal on Mathematics Education
Doctoral Program on Mathematics Education
Faculty of Teacher Training and Education, Universitas Sriwijaya
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