A Phenomenographic Investigation of Middle School Pre-service Mathematics Teachers' Conceptions of Proof

Yıl 2021, Cilt: 15 Sayı: 1, 118 - 143, 27.06.2021

Yasemin Yılmaz Akkurt , Selda Yıldırım

https://doi.org/10.17522/balikesirnef.939068

Öz

The capability of pre-service teachers to teach mathematical reasoning depends on the quality of their proof conceptions. This qualitative study focuses proof conceptions of middle school pre-service mathematics teachers. To this end, this study employed a phenomenographic approach to identify the variation in pre-service teachers’ experience of proof. Analysis of semi-structured interviews revealed five qualitatively different categories: proof is (a) a way of problem-solving, (b) a means for understanding, (c) explaining thinking in a convincing way, d) validating conjectures using logical arguments, and (e) a means for discovery of mathematics. This study contributes to the pedagogical knowledge about a framework of proof conceptions. Results may be used to promote the quality of the mathematics teacher preparation programs.

Anahtar Kelimeler

phenomenography, pre-service teachers, proof conception

Ortaokul Matematik Öğretmen Adaylarının İspat Kavramlarının Fenomenografik Bir İncelemesi

Öz

Öğretmen adaylarının matematiksel akıl yürütmeyi öğretme yetenekleri sahip oldukları ispat kavramlarının kalitesine bağlıdır. Bu nitel çalışma, ortaokul matematik öğretmen adaylarının ispat kavramlarına odaklanmaktadır. Bu amaçla, bu çalışma öğretmen adaylarının ispat deneyimlerindeki farklılıkları belirlemek için fenomenografik bir yaklaşım kullanmıştır. Yarı yapılandırılmış görüşmelerin analizi, niteliksel olarak farklı beş kategori ortaya çıkarmıştır. Buna göre, ispat (a) bir problem çözme yoludur, (b) anlamanın bir aracıdır, (c) düşünmeyi ikna edici bir şekilde açıklamaktır, d) mantıksal argümanlar kullanarak varsayımları doğrulamaktır ve (e) matematiğin keşfi için bir araçtır. Bu çalışma, ispat kavramlarıyla ilgili pedagojik bilgiye katkıda bulunmaktadır. Sonuçlar, matematik öğretmeni hazırlık programlarının kalitesini artırmak için kullanılabilir.

Anahtar Kelimeler

fenomenografi, ispat kavramı, öğretmen adayı

Kaynakça

• Akerlind, G. S. (2005). Learning about phenomenography: Interviewing, data analysis and the qualitative research paradigm. In J.A. Bowden and P. Green (Eds.) Doing developmental phenomenography (pp. 63-74). Melbourne: RMIT University Press.
• Almeida, D. (2001). Pupils' proof potential. International Journal of Mathematical Education in Science and Technology, 32(1), 53-60. https://doi.org/10.1080/00207390119535
• Aylar, E., & Şahiner, Y. (2014). A study on teaching proof to 7th grade students. Procedia - Social and Behavioral Sciences, 116, 3427-3431. https://doi.org/10.1016/j.sbspro.2014.01.777
• Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407. https://doi.org/10.1177/0022487108324554
• 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. https://doi.org/10.1016/j.jmathb.2017.08.002
• Baştürk, S. (2010). First-year secondary school mathematics students' conceptions of mathematical proofs and proving. Educational Studies, 36(3), 283-298. https://doi.org/10.1080/03055690903424964
• Cibangu, S. K., & Hepworth, M. (2016). The uses of phenomenology and phenomenography: A critical review. Library & Information Science Research, 38(2), 148-160. https://doi.org/10.1016/j.lisr.2016.05.001
• Davies, B., Alcock, L., & Jones, I. (2021). What do mathematicians mean by proof? A comparative-judgement study of students’ and mathematicians’ views. Journal of Mathematical Behavior, 61, 100824. https://doi.org/10.1016/j.jmathb.2020.100824
• de Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17-24.
• de Villiers, M. (1999). Rethinking Proof with Sketchpad. Key Curriculum Press.
• Dickerson D. D., & Doerr, H. M. (2014).High school mathematics teachers’ perspectives on the purposes of mathematical proof in school mathematics. Mathematics Education Research Journal, 26(4), 711–733. https://doi.org/10.1007/s13394-013-0091-6
• González, C. (2010). What do university teachers think eLearning is good for in their teaching? Studies in Higher Education, 35(1), 61-78. https://doi.org/10.1080/03075070902874632
• Han, F., & Ellis, R. A. (2019). Using phenomenography to tackle key challenges in science education. Frontiers in Psychology, 10, 1-10. https://doi.org/10.3389/fpsyg.2019.01414
• Hanna, G. (1995). Challenges to the importance of proof. For the Learning of Mathematics, 15(3), 42-49. https://www.jstor.org/stable/40248188
• Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44(1-2), 5-23. https://doi.org/10.1023/A:1012737223465
• Herbst, P., & Balacheff, N. (2009). Proving and knowing in public: The nature of proof in a classroom. In D. A. Stylianou, M. L. Blanton, & E. J. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (pp. 40-63). Routledge.
• Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24 (4), 389-399. https://www.jstor.org/stable/3482651
• 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
• Lesseig, K., Hine, G., Na, G., & Boardman, K. (2019). Perceptions on proof and the teaching of proof: a comparison across pre-service secondary teachers in Australia, USA and Korea. Mathematics Education Research Journal, 31(4), 393–418. https://doi.org/10.1007/s13394-019-00260-7
• Likando, K. M., & Ngoepe, M. G. (2014).Investigating mathematics trainee teachers’ conceptions of proof writing in algebra: A case of one college of education in Zambia. Mediterranean Journal of Social Sciences, 5(14), 331-338. https://doi.org/10.5901/mjss.2014.v5n14p331
• Limberg, L. (2008). Phenomenography. In L. M. Given (Ed.), The SAGE encyclopedia of qualitative research methods (pp. 611-614).SAGE Publications.
• Makowski, M. B. (2020). The written and oral justifications of mathematical claims of middle school pre-service teachers. Research in Mathematics Education. https://doi.org/10.1080/14794802.2020.1777190
• 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/10.5951/jresematheduc.20.1.0041
• Marton, F. (1981). Phenomenography-describing conceptions of the world around us. Instructional Science, 10(2), 177-200. https://doi.org/10.1007/BF00132516
• Marton, F. (2000).The structure of awareness. In J. A. Bowden, & E. Walsh (Eds.), Phenomenography (pp. 102-116). RMIT Publishing.
• Marton, F., & Booth, S. (1997). Learning and awareness. Lawrence Erlbaum Associates.
• Marton, F., & Pang, M. F. (2008).The idea of phenomenography and the pedagogy of conceptual change. In S. Vosniadou (Ed.), International Handbook of Research on Conceptual Change (pp. 533-559). Routledge.
• Marton, F., & Pong, W. Y. (2005). On the unit of description in phenomenography. Higher Education Research and Development, 24(4), 335-348. https://doi.org/10.1080/07294360500284706
• Mingus, T. T. Y., & Grassl, R. M. (1999). Preservice teacher beliefs about proofs. School Science and Mathematics, 99(8), 438-444. https://doi.org/10.1111/j.1949-8594.1999.tb17506.x
• Miyazaki, M., Fujita, T., & Jones, K. (2017). Students' understanding of the structure of deductive proof. Educational Studies in Mathematics, 94(2), 223-239. https://doi.org/10.1007/s10649-016-9720-9
• Morris, A. (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
• Mueller, M. F. (2009). The co-construction of arguments by middle-school students. The Journal of Mathematical Behavior, 28(2-3), 138-149. https://doi.org/10.1016/j.jmathb.2009.06.003
• Pang, M. F. (2003). Two faces of variation: On continuity in the phenomenographic movement. Scandinavian Journal of Educational Research, 47(2), 145–156. https://doi.org/10.1080/00313830308612
• Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7(1), 5–41. https://doi.org/10.1093/philmat/7.1.5
• Sears, R. (2019). Proof schemes of pre-service middle and secondary mathematics teachers. Investigations in Mathematics Learning, 11(4), 258-274. https://doi.org/10.1080/19477503.2018.1467106
• Sfard, A. (2000). On reform movement and the limits of mathematical discourse. Mathematical Thinking and Learning, 2(3), 157-189.https://doi.org/10.1207/S15327833MTL0203_1
• Shifter, D. (2009).Representation-based proof in the elementary grades. In D. A. Stylianou, M. L. Blanton, & E. J. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (pp. 71-86). Routledge.
• Son, J.W., & Lee, M. Y. (2021). Exploring the relationship between preservice teachers’ conceptions of problem solving and their problem-solving performances. International Journal of Science and Mathematics Education, 19, 129-150. https://doi.org/10.1007/s10763-019-10045-w
• Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289-321. https://doi.org/10.2307/30034869
• Stylianides, G. J., & Stylianides, A. J. (2009). Facilitating the transition from empirical arguments to proof. Journal for Research in Mathematics Education, 40(3), 314-352. https://doi.org/10.5951/jresematheduc.40.3.0314
• Stylianides, G., Stylianides, A., & Shilling-Traina, L. N. (2013).Prospective teachers’ challenges in teaching reasoning and-proving. International Journal of Science and Mathematics Education, 11(6), 1463–1490. https://doi.org/10.1007/s10763-013-9409-9
• Stylianou, D. A., Blanton, M. L., & Rotou, O. (2015). Undergraduate students’ understanding of proof: Relationships between proof conceptions, beliefs, and classroom experiences with learning proof. International Journal of Research in Undergraduate Mathematics Education, 1(1), 91-134. https://doi.org/10.1007/s40753-015-0003-0
• Tanışlı, D. (2016). How do students prove their learning and teachers their teaching? Do teachers make a difference? Eurasian Journal of Educational Research, (66), 47-70. http://dx.doi.org/10.14689/ejer.2016.66.3
• Uygan, C.,Tanışlı, D., & Köse, N. Y. (2014).Research of pre-service elementary mathematics teachers’ beliefs in proof, proving processes and proof evaluation processes. Turkish Journal of Computer and Mathematics Education (TURCOMAT), 5(2), 137-157. https://doi.org/10.16949/turcomat.33155
• Varghese, T. (2009). Secondary-level student teachers' conceptions of mathematical proof. Issues in the Undergraduate Mathematics Preparation of School Teachers. http://www.k-12prep.math.ttu.edu
• Weber, K. (2010). Proofs that develop insight. For the Learning of Mathematics, 30(1), 32–36.
• Wilder, R.W. (1981). Mathematics as a cultural system, Pergamon, New York.
• Yates, C., Partridge, H. L., & Bruce, C. (2012). Exploring information experiences through phenomenography. Library and Information Research, 36(112), 96-119. https://doi.org/10.29173/lirg496
• Zaslavsky, O., Nickerson, S. D., Stylianides, A. J., Kidron, I., & Winicki-Landman, G. (2012). The need for proof and proving: Mathematical and Pedagogical Perspectives. In G. Hanna, & M. de Villiers (Eds.). Proof and Proving in Mathematics Education. The 19th ICMI study (pp. 215-229).Springer.
• Zeybek, Z. (2015). Prospective teachers' conceptions of proof. The Journal of Academic Social Sciences, 3(10),593-602. https://doi.org/10.16992/ASOS.583
• Zeybek, Z. (2017). Pre-service elementary teachers’ conceptions of counterexamples. International Journal of Education in Mathematics, Science and Technology (IJEMST), 5(4), 295-316. DOI:10.18404/ijemst.70986