Preservice Mathematics Teachers’ Ability to Perform the Mathematizing Process: The Cylinder Packing Problem

Yıl 2022, Cilt: 9 Sayı: 6, 130 - 155, 01.11.2022

Nuray Çalışkan Dedeoğlu

https://doi.org/10.17275/per.22.132.9.6

Öz

The present study aims to examine the preservice middle school mathematics teachers’ ability to perform the process of mathematizing and to identify their competencies within this context. For this purpose, the study was conducted with 43 preservice teachers attending a state university. The research method used is descriptive research. As the data collection tool, a real-life problem called "cylinder packing problem" was presented to preservice teachers, with a view to finding the optimal and lowest-cost packing scheme for a hazelnut grower. The solutions presented by the preservice teachers were subjected to content analysis as well as descriptive analysis. The preservice teachers’ competencies were examined to see their level of effectiveness in the use of notations that are indicators of mathematizing, in achieving mathematical results in the process of problem-solving. The research findings indicate that the preservice teachers generally lacked sufficient comprehension of the problem in the real-life contexts and could not perform the requirements of important skills such as using notations, mathematical models and generalization method which are important in making abstract inferences in the mathematizing process. In light of these findings, further studies to contribute to preservice teachers’ mathematical sophistication levels are recommended in order to improve their competencies in mathematizing.

Anahtar Kelimeler

mathematizing, real-life problem, middle school mathematics, preservice mathematics teachers

Teşekkür

I would like to thank to the preservice mathematics teachers that participated in this study.

Kaynakça

• Alkan, H., & Bukova Güzel, E. (2005). Öğretmen adaylarında matematiksel düşünmenin gelişimi [Development of mathematical thinking in the student teachers]. Gazi University Journal of Gazi Educational Faculty, 25(3), 221-236.
• Archambeault, B. (1993). Holistic mathematics instruction: interactive problem solving and real-life situations help learners understand math concepts. Adult Learning, 5(1), 21-23. doi:10.1177/104515959300500111
• Basden, J., Boone, S., Fetter, A., Koenig, J., Lanius, C., Mabbott, A., et al. (n. d.). Encouraging mathematical thinking: Discourse around a rich problem. Retrieved from http://www.toolkitforchange.org/toolkit/documents/569_43_ttl_encouragemath.pdf
• Birgin, E.G., Martı́nez J.M., & Ronconi, D.P. (2005). Optimizing the packing of cylinders into a rectangular container: A nonlinear approach. European Journal of Operational Research, 160(1), 19-33. doi:10.1016/j.ejor.2003.06.018
• Boaler, J. (1993). The role of contexts in the mathematics classroom: Do they make mathematics more "real"?. For the Learning of Mathematics, 13(2), 12-17.
• Bozkurt, İ., Kozaklı Ülger, T., & Altun, M. (2019). Öğretmen adaylarının benzerlik konusu uygulamaları: Realistik matematik eğitimi açısından bakış [Prospective teachers' similarity applications: A perspective on realistic mathematics education]. Journal of Science, Mathematics, Entrepreneurship and Technology Education, 2(2), 115-133.
• Castillo, I, Kampas, F. J., & Pintér, J. D. (2008). Solving circle packing problems by global optimization: Numerical results and industrial applications. European Journal of Operational Research, 191(3), 786–802. doi:10.1016/j.ejor.2007.01.054
• Clarke, D., & Roche, R. (2009). Using mathematical tasks built around “real” contexts: Opportunities and challenges for teachers and students. Australian Primary Mathematics Classroom, 14(2), 24-31.
• Cohen, L., Manion, L., & Morrison, K. (2007). Research methods in education (6th Ed.). New York: Routledge.
• Davydov, V. V. (1990). Types of generalization in instruction: logical and psychological problems in the structuring of school curricula (J. Teller, Trans.). Soviet Studies in Mathematics Education (Volume 2). Reston VA: National Council of Teachers of Mathematics.
• Dumitrascu, G. (2017). Understanding the process of generalization in mathematics through activity theory. International Journal of Learning, Teaching and Educational Research, 16(12), 46-69. doi:10.26803/ijlter.16.12.4
• Fasano, G. (2014). Solving non-standard packing problems by global optimization and heuristics. Cham: Springer International Publishing. doi:10.1007/978-3-319-05005-8
• Freudenthal, H. (1968). Why to teach mathematics so as to be useful. Educational Studies in Mathematics, 1(1), 3-8.
• Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel Publishing Company.
• Freudenthal, H. (2002a). Didactical phenomenology of mathematical structures. New York: Kluwer Academic Publishers.
• Freudenthal, H. (2002b). Revisiting mathematics education. China lectures. New York: Kluwer Academic Publishers.
• Gordon, P. J. (1962). Heuristic problem solving: You can do it. Business Horizons, 5(1), 43-53.
• Gravemeijer, K. (2008). RME theory and mathematics teacher education, In D. Tirosh & T. Wood (Eds.), The International Handbook of Mathematics Teacher Education: Tools and Processes in Mathematics Teacher Education (pp. 283–302). Rotterdam: Sense Publishers. doi:10.1163/9789087905460_014
• Gravemeijer, K., & Terwel, J (2000). Hans Freudenthal: A mathematician on didactics and curriculum theory. Journal of Curriculum Studies, 32(6), 777-796. doi:10.1080/00220270050167170
• Hammack, R. (2018). İspat yöntemleri [Book of proof]. (M. Dağlı & O. Ölmez, Trans.). Retrieved from https://www.people.vcu.edu/~rhammack/BookOfProof/Translations/Turkish.pdf
• Joye-Bortolotti, R., & Vilmart, F. (2013). La feuille de format A4 et ses cylindres [A4 format sheet and its cylinders]. Retrieved from https://www.unige.ch/semainedesmaths/semaine-des-maths-2014/activites-pour-le-cycle-3/la-feuille-de-format-a4-et-ses-cylindres/
• Kabael, T., & Ata Baran, A. (2019). Ortaokul matematik öğretmeni adaylarının matematik okuryazarlığı performanslarının ve matematik okuryazarlığına ilişkin görüşlerinin incelenmesi [Investigation of prospective middle school mathematics teachers’ performances in, and conceptions of mathematical literacy]. Journal of Education in Eskisehir Osmangazi University Turkic World Apply and Research Center, 4(2), 51-67.
• Kabael, T., & Barak, B. (2016). Ortaokul matematik öğretmeni adaylarının matematik okuryazarlık becerilerinin PİSA soruları üzerinden incelenmesi [Research of middle school pre-service mathematics teachers’ mathematical literacy on PISA items]. Turkish Journal of Computer and Mathematics Education, 7(2), 321-349.
• Kant, D., & Sarikaya, D. (2021). Mathematizing as a virtuous practice: Different narratives and their consequences for mathematics education and society. Synthese, 199, 3405–3429. doi:10.1007/s11229-020-02939-y
• Knupfer, N. N., & McLellan, H. (1996). Descriptive Research Methodologies. In D. H. Jonassen (Ed.), Handbook of Research for Educational Communications and Technology: a Project of the Association for Educational Communications and Technology (pp. 1196-1212). New York: Macmillan Library Reference USA.
• Lady, A., Utom, B., & Lovi, C. (2018). Improving mathematical ability and student learning outcomes through realistic mathematic education (RME) approach. International Journal of Engineering & Technology, 7(2.10), 55-57. doi:10.14419/ijet.v7i2.10.10954
• Lai, M. Y., Kinnear, V., & Fung, C. I. (2019). Teaching mathematics for understanding in primary schools: Could teaching for mathematising be a solution?. International Journal for Mathematics Teaching and Learning, 20(1), 1-17.
• Mason J. (1996). Expressing generality and roots of algebra. In N. Bernarz, C. Kieran, & L. Lee (Eds). Approaches to algebra. Mathematics Education Library, Vol 18. Dordrecht: Springer. doi:10.1007/978-94-009-1732-3_5
• Mason, J., Burton, L., & Stacey, K. (2010). Thinking mathematically (2nd Ed.). Harlow: Pearson Education.
• Mialaret, G. (2004). Méthodes de recherche en sciences de l’éducation [Research methods in educational sciences]. Paris: Presses Universitaires de France.
• Ministry of National Education [MoNE], (2008). Matematik öğretmeni özel alan yeterlikleri [Mathematics teacher special field competences]. Ankara: MEB OYEGM.
• Ministry of National Education [MoNE]. (2018). Matematik dersi öğretim programı (ilkokul ve ortaokul 1, 2, 3, 4, 5, 6, 7 ve 8. sınıflar) [Mathematics curriculum (primary and middle school grade levels: 1, 2, 3, 4, 5, 6, 7 and 8]. Ankara: MEB.
• Mousoulides, N., & Sriraman, B. (2014). Heuristics in mathematics education. In S. Lerman (Ed.), Encyclopedia of Mathematics Education (pp. 253-255). Dordrecht: Springer. doi:10.1007/978-94-007-4978-8_172.
• Polya, G. (1973). How to solve it (2nd Ed.). Princeton, NJ: Princeton University Press.
• Rasmussen, C., Zandieh, M., King, K., & Teppo, A. (2005). Advancing mathematical activity: A practice-oriented view of advanced mathematical thinking. Mathematical Thinking and Learning, 7(1), 51-73. doi:10.1207/s15327833mtl0701_4
• Rosales, A. C. (2015). Mathematizing: an emergent math curriculum approach for young children. St. Paul, MN: Redleaf Press.
• Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp. 334-370). New York: MacMillan. doi:10.1177/002205741619600202
• Seaman, C. E., & Szydlik, J. E. (2007). Mathematical sophistication among preservice elementary teachers. Journal of Mathematics Teacher Education, 10, 167-182. doi:10.1007/s10857-007-9033-0
• Suh, J.M., Matson, K., & Seshaiyer, P. (2017). Engaging elementary students in the creative process of mathematizing their world through mathematical modeling. Education Sciences. 7(2), 62. doi:10.3390/educsci7020062
• Tabak, S. (2019). Türkiye’de “gerçekçi matematik eğitimi”ne ilişkin araştırma eğilimleri: Tematik içerik analizi çalışması [Research tendencies in Turkey on the “realistic mathematics education”: A research of thematic content analysis]. Ahi Evran University Journal of Kırşehir Education Faculty, 20(2), 481-526.
• Tall, D. (2006). A theory of mathematical growth through embodiment, symbolism and proof. 11. Annales de Didactique et de Sciences Cognitives (pp. 195-215). Strasbourg: IREM.
• Tall, D. (Ed.). (1991). Advanced mathematical thinking (Vol. 11). Dordrecht: Springer.
• Uygur-Kabael, T. (2017). Transfer skills of middle school pre-service mathematics teachers from informal to formal mathematical language: Turkey and United States cases. Hacettepe University Journal of Education, 32(4), 1013-1031. doi:10.16986/HUJE.2016023349
• Van den Heuvel-Panhuizen M., & Drijvers, P. (2014). Realistic mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education: With 67 figures and 9 tables (pp. 253-255). Dordrecht: Springer. doi:10.1007/978-3-030-15789-0_170
• Yerushalmy, M. (1997). Mathematizing verbal descriptions of situations: A language to support modeling. Cognition and Instruction, 15(2), 207-264. doi:10.1207/s1532690xci1502_3
• Yıldırım, C. (2008). Matematiksel düşünme [Mathematical thinking] (5th Ed.). Istanbul: Remzi Kitabevi.