Öğrencilerin Matematiksel Anlamalarını Geliştirmek için Ön Bilgilerin Kullanımı: Bir Fonksiyon olarak Öteleme

Year 2022, , 467 - 494, 30.06.2022

Hilal Gülkılık

https://doi.org/10.53444/deubefd.1079496

Abstract

Matematik eğitimi araştırmacıları ve öğretim programı geliştiricileri, matematik öğretiminin öğrencilerin ön bilgileri dikkate alınarak tasarlanması gerektiği konusunda hemfikirdir. Bu araştırmanın amacı, öğrencilerin herhangi bir matematiksel konuyla ilgili matematiksel anlamalarını geliştirmek için, ön bilgilerin öğretim dizaynında nasıl kullanılabileceğine dair bir yaklaşım sunmaktır. Konu olarak, okul matematiği ve matematik öğretmen adaylarının eğitimi için önemli konularından birisi olan öteleme seçilmiş ve Pirie-Kieren teorisinde yer alan geriye katlamalar ile öğretim müdahaleleri temel alınarak bir eylem araştırması dizayn edilmiştir. Katılımcılar, bir devlet üniversitesinin Matematik Öğretmenliği programının dördüncü sınıfında öğrenim gören 28 matematik öğretmen adayıdır. Araştırmanın verileri, öteleme dersleri başlamadan önce uygulanan ön teste katılımcıların verdiği yazılı cevaplar, dersler boyunca gerçekleşen matematiksel tartışmalara yönelik kayıtlar, ders gözlem notları ile dersi yürüten eğitimcilerin ders öncesi veya sonrası görüşmelerinde alınan notlardan oluşturmaktadır. Dersler boyunca, öğretmen adaylarının ön bilmeleri üzerinde çalışmalarını sağlayan geriye katlamaları teşvik eden öğretim müdahalelerin, adayların ötelemeyi bir fonksiyon olarak anlamlandırmalarına yardımcı olduğu belirlenmiştir. Araştırmanın, öğrencilerin matematiksel anlamalarını desteklemek amacıyla ön bilmeleri referans alarak öğretim tasarlamak isteyen matematik eğitimcilerine fikir vereceği düşünülmektedir.

Keywords

Matematik eğitimi, matematiksel anlama, ön bilgi

Using Prior Knowledge to Advance Students’ Mathematical Understanding: Geometric Translation as a Function

Abstract

Mathematics education researchers and curriculum developers agree that instruction should be designed by considering students’ prior knowledge. The purpose of this research was to present an approach to how prior knowledge can be used in instructional design to advance students’ mathematical understanding of any mathematical subject. Translation was chosen as one of the crucial subjects for school mathematics and the education of pre-service mathematics teachers. An action research was designed drawing on the folding backs and interventions in the Pirie-Kieren theory. The participants were 28 pre-service secondary mathematics teachers studying in the fourth year of the Mathematics Teaching program at a state university. The data consists of the written answers given by the participants to the pre-test applied before the translation lessons, the recordings of the mathematical discussions that took place during the lessons, the lesson observation notes, and the notes taken during the pre-or post-lesson interviews of the instructors. Interventions that encouraged pre-service teachers to work on their prior knowings through folding backs helped them understand translation as a function. The research has the potential to provide an approach to mathematics educators who want to design instruction based on prior knowledge to support students’ mathematical understanding.

Keywords

Mathematics education, mathematical understanding, prior knowledge, transformations

References

• Ada, T., & Kurtulus, A. (2010). Students’ misconceptions and errors in transformation geometry. International Journal of Mathematical Education in Science and Technology, 41(7), 901–909. https://doi.org/10.1080/0020739X.2010.486451
• Avcu, S. (2019). Prospective Middle School Mathematics Teachers’ Use of Parameters in Explaining Geometric Transformations. Ihlara Journal of Educational Research, 4(1), 102-111. http://ihead.aksaray.edu.tr/en/pub/issue/42161/519518 adresinden 10.01.2022 tarihinde alınmıştır.
• Avcu, S., & Çetinkaya, B. (2021). An instructional unit for prospective teachers’ conceptualization of geometric transformations as functions. International Journal of Mathematical Education in Science and Technology, 52(5), 669-698. https://doi.org/10.1080/0020739X.2019.1699966
• Creswell, J. W. (2012). Educational research: Planning, conducting, and evaluating quantitative and qualitative research. (4. Baskı). Boston, MA: Pearson.
• Edwards, L. D. (2003). The nature of mathematics as viewed from cognitive science. The Third Congress of the European Society for Research in Mathematics, Bellaria. http://www.mathematik.tu-dortmund.de/~erme/CERME3/Groups/TG1/TG1_edwards_cerme3.pdf adresinden 10.01.2022 tarihinde alınmıştır.
• Gokalp, N. D., & Bulut, S. (2018). A new form of understanding maps: multiple representations with Pirie and Kieren model of understanding. International Journal of Innovation in Science and Mathematics Education, 26(6), 1–21. https://openjournals.library.sydney.edu.au/index.php/CAL/article/view/12454/0 adresinden 10.01.2022 tarihinde alınmıştır.
• Gulkilik, H. (2016). The role of virtual manipulatives in high school students’ understanding of geometric transformations. P. S. Moyer-Packenham (Ed.), International perspectives on teaching and learning mathematics with virtual manipulatives içinde (ss. 213–243). Cham: Springer International Publishing.
• Gulkilik, H., Ugurlu, H. H., & Yürük, N. (2015). Examining students’ mathematical understanding of geometric transformations using the Pirie-Kieren model. Educational Sciences: Theory and Practice, 15(6), 1531–1548.
• Gulkilik, H., Moyer-Packenham, P. S., Ugurlu, H. H., & Yuruk, N. (2020). Characterizing the growth of one student’s mathematical understanding in a multi-representational learning environment. The Journal of Mathematical Behavior, 58, 100756. https://doi.org/10.1016/j.jmathb.2020.100756
• Güner, P., & Uygun, T. (2020). Examining students’ mathematical understanding of patterns by Pirie-Kieren model. Hacettepe University Journal of Education, 35 (3), 644-661. https://acikerisim.istanbulc.edu.tr/xmlui/handle/20.500.12831/1564 adresinden 10.01.2022 tarihinde alınmıştır.
• Harper, S. (2003). Enhancing elementary pre-service teachers’ knowledge of geometric transformations through the use of dynamic geometry computer software. C. Crawford et al. (Eds.), Proceedings of Society for Information Technology & Teacher Education International Conference içinde (ss. 2909–2916). Chesapeake: AACE. https://www.researchgate.net/publication/317571398 adresinden 10.01.2022 tarihinde alınmıştır.
• Hollebrands, K. F. (2003). High school students’ understandings of geometric transformations in the context of a technological environment. Journal of Mathematical Behavior, 22(1), 55-72. ://doi.org/10.1016/S0732-3123(03)00004-X
• Hollebrands, K. F. (2007). The role of a dynamic software program for geometry in the strategies high school mathematics students employ. Journal for Research in Mathematics Education, 38(2), 164-192. https://doi.org/10.2307/30034955
• Kaba, Y., & Şengül, S. (2018). The relationship between middle school students’ mathematics anxiety and their mathematical understanding. Pegem Journal of Education and Instruction, 8(3), 599–622. https://doi.org/10.14527/pegegog.2018.023
• Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up. Helping children learn mathematics. Washington, DC: National Academy Press.
• Komatsu, K., Jones, K. (2021). Generating mathematical knowledge in the classroom through proof, refutation, and abductive reasoning. Educational Studies in Mathematics. https://doi.org/10.1007/s10649-021-10086-5
• Mabotja, S., Chuene, K., Maoto, S., & Kibirige, I. (2018). Tracking grade 10 learners’ geometric reasoning through folding back. Pythagoras, 39(1), 1–10. https://pdfs.semanticscholar.org/37e9/e64f4c15058caf721287d7b91b014a04fcc7.pdf adresinden 10.01.2022 tarihinde alınmıştır.
• Mack, N. K. (2001). Building on informal knowledge through instruction in a complex content domain: Partitioning, units, and understanding multiplication of fractions. Journal for Research in Mathematics Education, 32(3), 267–295. https://doi.org/10.2307/749828
• Martin, G. (1982). Transformation geometry: an introduction to symmetry. New York: Springer.
• Martin, L. C. (2008). Folding back and the growth of mathematical understanding: Elaborating the Pirie-Kieren theory. Journal of Mathematical Behavior, 27(1), 64–85. https://doi.org/10.1016/j.jmathb.2008.04.001
• Martin, L. C., & Towers, J. (2016). Folding back, thickening and mathematical met-befores. The Journal of Mathematical Behavior, 43, 89–97. https://doi.org/10.1016/j.jmathb.2016.07.002
• Meel, D. E. (2003). Models and theories of mathematical understanding: Comparing Pirie and Kieren’s theory for the growth of mathematical understanding and APOS theory. A. Selden, E. Dubinsky, G. Harel, & F. Hitt (Eds.), Conference Board of the Mathematical Sciences, Issues in Mathematics Education: Vol. 12. Issues in Collegiate Mathematics Education V içinde (pp. 132–181). Providence, RI: American Mathematical Society.
• Milli Eğitim Bakanlığı (2018a). Matematik dersi öğretim programı (İlkokul ve Ortaokul 1, 2, 3, 4, 5, 6, 7 ve 8. Sınıflar). Ankara: MEB Yayınları. http://mufredat.meb.gov.tr/Programlar.aspx
• Milli Eğitim Bakanlığı (2018b). Ortaöğretim matematik dersi (9, 10, 11 ve 12. sınıflar) öğretim programı. Ankara: MEB Yayınları. http://mufredat.meb.gov.tr/Programlar.aspx
• Mills, G. E. 2014. Action research: A guide for the teacher researcher. (5. Baskı). Boston, MA: Pearson. National Council of Teachers of Mathematics (NCTM) (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
• National Council of Teachers of Mathematics (NCTM) (2020a). Standards for the preparation of secondary mathematics teachers. https://www.nctm.org/Standards-and-Positions/CAEP-Standards/ adresinden 10.01.2022 tarihinde alınmıştır.
• National Council of Teachers of Mathematics (NCTM) (2020b). Standards for the preparation of middle level mathematics teachers. https://www.nctm.org/Standards-and-Positions/CAEP-Standards/ adresinden 10.01.2022 tarihinde alınmıştır.
• Pirie, S. E. B., & Kieren, T. (1994). Growth in mathematical understanding: how can we characterise it and how can we represent it? Educational Studies in Mathematics, 26(2–3), 165–190. https://doi.org/10.1007/BF01273662
• Portnoy, N., Grundmeier, T., & Graham, K. J. (2006). Students’ understanding of mathematical objects in the context of transformational geometry: Implications for constructing and understanding proofs. Journal of Mathematical Behavior, 25, 196–207. https://doi.org/10.1016/j.jmathb.2006.09.002
• Powell, A., Francisco, J., & Maher, C. (2003). An analytical model for studying the development of learners’ mathematical ideas and reasoning using videotape data. Journal of Mathematical Behavior, 22, 405–435. https://doi.org/10.1016/j.jmathb.2003.09.002
• Sengul, S., & Goktepe-Yildiz, S. (2016). An Examination of the Domain of Multivariable Functions Using the Pirie-Kieren Model. Universal Journal of Educational Research, 4(7), 1533-1544. DOI: 10.13189/ujer.2016.040706
• Thaqi, X., Gimenez, J., & Rosich, N. (2011, February). Geometrical transformation as viewed by prospective teachers. M. Pytlak, T.Rowland, & E. Swoboda (Eds.), Proceedings of the seventh congress of the European society for research in mathematics education içinde (pp. 578–587). Poland: Univerity of Rszeszów. https://hal.archives-ouvertes.fr/hal-02158191/ adresinden 10.01.2022 tarihinde alınmıştır.
• Thom, J. S., & Pirie, S. E. (2006). Looking at the complexity of two young children’s understanding of number. The Journal of Mathematical Behavior, 25(3), 185-195. https://doi.org/10.1016/j.jmathb.2006.09.004
• Uygun, T. (2020). An inquiry-based design research for teaching geometric transformations by developing mathematical practices in dynamic geometry environment. Mathematics Education Research Journal, 32(3), 523-549. https://doi.org/10.1007/s13394-020-00314-1
• Vamvakoussi, X. (2017). Using analogies to facilitate conceptual change in mathematics learning. ZDM Mathematics Education, 49, 497–507. https://doi.org/10.1007/s11858-017-0857-5
• Wright, V. (2014). Frequencies as proportions: using a teaching model based on Pirie and Kieren’s model of mathematical understanding. Mathematics Education Research Journal, 26(1), 101–128. https://doi.org/10.1007/s13394-014-0118-7
• Yanik, H. B. (2011). Prospective middle school mathematics teachers’ preconceptions of geometric translations. Educational Studies in Mathematics,78(2), 231-260. https://www.jstor.org/stable/41486163
• Yanık, H.B. (2013). Learning geometric translations in a dynamic geometry environment. Education and Science, 38(168), 272–287.
• Yanık, H. B. (2014). Middle-school students’ concept images of geometric translations. The Journal of Mathematical Behavior, 36(1), 33–50. https://doi.org/10.1016/j.jmathb.2014.08.001
• 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. https://doi.org/10.1016/j.jmathb.2009.04.003
• Yao, X. (2020a). Unpacking learner’s growth in geometric understanding when solving problems in a dynamic geometry environment: Coordinating two frames. The Journal of Mathematical Behavior, 60, 100803. https://doi.org/10.1016/j.jmathb.2020.100803
• Yao, X. (2020b). Characterizing learners’ growth of geometric understanding in dynamic geometry environments: a perspective of the Pirie–Kieren theory. Digital Experiences in Mathematics Education, 6, 293-319. https://doi.org/10.1007/s40751-020-00069-1
• Yao, X., & Manouchehri, A. (2019). Middle school students’ generalizations about properties of geometric transformations in a dynamic geometry environment. The Journal of Mathematical Behavior, 55, 100703. https://doi.org/10.1016/j.jmathb.2019.04.002
• Yao, X., & Manouchehri, A. (2020a). Folding back in students’ construction of mathematical generalizations within a dynamic geometry environment. Mathematics Education Research Journal. https://doi.org/10.1007/s13394-020-00343-w
• Yao, X., & Manouchehri, A. (2020b). Teacher Interventions for Advancing Students’ Mathematical Understanding. Education Sciences, 10(6), 164. https://doi.org/10.3390/educsci10060164