Covariational Reasoning of Prospective Mathematics Teachers: How Do Dynamic Animations Affect?

Yıl 2020, Cilt: 11 Sayı: 2, 312 - 342, 31.08.2020

Mahmut Kertil

https://doi.org/10.16949/turkbilmat.652481

Öz

Covariational reasoning is about the ability of coordinating the variation in two simultaneously and dynamically changing quantities and being able to see these quantities at the same time by forming a multiplicative unit. Covariational reasoning ability has been considered as necessary and foundational to the understanding of many mathematical concepts ranging from elementary to tertiary levels. In this study, covariational reasoning abilities of prospective elementary school mathematics teachers and the effects of dynamic animations created in computer-based environments on these abilities have been investigated. Case study was used as a research design one of which is a qualitative research methodology. The participants of the study were 19 prospective elementary school mathematics teachers attending to an elective Computer-Assisted Mathematics Education course and four of them were selected for semi-structured interviews. The results showed the weakness in prospective elementary school mathematics teachers’ covariational reasoning abilities and the potential of dynamic animations in supporting covariational reasoning. The animations in dynamic computer environments seem to have minimal effect on paper and pencil solutions in general. However, these animations, if they were used with their data collection and graph drawing properties, affected prospective teachers ways of reasoning in two ways: (i) forcing them to revise and rethink about the current ways of reasoning used during paper and pencil solutions, and (ii) lowering the cognitive load or removing the necessity of deep thinking on the situation. For the first case, the activities supported with the dynamic animations play a supportive role in developing covariational reasoning. For the second case, the dynamic animations did not contribute to prospective teachers’ covariational reasoning, rather they just played a mediating tool role that helps them to find a result.

Anahtar Kelimeler

Change, rate of change, covariation, covariational reasoning, quantitative reasoning

Matematik Öğretmen Adaylarının Kovaryasyonel Düşünme Becerileri: Dinamik Animasyonlar Nasıl Etkiliyor?

Öz

Kovaryasyonel düşünme eş zamanlı ve dinamik olarak değişen iki niceliğin birlikte değişimini düşünerek koordine edebilme ve değişimler arasındaki ilişkiyi bir bütün olarak yorumlayabilme becerisidir. Kovaryasyonel düşünme becerisi oran-orantı, türev ve integral gibi ilköğretim ve daha ileri düzeyde birçok matematiksel kavramın anlaşılmasında önemlidir. Bu çalışmada, ilköğretim matematik öğretmen adaylarının kovaryasyonel düşünme becerileri ve bilgisayar destekli ortamlarında oluşturulan dinamik animasyonların bu becerileri nasıl etkilediği incelenmiştir. Nitel araştırma yöntemlerinden özel durum çalışması kullanılmıştır. Çalışmanın katılımcıları, Bilgisayar Destekli Matematik Öğretimi dersine kayıtlı son sınıf 19 ilköğretim matematik öğretmen adayı olup dört öğretmen adayı ile yarı-yapılandırılmış görüşmeler yapılmıştır. Elde edilen bulgular öğretmen adaylarının kovaryasyonel düşünme becerilerinin yeterli düzeyde olmadığını ve dinamik geometri yazılımları ile elde edilen animasyonların kovaryasyonel düşünme becerisine katkı sağlayabileceğini göstermektedir. Sadece dinamik animasyon oluşturmanın ve onu izlemenin öğretmen adaylarının kâğıt-kalem çözümlerine etkisi çok azdır. Fakat animasyonlar, dinamik geometri programının veri alma ve grafik çizdirme özellikleri ile birlikte kullanıldığında, öğretmen adaylarının kovaryasyonel düşünme becerilerini iki şekilde etkilemiştir: (i) statik (kâğıt-kalem) bağlamlardaki düşünme biçimlerinden farklı sonuçlar vererek tekrar düşünmeye sevk etme veya (ii) zihinsel iş yükünü alarak durum üzerinde derin düşünme gereksinimini ortadan kaldırma. Birinci durumda dinamik animasyonlar öğretmen adayları için kovaryasyonel düşünmeyi destekleyici bir unsur olurken ikinci durumda ise çözüme odaklı ve durum üzerinde derinlemesine düşünme ihtiyacını ortadan kaldıran bir araç rolünü almıştır.

Anahtar Kelimeler

Değişim, değişim oranı, kovaryasyon, kovaryasyonel düşünme, niceliksel düşünme

Kaynakça

• Ärlebäck, J. B., Doerr, H. M., & O’Neil, A. M. (2013). A modeling perspective on interpreting rates of change in context. Mathematical Thinking and Learning, 15(4), 314–336.
• Carlson, M. P. (1998). A cross-sectional investigation of the development of the function concept. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), CBMS Issues in Mathematics Education: Research in Collegiate Mathematics Education III (Vol.7, pp. 114–162). Washington DC: American Mathematical Association.
• Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events. Journal for Research in Mathematics Education, 33(5), 352-378.
• Carlson, M., Larsen, S., & Lesh, R. (2003). Integrating a models and modeling perspective with existing research and practice. In R. Lesh, & H. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 465–478). Mahwah, NJ: Lawrence Erlbaum Associates.
• Castillo-Garsow, C. (2012). Continuous quantitative reasoning. In R. L. Mayes, & L. Hatfield (Eds.), Quantitative reasoning and mathematical modeling: A driver for STEM integrated education and teaching in context (pp. 55–73). Laramie: University of Wyoming Press.
• Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics, 26(2/3), 134-165.
• De Bock, D., Van Dooren, W., Janssens, D., & Verschaffel, L. (2002). Improper use of linear reasoning: An in-depth study of the nature and the irresistibility of secondary school students’ errors. Educational Studies in Mathematics, 50(3), 311-334.
• Donevska-Todorova, A. (2018). Recursive exploration space for concepts in linear algebra. In L. Ball, P. Drijvers, S. Ladel, HS. Siller, M. Tabach, C. Vale (Eds.), Uses of technology in primary and secondary mathematics education. ICME-13 Monographs (pp. 351-361). Cham, Switzerland: Springer International Publishing.
• Herbert, S., & Pierce, R. (2011). What is rate? Does context or representation matter? Mathematics Education Research Journal, 23(4), 455-477.
• Herbert, S., & Pierce, R. (2012). Revealing educationally critical aspects of rate. Educational Studies in Mathematics, 81(1), 85–101.
• Hoffkamp, A. (2011). The use of interactive visualizations to foster the understanding of concepts of calculus: Design principles and empirical results. ZDM–Mathematics Education, 43(3), 359–372.
• Johnson, H. L. (2012). Reasoning about variation in the intensity of change in covarying quantities involved in rate of change. The Journal of Mathematical Behavior, 31(3), 313–330.
• Johnson, H. L., McClintock, E., & Hornbein, P. (2017). Ferris wheels and filling bottles: A case of a student’s transfer of covariational reasoning across tasks with different backgrounds and features. ZDM–Mathematics Education, 49(6), 851–864.
• Jones, S. R. (2017). An exploratory study on student understandings of derivatives in real-world, non-kinematics contexts. The Journal of Mathematical Behavior, 45, 95–110. Keene, K. A. (2007). A characterization of dynamic reasoning: Reasoning with time as parameter. Journal of Mathematical Behavior, 26, 230-246.
• Kertil, M. (2014). Pre-service elementary mathematics teachers' understanding of derivative through a model development unit (Unpublished doctoral dissertation). Middle East Technical University, Graduate School of Natural and Applied Sciences, Ankara.
• Kertil, M., Erbaş, A. K., & Çetinkaya, B. (2019). Developing prospective teachers’ covariational reasoning through a model development sequence. Mathematical Thinking and Learning, 21, 207-233.
• Lesh, R., Hoover, M., Hole, B., Kelly, A., & Post, T. (2000). Principles for developing thought-revealing activities for students and teachers. In R. Lesh, & A. Kelly (Eds.), Handbook of research design in mathematics and science education (pp. 591–645). Hillsdale, NJ: Lawrence Erlbaum.
• Lobato, J., & Ellis, A. B. (2010). Developing essential understandings of ratios, proportions, and proportional reasoning for teaching mathematics in grades 6-8. Reston, VA: National Council of Teachers of Mathematics.
• Lobato, J., & Thanheiser, E. (2002). Developing understanding of ratio-as-measure as a foundation for slope. In B. Litwiller, & G. Bright (Eds.), Making sense of fractions, ratios, and proportions: 2002 yearbook (pp. 162-175). Reston, VA: National Council of Teachers of Mathematics.
• Monk, S. (1992). Students’ understanding of a function given with a physical model. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 175–193). Washington, DC: Mathematical Association of America.
• Oehrtman, M. C., Carlson, M. P., & Thompson, P. W. (2008). Foundational reasoning abilities that promote coherence in students’ understandings of function. In M. P. Carlson, & C. Rasmussen (Eds.), Making the connection: Research and practice in undergraduate mathematics (pp. 27-42). Washington, DC: Mathematical Association of America.
• Saldanha, L., & Thompson, P. (1998). Re-thinking co-variation from a quantitative perspective: Simultaneous continuous variation. In S. B. Berensen, & W. N. Coulombe (Eds.), Proceedings of the Twentieth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 298-304). Raleigh, NC: North Carolina State University.
• Stalvey, H. E., & Vidakovic, D. (2015). Students’ reasoning about relationships between variables in a real-world problem. The Journal of Mathematical Behavior, 40, 192–210.
• Strauss, A., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory (2nd ed.). Thousand Oaks, CA: Sage.
• Stroup, W. (2002). Understanding qualitative calculus: A structural synthesis of learning research. International Journal of Computers for Mathematical Learning, 7(2), 167-215.
• Şen-Zeytun, A., Cetinkaya, B. ve Erbas, A. K. (2010). Matematik öğretmenlerinin kovaryasyonel düşünme düzeyleri ve öğrencilerinin kovaryasyonel düşünme becerilerine ilişkin tahminler. Educational Sciences: Theory and Practice, 10(3), 1601–1612.
• Thompson, P. W. (1994). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics, 26, 229-274.
• Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In L. L. Hatfield, S. Chamberlain, & S. Belbase (Eds.), New perspectives and directions for collaborative research in mathematics education (Vol. 1, pp. 33-57). Laramie, WY: University of Wyoming.
• Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 421-456). Reston, VA: National Council of Teachers of Mathematics.
• Thompson, P. W., Hatfield, N. J., Yoon, H., Joshua, S., & Byerley, C. (2017). Covariational reasoning among U.S. and South Korean secondary mathematics teachers. The Journal of Mathematical Behavior, 48, 95–111.
• Yemen-Karpuzcu, S.,Ulusoy, F., Işıksal- Bostan, M. (2017). Prospective middle school mathematics teachers’ covariational reasoning for ınterpreting dynamic events during peer ınteractions. International Journal of Science and Mathematics Education, 15, 89–108.
• Yıldırım, A. ve Şimşek, H. (2011). Sosyal bilimlerde nitel araştırma yöntemleri. Ankara: Seçkin Yayıncılık.
• Zbiek, R. M., Heid, M. K., Blume, G. W., & Dick, T. P. (2007). Research on technology in mathematics education: The perspective of constructs. In F. Lester (Ed.), Handbook of research on mathematics teaching and learning (Vol. 2, pp. 1169-1207). Charlotte, NC: Information Age Publishing.