An Instructional Sequence Triggering Students’ Quantitative Reasoning during Learning of Quadratic Functions

Abstract

The purpose of this study is to design an instructional sequence triggering students’ quantitative reasoning in the process of learning quadratic functions. The study was conducted as a design-based research, following a cyclical process. The study consisted of three phases of the design, implementation, and analysis phases and the implementation phase consisted of two consecutive cycles. While the first cycle was carried out to evaluate the success of the instructional sequence in supporting student learning in a class with ten 10th students, the second cycle was carried out with two 10th grade students. The video recordings taken during the teaching experiments, the researchers’ observation notes, the clinical interviews, and the students’ reflective journals constituted the data sources of the study. In the data analysis process of the study, the constant comparison method simultaneously conducted with the data collection process was used and also a retrospective analysis of the teaching experiments data was conducted after completing the cycles. Because the tasks were grounded in real-life contexts involving dynamic situations, they contributed to the students’ understanding of functional relations and helped them construct the idea of covarying change in the functions. It is suggested that mathematics teachers revise and use the instructional sequence according to their own classroom context with the aim of supporting their students’ understanding of quadratic functions.

Keywords

• logico-mathematical activity,mathematical modeling,quantitative reasoning,Quadratic function,instructional sequence

İkinci Dereceden Fonksiyonların Öğrenilmesi Sürecinde Öğrencilerin Nicel Muhakemelerini Tetikleyen Bir Öğretim Dizisi

Öz

Bu çalışmanın amacı ikinci dereceden fonksiyonların öğrenilmesi sürecinde nicel muhakemeyi tetikleyen bir öğretim dizisi tasarlamaktır. Çalışma, döngüsel bir süreç olan tasarım tabanlı araştırma modeline dayandırılmıştır. Tasarı, uygulama ve analiz olacak şekilde üç aşamada gerçekleştirilen tasarım tabanlı araştırmanın uygulama aşaması iki ardıl döngüyü kapsamıştır. İlk döngü, öğretim dizisinin bir sınıf ortamındaki farklı öğrencilerin öğrenmelerini destekleyip desteklemediğini değerlendirmek amacıyla on öğrenci ile, ikinci döngü ise iki onuncu sınıf öğrencisi ile gerçekleştirilmiştir. Çalışmadaki veri toplama araçları öğretim deneyleri boyunca alınan video kamera kayıtları, araştırmacı gözlem notları, öğrencilerle yapılan klinik mülakatlar ve öğrencilerin yansıtıcı günlükleridir. Çalışmanın veri analiz sürecinde veri toplama süreci ile eş zamanlı bir şekilde sürekli karşılaştırmalı olarak devam eden analizler ve birinci ve ikinci döngülerin sonunda öğrencilerin nicel muhakemeleri bağlamında geriye dönük analizler yapılmıştır. Öğretim dizisindeki gerçek yaşam bağlamlı etkinlikler dinamik durumları içerdiği için öğrencilerin fonksiyonel ilişkileri anlamlandırmalarına ve fonksiyonun eş zamanlı değişim fikrini oluşturmalarına imkan vermiştir. Matematik öğretmenlerinin öğrencilerin ikinci dereceden fonksiyonları öğrenmelerini desteklemek amacıyla öğretim dizisini kendi sınıf ortamlarına uygun şekilde revize ederek kullanmaları önerilmektedir.

Anahtar Kelimeler

• İkinci dereceden fonksiyon,mantıksal-matematiksel öğrenme etkinliği,matematiksel modelleme,nicel muhakeme,öğretim dizisi

Kaynakça

1. Baki, A. (2018). Matematiği öğretme bilgisi (1. baskı). Ankara: Pegem Akademi.
2. Carlson, M. P., & Oehrtman, M. (2005). Key aspects of knowing and learning the concept of function. Washington, DC: Mathematical Association of America.
3. Charmaz, K. (2011). Grounded theory methods in social justice research. The Sage Handbook of Qualitative Research, 4(1), 359-380.
4. Common Core State Standards Initiative [CCSSI]. (2010). The common core state standards for mathematics. Washington, DC: National Governors Association Center for Best Practices and Council of Chief State School Officers. Retrieved April 4, 2016 from http://www.corestandards. org/the-standards/mathematics.
5. Design-Based Research Collective [DRBC]. (2003). Design based research: An emerging paradigm for educational inquiry. Educational Researcher, 32(1), 5–8.
6. Ellis, A. B. (2011). Algebra in the middle school: Developing functional relationships through quantitative reasoning. In J. Cai, & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 215-235). New York: Springer.
7. Ellis, A. B., & Grinstead, P. (2008). Hidden lessons: How a focus on slope-like properties of quadratic functions encouraged unexpected generalizations. The Journal of Mathematical Behavior, 27(4), 277-296.
8. Eraslan, A. (2005). A qualitative study: Algebra honor students’ cognitive obstacles as they explore concepts of quadratic functions (Unpublished doctoral dissertation). The Florida State University College of Education, USA.

9. Eraslan, A. (2007). The notion of compartmentalization: The case of Richard. International Journal of Mathematical Education in Science and Technology, 38(8), 1065-1073.
10. Gravemeijer, K., & Cobb, P. (2006). Design research from the learning design perspective. In J. van den Akker, K. Gravemeijer, S. McKenney, & N. Nieveen (Eds.), Educational design research (pp. 17-51). London: Routledge.
11. Gravemeijer, K., & van Eerde, D. (2009). Design research as a means for building a knowledge base for teachers and teaching in mathematics education. The Elementary School Journal, 109(5), 510–524.
12. Hohensee, C. (2016). Student noticing in classroom settings: A process underlying influences on prior ways of reasoning. The Journal of Mathematical Behavior, 42, 69-91.
13. Hunting, R. P. (1997). Clinical interview methods in mathematics education research and practice. The Journal of Mathematical Behavior, 16(2), 145-165.
14. Johnson, H. L. (2013). Reasoning about quantities that change together. Mathematics Teacher, 106(9), 704-708.
15. Konold, C., & Johnson, D. K. (1991). Philosophical and psychological aspects of constructivism. In L. P. Steffe (Ed.), Epistemological foundations of mathematical experience (pp. 1-13). New York: Springer.
16. Kotsopoulos, D. (2007). Unravelling student challenges with quadratics. Australian Mathematics Teacher, 63(2), 19-24. Lobato, J., & Siebert, D. (2002). Quantitative reasoning in a reconceived view of transfer. The Journal of Mathematical Behavior, 21(1), 87-116.
17. Lobato, J., Hohensee, C., Rhodehamel, B., & Diamond, J. (2012). Using student reasoning to inform the development of conceptual learning goals: The case of quadratic functions. Mathematical Thinking and Learning, 14(2), 85-119.
18. Metcalf, R. C. (2007). The nature of students’ understanding of quadratic functions (Unpublished doctoral dissertation). The State University of New York at Buffalo, USA.
19. Milli Eğitim Bakanlığı [MEB]. (2017). Ortaöğretim 10. sınıf matematik ders kitabı. Ankara: Milli Eğitim Bakanlığı.
20. Milli Eğitim Bakanlığı [MEB]. (2018). Ortaöğretim matematik dersi (9, 10, 11 ve 12. sınıflar) öğretim programı. Ankara: Milli Eğitim Bakanlığı. Mitchelmore, M., & Cavanagh, M. (2000). Students' difficulties in operating a graphics calculator. Mathematics Education Research Journal, 12(3), 254-268.
21. Mojica, G. (2010). Preparing pre-service elementary teachers to teach mathematics with learning trajectories (Unpublished doctoral dissertation). North Carolina State University, USA.
22. Moore, K. C. (2014). Quantitative reasoning and the sine function: The case of Zac. Journal for Research in Mathematics Education, 45(1), 102-138.
23. Moore, K. C., Carlson, M. P., & Oehrtman, M. (2009). The role of quantitative reasoning in solving applied precalculus problems. Proceedings of the Twelfth Annual Conference on Research in Undergraduate Mathematics Education. Raleigh, NC: North Carolina State University.
24. National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Reston, VA: Author.
25. National Research Council [NCR]. (2007). Taking science to school. Washington, DC: National Academy Press.
26. Nielsen, L. E. J. (2015). Understanding quadratic functions and solving quadratic equations: An analysis of student thinking and reasoning (Unpublished doctoral dissertation). Washington University, Missouri, USA.
27. 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. 150-171). Washington, DC: Mathematical Association of America.
28. Özaltun-Çelik, A., & Bukova-Güzel, E. (2017). Revealing Ozgur’s thoughts of a quadratic function with a clinical interview: Concepts and their underlying reasons. International Journal of Research in Education and Science, 3(1), 122-134.
29. Sevim, V. (2011). Students’ understanding of quadratic functions: A multiple case study (Unpublished doctoral dissertation). The University of North Carolina, USA.
30. Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26,114-145.
31. Simon, M. (2006). Pedagogical concepts as goals for teacher education: Towards an agenda for research in teacher development. In S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of the Twenty-Eighth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 730-735). Mérida, Mexico: Universidad Pedagógica Nacional.
32. Steffe, L. P. (2002). A new hypothesis concerning children’s fractional knowledge. Journal of Mathematical Behavior, 102, 1–41.
33. Steffe, L. P., & Kieren, T. (1994). Radical constructivism and mathematics education. Journal for Research in Mathematics Education, 25(6), 711-733.
34. Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh, & A. E. Kelly (Eds.), Research design in mathematics and science education (pp. 267- 307). Hillsdale, NJ: Erlbaum.
35. Stephan, M. L. (2015). Conducting classroom design research with teachers. ZDM Mathematics Education, 47(6), 905-917.
36. Thompson, P. W. (1990). A theoretical model of quantity-based reasoning in arithmetic and algebraic. Progress report to the National Science Foundation. San Diego State University, Center for Research in Mathematics and Science Education.
37. Thompson, P. W. (1991). To experience is to conceptualize: Discussions of epistemology and experience. In L. P. Steffe (Ed.), Epistemological foundations of mathematical experience (pp. 260–281). New York: Springer-Verlag.
38. Thompson, P. W. (1994). Students, functions, and the undergraduate curriculum. In E. Dubinsky, A. H. Schoenfeld, & J. J. Kaput (Eds.), Research in collegiate mathematics education (Vol. 4, pp. 21–44). Providence, RI: American Mathematical Society.
39. Thompson, P. W. (2013). Why use f (x) when all we really mean is y. OnCore, The Online Journal of the AAMT, 18-26. van den Heuvel-Panhuizen, M. (1996). Assessment and realistic mathematics education. Utrecht: CD-β Press.
40. van den Heuvel-Panhuizen, M., & Drijvers, P. (2014). Realistic mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 521-525). Netherlands: Springer.
41. von Glasersfeld, E. (1989). Cognition, construction of knowledge, and teaching. Synthese, 80(1), 121-140.
42. von Glasersfeld, E. (2002). Radical constructivism: A way of knowing and learning. Bristol, PA: Routledge Falmer.
43. Weber, E., Ellis, A., Kulow, T., & Ozgur, Z. (2014). Six principles for quantitative reasoning and modeling. Mathematics Teacher, 108(1), 24-30.
44. Yıldırım, A. ve Şimşek H. (2011). Sosyal bilimlerde nitel araştırma yöntemleri (8. baskı). Ankara: Seçkin Yayıncılık.
45. Zaslavsky, O. (1997). Conceptual obstacles in the learning of quadratic functions. Focus on Learning Problems in Mathematics, 19(1), 20-45.
46. Zazkis, R., Liljedahl, P., & Gadowsky, K. (2003). Conceptions of function translation: Obstacles, intuitions, and rerouting. Journal of Mathematical Behavior, 22(4), 435–448.