Investigating Preservice Teachers’ Determination and Representation of Proportional and Nonproportional Relationships in Terms of Problem Contexts

Yıl 2020, Cilt: 14 Sayı: 1, 629 - 660, 30.06.2020

Muhammet Arıcan

https://doi.org/10.17522/balikesirnef.683225

Cited By: 1

Öz

This study investigated 46 preservice middle school mathematics teachers’ solution strategies and determination and representation of proportional and nonproportional relationships in terms of problem contexts. The preservice teachers were given a paper-pencil test with two mathematical tasks (Bicycle and Candle). The preservice teachers’ responses were analyzed using a content analysis method. Based on the analysis, semi-structured interviews were conducted with eight preservice teachers. The findings indicated that the preservice teachers’ solution strategies and determination and representation of relationships were affected by the problem contexts. The preservice teachers were better at determining and representing inversely proportional relationship than directly proportional relationship, which was quite opposite of the findings usually cited in the literature. Determining and representing nonproportional relationship appeared to be the most challenging task for them. Problems that required in-depth examinations elicited the use of more sophisticated solution strategies and helped the preservice teachers to avoid using rote computations.

Anahtar Kelimeler

mathematical representations, preservice teachers, problem context, proportional reasoning, proportional relationships

Destekleyen Kurum

This study was supported by the Ahi Evran University Scientific Research Projects Coordination Unit.

Proje Numarası

EGT.A4.18.014

Teşekkür

I would like to thank Dr. Wim Van Dooren and Dr. Lieven Verschaffel for their valuable feedback on earlier drafts of this paper. I also thank to TÜBİTAK- Directorate of Science Fellowships and Grant Programmes (BİDEB) for their support with providing me a research funding to stay in KU Leuven during which I was able to write this paper.

Öğretmen Adaylarının Orantısal Olan ve Olmayan İlişkileri Belirleyebilme ve Temsil Edebilmelerinin Problem İçerikleri Açısından İncelenmesi

Öz

Bu çalışmada, 46 ortaokul matematik öğretmen adayının çözüm yöntemleri, orantısal olan ve olmayan ilişkileri belirleyebilmeleri ve temsil edebilmeleri problem içerikleri bağlamında incelenmiştir. Öğretmen adaylarına iki adet sorudan (Bisiklet ve Mum) oluşan bir kağıt-kalem testi verilmiştir. Adayların kağıt-kalem testine verdikleri cevaplar içerik analizi yöntemi kullanılarak analiz edilmiştir. Analizler sonucunda sekiz öğretmen adayı ile yarı yapılandırılmış görüşmeler gerçekleştirilmiştir. Elde edilen bulgular, öğretmen adaylarının çözüm yöntemlerinin ve orantısal olan ve olmayan ilişkileri belirleyebilmelerinin ve temsil edebilmelerinin problem içeriklerinden etkilendiğini göstermiştir. Alan yazında belirtilenin aksine, öğretmen adayları ters orantılı ilişkiyi belirleme ve temsil etme konusunda doğru orantılı ilişkiyi belirleme ve temsil etmeye göre daha başarılı olmuşlardır. Öte yandan, adaylar en çok orantısal olmayan ilişkinin belirlenmesi ve temsil edilmesinde zorlanmışlardır. Derinlemesine inceleme gerektiren problemler daha gelişmiş çözüm yöntemlerinin ortaya çıkmasını sağlayıp, öğretmen adaylarının ezbere hesaplamaları kullanmaktan kaçınmasına yardımcı oldu.

Anahtar Kelimeler

matematiksel temsiller, orantısal akıl yürütme, orantısal ilişkiler, öğretmen adayları, problem içeriği

Proje Numarası

EGT.A4.18.014

Kaynakça

• Author (2018). International Journal of Science and Mathematics Education.
• Author (2019). International Journal of Science and Mathematics Education.
• Cramer, K., & Post, T. (1993). Making connections: A case for proportionality. Arithmetic Teacher, 60(6), 342–346.
• Cramer, K., Post, T., & Currier, S. (1993). Learning and teaching ratio and proportion: Research implications. In D. Owens (Ed.), Research ideas for the classroom: Middle grades mathematics (pp. 159–178). New York, NY: Macmillan.
• Common Core State Standards Initiative (2010). The common core state standards for mathematics. Washington, D.C.: Author. http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
• Degrande, T., Van Hoof, J., Verschaffel, L., & Van Dooren, W. (2017). Open word problems: Taking the additive or the multiplicative road?. ZDM, 50(1-2), 91–102. https://doi.org/10.1007/s11858-017-0900-6
• Fernández, C., Llinares, S., Modestou, M., & Gagatsis, A. (2010). Proportional reasoning: How task variables influence the development of students’ strategies from primary to secondary school. Acta Didactica Universitatis Comenianae Mathematics, 10, 1–18. http://hdl.handle.net/10045/16588
• Fisher, L. C. (1988). Strategies used by secondary mathematics teachers to solve proportion problems. Journal for Research in Mathematics Education, 19(2), 157–168. http://www.jstor.org/stable/749409
• Fraenkel, J. R., & Wallen, N. E. (2006). How to design and evaluate research in education (6th ed.). New York: NY, McGraw-Hill.
• Harel, G., & Behr, M. (1995). Teachers' solutions for multiplicative problems. Hiroshima Journal of Mathematics Education, 3, 31–51.
• Hsieh, H. F., & Shannon, S. E. (2005). Three approaches to qualitative content analysis. Qualitative health research, 15(9), 1277–1288.
• Izsák, A., & Jacobson, E. (2017). Preservice teachers’ reasoning about relationships that are and are not proportional: A knowledge-in-pieces account. Journal for Research in Mathematics Education, 48(3), 300–339. https://doi.org/10.5951/jresematheduc.48.3.0300
• Johnson, K. (2017). A study of pre-service teachers use of representations in their proportional reasoning. In Galindo, E., & Newton, J., (Eds.), Proceedings of the 39th North American Chapter of the International Group for the Psychology of Mathematics Education conference (pp. 551–558). Indianapolis, IN. https://files.eric.ed.gov/fulltext/ED581310.pdf
• Kaput, J. J., & West, M. M. (1994). Missing-value proportional reasoning problems: Factors affecting informal reasoning patterns. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 235–287). Albany, NY: State University of New York Press.
• Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
• Lamon, S. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (Vol 1, pp. 629–667). Charlotte, NC: Information Age Publishing.
• Lesh, R., Post, T., & Behr, M. (1988). Proportional reasoning. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 93–118). Reston, VA: National Council of Teachers of Mathematics.
• Lim, K. (2009). Burning the candle at just one end: Using nonproportional examples helps students determine when proportional strategies apply. Mathematics Teaching in the Middle School, 14(8), 492–500.
• Lo, J. J. (2004). Prospective elementary school teachers' solution strategies and reasoning for a missing value proportion task. In M. J. Høines & A. B. Fuglestad (Eds.), Proceedings of the 28th International Group for the Psychology of Mathematics Education Conference (pp. 265–272). Bergen, Norway. http://emis.ams.org/proceedings/PME28/RR/RR207_Lo.pdf
• Lobato, J., & Ellis, A. (2010). Developing essential understanding of ratios, proportions, and proportional reasoning for teaching mathematics: Grades 6-8. National Council of Teachers of Mathematics. 1906 Association Drive, Reston, VA 20191-1502. https://eric.ed.gov/?id=ED511861
• Modestou, M., & Gagatsis, A. (2007). Students’ improper proportional reasoning: A result of the epistemological obstacle of “linearity”. Educational Psychology, 27(1), 75–92. https://doi.org/10.1080/01443410601061462
• National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author.
• Orrill, C. H., & Brown, R. E. (2012). Making sense of double number lines in professional development: Exploring teachers’ understandings of proportional relationships. Journal of Mathematics Teacher Education, 15(5), 381–403. https://doi.org/10.1007/s10857-012-9218-z
• Patton, M. Q. (2005). Qualitative research. John Wiley & Sons, Ltd.
• Riley, K. R. (2010). Teachers’ understanding of proportional reasoning. In P. Brosnan, D. B. Erchick, & L. Flevares (Eds.), Proceedings of the 32nd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1055–1061). Columbus, OH: The Ohio State University
• Van Dooren, W., De Bock, D., & Verschaffel, L. (2010). From addition to multiplication… and back: The development of students’ additive and multiplicative reasoning skills. Cognition and Instruction, 28, 360–381. https://doi.org/10.1080/07370008.2010.488306