Pre-service Mathematics Teachers’ Modes of Thinking in Linear Algebra: The Case of Linear Transformation

Year 2024, Issue: 62, 2988 - 3004, 30.12.2024

Meltem Coşkun Şimşek , Necla Turanlı

https://doi.org/10.53444/deubefd.1481905

Abstract

The aim of this research was to determine the modes of thinking that pre-service mathematics teachers’ employ to solve problems related to the concept of linear transformation in linear algebra. A study was conducted with 22 pre-service mathematics teachers’ using the case study method - a qualitative research method. The data of the research were collected through four problems defined in the context of the “definition of linear transformation” and “matrix representation of linear transformation”. 10 codes were created upon the descriptive analysis of the data collected, and those codes were classified in the context of Sierpinska’s (2000) theoretical framework modes of thinking (analytical-structural, analytical-arithmetic, synthetic-geometric). According to the study, pre-service mathematics teachers’ had different modes of thinking in “definition” and “matrix representation” but they could not switch between modes of thinking. It was found that analytical-arithmetic thinking was more common than analytical-structural and synthetic-geometric thinking throughout the study. The concept of linear transformation could not be internalized with all its components and it was a challenging process for pre-service teachers’ to switch to the matrix representation of linear transformation.

Keywords

Linear algebra, linear transformation, modes of thinking, pre-service mathematics teachers

Matematik Öğretmeni Adaylarının Lineer Cebirde Düşünme Biçimleri: Lineer Dönüşüm Örneği

Abstract

Bu araştırmanın amacı matematik öğretmeni adaylarının lineer cebirde, lineer dönüşüm kavramına ilişkin problemleri çözerken sahip oldukları düşünme biçimlerini belirlemektir. Nitel araştırma yöntemlerinden durum çalışması benimsenerek, 22 matematik öğretmeni adayı ile araştırma gerçekleştirilmiştir. Araştırmanın verileri “lineer dönüşümün tanımı” ve “lineer dönüşümün matris temsili” bağlamında hazırlanan dört adet problem aracılığıyla toplanmıştır. Elde edilen verilerin betimsel analize tabi tutulmasıyla 10 adet kod oluşturulmuş ve bu kodlar Sierpinska’nın (2000) düşünme biçimleri (analitik-yapısal, analitik-aritmetik, sentetik-geometrik) kuramsal çerçevesi bağlamında sınıflandırılmıştır. Araştırmanın sonucunda öğretmen adaylarının lineer dönüşüm kavramını “tanım” ve “matris temsili” bağlamında farklı düşünme biçimlerine sahip oldukları ancak düşünme biçimleri arasında geçiş yapamadıkları belirlenmiştir. Tüm süreçte analitik-aritmetik düşünme biçiminin analitik-yapısal ve sentetik-geometrik düşünme biçimine kıyasla daha baskın olduğu belirlenmiştir. Lineer dönüşüm kavramı tüm bileşenleri ile içselleştirilememiş ve lineer dönüşümün matris temsiline geçme fikri öğretmen adayları için zorlayıcı bir süreç olduğu belirlenmiştir.

Keywords

Lineer cebir, lineer dönüşüm, düşünme biçimi, matematik öğretmeni adayı

References

• Andrews-Larson, C., Wawro M., & Zandieh, M. (2017). A hypothetical learning trajectory for conceptualizing matrices as linear transformations. International Journal of Mathematical Education in Science and Technology, 48(6), 809-829.
• Bagley, S., Rasmussen, C., & Zandieh, M. (2015). Inverse, composition, and identity: The case of function and linear transformation. The Journal of Mathematical Behavior, 37, 36-47.
• Bogomolny, M. (2006). The role of example-generation tasks in students' understanding of linear algebra [Unpublished Doctoral Thesis]. Simon Fraser University.
• Brown. A. P. (2008). A review of the literature on case study research. Canadian Journal for New Scholars in Education, 1(1), 1-13.
• Çelik, D. (2015). Investigating students’ modes of thinking in linear algebra: The case of linear independence. International Journal for Mathematics Teaching and Learning. Retrieved from https://www.cimt.org.uk/journal/celik.pdf
• Dorier, J. L., Robert, A., Robinet, J., & Rogalsiu, M. (2000). The obstacle of formalism in linear algebra: A variety of studies from 1987 until 1995. In J. L. Dorier (Ed.), On the teaching of linear algebra (pp. 85-124). Dordrecht: Springer Netherlands.
• Dubinsky, E. (1997). Some thoughts on a first course in linear algebra at the college level. In D. Carlson, C. Johnson, D. Lay, A. D. Porter, A. Watkins, & W. Watkins (Eds.), Resources for teaching linear algebra (pp. 85–106). Washington: The Mathematical Association of America.
• González-Rojas, E., & Roa-Fuentes, S. (2017). Un esquema de transformación lineal: Construcción de objetos abstractos a partir de la interiorización de acciones concretas. Enseñanza de las Ciencias, 35, 89–107.
• Harel, G. (1987). Variations in linear algebra content presentations. For the Learning of Mathematics, 7(3), 29-32.
• Lamb, M., Leong, S., & Malone, J. A. (2002). Patterns of misperception in linear transformations: Four illustrations. In Proceedings of the 25th Annual Conference of the Mathematics Education Research Group of Australasia Inc. 7-10 July 2002 (pp. 407-414). Sydney: MERGA.
• Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis: An expanded sourcebook. Sage.
• Oktaç, A. (2018). Understanding and visualizing linear transformations. In G. Kaiser et al. (Eds.), Invited Lectures from the 13th International Congress on Mathematical Education (pp. 436–481). Cham: SpringerOpen.
• Roa-Fuentes, S., & Oktaç, A. (2010). Construcción de una descomposición genética: Análisis teórico del concepto transformación lineal. Revista Latinoamericana de Investigación en Matemática Educativa, 13(1), 89-112.
• Sierpinska, A., Dreyfus, T., & Hillel, J. (1999). Evaluation of a teaching design in linear algebra: The case of linear transformations. Recherches en Didactique des Mathématiques, 19(1), 7–40.
• Sierpinska, A. (2000). On some aspects of students thinking in linear algebra. In J. L. Dorier (Ed.), On the teaching of linear algebra (pp. 209-246). Dordrecht: Kluwer Academic Publishers.
• Turgut, M. (2022). Reinventing geometric linear transformations in a dynamic geometry environment: Multimodal analysis of student reasoning. International Journal of Science and Mathematics Education, 20(6), 1203-1223.
• Viirman, O. (2011). Discursive practices of two mathematics teachers on the concept of ‘linear trans-formation’. In Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education 10-15 July 2011 (pp. 313-320). Ankara, Türkiye.
• Villabona, D., Camacho, G., Vázquez, R., Ramírez, O., & Oktaç, A. (2020). Process conception of linear transformation from a functional perspective. In T. Hausberger, M. Bosch, & F. Chellougui (Eds.), Proceedings of the third conference of the INDRUM 12-19 September 2020 (pp. 388–396). University of Carthage, Bizerte, Tunisia.
• Yıldırım, A., & Şimşek, H. (2016). Sosyal bilimlerde nitel araştırma yöntemleri [Qualitative research methods in the social sciences]. Ankara: Seçkin Publishing.
• Zandieh, M., Ellis, J., & Rasmussen, C. (2017). A characterization of a unified notion of mathematical function: The case of high school function and linear transformation. Educational Studies in Mathematics, 95(1), 21-38.