Creating Design Principles of a Learning Environment for Teaching Vector Spaces

Yıl 2021, Cilt: 12 Sayı: 1, 244 - 289, 05.02.2021

Gökay Açıkyıldız Temel Kösa

https://doi.org/10.16949/turkbilmat.860627

Öz

In this study, determining the design principles of a technology-supported learning environment for teaching vector spaces by taking into account the representation languages defined by Hillel (2000), Harel's (2000) pedagogical principles and Sierpinska's (2000) thinking modes on Linear Algebra teaching were intended to be established. The research is a design-based research and three cycles were conducted to determine the design principles for the learning environment. The study group of the first cycle consists of 51, the second cycle's working group was 44, and the third cycle's study group consisted of 11 teacher candidates. The data of the research were obtained by field notes and video recordings. By analyzing the field notes and video recordings, design principles for the learning environment were determined after the first two cycles in light of the literature. The third cycle was carried out with the determined design principles, and the design principles were revised in line with the opinions of the teacher candidates and the course teacher after the application with the reports obtained during the application process. Design principles in the light of the results obtained from the research are as follows ; the use of technology, usage of modes of description, tasks, worksheets, the role of the teacher and group work. It is thought that a learning environment that will be created by paying attention to these design principles will contribute to the pre-service teachers' differentiation and use of different languages, the development of thinking styles, and to meet the principles of concreteness, necessity and generalizability.

Anahtar Kelimeler

Linear algebra, vector spaces, learning environment, design principles, technology, desing-based research

Vektör Uzaylarının Öğretimine Yönelik Bir Öğrenme Ortamının Tasarım İlkelerini Oluşturma

Öz

Bu çalışmada Lineer Cebir öğretimi üzerine Hillel’in (2000) tanımladığı temsil dilleri, Harel’in (2000) pedagojik prensipleri ve Sierpinska’nın (2000) düşünme biçimleri göz önünde bulundurularak vektör uzaylarının öğretimine yönelik teknoloji destekli bir öğrenme ortamının tasarım ilkelerinin belirlenmesi amaçlanmıştır. Araştırma, tasarım tabanlı bir araştırma olup öğrenme ortamına yönelik tasarım ilkelerinin belirlenmesi için üç döngü gerçekleştirilmiştir. Birinci döngünün çalışma grubunu 51, ikinci döngünün çalışma grubunu 44 ve üçüncü döngünün çalışma grubunu 11 öğretmen adayı oluşturmaktadır. Araştırmanın verileri alan notları ve video kayıtları ile elde edilmiştir. Alan notları ve video kayıtları analiz edilerek literatür ışığında ilk iki döngü sonrasında öğrenme ortamına yönelik tasarım ilkeleri belirlenmiştir. Belirlenen tasarım ilkeleriyle üçüncü döngü gerçekleştirilmiş, uygulama sürecinde elde edilen raporlar ile uygulama sonrasında öğretmen adayları ve ders öğretmenin görüşleri doğrultusunda tasarım ilkeleri revize edilerek son halini almıştır. Araştırmadan elde edilen sonuçlar ışığında tasarım ilkeleri; teknoloji kullanımı, temsil dillerinin kullanımı, ödevler, çalışma yaprakları, öğretmenin rolü ve grup çalışması şeklinde ortaya çıkmıştır. Belirlenen bu tasarım ilkelerine dikkat edilerek oluşturulacak bir öğrenme ortamının öğretmen adaylarının farklı dilleri ayırt etmesine ve kullanmasına, düşünme biçimlerinin gelişimine katkı sağlayacağı ve somutluk, gereklilik ve genellenebilirlik prensiplerinin karşılanmasında yardımcı olacağı düşünülmektedir.

Anahtar Kelimeler

Lineer cebir, vektör uzayları, öğrenme ortamı, tasarım ilkeleri, teknoloji, tasarım tabanlı araştırma

Kaynakça

• Aydın, S. (2009b). The factors effecting linear algebra. Procedia Social and Behavioral Sciences, 1, 1549–1553.
• Baki, A., Güven, B. ve Karataş, İ., (2002, Eylül). Dinamik geometri yazılımı cabri ile keşfederek öğrenme. V. Ulusal Fen Bilimleri ve Matematik Eğitimi Kongresi’nde sunulan bildiri, ODTÜ, Ankara.
• Britton, S. and Henderson, J. (2009). Linear algebra revisited: An attempt to understand students’ conceptual difficulties. International Journal of Mathematical Education in Science and Technology, 40(7), 963–974.
• Carlson, D. (1993). Teaching linear algebra: Must the fog always roll in? The College Mathematics Journal, 24(1), 29–40.
• Çelik, D. (2015). Investigating students’ modes of thinking in linear algebra: The case of linear independence. International Journal for Mathematics Teaching and Learning, 16(1). Retrieved January 20, 2015 from http://www.cimt.org.uk/journal/index.htm.
• Dikovic, L. (2007). Interactive learning and teaching of linear algebra by webtechnologies: Some examples. The Teaching of Mathematics, 10, 109-116.
• Dogan, H. (2001). A comparison study between a traditional and experimental first-year linear algebra program (Unpublished doctoral dissertation). University of Oklahoma, USA
• Doğan-Dunlap, H. (2010). Linear algebra students’ modes of reasoning: Geometric representations. Linear Algebra and Its Applications, 432, 2141–2159.
• Donevska-Todorova, A. (2018). Fostering students’ competencies in linear algebra with digital resources. In S. Stewart, C. Andrews-Larson, A. Berman & M. Zandieh (Ed.) Challenges and Strategies in Teaching Linear Algebra (pp. 261-276). Hamburg: Springer International Publishing.
• Dorier, J. L. (1995). Meta level in the teaching of unifying and generalizing concepts in mathematics. Educational Studies in Mathematics, 29, 175–197.
• Dorier, J. L. (1998). The role of formalism in the teaching of the theory of vector spaces. Linear Algebra and its Applications (275), 14, 141–160.
• Dorier, J. L. (2000). On the teaching of linear algebra. Dordrecht: Kluwer Academic.
• Dorier, J. L., Robert, A., Robinet, J. and Rogalski, M. (2000). The obstacle of formalism in linear algebra. In J. L. Dorier (Ed.), On the teaching of linear algebra (pp. 85–124). Dordrecht: Kluwer Academic.
• Dorier, J. L. (2002). Teaching linear algebra at university. Proceedings of ICM, 3, 875-884.
• Dubinsky, E. (1997). Some thoughts on a first course in linear algebra at the college level. In D. Carlson, C. R. Johnson, D. Lay, A. D. Porter, A. E. Watkins & W. Watkins (Eds.), Resources for teaching linear algebra (pp. 85-105). Washington: Mathematical Association of America.
• Glesne, C. (2012). Nitel araştırmaya giriş (A. Ersoy ve P. Yalçınoğlu, Çev.). Ankara: Anı Yayıncılık.
• Harel, G. (1987). Variations in linear algebra content presentations. For the Learning of Mathematics, 7(3), 29-32.
• Harel, G. (1989a). Learning and teaching linear algebra: Difficulties and an alternative approach to visualizing concepts and processes, Focus on Learning Problems in Mathematics 11, 139-148.
• Harel, G. (1989b). Applying the principle of multiple embodiments in teaching linear algebra: Aspects of familiarity and mode of representation, School Science and Mathematics, 89, 49-57.
• Harel, G. (1990). Using geometric models and vector arithmetic to teach high school students basic notions in linear algebra, International Journal for Mathematics Education in Science and Technology, 21, 387-392.
• Harel, G. (2000). Principles of learning and teaching of linear algebra: Old and new observations. In J. L. Dorier (Ed.), On the teaching of linear algebra (pp. 177-189). Dordrecht: Kluwer Academic Publishers.
• Herrington, J. A., McKenney, S., Reeves, T. C. and Oliver, R. (2007). Design-based research and doctoral students: Guidelines for preparing a dissertation proposal. In C. Montgomerie & J. Seale (Eds.), Proceedings of EdMedia 2007: World Conference on Educational Multimedia, Hypermedia & Telecommunications (pp. 4089-4097). Chesapeake, VA: AACE.
• Hillel, J. and Sierpinska, A. (1994). On one persistent mistake in linear algebra. The Proceedings PME 18, 2, 65-72.
• Hillel, J. (2000). Modes of description and the problem of representation in linear algebra. In J. L. Dorier (Ed.), On the teaching of linear algebra (pp. 191-207). Dordrecht: Kluwer Academic Publishers.
• Klasa, J. (2009). A few pedagogical designs in linear algebra with Cabri and Maple. Linear Algebra and its Applications, 432, 2100–2111.
• Kolman, B. and Hill, D.R. (2008). Elementary linear algebra and its applications (9th ed.). New Jersey: Pearson Prentice Hall.
• Kuzu, A., Çankaya, S. ve Mısırlı, A. (2011). Tasarım tabanlı araştırma ve öğrenme ortamlarının tasarımı ve geliştirilmesinde kullanımı. Anadolu Journal of Educational Sciences International, 1(1), 19-35.
• National Council of Teachers of Mathematics [NTCM]. (2000). Principles and standards for school mathematics. Reston: VA.
• Nardi, E. (1997). The novice mathematician’s encounter with mathematical abstraction: A concept image of spanning sets in vectorial analysis. Educación Matemática, 91(1), 47-60.
• Pecuch-Herrero, M. (2000). Strategies and computer projects for teaching linear algebra. International Journal of Mathematical Education in Science and Technology, 31, 181-186.
• Reeves, T. C. (2000, April). Enhancing the worth of instructional technology research through "design experiments" and other development research strategies. Paper presented at the International Perspectives on Instructional Technology Research for the 21st Century, New Orleans, LA.
• Robert, A. and Robinet, J. (1989). Quelques résultats sur l'apprentissage de l'algèbre linéaire en première année de DEUG, Cahier de Didactique des Mathématiques nº53, IREM de Paris VII.
• Sierpinska, A., Dreyfus, T. and Hillel, J. (1999). Evaluation of a teaching design in linear algebra: The case of linear transformations. Recherches en Didactique des Mathématiques 19(1), 7-41.
• 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.
• Stewart, S. and Thomas, M. O. J. (2010). Student learning of basis, span and linear independence in linear algebra. International Journal of Mathematical Education in Science and Technology, 41(2), 173–188.
• Stringer, E. (1999). Action research: A hand book for practitioners (2nd ed.). Thousand Oaks, CA: Sage Publications.
• Turğut, M. (2010). Teknoloji destekli lineer cebir öğretiminin ilköğretim matematik öğretmen adaylarının uzamsal yeteneklerine etkisi (Yayınlanmamış doktora tezi). Dokuz Eylül Üniversitesi, Eğitim Bilimleri Enstitüsü, İzmir.
• Wang, F. and Hannafin, M. J. (2005). Design-based research and technology-enhanced learning environments. Educational Technology Research and Development, 53(4), 5-23.
• Wu, H. (2004). Computer aided teaching in linear algebra. The China Papers, July, 100-102.
• Yıldırım, A. ve Şimşek, H. (2005). Sosyal bilimlerde nitel araştırma yöntemleri. Ankara: Seçkin Yayınları.