ÖĞRETMEN ADAYLARININ ÜÇGEN KAVRAMINA YÖNELİK KAVRAM İMAJLARININ GÖRSELLEŞTİRİLMESİNDE SOM VE WARD KÜMELEME ALGORİTMALARININ KULLANIMI

Abstract

USING SOM (SELF-ORGANIZINGMAP) COMBINED WITH WARD’S CLUSTERING ALGORITHM FOR VISUALIZATION OF TEACHER CANDIDATES’ CONCEPT IMAGES ABOUT TRIANGLE CONCEPT

The purpose of this study is determining the concept images of primary school mathematics teacher candidates on a geometrical concept. Tall and Vinner (1981) defined the concept image as the mental structures encoded by the person to his/her mind about a mathematical thought. The concept image of an individual for a concept develops with his/her experiences that are related and not related with that concept. The concept image is considered as an efficient method in revealing the knowledge of students on a concept and in revealing their misconceptions. With this study, the use of a new method is suggested in visualizing and modeling the cognitive structures of students on the triangle concept. The SOM (Self-Organizing Map) and Ward Cluster Analysis were used in the study to analyze the data. SOM is a specific form of artificial neural network; and during the training, non-supervised training is used. Basically, the working logic is a process that aims to simplify the problem based on reducing the multi-dimensional inputs into outputs with less size. Open-ended questions were used as data collection tools in the study in which 160 teacher candidates were included. By using the data obtained in the study, the cognitive structures of triangle concept of the teacher candidates were visualized with the SOM Method. Students' answers are transformed into numerical data by a series of operations. The data obtained is analyzed with Viscovery SOMine software. It has been seen that students are divided into 4 groups according to concept images.

Keywords: Concept image, Concept of triangle, SOM, Ward’s clustering algorithm.

Özet

Bu çalışmanın amacı ilköğretim matematik öğretmen adaylarının bir geometrik kavramailişkin kavram imajlarını belirlemektir. Tall ve Vinner (1981) kavram imajını bir matematiksel düşünceye ilişkin kişinin zihnine kodlamış olduğu zihinsel yapılar olarak tanımlamıştır. Bir bireyin bir kavrama ilişkin kavram imajı o kavramla ilişkili olan ve olmayan deneyimleri ile gelişir. Kavram imajı öğrencilerin kavrama ilişkin bilgilerini ve kavram yanılgılarını ortay çıkarmada etkili bir yöntem olarak görülmektedir. Bu çalışma ile öğrencilerin üçgen kavramına ilişkin bilişsel yapılarını görselleştirerek modellemek için yeni bir yöntem kullanılması önerilmektedir. Çalışmada SOM (Self-OrganizingMap) ile birlikte Wardkümeleme analizi kullanılarak veriler analiz edilmiştir. SOM, yapay sinir ağlarının özel bir biçimidir ve eğitimleri sırasında gözetimsiz eğitim kullanılmaktadır. Temel olarak çok boyutlu girdilerin daha az boyuttaki çıktılara indirgenmesine dayanan çalışma mantığı problemin basitleştirilmesini amaçlayan bir boyut azaltma işlemidir. 160 öğretmen adayının yer aldığı çalışmada veri toplama aracı olarak açık uçlu soru kullanılmıştır.Öğrencilerin cevapları bir dizi işlem ile numerik verilere dönüştürülmüştür. Elde edilen veriler ViscoverySOMine yazılımında analiz edilmiştir. Öğrencilerin kavram imajlarına göre 4 gruba ayrıldığı görülmüştür.

Anahtar Kelimeler:Kavram imajı, Üçgen Kavramı, SOM, Ward kümeleme algoritması.

KAYNAKLAR

Thuneberg, H. &Hotulainen, R. (2006). Contributions of data miningfor psycho‐educational research: what self‐organizing maps tell us about the well‐being ofgifted learners, High Ability Studies, 17:1, 87-100

Nordlander, M.C. &Nordlander, E. (2012). On the conceptimage of complex numbers, International Journal of Mathematical Education in Science andTechnology, 43:5, 627-641,

Bingolbali E. & Monaghan, J. (2008). Concept image revisited, Educ. Stud. Math. 68(1) pp. 19–35.

Tall, D.&Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to Limits and Continuity, Educ. Stud. Math., 12, pp. 151–169.

Heeffer, A.(2007). Learning concepts through the history of mathematics: The case of symbolic algebra, in Philosophical Dimensions in Mathematics Education, K. Franc¸ois and J.P. VanBendegem, eds., Springer, New York, pp. 83–103.

Fischbein, E. (2007). Intuitions in Science and Mathematics: An Educational Approach, Reidel, Dordrecht, The Netherlands.

Fischbein, E., Tirosh, D.& Hess, P. (1979).The intuition of infinity, Educ. Stud.Math. 10(1), pp. 3–40.

Biza, I., Christou, C.&Zachariades, T. (2008).Student perspectives on the relationship between a curve and its tangent in the transition from Euclidean Geometry to Analysis, Res. Math. Educ. 10(1), pp. 53–70.

Vinner, S. (1991).The role of definitions in the teaching and learning of mathematics. In Advanced mathematical thinking (pp. 65-81).Springer, Netherlands.

Vinner, S.&Hershkowitz, R. (1980).Concept images and some common cognitive paths in thedevelopment of some simple geometric concepts, Proceedings of the 4th InternationalConference of the International Group for the Psychology of Mathematics Education,R. Karplus, ed., Berkeley, CA, vol. 1, pp. 177–184.

Marton, F. &Booth, S. (1997). Learning and Awareness, Lawrence Erlbaum, Mahwah, NJ

Vincent, B., Larue, R., Sealey, V. &Engelke, N. (2015).Calculus students' early concept images of tangent lines, International Journal of MathematicalEducation in Science and Technology, 46:5, 641-657

Kohonen, T. (1988) Self-organization and associative memory (New York, Springer-Verlag).

Gan, G., Ma, C., &Wu, J. (2007).Data Clustering Theory, Algorithms,and Applications, American Statistical Association, Philadelphia, Pa, USA.

Kohonen, T. (1995).Self-Organization Maps, Springer, Berlin, Germany

Ward, J.H. (1963). “Hierarchical grouping to optimize an objectivefunction,” Journal of theAmericanStatistical Association, vol. 58,pp. 236–244

Vesanto, J.&Alhoniemi, E. (2000). “Clustering of the self-organizingmap,” IEEE Transactions on Neural Networks, vol. 11, no. 3, pp.586–600