Öz
Limit, güçlü bir matematiksel düşünme becerisi gerektiren, öğrencilerin öğrenmede
zorlandıkları bir kavramdır. Öğrencilerde limit kavramına yönelik, kavram imajlarının
ve kavram tanımlarının araştırılmasının, limit kavramının öğrenilme süreçlerinin
belirlenmesine katkı getireceği düşünülmektedir. Bu çalışmada, tek değişkenli reel
değerli fonksiyonlar için limit kavramına yönelik öğrencilerin kavram imajları, kavram
tanımları ve öğrencilerin kavram imajları ile limitin formal tanımını ilişkilendirme
durumlarının belirlenmesi amaçlanmıştır. Çalışma, Analiz I dersini almakta olan 31
öğrenci arasından amaçlı örnekleme yöntemlerinden biri olan ölçüt örnekleme yöntemi
kullanılarak seçilen 11 öğrenci ile gerçekleştirilmiştir. Analiz I dersinde limit
konusunun öğretimi tamamlandıktan sonra, açık uçlu sorulardan oluşan bir test 31
öğrenciye uygulanmıştır. Öğrencilerin yanıtları araştırmacılar tarafından nitel olarak
analiz edilmiştir. Öğrencilerin kavram imajları, kavram tanımları ve kavram imajları ile
limitin formal tanımı arasında kurdukları ilişkiler ölçüt olarak alınarak öğrenciler sekiz
gruba ayrılmıştır. Her gruptan grup büyüklüğüne göre öğrenci seçilerek 11 öğrenciyle
klinik görüşmeler gerçekleştirilmiştir. Yapılan görüşmeler araştırmacılar tarafından
nitel olarak analiz edilmiştir. Çalışmanın sonucunda öğrencilerin hem limit kavramına
ilişkin kavram imajlarında hem de kavram tanımlarında sağ-sol limit eşitliği teoremi ile limitin dinamik formunun baskın olduğu görülmüştür. Öğrencilerin limitin formal tanımını açıklama ve kullanmada zorlandığı tespit edilmiştir. Ayrıca öğrencilerin
uygulama sorularını çözmek için kavram imajlarını kullandığı görülmüştür. Diğer
yandan, öğrencilerin çoğunun kavram imajları ile limitin formal tanımı arasında ilişki
kuramadığı sonucuna ulaşılmıştır.
Anahtar Kelimeler
Limit kavramı Kavram tanımı Kavram imajı Formal limit tanımı
Kaynakça
- Bezuidenhout, J. (2001). Limits and continuity: Some conceptions of first-year students. International Journal of Education, Science, and Technology, 32(4), 487 – 500.
- Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. A. E. Kelly & R. A. Lesh, (Eds.), Handbook of research design in mathematics and science education (pp. 547–589). London: Lawrence Erlbaum.
- Cornu, B. (1991). Limits. In Tall, D. (Ed.), Advanced Mathematical Thinking (pp. 153-166). Boston: Kluwer.
- Cottrill, J., Dubinsky, E., Nichols, D., Schwinngendorf, K., Thomas, K., & Vidakovic, D. (1996). Understanding the Limit concept: Beginning with a coordinated process schema. Journal of Mathematical Behavior, 15, 17-192.
- Juter, K., & Grevholm, B. (2006). Limits and infinity: A study of university students' performance. To appear in C. Bergsten, B. Grevholm, H. Måsøval, & F. Rønning (Eds.), Relating practice and research in mathematics education. Fourth Nordic Conference on Mathematics Education, Trondheim, 2nd-6th of September 2005. Trondheim: Sør-Trøndelag University College.
- Mamona-Downs, J. (2001). Letting the intuitive bear on the formal: A didactical approach for the understanding of the limit of a sequence. Educational Studies in Mathematics, 48, 259-288.
- Miles, M., & Huberman, A. M. (1994). Qualitative data analysis: an expanded sourcebook. Thousand Oaks, CA: Sage.
- Przenioslo, M. (2004). Images of the limit of function formed in the course of mathematical studies at the university. Educational Studies in Mathematics, 55, 103-132.
- Roh, H. K. (2007). An activity for development of the understanding of the concept of Limit. In Woo, J. H., Lew, H. C., Park, K. S. & Seo, D. Y. (Eds.).Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, pp. 105-112. Seoul: PME.
- Sierpinska, A. (1987). Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics, 18, 371-397.
- Szydlik, J. E. (2000). Mathematical beliefs and conceptual understanding of the limit of a function, Journal for Research in Mathematics Education, 31(3), 258–276.
- Tall, D. (1980). Mathematical intuition, with special reference to limiting processes. In R. Karplus (Ed.), Proceedings of the Fourth International Conference for the Psychology of Mathematics Education (pp. 170-176). Berkeley, CA: PME.
- Tall, D. O. & Vinner, S. (1981). Concept image and concept definition in Mathematics with particular reference to limit and continuity. Educational Studies in Mathematics, 12, 151-169.
- Tall, D. O. (1992). The Transition to Advanced Mathematical Thinking: Functions, Limits, Infinity, and Proof, in Grouws D.A. (ed.) Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York, 495– 511.
- Tall, D. O. (2001). Natural and formal infinities. Educational Studies in Mathematics, 48, 199-238.
- Williams, S. R. (1991). Models of limit held by college calculus students. Journal for Research in Mathematics Education, 22, 219-236.
- Vinner, S. (1983). Concept definition, concept image and the notion of function. International Journal of Mathematical Education in Science and Technology, 14, 293-305.
- Vinner, S. (1991). The Role of Definitions in the Teaching and Learning of Mathematics. In Tall, D. (Ed.), Advanced Mathematical Thinking (pp. 65-81). Boston: Kluwer.
- Yıldırım, A. & Şimşek, H. (2003). Sosyal Bilimlerde Nitel Araştırma Yöntemleri. Ankara: Seçkin Yayıncılık.
Öz
Limit is the concept that students have difficulties and is required to powerful mathematical thinking skill. To investigate students’ concept definitions and concept images about limit concept is important to reveal learning processes of limit concept. The purpose of the study is to determine students’ concept images, concept definitions and relationships between students’ concept images and formal definition about limit concept for one variable real valued functions. The study was conducted with 11 students who were selected from 31 students attending Analysis I course. Criterion sampling method that is a purposive sampling method was used in selection of the students. After teaching process of limit concept, a test which had open-ended questions was carried out to 31 students. Responses of students were analyzed qualitatively by researchers. With regard to analyses, the students were classified into eight groups using their concept images, concept definitions and relationships between their concept images and formal definition of limit as criterions. Eleven students were selected from groups according to group sizes to carry out clinical interviews. The clinical interviews were analyzed qualitatively by researchers. Consequently, it was found that both students’ concept images and concept definitions focused on the theorem about equality of right and left hand limits and dynamic form of limit. Moreover, the students had difficulties in explaining and using the formal definition of limit. In addition, students used their concept images to solve questions and most of them could not establish the relationship between their concept images and formal definition of limit
Anahtar Kelimeler
Limit concept Concept definition Concept image Formal definition of limit
Kaynakça
- Bezuidenhout, J. (2001). Limits and continuity: Some conceptions of first-year students. International Journal of Education, Science, and Technology, 32(4), 487 – 500.
- Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. A. E. Kelly & R. A. Lesh, (Eds.), Handbook of research design in mathematics and science education (pp. 547–589). London: Lawrence Erlbaum.
- Cornu, B. (1991). Limits. In Tall, D. (Ed.), Advanced Mathematical Thinking (pp. 153-166). Boston: Kluwer.
- Cottrill, J., Dubinsky, E., Nichols, D., Schwinngendorf, K., Thomas, K., & Vidakovic, D. (1996). Understanding the Limit concept: Beginning with a coordinated process schema. Journal of Mathematical Behavior, 15, 17-192.
- Juter, K., & Grevholm, B. (2006). Limits and infinity: A study of university students' performance. To appear in C. Bergsten, B. Grevholm, H. Måsøval, & F. Rønning (Eds.), Relating practice and research in mathematics education. Fourth Nordic Conference on Mathematics Education, Trondheim, 2nd-6th of September 2005. Trondheim: Sør-Trøndelag University College.
- Mamona-Downs, J. (2001). Letting the intuitive bear on the formal: A didactical approach for the understanding of the limit of a sequence. Educational Studies in Mathematics, 48, 259-288.
- Miles, M., & Huberman, A. M. (1994). Qualitative data analysis: an expanded sourcebook. Thousand Oaks, CA: Sage.
- Przenioslo, M. (2004). Images of the limit of function formed in the course of mathematical studies at the university. Educational Studies in Mathematics, 55, 103-132.
- Roh, H. K. (2007). An activity for development of the understanding of the concept of Limit. In Woo, J. H., Lew, H. C., Park, K. S. & Seo, D. Y. (Eds.).Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, pp. 105-112. Seoul: PME.
- Sierpinska, A. (1987). Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics, 18, 371-397.
- Szydlik, J. E. (2000). Mathematical beliefs and conceptual understanding of the limit of a function, Journal for Research in Mathematics Education, 31(3), 258–276.
- Tall, D. (1980). Mathematical intuition, with special reference to limiting processes. In R. Karplus (Ed.), Proceedings of the Fourth International Conference for the Psychology of Mathematics Education (pp. 170-176). Berkeley, CA: PME.
- Tall, D. O. & Vinner, S. (1981). Concept image and concept definition in Mathematics with particular reference to limit and continuity. Educational Studies in Mathematics, 12, 151-169.
- Tall, D. O. (1992). The Transition to Advanced Mathematical Thinking: Functions, Limits, Infinity, and Proof, in Grouws D.A. (ed.) Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York, 495– 511.
- Tall, D. O. (2001). Natural and formal infinities. Educational Studies in Mathematics, 48, 199-238.
- Williams, S. R. (1991). Models of limit held by college calculus students. Journal for Research in Mathematics Education, 22, 219-236.
- Vinner, S. (1983). Concept definition, concept image and the notion of function. International Journal of Mathematical Education in Science and Technology, 14, 293-305.
- Vinner, S. (1991). The Role of Definitions in the Teaching and Learning of Mathematics. In Tall, D. (Ed.), Advanced Mathematical Thinking (pp. 65-81). Boston: Kluwer.
- Yıldırım, A. & Şimşek, H. (2003). Sosyal Bilimlerde Nitel Araştırma Yöntemleri. Ankara: Seçkin Yayıncılık.
Ayrıntılar
| Birincil Dil | Türkçe |
|---|---|
| Bölüm | Araştırma Makalesi |
| Yazarlar | |
| Yayımlanma Tarihi | 8 Temmuz 2015 |
| Gönderilme Tarihi | 8 Temmuz 2015 |
| Yayımlandığı Sayı | Yıl 2015 Cilt: 5 Sayı: 1 |
Kaynak Göster
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