Öğretmenlerin Fonksiyon Kavramı ve Öğretimine İlişkin Pedagojik Görüşleri
Yıl 2010,
Cilt: 9 Sayı: 3, 697 - 723, 01.12.2010
İbrahim Bayazıt
Yılmaz Aksoy
Öz
Bu araştırmada öğretmenlerin fonksiyonlar konusunda sahip oldukları pedagojik alan bilgileri iki boyutu itibariyle incelenmektedir. Araştırmanın örneklem uzayı meslekte yeterli tecrübeye sahip iki öğretmenden oluşmaktadır. Veri toplama ve analiz aşamalarında, nitel araştırma yönteminin araçları kullanılmış ve çalışmaya teorik alt yapı oluşturmak üzere Shulman (1986) tarafından geliştirilmiş olan „pedagojk alan bilgisi‟ düşüncesinden yararlanılmıştır. Çalışmanın bulguları öğretmenlerin, öğrencilerinin fonksiyon kavramını öğrenirken karşılaştıkları zorluklar ve geliştirdikleri kavram yanılgılarını teşhis etme ve bunların zihinsel sebeplerini anlamada oldukça benzer düşüncelere sahip olduklarını, ancak bu zorlukların ve yanılgıların giderilmesi için farklı öğretim yaklaşımları sergilediklerini göstermektedir
Kaynakça
- Ball, D. L. (1991). Research on Teaching Mathematics: Making Subject-Matter Knowledge Part of the Equation. In J. Brophy (Ed.), Advances in Research on Teaching (Vol. 2, pp. 1-48). Greenwich: JAI Press.
- Bayazit, I., & Gray, E. (2004). Understanding Inverse Functions: The Relationship between Teaching Practice and Student Learning. In M. J. Honies & A. B. Fuglestad (Eds.), Proceedings of 28th Conference of the International Group for the Psychology of Mathematics Education, Bergen, Norway, Vol. 2, pp. 103- 110.
- Breidenbach, D., Dubinsky, Ed., Hawks, J., & Nichols, D. (1992). Development of the Process Conception of Function. Educational Studies in Mathematics, 23(3), pp. 247-285.
- Bromme, R. (1995). What Exactly Is Pedagogic Content Knowledge? Critical Remarks Regarding a Fruitful Research Program. In S. Hopmann & K. Riquarts (Eds.), Didactic and/or Curriculum (pp. 205-216). Schriftenreihe:Kiel.
- Dede, Y., Bayazit, İ., & Soybaş, D. (2010). Öğretmen Adaylarının Denklem, Fonksiyon ve Polinom Kavramlarını Anlamaları. Kastamonu Eğitim Dergisi, 18(1), 67-88.
- DeMarois, P. & Tall, D.O. (1996). Facets and Layers of the Function Concept. Proceedings of 20th Conference of the International Group for the Psychology of Mathematics Education, Valencia, Vol. 2, pp.297-304.
- Dubinsky, Ed. & Harel, G. (1992). The Nature of the Process Conception of Function. In G. Harel & Ed. Dubinsky (Eds.), The Concept of Function: Aspects of Epistemology and Pedagogy (pp. 85-107). United States of America: Mathematical Association of America.
- Eisenberg, T. (1991). Function and Associated Learning Difficulties. In D.O.Tall (Ed.), Advanced Mathematical Thinking (pp. 140-152). Dordrecht: Kluwer Academic Publishers.
- Escudero, I., & Sanchez, V. (2002). Integration of Domains of Knowledge in Mathematics Teachers‟ Practice. In Cockburn & Nardi (Eds.,) Proceedings of the 26 Conference of International Group of PME, Vol. 2, pp. 177-184.
- Even, R. (1988). Pre-service Teachers conceptions of the Relationships between Functions and Equations. Proceedings of the International Group for the Psychology of Mathematics Education XI (PME XII), Hungary.
- Even, R. (1992). The Inverse Function: Prospective Teachers‟ Use of „Undoing‟. International Journal of Mathematical Education in Science and Technology, 23(4), pp. 557-562
- Ginsburg, H. (1981). The Clinical Interview in Psychological Research on Mathematical Thinking: Aims, Rationales, Techniques. For the Learning of Mathematics, 1(3), pp. 57-64.
- Heibert, J., & Lefevre, P. (1986). Conceptual and Procedural Knowledge: The Case of Mathematics. New Jersey: Lawrence Erlbaum Assocıates Inc.
- Heibert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., Oliver, A., & Human, P. (1997). A Day in teh Life of a Conceptually Based Instruction Classroom. In Paeke, L. (Ed.), Making Sense: Teaching and Learning Mathematics with Understanding (pp. 101-114). United States of America: Heinemann.
- Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, Graphs, and Graphing: Tasks, Learning, and Teaching. Review of Educational Research, 60(1), 1-64.
- Lloyd, G. M., & Wilson, M. (1998). Supporting Innovation: The Impact of a Teacher‟s Conceptions of Functions on His Implementation of a Reform Curriculum. Journal for Research in Mathematics Education, 29(3), 248-274.
- Malik, M. A. (1980). Historical and Pedagogical Aspects of the Definition of Function. International Journal of Mathematical Education in Science and Technology, 11(4), 489-492.
- Markovits, R., Eylon, B. S., & Brukheimer, M. (1986). Functions Today and Yesterday. For the Learning of Mathematics, 6(2), 18-28.
- Miles, M. B., & Huberman, A. M. (1994). Qualitative Data Analysis: An Expanded Sourcebook. London: Sage Publications.
- Özmantar. M. F., & Bingölbali, E. (2010). Sınıf Öğretmenleri ve Matematiksel Zorlukları. Gaziantep Üniversitesi Sosyal Bilimler Dergisi, 8(2), 401-427.
- Phillips, N. & Hardy, C. (2002). Discourse Analysis: Investigating Processes of Social Construction. United Kingdom: Sage Publications Inc.
- Schwingendorf, K., Hawks, J., & Beineke, J. (1992). Horizontal and Vertical Growth of the Students‟ Conception of Function. In G. Harel & Ed. Dubinsky (Eds.), The Concept of Function Aspects of Epistemology and Pedagogy (pp. 133-151). United States of America: Mathematical Association of America.
- Sfard, A. (1992). Operational Origins of Mathematical Objects and the Quandary of Reification-The Case of Function. In Harel & Ed. Dubinsky (Eds.), The Concept of Function Aspects of Epistemology and Pedagogy (pp. 59-85). United States of America: Mathematical Association of America.
- Shulman, L. (1986). Those Who Understand: Knowledge Growth in Teaching. Educational Researcher, 15, pp. 4-14.
- Smart, T. (1995). Visualising Quadratic Functions: A Study of Thirteen-Year-Old Girls Learning Mathematics with Graphic Calculators. In L. Meira & Carraher (Eds.) Proceeding of the 19th International Conference for the Psychology of Mathematics Education. Brazil: Atual Editora Ltda, Vol. 2, pp. 272-279.
- Tall, D. & Bakar, M. (1992). Students‟ Mental Prototypes for Function and Graphs. International Journal of Mathematics Education in Science and Technology, 23(1), pp. 39-50.
- Tall, D. & Vinner, S. (1981). Concept Image and Concept Definition in Mathematics with Particular Reference to Limits and Continuity. Educational Studies in Mathematics, 12, pp. 151-169.
- Tirosh, D., Even, R., & Robinson, N. (1998). Simplifying Algebraic Expressions: Teacher Awareness and Teaching Approaches. Educational Studies in Mathematics, 35, 51-64.
- Vinner, S. (1983). Concept Definition, Concept Image and the Notion of Function. International Journal of Mathematical Education in Science and Technology, 14(3), pp. 293-305.
- Watkins, C. & Mortimore, P. (1999). Pedagogy: What Do We Know? In P. Mortimore (Ed.), Understanding Pedagogy and its Impact on Learning (pp. 1- 20). London: Paul Chapman Publishing Ltd.
- Wilson, S. M., Shulman, L. S. & Richert, A. E. (1987). „150 Different Ways‟ of Knowing: Representations of Knowledge in Teaching‟. In J. Calderhead (Ed.), Exploring Teachers’ Thinking (pp. 104-124). London: Cassel Education Ltd.
- Yerushalmy, M. & Schwartz, J. L. (1993). Seizing the Opportunity to make Algebra Mathematically and Pedagogically Interesting. In Romberg, T. A., Fennema, E., & Carpenter, T. P. (Eds.), Integrating Research on the Graphical Representation of Functions (pp. 41-68). Hillsdale, NJ: Lawrence Erlbaum Associates.
- Yeşildere, S. & Akkoç, H. (2010). Matematik Öğretmen Adaylarının Sayı Örüntülerine İlişkin Pedagojik Alan Bilgilerinin Konuya Özel Stratejiler Bağlamında İncelenmesi. Ondokuz Mayıs Üniversitesi Eğitim Fakültesi Dergisi, 29(1), 125-149.
- Yin, R. K. (2003). Case Study research: Design and methods. United Kingdom: Sage Publications Ltd.
Teachers’ Pedagogical Indications about The Concept of Function and Its Teaching
Yıl 2010,
Cilt: 9 Sayı: 3, 697 - 723, 01.12.2010
İbrahim Bayazıt
Yılmaz Aksoy
Öz
This paper examines two aspects of teacher‟s pedagogical content knowledge within the function context. The participants were two experienced mathematics teachers. The research employed a qualitative case study using semi-structured interviews as the main source of data. Data was analysed using qualitative methods which included content and discourse analysis methods. The notion of „pedagogical content knowledge‟ provided a theoretical framework for the study. The research findings indicated that there was no difference between the teachers in diagnosing their students‟ diffculties and misconceptions with the function concept and their sources; yet the teachers differed remarkably in proposing pedagogical treatments to help their students overcome these obstacles
Kaynakça
- Ball, D. L. (1991). Research on Teaching Mathematics: Making Subject-Matter Knowledge Part of the Equation. In J. Brophy (Ed.), Advances in Research on Teaching (Vol. 2, pp. 1-48). Greenwich: JAI Press.
- Bayazit, I., & Gray, E. (2004). Understanding Inverse Functions: The Relationship between Teaching Practice and Student Learning. In M. J. Honies & A. B. Fuglestad (Eds.), Proceedings of 28th Conference of the International Group for the Psychology of Mathematics Education, Bergen, Norway, Vol. 2, pp. 103- 110.
- Breidenbach, D., Dubinsky, Ed., Hawks, J., & Nichols, D. (1992). Development of the Process Conception of Function. Educational Studies in Mathematics, 23(3), pp. 247-285.
- Bromme, R. (1995). What Exactly Is Pedagogic Content Knowledge? Critical Remarks Regarding a Fruitful Research Program. In S. Hopmann & K. Riquarts (Eds.), Didactic and/or Curriculum (pp. 205-216). Schriftenreihe:Kiel.
- Dede, Y., Bayazit, İ., & Soybaş, D. (2010). Öğretmen Adaylarının Denklem, Fonksiyon ve Polinom Kavramlarını Anlamaları. Kastamonu Eğitim Dergisi, 18(1), 67-88.
- DeMarois, P. & Tall, D.O. (1996). Facets and Layers of the Function Concept. Proceedings of 20th Conference of the International Group for the Psychology of Mathematics Education, Valencia, Vol. 2, pp.297-304.
- Dubinsky, Ed. & Harel, G. (1992). The Nature of the Process Conception of Function. In G. Harel & Ed. Dubinsky (Eds.), The Concept of Function: Aspects of Epistemology and Pedagogy (pp. 85-107). United States of America: Mathematical Association of America.
- Eisenberg, T. (1991). Function and Associated Learning Difficulties. In D.O.Tall (Ed.), Advanced Mathematical Thinking (pp. 140-152). Dordrecht: Kluwer Academic Publishers.
- Escudero, I., & Sanchez, V. (2002). Integration of Domains of Knowledge in Mathematics Teachers‟ Practice. In Cockburn & Nardi (Eds.,) Proceedings of the 26 Conference of International Group of PME, Vol. 2, pp. 177-184.
- Even, R. (1988). Pre-service Teachers conceptions of the Relationships between Functions and Equations. Proceedings of the International Group for the Psychology of Mathematics Education XI (PME XII), Hungary.
- Even, R. (1992). The Inverse Function: Prospective Teachers‟ Use of „Undoing‟. International Journal of Mathematical Education in Science and Technology, 23(4), pp. 557-562
- Ginsburg, H. (1981). The Clinical Interview in Psychological Research on Mathematical Thinking: Aims, Rationales, Techniques. For the Learning of Mathematics, 1(3), pp. 57-64.
- Heibert, J., & Lefevre, P. (1986). Conceptual and Procedural Knowledge: The Case of Mathematics. New Jersey: Lawrence Erlbaum Assocıates Inc.
- Heibert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., Oliver, A., & Human, P. (1997). A Day in teh Life of a Conceptually Based Instruction Classroom. In Paeke, L. (Ed.), Making Sense: Teaching and Learning Mathematics with Understanding (pp. 101-114). United States of America: Heinemann.
- Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, Graphs, and Graphing: Tasks, Learning, and Teaching. Review of Educational Research, 60(1), 1-64.
- Lloyd, G. M., & Wilson, M. (1998). Supporting Innovation: The Impact of a Teacher‟s Conceptions of Functions on His Implementation of a Reform Curriculum. Journal for Research in Mathematics Education, 29(3), 248-274.
- Malik, M. A. (1980). Historical and Pedagogical Aspects of the Definition of Function. International Journal of Mathematical Education in Science and Technology, 11(4), 489-492.
- Markovits, R., Eylon, B. S., & Brukheimer, M. (1986). Functions Today and Yesterday. For the Learning of Mathematics, 6(2), 18-28.
- Miles, M. B., & Huberman, A. M. (1994). Qualitative Data Analysis: An Expanded Sourcebook. London: Sage Publications.
- Özmantar. M. F., & Bingölbali, E. (2010). Sınıf Öğretmenleri ve Matematiksel Zorlukları. Gaziantep Üniversitesi Sosyal Bilimler Dergisi, 8(2), 401-427.
- Phillips, N. & Hardy, C. (2002). Discourse Analysis: Investigating Processes of Social Construction. United Kingdom: Sage Publications Inc.
- Schwingendorf, K., Hawks, J., & Beineke, J. (1992). Horizontal and Vertical Growth of the Students‟ Conception of Function. In G. Harel & Ed. Dubinsky (Eds.), The Concept of Function Aspects of Epistemology and Pedagogy (pp. 133-151). United States of America: Mathematical Association of America.
- Sfard, A. (1992). Operational Origins of Mathematical Objects and the Quandary of Reification-The Case of Function. In Harel & Ed. Dubinsky (Eds.), The Concept of Function Aspects of Epistemology and Pedagogy (pp. 59-85). United States of America: Mathematical Association of America.
- Shulman, L. (1986). Those Who Understand: Knowledge Growth in Teaching. Educational Researcher, 15, pp. 4-14.
- Smart, T. (1995). Visualising Quadratic Functions: A Study of Thirteen-Year-Old Girls Learning Mathematics with Graphic Calculators. In L. Meira & Carraher (Eds.) Proceeding of the 19th International Conference for the Psychology of Mathematics Education. Brazil: Atual Editora Ltda, Vol. 2, pp. 272-279.
- Tall, D. & Bakar, M. (1992). Students‟ Mental Prototypes for Function and Graphs. International Journal of Mathematics Education in Science and Technology, 23(1), pp. 39-50.
- Tall, D. & Vinner, S. (1981). Concept Image and Concept Definition in Mathematics with Particular Reference to Limits and Continuity. Educational Studies in Mathematics, 12, pp. 151-169.
- Tirosh, D., Even, R., & Robinson, N. (1998). Simplifying Algebraic Expressions: Teacher Awareness and Teaching Approaches. Educational Studies in Mathematics, 35, 51-64.
- Vinner, S. (1983). Concept Definition, Concept Image and the Notion of Function. International Journal of Mathematical Education in Science and Technology, 14(3), pp. 293-305.
- Watkins, C. & Mortimore, P. (1999). Pedagogy: What Do We Know? In P. Mortimore (Ed.), Understanding Pedagogy and its Impact on Learning (pp. 1- 20). London: Paul Chapman Publishing Ltd.
- Wilson, S. M., Shulman, L. S. & Richert, A. E. (1987). „150 Different Ways‟ of Knowing: Representations of Knowledge in Teaching‟. In J. Calderhead (Ed.), Exploring Teachers’ Thinking (pp. 104-124). London: Cassel Education Ltd.
- Yerushalmy, M. & Schwartz, J. L. (1993). Seizing the Opportunity to make Algebra Mathematically and Pedagogically Interesting. In Romberg, T. A., Fennema, E., & Carpenter, T. P. (Eds.), Integrating Research on the Graphical Representation of Functions (pp. 41-68). Hillsdale, NJ: Lawrence Erlbaum Associates.
- Yeşildere, S. & Akkoç, H. (2010). Matematik Öğretmen Adaylarının Sayı Örüntülerine İlişkin Pedagojik Alan Bilgilerinin Konuya Özel Stratejiler Bağlamında İncelenmesi. Ondokuz Mayıs Üniversitesi Eğitim Fakültesi Dergisi, 29(1), 125-149.
- Yin, R. K. (2003). Case Study research: Design and methods. United Kingdom: Sage Publications Ltd.