Matematiksel İlişkilendirme Becerisinin Kuramsal Boyutta İncelenmesi: Türev Kavramı Örneği

Öz

Bu çalışmanın amacı öğretmen adaylarının matematiksel ilişkilendirme becerilerinin türev kavramı bağlamında ele alınarak yorumlanmasıdır. Bu kapsamda öncelikle matematiksel ilişkilendirme becerisi kuramsal olarak analiz edilerek bu becerinin değerlendirilmesinde kullanılabilecek bir kuramsal yapı oluşturulmaya çalışılmıştır. Daha sonra ise bir devlet üniversitesinin eğitim fakültesi son sınıf öğrencilerinden seçilen ve 51 kişiden oluşan matematik öğretmen adaylarına araştırmacı tarafından geliştirilmiş olan İlişkilendirme Beceri Testi (İBT) uygulanmıştır. İlişkilendirme Beceri Testi’nin temel bileşenleri farklı gösterimler arası ilişkilendirme, kavramlar arası ilişkilendirme, gerçek yaşamla ilişkilendirme ve farklı disiplinlerle İlişkilendirme olarak ifade edilebilir.  Çalışma sonucunda öğretmen adaylarının genel olarak türev kavramına yönelik ders kitaplarında yer alan ezberi bir takım bilgilere sahip oldukları fakat bu bilgileri birbiri ile ilişkili olarak anlamlandırmakta ve kullanmakta güçlük çektikleri gözlenmiştir.  Matematiğin daha anlamlı ve ilişkisel olarak öğrenilmesi anlamında, matematik eğitimcilerinin sınıflarında, kavramsal anlamaya odaklanmaları ve kavramların anlamlı öğrenilmesini ve gerçek yaşamla ilişkilendirilebilmesini sağlayacak etkinlik ve uygulamalara yer vermeleri önerilmiştir.

Anahtar Kelimeler

• Matematiksel ilişkilendirme becerisi,öğretmen adayları

A Theoretical Examination of the Mathematical Connection Skill: The Case of the Concept of Derivative

Öz

The aim of this study is to explore and interpret the mathematical connection skills of pre-service teachers within the context of the concept of derivative. In this regard, firstly the mathematical connection skill was analyzed theoretically, and an attempt was made to establish a theoretical structure that can be used in the evaluation of this skill. Then the Connection Skill Test (CST), developed by the researcher, was applied to the students in the study group composed of 51 people selected from among senior students of the Faculty of Education of a state university in Turkey. Four basic components of the test are as follows: connection between different representations, connection between concepts, connection with real life, and connection with different disciplines. The findings of the study indicate that most of the pre-service teachers have some rote-learning based pieces of knowledge from textbooks with regards to the concept of derivative, but they cannot, to a large extent, understand and use them in connection with each other. For a more meaningful and relational understanding of mathematics, it is suggested for mathematics educators to focus on conceptual understanding and include activities and practices that will enable concepts to be learned meaningfully and in connection with real life in their classes.

Anahtar Kelimeler

• Mathematical connection skill,teacher candidates

Kaynakça

1. Açıkyıldız, G. (2013). Matematik öğretmeni adaylarının türev kavramını anlamaları ve yaptıkları hatalar (Yayınlanmamış Yüksek Lisans Tezi). Karadeniz Teknik Üniversitesi, Eğitim Bilimleri Enstitüsü, Trabzon.
2. Ainsworth, S. (1999). The functions of multiple representations. Computers & Education, 33(2), 131-152.
3. Ainsworth, S., & Van Labeke, N. (2004). Multiple forms of dynamic representation. Learning and Instruction, 14(3), 241-255.
4. Akkoç, H. (2006). Fonksiyon kavramının çoklu temsillerinin çağrıştırdığı kavram görüntüleri. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 30, 1-10.
5. Akkuş, O. (2008). İlköğretim matematik öğretmeni adaylarının matematiği günlük yaşamla ilişkilendirme düzeyleri. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 35, 1-12.
6. Amit, M., & Vinner, S. (1990). Some misconceptions in calculus: Anecdotes or the tip of an iceberg? In G. Booker, P. Cobb, & T. N. De Mendicuti (Eds.), Proceedings of the 14th International Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 3–10). Cinvestav, Mexico: PME.
7. Amoah, V., & Laridon, P. (2004). Using multiple representations to assess students’ understanding of the derivative concept. Proceeding of the British Society for Research into Learning Mathematics, 24(1).
8. Aspinwell, L., & Miller, D. (1997). Students’ positive reliance on writing as a process to learn first semester calculus. Journal of Instructional Psychology, 24, 253–261.

9. Baker, B., Cooley, L., & Trigueros, M. (2000). A calculus graphing schema. Journal for Research in Mathematics Education, 31(5), 557-578.
10. Ball, D. L., Hill, H., & Bass, H., (2005). Knowing mathematics for teaching: who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 29(3), 14-46.
11. Barmby, P., Harries, T., Higgins, S., & Suggate, J. (2009). The array representation and primary children’s understanding and reasoning in multiplication. Educational Studies in Mathematics, 70(3), 217−241.
12. Berry, J. S., & Nyman, M. N. (2003). Promoting students’ graphical understanding of the calculus. Journal of Mathematical Behavior, 22, 481-497.
13. Bezuidenhout, J. (1998). First-year University Students’ Understanding of Rate of Change. International Journal of Mathematical Education in Science and Technology, 29, 389–399.
14. Billings, E. M., & Klanderman, D. (2000). Graphical representations of speed: obstacles preservice k‐8 teachers experience. School Science and Mathematics, 100(8), 440-450.
15. Bingölbali, E., & Coşkun, M. (2016). İlişkilendirme becerisinin matematik öğretiminde kullanımının geliştirilmesi için kavramsal çerçeve önerisi. Eğitim ve Bilim, 41(183).
16. Blum, W., Galbraith, P.L., Henn, H.-W., & Niss, M. (2007). Modelling and applications in mathematics education. New York: Springer.
17. Boaler, J. (1993). Encouraging the transfer of ‘school’ mathematics to the ‘real world’ through the integration of process and content, context and culture. Educational Studies in Mathematics, 25(4), 341-373.
18. Bosse, M. J. (2003). The beauty of “and” and “or”: Connections within mathematics for students with learning differences. Mathematics and Computer Education, 37(1), 105-114.
19. Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.) (1999). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press.
20. Businskas, A.M. (2008). Conversations about connections: how secondary mathematics teachers conceptualize and contend with mathematical connections (Unpublished PhD Thesis), Simon Fraser University, Canada.
21. Carpenter, T. P., & Lehrer, R. (1999). Teaching and learning mathematics with understanding. In E. Fennema & T. A. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 19–32). Mahwah, NJ: Lawrence Erlbaum.
22. Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in the elementary school. Portsmouth, NH: Heinemann.
23. Carpenter, T. P., Levi, L., Franke, M. L., & Zeringue, J. K. (2005). Algebra in elementary school: developing relational thinking. ZDM, 37(1), 53-59.
24. Carrejo, D. (2004). Mathematical modelling and kinematics: a study of emerging themes and their implications for learning mathematics through an inquiry-based approach (Unpublished PhD Thesis), University of Texas, Austin.
25. Carrejo, D. J., & Marshall, J. (2007). What is mathematical modelling? exploring prospective teachers’ use of experiments to connect mathematics to the study of motion. Mathematics Education Research Journal, 19(1), 45–76.
26. Chapman, O. (2012). Challenges in mathematics teacher education. Journal of Mathematics Teacher Education, 15(4), 263-270.
27. Cherniak, C., Changizi, M., & Kang, D. W. (1999). Large-scale optimization of neuron arbors. Physical Review, 59, 6001–6009.
28. Çelik, D., & Sağlam- Arslan, A. (2012). Öğretmen adaylarının çoklu gösterimleri kullanma becerilerinin analizi. İlköğretim-Online, 11(1), 239-250.
29. Davis, Z., & Johnson, Y. (2007). Failing by example: Initial remarks on the constitution school matematics, with special reference to the teaching and learning of mathematics in five seconday schools. In M. Setati, N. Chitera, & A. Essien (Eds.), Proceedings of the 13th Annual National Congress of the Association for Mathematics Education of South Africa, (Vol. 1, pp. 121−136). White River: AMESA
30. Delice, A., & Sevimli, E. (2010). Öğretmen adaylarının çoklu temsil kullanma becerilerinin problem çözme başarılarının incelenmesi: belirli integral örneği. Kuram ve Uygulamada Eğitim Bilimleri, 10 (1), 111-149.
31. Dervişoğlu, S., & Soran, H. (2003). Orta öğretim biyoloji eğitiminde disiplinler arası öğretim yaklaşımının değerlendirilmesi. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 25, 48-57.
32. Dilberoğlu, M. (2015). An investigation of pre-service middle school mathematics teachers’ ability to connect the mathematics in content courses with the middle school mathematics (Unpublished PhD Thesis). Middle East Technical University, Ankara.
33. Duran, M., & Kaplan, A. (2016). Lise matematik öğretmenlerinin türevin tanımına ve türev-süreklilik ilişkisine yönelik pedagojik alan bilgileri. Erzincan Üniversitesi Eğitim Fakültesi Dergisi, 18(2), 795-831.
34. Duru, A. (2006). Bir fonksiyon ve onun türevi arasındaki ilişkiyi anlamada karşılaşılan zorluklar (Yayınlanmamış doktora tezi), Atatürk Üniversitesi, Erzurum.
35. Eli, J. A. (2009). An exploratory mixed methods study of prospective middle grades teachers’ mathematical connections while completing investigative tasks in geometry (Unpublished PhD Thesis), University of Kentucky.
36. Empson, S. B., Levi, L., & Carpenter, T. P. (2010). The algebraic nature of fractions: Developing relational thinking in elementary school. J. Cai & E. Knuth (Ed.). Early algebraization: Cognitive, curricular, and ınstructional perspectives. New York: Springer.
37. Ferrini-Mundy, J., & Graham, K. G. (1991). An overview of the calculus curriculum reform effort: issues for learning, teaching, and curriculum development. The American Mathematical Monthly, 98 (7), 627-635.
38. Ferrini-Mundy, J., & Graham, K. G. (1994). Research in calculus learning: understanding of limits, derivatives, and integrals. In J. J. Kaput & E. Dubinsky (Eds.), Research issues in undergraduate mathematics learning (pp. 31-45). Washington, DC: Mathematics Association of America.
39. Gagatsis, A., & Elia, I. (2004). The effects of different modes of representation on mathematical problem solving. In M. J. Høines, & A.B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, (Vol. 2, pp. 447−454). Bergen: Bergen University College.
40. Goerdt, S. L. (2007). The effect of emphasizing multiple representations on calculus students’ understanding of derivative concept (Unpublished PhD dissertation), University of Minnesota.
41. Goldin, G. A. (2004). Representations in school mathematics: A unifying research perspectives. In J. Kilpatrick, W. G. Martin and D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 275-285). Reston, VA: NCTM.
42. Green, A. E. (2016). Creativity, within reason: semantic distance and dynamic state creativity in relational thinking and reasoning. Current Directions in Psychological Science, 25(1), 28-35.
43. Greeno, J. G., Colliins, A. M., & Resnick, L. (1996). Cognition and learning. – In D. C. Berliner & R. C. Calfee (Eds.), Handbook of educational psychology (pp. 15-46). New York: Macmillan.
44. Gülten, D. Ç., Ilgar, L., & Gülten, İ. (2009). Lise 1. sınıf öğrencilerinin matematik konularının günlük yaşamda kullanımı konusundaki fikirleri üzerine bir araştırma. Hasan Ali Yücel Eğitim Fakültesi Dergisi, 11(1), 51-62.
45. Hacıömeroğlu, E. S. (2007). Calculus students’ understanding of derivative graphs: problems of representations in calculus (Unpublished PhD Thesis), Florida State University.
46. Haynie, W. J., & Greenberg, D. (2001). Genetic disorders: An integrated curriculum project. The Technology Teacher, 60 (6), 10-13.
47. Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65-97). New York: Mcmillan.
48. Hunter, J. (2007). Relational or calculational thinking: students solving open number equivalence problems. In Proceedings of the 30th annual conference of the Mathematics Education Research Group of Australasia (Vol. 2, pp. 421-429).
49. İpek, A. S., & Okumuş, S. (2012). İlköğretim matematik öğretmen adaylarının matematiksel problem çözmede kullandıkları temsiller. Gaziantep Üniversitesi Sosyal Bilimler Dergisi, 11(3), 681-700.
50. İşleyen, T., & Akgün, L. (2009). Matematik öğretmen adaylarının türev ve diferansiyel kavramlarını algılama düzeyleri, XVIII. Ulusal Eğitim Bilimleri Kurultayı, Ege Üniversitesi, İzmir.
51. Ji, E. L. (2012). Prospective elementary teachers’ perceptions of real-life connections reflected in posing and evaluating story problems. Journal of Mathematics Teacher Education, 15, 429–452.
52. Karakoç, G., & Alacacı, C. (2015). real world connections in high school mathematics curriculum and teaching. Turkish Journal of Computer and Mathematics Education, 6(1), 31-46.
53. Karasar, N. (2007). Bilimsel araştırma yöntemi (17. Baskı), Ankara: Nobel Yayın Dağıtım.
54. Karslı, N. (2016). Buluş yoluyla öğrenme yaklaşımını esas alan matematik öğretiminin 8. sınıf öğrencilerinin akıl yürütme ve ilişkilendirme becerilerine etkisi (Yayınlanmamış Yüksek Lisans Tezi). Başkent Üniversitesi, Ankara.
55. Kendal, M. (2002). Teaching and learning introductory differential calculus (Unpublished PhD Thesis), The University of Melbourne, Australia.
56. Kertil, M. (2014). Pre-service elementary mathematics teachers' understanding of derivative through a model development unit (Unpublished PhD Thesis), Middle East Technical University, Ankara.
57. Kızıloğlu, F. N., & Konyalıoğlu, A. C. (2002). Matematik öğretmenlerinin sınıf içi davranışları. Kastamonu Eğitim Dergisi, 10, 119-124.
58. Kilpatrick, J., Swafford, J., & Findell, B. (2001). The strands of mathematical proficiency. In Adding it up: Helping children learn mathematics (pp. 115–155). Washington, DC: National Academy Press.
59. Leikin, R., & Levav-Waynberg, A. (2007). Exploring mathematics teacher knowledge to explain the gap between theory-based recommendations and school practice in the use of connecting tasks. Educational Studies in Mathematics, 66(3), 349-371.
60. MEB (2009). İlköğretim matematik dersi öğretim programı. Ankara: T.C. Milli Eğitim Bakanlığı, Talim ve Terbiye Kurulu Başkanlığı.
61. MEB (2013). Ortaöğretim matematik dersi öğretim programı. Ankara: T.C. Milli Eğitim Bakanlığı, Talim ve Terbiye Kurulu Başkanlığı.
62. Mhlolo, M. K., Schafer, M., & Venkat, H. (2012). The nature and quality of the mathematical connections teachers make. Pythagoras, 33(1), 1-9.
63. NCTM (2000). Principles and standards for school mathematics. Reston, VA: NCTM Publications.
64. Noss, R., & Hoyles, C. (1996). Windows on mathematical meaning: Learning cultures and computers (Vol. 17). Dordrecht, the Netherlands: Kluwer Academic Publishers.
65. Orton, A. (1983). Students’ Understanding of differentiation. Educational Studies in Mathematics, 14, 235–250.
66. Özgen, K. (2013a). Problem çözme bağlamında matematiksel ilişkilendirme becerisi: öğretmen adayları örneği. E-Journal of New World Sciences Academy, 590, 323-345.
67. Özgen, K. (2013b). İlköğretim matematik öğretmen adaylarının matematiksel ilişkilendirmeye yönelik görüş ve becerilerinin incelenmesi. Turkish Studies, 8(8), 2001-2020.
68. Özmantar, M. F., Akkoç, H., Bingölbali, E., Demir, S., & Ergene, B. (2010). Pre-service mathematics teachers' use of multiple representations in technology-rich environments. Eurasia Journal of Mathematics, Science & Technology Education, 6(1), 19-37.
69. Pang, J., & Good, R. (2000). A review of the integration of science and mathematics: Implications for further research. School Science and Mathematics, 100(2), 73-82.
70. Park, J. (2011). Calculus instructors' and students' discourses on the derivative (Unpublished PhD Thesis), Michigan State University.
71. Patton, M. Q. (1987). How to use qualitative methods in evaluation (No. 4). CA: Sage.
72. Prins, G., T., Bulte, A. M. W., Van Driel, J. H., & Pilot, A. (2009). Students’ involvement in authentic modelling practices as contexts in chemistry education. Research Science Education, 39, 681–700.
73. Rasila, A., Malinen, J., & Tiitu, H. (2015). On automatic assessment and conceptual understanding. Teaching Mathematics and Its Applications: An International Journal of the IMA, 34(3), 149-159.
74. Reead, S. K., & Jazo, L. (2002). Using multiple representations to improve conceptions of average speed. Journal of Educational Computing Research, 27(1/2), 147−166.
75. Rogers, M. A. P., & Abell, S. K. (2007). Perspectives: Connecting with other disciplines. Science and Children, 44(6), 58-59.
76. Saglam-Arslan, A., & Arslan, S. (2010). Mathematical models in physics: A study with prospective physics teacher. Scientific Research and Essays, 5 (7), 634-640.
77. Sahin, Z., Yenmez, A. A., & Erbas, A. K. (2015). Relational understanding of the derivative concept through mathematical modeling: a case study. Eurasia Journal of Mathematics, Science & Technology Education, 11(1), 177-188.
78. Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20-26.
79. Star, J. R. (2014). Relational and instrumental understanding in mathematics education. In S.Lerman (Ed.), Encyclopedia of mathematics education. Netherlands: Springer.
80. Taşdan, B. T., Uğurel, I., & Koyunkaya, M. Y. (2017). Matematik öğretmen adaylarının geliştirdikleri matematik öğrenme etkinliklerinin matematik içi ilişkilendirmeye ilişkin görüşleri kapsamında incelenmesi. 3. Türk Bilgisayar ve Matematik Eğitimi Sempozyumu (Türkbilmat–3), 87-89.
81. Tchoshanov, M. A. (2011). Relationship between teacher knowledge of concepts and connections, teaching practice, and student achievement in middle grades mathematics. Educational Studies in Mathematics, 76, 141-164.
82. Thompson, P. W. (1994). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics, 26, 229–274.
83. Tsamir, P., Tirosh, D., & Levenson, E. (2008). Intuitive nonexamples: The case of triangles. Educational Studies in Mathematics, 69 (2), 81-95.
84. Ubuz, B. (2001). First year engineering students’ learning of point of tangency, numerical calculation of gradients, and the approximate value of a function at a point through computers. Journal of Computers in Mathematics and Science Teaching, 20(1), 113–137.
85. Ubuz, B. (1996). Evaluating the impact of computers on the learning and teaching of calculus (Unpublished PhD Thesis). University of Nottingham, United Kingdom.
86. Van Den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: an example from a longitudinal trajectory on percentage. Educational Studies in Mathematics, 54(1), 9-35.
87. Vinner, S. (1992). The function concept as a prototype for problems in mathematics learning. In G. Harel & E. Dubinsky (Eds.), The concept of a function: Aspects of epistemology and pedagogy (pp. 195-213). Washington, DC: Mathematical Association of America.
88. Whitacre, I., Schoen, R. C., Champagne, Z., & Goddard, A. (2017). Relational thinking: what's the difference?. Teaching Children Mathematics, 23(5), 302-308.
89. Wiersma, W., & Jurs, S. G. (2005). Research methods in education (8 th). Boston, MA.
90. Yıldırım, C. (1996). Matematiksel düşünme. İstanbul: Remzi Kitabevi.
91. Yiğitcan Nayir, Ö. (2013). İlköğretim matematik öğretmenliği adaylarının türevi kavrayışlarının bilişe iletişimsel yaklaşım açısından incelenmesi (Yayınlanmamış Yüksek Lisans tezi), Gazi Üniversitesi, Ankara.