Investigation of the Concept Images of Mathematics Educators and Preservice Mathematics Teachers Regarding Slope, Rate of Change, and Derivative

Abstract

The aim of this study is to examine the concept images of mathematics educators and pre-service elementary mathematics teachers regarding the relationships between the concepts of slope, rate of change and derivative. Four mathematics educators working in different state universities and responsible for Analysis courses, three pre-service elementary mathematics teachers who successfully completed the Calculus 1 course at a state university, and four pre-service elementary mathematics teachers who successfully completed the Calculus 1 and Calculus 2 courses participated in the study. In order to determine the concept images of the participants about the concepts of slope, rate of change and derivative, the participants were asked semi-structured interview questions prepared by the researcher and application questions including the second stage questions. The audio recordings obtained from the interviews were transcribed and then the second stage questions were applied to the participants. The second stage questions were received in writing from the participants and analysed by the researcher through descriptive content analysis. As a result of this study, mathematics educators were able to use the concept images they had in the first stage questions in the second stage questions. On the other hand, pre-service elementary mathematics teachers were able to use the concept images they had in the first stage questions in the second stage questions.

Keywords

• concept image
• slope
• rate of change
• derivative
• mathematics educator

Kaynakça

1. Açıkyıldız, G., & Gökçek, T. (2015). Matematik öğretmeni adaylarının türev teğet ilişkisi ile ilgili yaptıkları hatalar. Journal of Instructional Technologies and Teacher Education, 4(2), 29-42.
2. Aksu, H. H. (2016). Eğitim fakültesinde öğrenim gören öğrencilerin bölümleri hakkındaki görüşleri: Giresun Üniversitesi Örneği. Kastamonu Eğitim Dergisi, 24(1), 299-316.
3. Amit, M., & Vinner, S. (1990). Some misconceptions in calculus: Anecdotes or the tip of an iceberg?. In G. Booker, P. Cobb, ve T. N. de Mendicuti (Eds.), PME 14, (1, pp. 3-10). Cinvestav, Mexico.
4. Artigue, M. (2002). Analysis. In Tall, D. (Ed.), Advanced mathematical thinking (pp. 167-198). Springer.
5. Ashcraft, M. H., & Ridley, K. S. (2005). Math anxiety and its cognitive consequences. Handbook of mathematical cognition, 315-327.
6. Bartell, T. G., Webel, C., Bowen, B., & Dyson, N. (2013). Prospective teacher learning: Recognizing evidence of conceptual understanding. Journal of Mathematics Teacher Education, 16(1), 57-79. https://doi.org/10.1007/s10857-012-9205-4
7. Bassey, M. (1999). Case study research in educational settings. McGraw-Hill Education (UK).
8. Bingölbali, E. (2010). Türev kavramına ilişkin öğrenme zorlukları ve kavramsal anlama için öneriler. MF Özmantar, E. Bingölbali & H. Akkoç (Ed.). Matematiksel kavram yanılgıları ve çözüm önerileri (ss. 223-255), Pegem Akademi.

9. Bingölbali, E., Arslan, S., & Zembat, İ. Ö. (2016). Matematik eğitiminde teoriler. Pegem Akademi.
10. Bingölbali, E., & Monaghan, J. (2008). Concept image revisited. Educational Studies in Mathematics, 68(1), 19-35. https://doi.org/10.1007/s10649-007-9112-2
11. Breen, S., Larson, N., O’Shea, A., & Pettersson, P. (1992). Students’ concept images of inverse functions. CERME 9 - Ninth Congress of the European Society for Research in Mathematics Education, Charles University in Prague, Faculty of Education; ERME, Feb 2015, Prague, Czech Republic. pp.2228-2234. ⟨hal-01288621⟩
12. Bukova, E. (2006). Öğrencilerin limit kavramını algılamasında ve diğer kavramların ilişkilendirilmesinde karşılaştıkları güçlükleri ortadan kaldıracak yeni bir program geliştirme [Doktora Tezi]. Dokuz Eylül Üniversitesi.
13. Cornu, B. (1981). Apprentissage de la notion de limite: modèles spontanés et modèles propres. In Actes du Cinquième Colloque du Groupe Internationale PME (pp. 322-326).
14. Cornu, B. (1991). Limits. In. Tall, D. (Eds), Advanced mathematical thinking. Kluwer, Boston.
15. Çakıcı, D., Alver, B., & Ada, Ş. (2006). Anlamlı öğrenmenin öğretimde uygulanması. Atatürk Üniversitesi Kazım Karabekir Eğitim Fakültesi Dergisi, 13, 71-80.
16. Çekmez, E. (2013). Dinamik matematik yazılımı kullanımının öğrencilerin türev kavramının geometrik boyutuna ilişkin anlamalarına etkisi [Doktora tezi]. Karadeniz Teknik Üniversitesi.
17. Çekmez, E., & Baki, A. (2019). Dinamik matematik yazılımı kullanımının öğrencilerin türev kavramının geometrik boyutuna yönelik anlamalarına etkisi. Türk Bilgisayar ve Matematik Eğitimi Dergisi, 10(1), 30-58.
18. Cohen, B. J. (2012). The benefits and costs of an international currency: Getting the calculus right. Open Economies Review, 23(1), 13–31. https://doi.org/10.1007/s11079-011-9216-2
19. Creswell, J. W., Hanson, W. E., Clark, V. L. P., & Morales, A. (2007). Qualitative research designs: Selection and implementation. The Counseling Psychologist, 35, 236-264. https://doi.org/10.1177/0011000006287390
20. Doruk, M., Duran, M., & Kaplan, A. (2018). Lisans öğrencilerinin limit tanımını yorumlama becerileri. Journal of Education, 8(1), 177-194. https://doi.org/10.19126/suje.356518
21. Dreyfus, T. (2022). Hypertranscendence and linear difference equations, the exponential case. https://arxiv.org/abs/2212.00388v1
22. Dreyfus, T., & Eisenberg, T. (1982). Intuitive functional concepts: A baseline study on intuitions. Journal for Research in Mathematics Education, 13(5), 360-380. https://doi.org/10.5951/jresematheduc.13.5.0360
23. Dündar, S. (2015). Matematik öğretmeni adaylarının eğim kavramına ilişkin bilgileri. Eğitimde Kuram ve Uygulama, 11(2),673-693.
24. Duru, A. (2006). Bir fonksiyon ve onun türevi arasındaki ilişkiyi anlamada karşılaşılan zorluklar [Doktora Tezi]. Atatürk Üniversitesi.
25. Engin, A. (2016). İlköğretim matematik öğretmeni adaylarının analiz alan dilini kullanma becerileri ve tutumlarının incelenmesi (Yayımlanmamış yüksek lisans tezi). Gazi Üniversitesi, Ankara.
26. Erdoğan, G. (2017). Lise matematik öğretmenlerinin noktada türev ve türev fonksiyonu hakkındaki kavram imajları [Yüksek Lisans Tezi]. Necmettin Erbakan Üniversitesi.
27. Ergene, Ö. (2019). Matematik öğretmeni adaylarının Riemann toplamlarını kullanarak modelleme yoluyla belirli integrali anlama durumlarının incelenmesi [Doktora tezi]. Marmara Üniversitesi.
28. Erol, B. (2013). İlköğretim matematik öğretmenliği 2. sınıf öğrencilerinin fizik dersine yönelik tutumları ile öğrenme stilleri arasındaki ilişki (Yüksek Lisans Tezi). Dokuz Eylül Üniversitesi.
29. Ervynck, G. (1983). Conceptual difficulties for first year university students in the acquisition of the notion of limit of a function. In Proceedings of the Fifth Conference of the International Group for the Psychology of Mathematics Education (pp. 330-333).
30. Fahy, P. J. (2001). Addressing some common problems in transcript analysis. The International Review of Research in Open and Distributed Learning, 1(2). https://doi.org/10.19173/irrodl.v1i2.321
31. Gleason, M. A., & Hallett, H. D. (1992). The calculus consortium based at Harvard university. Focus on Calculus, 1, 1-4.
32. Goldsmith, L. T., Doerr, H. M., & Lewis, C. C. (2014). Mathematics teachers’ learning: A conceptual framework and synthesis of research. Journal of Mathematics Teacher Education, 17(1), 5-36. https://doi.org/10.1007/s10857-013-9245-4
33. Gözen, Ş. (2001). Matematik ve öğretimi. Evrim.
34. Grant, M. R., Crombie, W., Enderson, M., & Cobb, N. (2016). Polynomial calculus: Rethinking the role of calculus in high schools. International Journal of Mathematical Education in Science and Technology, 47(6), 823-836. https://doi.org/10.1080/0020739X.2015.1133851
35. Grover, R. (2015). Student conceptions of functions: how undergraduate mathematics students understand and perceive functions. School of Education Graduate Theses & Dissertations. 80 Retrieved on 5 December 2018, from https://scholar.colorado.edu/educ_gradetds/80
36. Habre, S., & Abboud, M. (2006). Students’ conceptual understanding of a function and its derivative in an experimental calculus course. The Journal of Mathematical Behavior, 25(1), 57–72. https://doi.org/10.1016/j.jmathb.2005.11.004
37. Hart, D. K. (1991). Building concept images--supercalculators and students’ use of multiple representations in calculus [Doctoral Dissertation]. Oregon State University.
38. Hauger, G. S. (1995). Rate of change knowledge in high school and college students. https://eric.ed.gov/?id=ED392598
39. Heinze, A., Star, J. R., & Verschaffel, L. (2009). Flexible and adaptive use of strategies and representations in mathematics education. ZDM, 41(5), 535–540. https://doi.org/10.1007/s11858-009-0214-4
40. Herman, M. F. (2002). Relationship of college students’ visual preference to use of representations: Conceptual understanding of functions in algebra [Doctoral Dissertation]. The Ohaio State University.
41. Kaymakçı, K., Keskin, E., & Ev Çimen, E. (2018). Eskişehir ilindeki ilköğretim matematik öğretmenleri ve öğretmen adaylarının lisans eğitiminde aldıkları dersler üzerine görüşleri. Eskişehir Osmangazi Üniversitesi Türk Dünyası Uygulama ve Araştırma Merkezi Eğitim Dergisi, 3(1), 23–41.
42. Konyalıoğlu, A. C., Tortumlu, N., Kaplan, A., Işık, A., & Hızarcı, S. (2011). Matematik öğretmen adaylarının integral kavramını kavramsal anlamaları üzerine. Bayburt Eğitim Fakültesi Dergisi, 6(1), 1-8.
43. Kuzu, O. (2021). Matematik ve fen bilgisi öğretmeni adaylarının integral konusundaki yeterliklerinin tanısal değerlendirilmesi. Yüzüncü Yıl Üniversitesi Eğitim Fakültesi Dergisi, 16(1), 1402-1418. https://doi.org/10.33711/yyuefd.859592
44. Lewin, K. (1951). Field theory in social science: Selected theoretical papers (Edited by Dorwin Cartwright.). Harpers.
45. Likwambe, B., & Christiansen, I. M. (2008). A case study of the development of in-service teachers’ concept images of the derivative. Pythagoras, 2008(68), 22-31.
46. Lithner, J. (2004). Mathematical reasoning in calculus textbook exercises. The Journal of Mathematical Behavior, 23(4), 405-427. https://doi.org/10.1016/j.jmathb.2004.09.003
47. Merriam, S. B., & Tisdell, E. J. (2015). Qualitative research: A guide to design and implementation. John Wiley & Sons.
48. Ministry of National Education [MoNE]. (2018). İlköğretim matematik öğretim programı. Milli Eğitim Bakanlığı.
49. Ministry of National Education [MoNE]. (2024). İköğretim matematik öğretim programı. Milli Eğitim Bakanlığı.
50. Moore, L. C., & Smith, D. A. (1987). Review of Toward a Lean and Lively Calculus by Ronald G. Douglas. The College Mathematics Journal, 18(5), 439-442. https://doi.org/10.2307/2686974
51. Moore-Russo, D., Conner, A., & Rugg, K. I. (2011). Can slope be negative in 3-space? Studying concept image of slope through collective definition construction. Educational Studies in Mathematics, 76(1), 3–21. https://doi.org/10.1007/s10649-010-9277-y
52. Mueller, E. T. (2004). Event calculus reasoning through satisfiability. Journal of Logic and Computation, 14(5), 703-730. https://doi.org/10.1093/logcom/14.5.703
53. National Council of Teachers of Mathematics [NCTM]. (1987). Principles and standards for school mathematics. Reston, VA.
54. Nayir, Ö. (2013). İlköğretim matematik öğretmenliği adaylarının türevi kavrayışlarının bilişe iletişimsel yaklaşım açısından incelenmesi [Doktora Tezi]. Orta Doğu Teknik Üniversitesi.
55. Ndlovu, M., Wessels, D., & De Villiers, M. (2011). An instrumental approach to modelling the derivative in Sketchpad. Pythagoras, 32(2), a52. http://dx.doi.org/10.4102/pythagoras.v32i2.52
56. Orton, A. (1983). Students’ understanding of differentiation. Educational Studies in Mathematics, 14(3), 235–250. https://doi.org/10.1007/BF00410540
57. Öztoprakçı, S. (2014). Pre-service middle school mathematics teachers’ understanding of quadrilaterals through the definitions and their relationships [Doctoral Dissertation]. Middle East Technical University.
58. Poincaré, H. (1908). ‘Science et Méthode’ translated by Francis Maitland 2007, Courier Corporation, New York.
59. Punch, K. F. (2005). Sosyal araştırmalara giriş: Nicel ve nitel yaklaşımlar. (Etöz, Z. Çev.) Siyasal kitabevi.
60. Sağırlı, M. Ö., Kırmacı, U., & Bulut, S. (2010). Türev konusunda uygulanan matematiksel modelleme yönteminin ortaöğretim öğrencilerinin akademik başarılarına ve öz-düzenleme becerilerine etkisi. Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 3(2), 221-247.
61. Sağlam, Y. (2011). Üniversite öğrencilerinin integral konusunda görsel ve analitik stratejileri̇ [Doktora Tezi]. Hacettepe Üniversitesi.
62. Sfard, A. (2000). On reform movement and the limits of mathematical discourse. Mathematical Thinking and Learning, 2(3), 157-189. https://doi.org/10.1207/S15327833MTL0203_1
63. Sfard, A. (2005). What could be more practical than good research?. Educational Studies in Mathematics, 58(3), 393-413. https://doi.org/10.1007/s10649-005-4818-5
64. Strasser, N. (2010). Who wants to pass math? Using clickers in calculus. Journal of College Teaching & Learning (TLC), 7(3). https://doi.org/10.19030/tlc.v7i3.102
65. Tall, D. (1988). The nature of advanced mathematical thinking. In Proceedings of Psychology of Mathematics Education Conference, Hungary.
66. Tall, D. (1977). ‘Cognitive conflict and the learning of mathematics’. The International Group for the Psychology of Mathematics Education, Utrecht, Holland.
67. Tall, D. (1993). Students’ difficulties in calculus. In proceedings of working group (Vol. 3, pp. 13-28).
68. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151-169. https://doi.org/10.1007/BF00305619
69. Thomas, G. B., & Finney, R. L. (2001). Calculus ve analitik geometri. İstanbul Beta.
70. Turner, K. R., & Álvarez, J. A. (2021). Supporting connections to teaching in an undergraduate calculus course. Proceedings of the 48th Annual Meeting of the Research Council on Mathematics Learning.
71. Tzur, R., & Simon, M. (2004). Distinguishing two stages of mathematics conceptual learning. International Journal of Science and Mathematics Education, 2, 287-304. https://doi.org/10.1007/s10763-004-7479-4
72. Ubuz, B. (1996). Evaluating the impact of computers on the learning and teaching of calculus [Doctoral Dissertation]. University of Nottingham.
73. 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.
74. Ubuz, B. (2007). Interpreting a graph and constructing its derivative graph: Stability and change in students’ conceptions. International Journal of Mathematical Education in Science and Technology, 38(5), 609-637. https://doi.org/10.1080/00207390701359313
75. Vinner, S. (1983). Concept definition, concept image and the notion of function. International Journal of Mathematical Education in Science and Technology, 14(3), 293-305. https://doi.org/10.1080/0020739830140305
76. Vinner, S. (2002). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.) Advanced mathematical thinking. Springer.
77. White, P., & Mitchelmore, M. (1996). Conceptual knowledge in introductory calculus, Journal for Research in Mathematics Education JRME, 27(1), 79-95.
78. Yeşildere, S. (2007). İlköğretim matematik öğretmen adaylarının matematiksel alan dilini kullanma yeterlilikleri. Boğaziçi Üniversitesi Eğitim Dergisi, 24(2), 61-70.
79. Yıldırım, A., & Şimşek, H. (2016). Sosyal bilimlerde nitel araştırma yöntemleri. Seçkin Yayıncılık.
80. Zandieh, M. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. CBMS issues in mathematics education, 8, 103-127.
81. Zazkis, R., & Leikin, R. (2008). Exemplifying definitions: A case of a square. Educational Studies in Mathematics, 69(2), 131-148. https://doi.org/10.1007/s10649-008-9131-7
82. Zimmermann, W. (1991). Visual thinking in calculus. In Visualization in teaching and learning mathematics (pp. 127-137). Mathematical Association of America.