Examination of Prospective Mathematics Teachers' Concept Images of Prism and Cylinder in the Scope of Their Definitions, Drawings, and Grouping Skills

Öz

In this study, it was aimed to reveal the concept images of prospective mathematics teachers (known hereafter as PMTs) about prisms and cylinders. For this purpose, PMTs’ definitions of these geometric objects, their different drawings, and the way they grouped the given geometric objects were examined. The research was conducted based on the case study model. The study group of the research consisted of 45 prospective teachers studying at the first grade level in the Department of Primary Education Mathematics Teaching at the Faculty of Education of a state university in the north of Turkey. Firstly, the geometric objects test, which includes the skills of defining, drawing, and grouping cylinders and prisms, was applied to the PMTs. Then interviews were conducted with 6 prospective teachers. The analysis of the data was carried out based on the content analysis technique. The descriptive analysis technique was used in the analysis of the interview data. The results obtained from the research were presented under the headings of the PMTs’ definitions, drawings and grouping styles. As a result of the research, it was found that the PMTs’ definitions of cylinder and prism were not fully sufficient and they had difficulty in distinguishing the critical features. Concept images are based on prototype examples-usually in the form of right objects with a circular region base for a cylinder and a polygon base for a prism. It was seen that the concept images of the PMTs were more dominant than the concept definitions. In addition, it was determined that they made mistakes in using the mathematical language and that there were deficiencies in the content knowledge about the subject. When considering the hierarchical relationship between the cylinder and the prism, it was concluded that the PMTs generally thought of these objects as discrete sets, but had different ideas. Various suggestions were made based on the results obtained from the research.

Anahtar Kelimeler

• Cylinder
• Concept image
• Mathematics education
• Prism
• Prospective mathematics teachers

Matematik Öğretmen Adaylarının Prizma ve Silindire Yönelik Kavram İmajlarının Tanımları, Çizimleri ve Gruplandırma Becerileri Kapsamında İncelenmesi

Öz

Bu araştırmada ilköğretim matematik öğretmen adaylarının prizma ve silindire yönelik kavram imajlarını ortaya çıkarmak amaçlanmıştır. Bu amaç doğrultusunda öğretmen adaylarının bu geometrik cisimlere yönelik tanımları, farklı çizimleri ve verilen geometrik cisimleri gruplandırma biçimleri incelenmiştir. Araştırma durum çalışması modeline dayalı olarak yürütülmüştür. Araştırmanın çalışma grubunu Türkiye’nin kuzeyinde bulunan bir devlet üniversitesinin Eğitim Fakültesi İlköğretim Matematik Öğretmenliği Bölümünde birinci sınıf düzeyinde öğrenim gören 45 öğretmen adayı oluşturmaktadır. Öncelikle uygun örnekleme yöntemi kullanılarak belirlenen öğretmen adaylarına silindir ve prizmaya yönelik tanımlama, çizim yapma ve gruplama becerilerini içeren geometrik cisimler bilgi testi uygulanmıştır. Ardından maksimum çeşitlilik örneklemesi yöntemine dayalı olarak öğretmen adaylarının verdikleri cevapların incelenmesi sonucunda 6 öğretmen adayı ile görüşmeler gerçekleştirilmiştir. Verilerin analizi içerik analizi tekniğine dayalı olarak gerçekleştirilmiştir. Görüşme verilerinin analizinde ise betimsel analiz tekniği kullanılmıştır. Araştırmadan elde edilen sonuçlar öğretmen adaylarının tanımları, çizimleri ve gruplandırma biçimleri başlıkları altında sunulmuştur. Araştırma sonucunda öğretmen adaylarının silindir ve prizma tanımlarının tam olarak yeterli olmadığı, bu geometrik cisimlere yönelik kritik özellikleri ayırt etmekte güçlük yaşadıkları bulunmuştur. Kavram imajları genellikle silindir için dairesel tabanlı ve prizma için çokgen tabanlı ve dik cisimler şeklindedir. Çizimlerinde genellikle prototip algıya dayalı çizim yapmışlar, gruplandırmada da prototip örnekleri ayırt etmekte zorlanmamışlar ancak prototip olmayan örnekleri ayırt etmekte zorlanmışlardır. Öğretmen adaylarının kavram imajlarının kavram tanımlarına göre daha baskın olduğu görülmüştür. Ayrıca matematiksel dili kullanmada hatalar yaptıkları ve konu ile ilgili alan bilgilerinde de eksikler olduğu tespit edilmiştir. Silindir ve prizma arasındaki hiyerarşik ilişki düşündüğünde öğretmen adaylarının genellikle bu cisimleri ayrık kümeler olarak düşünmekle birlikte farklı fikirlere sahip olduğu sonucuna ulaşılmıştır. Araştırmadan elde edilen sonuçlara dayalı olarak çeşitli öneriler getirilmiştir.

Anahtar Kelimeler

• Kavram imajı
• Matematik eğitimi
• Matematik öğretmen adayları
• Prizma
• Silindir

Kaynakça

1. Accascina, G., & Rogora, E. (2006). Using cabri 3D diagrams for teaching geometry. International Journal for Technology in Mathematics Education, 13(1), 11-22.
2. Alkış Küçükaydın, M., & Gökbulut, Y. (2013). Prospective primary teachers’ misconceptions about definition of geometric shapes and unfolding process. Cumhuriyet International Journal of Education, 2(1), 102-117.
3. Altaylı, D., Konyalıoğlu A. C., Hızarcı, S., & Kaplan, A. (2014). The investigation of pre-service elementary mathematics teachers’ pedagogical content knowledge on three dimensional objects. Middle Eastern & African Journal of Educational Research, 10(1), 4-24.
4. Attneave, F. (1957). Transfer of experience with a class schema to identification of patterns and shapes. Journal of Experimental Psychology, 54(2), 81–88.
5. Avgören, S. (2011). Farklı sınıf seviyelerindeki öğrencilerin katı cisimler (prizma, piramit, koni, silindir, küre) ile ilgili sahip oldukları kavram imajı. Yayımlanmamış Yüksek Lisans Tezi, Gazi Üniversitesi, Ankara.
6. Baki, M. (2013). Pre-service classroom teachers’ mathematical knowledge and instructional explanations associated with division. Education and Science, 38(167), 300-311.
7. Battista, M. T., & Clements, D. H. (1996). Students' understanding of three-dimensional rectangular arrays of cubes. Journal for Research in Mathematics Education, 27(3), 258- 292.
8. Baykul, Y. (2014). Ortaokulda matematik öğretimi (5-8 sınıflar). Ankara: Pegem Akademi.

9. Böge, H., & Akıllı, R. (2018). Ortaokul ve imam hatip ortaokulu matematik 8 ders kitabı. Milli Eğitim Bakanlığı.
10. Bozkurt, A., & Koc, Y. (2012). Investigating first year elementary mathematics teacher education students' knowledge of prism. Educational Sciences: Theory and Practice, 12(4), 2949-2952.
11. Clements, D. H. (2003). Teaching and learning geometry. In J. Kilpatrick, G. Martin, & D. Schifter (Eds.), Research companion to principles and standards for school mathematics (pp. 15–78). Reston, VA: National Council of Teachers of Mathematics.
12. Clements, D. H., & Battista, M. T. (1992). Geometry and spatial understanding. In D. A. Grouws. (Ed.), Handbook of research mathematics teaching and learning (pp. 420-465). New York: McMillan Publishing Company.
13. Clements, D. H., & Sarama, J. (2011). Early childhood teacher education: the case of geometry. Journal of Mathematics Teacher Education, 14(2), 133–148.
14. Cohen, L. M., & Manion, L. (1998). L. (1989). Research methods in education. New York: Routledge.
15. Çakmak, Z., Konyalıoğlu, A. C., & Işık, A. (2014). The investigation of pre-service elementary mathematics teachers’ content knowledge on three dimensional objects. Middle Eastern & African Journal of Educational Research, 8(1), 28-44.
16. De Villiers, M. (1998). To teach definıtıons in geometry or teach to defıne?, In A. Olivier & K. Newstead (Eds), Proceedings of the 22nd international conference of the international group for psychology of mathematics education: Vol. 2. (pp. 248-255). Univ Stellenbosch: South Africa.
17. Ergin, A. S., & Türnüklü, E. (2015). Investigation of middle school students’ images of solids: geometric and spatial thinking relations. Journal of Research in Education and Teaching, 4(2), 188-199.
18. Ertekin, E., Yazici, E., & Delice, A. (2014). Investigation of primary mathematics student teachers’ concept images: Cylinder and cone. International Journal of Mathematical Education in Science and Technology, 45(4), 566-588.
19. Forman, J., & Damschroder, L. (2008). Qualitative content analysis. In L. Jacoby, & L. A. Siminoff (Eds.), Empirical methods for bioethics: A primer (pp. 39–62). Elsevier.
20. Fujita, T., & Jones, K. (2007). Learners’ understanding of the definitions and hierarchical classification of quadrilaterals: Toward a theoretical framing. Research in Mathematics Education, 9(1), 3-20.
21. Ginsburg, H. P., Lee, J. S., & Boyd, J. S. (2008). Mathematics education for young children: What it is and how to promote it. Society for Research in Child Development, Social Policy Report, 22, 3–22.
22. Gökbulut, Y. (2010). Sınıf öğretmeni adaylarının geometrik cisimler konusundaki pedagojik alan bilgileri. Yayımlanmamış Doktora Tezi, Gazi Üniversitesi, Ankara.
23. Gökbulut, Y., & Ubuz, B. (2013). Prospective primary teachers' knowledge on prism: Generating definitions and examples. Elementary Education Online, 12(2), 401-412.
24. Gökkurt, B. (2014). Ortaokul matematik öğretmenlerinin geometrik cisimler konusuna ilişkin pedagojik alan bilgilerinin incelenmesi. Yayımlanmamış Doktora Tezi, Atatürk Üniversitesi, Erzurum.
25. Gökkurt, B., & Soylu, Y. (2016). Examination of middle school mathematics teachers’ mathematical content knowledge: The sample of prism. Abant İzzet Baysal University Journal of the Faculty of Education, 16(2), 451- 481.
26. Gökkurt, B., Şahin, Ö., Soylu, Y., & Doğan, Y. (2015). Pre-service teachers’ pedagogical content knowledge regarding student mistakes on the subject of geometric shapes. Elementary Education Online, 14(1), 55-71.
27. Hershkowitz, R. (1989). Visualization in geometry: Two sides of the coin. Focus on Learning Problems in Mathematics, 11(1), 61–76.
28. Horzum, T., & Ertekin, E. (2018). Prospective mathematics teachers’ understanding of the base concept. International Journal of Mathematical Education in Science and Technology, 49(2), 176-199.
29. Işıksal Bostan, M., & Yemen Karpuzcu, S. (2017). The role of definitions on classification of solids including (non)prototype examples: The case of cylinder and prism. In T. Dooley, & G. Gueudet (Eds.), Proceedings of the tenth congress of the european society for research in mathematics education: Vol.5. (pp. 3320-3327). Dublin: CERME.
30. Karakuş, F. (2018). Investigation of primary pre-service teachers’ concept images on cylinder and cone. Elementary Education Online, 17(2), 1033-1050.
31. Koç, Y., & Bozkurt, A. (2011). Evaluating pre-service mathematics teachers’ comprehension level of geometric concepts. In B. Ubuz, (Ed.), The proceedings of the 35th annual meeting of the ınternational group for the psychology of mathematics education: Vol. 2. (pp. 335). Ankara, Turkey: PME.
32. Kocak, M., Gökkurt Özdemir, B., & Soylu, Y. (2017). An investigation the pedagogical content knowledge of primary mathematics prospective teachers about the concept of cylinder. Cukurova University Faculty of Education Journal, 46(2), 711-765.
33. Levenson, E., Tirosh, D., & Tsamir, P. (2011). Theories and research related to concept formation in geometry. In E. Levenson, D. Tirosh, & P. Tsamir (Eds.), Preschool geometry (pp. 3-18). AW Rotterdam: Sense Publishers. Marshall, M. N. (1996). Sampling for qualitative research. Family Practice, 13(6), 522-526.
34. Maviş, M., Gül, G., Solaklıoğlu, H., Tarku, H., Bulut, F., & Gökşen, M. (2021). Ortaöğretim matematik 10 ders kitabı. Milli Eğitim Bakanlığı.
35. McMillan, J. H. (2000). Educational research: Fundamentals for the consumer. New York: Longman.
36. Miles, M, B., & Huberman, A. M. (1994). Qualitative data analysis: An expanded Sourcebook. (2nd ed). Thousand Oaks, CA: Sage.
37. Milli Eğitim Bakanlığı (MEB). (2017). Matematik dersi öğretim programı (İlkokul ve ortaokul 1, 2, 3, 4, 5, 6, 7 ve 8. sınıflar). Ankara: MEB.
38. Monaghan, F. (2000). What difference does it make? Children’s views of the differences between some quadrilaterals. Educational Studies in Mathematics, 42(2),179-196.
39. National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Reston, VA: Author.
40. National Council of Teachers of Mathematics [NCTM] (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. National Council of Teachers of Mathematics.
41. Oberdorf, C. D., & Taylor-Cox, J. (1999). Shape up!. Teaching Children Mathematics, 5(6), 340-345.
42. Olkun, S., & Toluk U.Z., (2007). İlköğretimde etkinlik temelli matematik öğretimi. Ankara: Maya Akademi.
43. Putnam, R.T., Heaton, R.M., Prawat, R.S., & Remillard, J. (1992). Teaching mathematics for understanding: Discussing case studies of four fifth-grade teachers. The Elementary School Journal, 93(2), 213–228.
44. Reed, S. K. (1972). Pattern recognition and categorization. Cognitive Psychology, 3(3), 382–407.
45. Rosch, E. H. (1973). Natural categories. Cognitive Psychology, 4(3), 328-350.
46. Sarfaty, Y., & Patkin, D. (2013). The ability of second graders to identify solids in different positions and to justify their answer. Pythagoras, 34(1), 212-222.
47. Seymen, E., Gazioğlu, G., Yıldırım, S., & Meral, Y. (2021). Ortaöğretim matematik 11 ders kitabı. Milli Eğitim Bakanlığı.
48. Tsamir, P., Tirosh, D., & Levenson, E. (2008). Intuitive non-examples: the case of triangles. Educational Studies in Mathematics, 69(2), 81–95.
49. Tsamir, P., Tirosh, D., Levenson, E., Barkai, R., & Tabach, M. (2015). Early-years teachers’ concept images and concept definitions: Triangles, circles, and cylinders. ZDM, 47(3), 497-509.
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(2), 151-169.
51. Türnüklü, E., & Ergin, A. S. (2016). 8th year students’ definitions and figural recognitions of solids: Concept images. Elementary Education Online, 15(1), 40-52.
52. Ubuz, B., & Gökbulut, Y. (2015). Primary prospective teachers’ knowledge on pyramid: Generating definitions and examples. Journal of Kırsehir Education Faculty, 16(2), 335-351.
53. Ulusoy, F. (2019). Early-years prospective teachers' definitions, examples and non-examples of cylinder and prism. International Journal for Mathematics Teaching and Learning, 20(2), 149-169.
54. Unlu, M., & Horzum, T. (2018). Mathematics teacher candidates' definitions of prism and pyramid. International Journal of Research in Education and Science, 4(2), 670-685.
55. Van de Walle, J., Karp, K. S., & Bay-Williams, J. M. (2014). Elementary and middle school mathematics: Teaching developmentally (Eight international edition). Essex: Pearson.
56. Van Dormolen, J., & Zaslavsky, O. (2003). The many facets of a definition: The case of periodicity. Journal of Mathematical Behavior, 22(1), 91–106.
57. 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.
58. Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In Tall D. (Ed.), Advanced Mathematical Thinking (pp. 65-81). Dordrecht, Kluwer Academic.
59. Vinner, S. (2011). The role of examples in the learning of mathematics and in everyday thought processes. ZDM, 43(2), 247–256.
60. Weber, K., Porter, M., & Housman, D. (2008). Worked examples and conceptual usage in understanding mathematical concepts and proofs. In M. P. Carlson and C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics (pp. 245-252). Washington, DC: Mathematical Association of America.
61. Wilson, P. S. (1990). Inconsistent ideas related to definitions and examples. Focus on Learning Problems in Mathematics, 12(3&4), 31-47.
62. Yemen Karpuzcu, S., & Işıksal Bostan, M. (2013). Geometrik cisimler: Silindir, prizma, koni, piramit ve kürenin matematiksel anlamı. In İ. Ö Zembat, M. F Özmantar, E. Bingölbali, Şandır, H, & A. Delice (Eds.), Tanımları ve tarihsel gelişimleriyle matematiksel kavramlar (pp. 278- 279). Ankara: Pegem.
63. Zazkis, R., & Chernoff, E. J. (2008). What makes a counterexample exemplary?. Educational Studies in Mathematics, 68(3), 195-208.
64. Zazkis, R., & Leikin, R. (2008). Exemplifying definitions: a case of a square. Educational Studies in Mathematics, 69(2), 131-148.
65. Zeybek Şimşek, Z. (2019). Investigating pre-service teachers’ ability to recognize and classify geometric concepts hierarchically. Turkish Journal of Computer and Mathematics Education (TURCOMAT), 10(3), 680-710.
66. Zodik, I., & Zaslavsky, O. (2008). Characteristics of teachers’ choice of examples in and for the mathematics classroom. Educational Studies in Mathematics, 69(2), 165-182.