Matematik Öğretmen Adaylarının Kesirlerde Bölme İşlemi Konusunda Problem Kurma ve Kavramsal Açıklama Becerilerinin Değerlendirilmesi

Abstract

Kesirler, öğrencilerin matematiğin soyut yapısına geçişindeki ilk önemli adımlardan biri olarak görülmektedir. Ancak öğretmen adayları ve öğretmenler, kesirlerde bölme işlemini kavramsal olarak anlamakta ve bunu gerçek yaşam bağlamlarıyla ilişkilendirmekte güçlük çekmektedir. Araştırmalar öğretmen ve öğretmen adaylarının kesirler konusundaki kavrayışlarını derinlemesine incelemek amacıyla matematiksel problem kurma çalışmalarının faydalı olabileceği göstermektedir. Bu araştırmanın amacı matematik öğretmen adaylarının kesirlerde bölme işlemiyle ilgili problem kurma becerilerini incelemek, bu süreçte yaptıkları hataları belirlemek ve kurdukları problemleri kavramsal düzeyde nasıl açıkladıklarını analiz etmektir. Bir devlet üniversitesinde İlköğretim Matematik Öğretmenliği programının üçüncü sınıfında öğrenim gören 53 öğretmen adayı bu araştırmanın katılımcılarını oluşturmaktadır. Bu araştırmada öğrencilerden bir kesri bir tam sayıya, bir tam sayıyı bir kesre ve bir kesri bir diğer kesre bölmelerini gerektiren problemler kurmaları ve kurdukları bu problemin çözümünü anlatmaları istenmiştir. Problemi doğru kuramayan ve gönüllü olan altı öğretmen adayı ile görüşme gerçekleştirilmiştir. Araştırmanın bulguları, matematik öğretmen adaylarının kesirlerde bölme işlemiyle ilgili problem kurma becerilerinde çeşitli zorluklar yaşadıklarını ve kavramsal anlamda eksikliklere sahip olduklarını göstermektedir. Öğretmen adayları en çok bir kesri başka bir kesre bölmeyi gerektiren işleme ilişkin problem kurmada zorluk yaşamaktadır. Bulgular, öğretmen adaylarının matematiksel anlayışlarını teşhis ve değerlendirmede problem kurma görevlerinin yararlılığını vurgulamaktadır.

Keywords

• Kesirlerde bölme işlemi
• problem kurma
• öğretmen adayları.

Evaluation of Mathematics Teacher Candidates' Problem Posing and Conceptual Explanation Skills on Division of Fractions

Abstract

Fractions are seen as one of the first important stages in students' transition to the abstract structure of mathematics. However, pre-service teachers and teachers have difficulty in conceptually understanding the division of fractions and relating it to real life contexts. Research has shown that mathematical problem posing activities can be useful in order to examine teachers' and pre-service teachers' comprehension of fractions in depth. The purpose of this study was to examine pre-service mathematics teachers' problem posing skills related to division of fractions, to identify the errors they made in this process, and to analyze how they explained the problems they posed at the conceptual level. The participants of this study were 53 pre-service mathematics teachers studying in the third year of the Primary Mathematics Teacher Education program at a state university. Students were asked to pose problems that required them to divide a fraction by a whole number, a whole number by a fraction, and a fraction by another fraction, and to explain the solution of the problem they posed. Interviews were conducted with six pre-service mathematics teachers who could not construct the problem correctly and volunteered. The findings of the study show that pre-service mathematics teachers have various difficulties and conceptual deficiencies in their problem posing skills related to division of fractions. The pre-service mathematics teachers had the most difficulties in constructing problems related to the operation that requires dividing one fraction by another fraction. The findings emphasize the usefulness of problem posing tasks in diagnosing and assessing pre-service teachers' mathematical understanding.

Keywords

• Division of fractions
• problem posing
• pre-service teachers.

References

1. Ashlock, R. D. (1990). Error patterns in computation. Macmillan.
2. Aytekin, C., ve Şahiner, Y. (2020). An investigation of preservice mathematics teachers' teaching processes about" procedural and conceptual knowledge" related to division with fractions. İlköğretim Online, 958-981.
3. Bailey, D. H., Hoard, M. K., Nugent, L., ve Geary, D. C. (2012). Competence with fractions predicts gains in mathematics achievement. Journal of Experimental Child Psychology, 113(3), 447–455. https://doi.org/10.1016/j.jecp.2012.06.004
4. Ball, D. L. (1990). Prospective elementary and secondary teachers’ understanding of division. Journal for Research in Mathematics Education, 21(2), 132-144.
5. Ball, D. L., Lubienski, S. T., ve Mewborn, D. (2001). Research on teaching mathematics: The unsolved problem of teachers’ mathematical knowledge. İçinde V. Richardson (Ed.), Handbook of research on teaching (4th ed., pp. 433–456). Macmillan.
6. Booth, J. L., ve Newton, K. J. (2012). Fractions: Could they really be the gatekeeper's doorman? Contemporary Educational Psychology, 37(4), 247–253. https://doi.org/10.1016/j.cedpsych.2012.07.001
7. Borko, H., Eisenhart, M., Brown, C. A., Underhill, R. G., Jones, D., ve Agard, P. C. (1992). Learning to teach hard mathematics: Do novice teachers and their instructors give up too easily? Journal for Research in Mathematics Education, 23(3), 194-222.
8. Braithwaite, D. W., ve Siegler, R. S. (2021). Putting fractions together. Journal of Educational Psychology, 113(3), 556-571. https://doi.org/10.1037/edu0000477

9. Bulgar, S. (2003). Children’s sense-making of division of fractions. Journal of Mathematical Behavior, 22(3), 319-334. https://doi.org/10.1016/S0732-3123(03)00024-5
10. Büyüköztürk, Ş., Akgün, Ö. E., Demirel, F., Karadeniz, Ş., ve Çakmak, E. K. (2015). Bilimsel araştırma yöntemleri (20. baskı). Pegem Akademi.
11. Cai, J., ve Hwang, S. (2020). Learning to teach through mathematical problem posing: Theoretical considerations, methodology, and directions for future research. International Journal of Educational Research, 102, 101391. https://doi.org/10.1016/j.ijer.2019.01.001
12. Cai, J., Hwang, S., Jiang, C., ve Silber, S. (2015). Problem-posing research in mathematics education: Some answered and unanswered questions. İçinde F. M. Singer, N. Ellerton ve J. Cai (Ed.), Mathematical problem posing: From research to effective practice (ss. 3–34). Springer.
13. Cengiz, N. (2013). Doğal sayılarla dört işlem ve tarihçesi. İçinde İ. Ö. Zembat, M. F. Özmantar, E. Bingölbali, H. Şandır ve A. Delice (Ed.), Tanımları ve tarihsel gelişimleriyle matematiksel kavramlar (s. 59-80). Pegem Akademi.
14. Chabe, A. M. (1963). Rationalizing, inverting and multiplying. Arithmetic Teacher, 10, 221-273. https://doi.org/10.5951/AT.10.5.0272
15. Christensen, L., Johnson, R. B., ve Turner, L. (2014). Research methods: Design and analysis (12th ed.). Pearson.
16. Clemson, D., ve Clemson, W. (2006). Mathematics in the early years. Routledge.
17. Collopy, R. (2003). Curriculum materials as a professional development tool: How a mathematics textbook affected two teachers' learning. The Elementary School Journal, 103(3), 287-311.
18. Copur-Gencturk, Y. (2021). Teachers’ conceptual understanding of fraction operations: Results from a national sample of elementary school teachers. Educational Studies in Mathematics, 107(3), 525-545. https://doi.org/10.1007/s10649-021-10033-4
19. Davis, G. E., ve Pearn, C. A. (2009). Division of fractions. Republic of Mathematics Publications.
20. Elgrably, H., ve Leikin, R. (2021). Creativity as a function of problem-solving expertise: Posing new problems through investigations. ZDM–Mathematics Education, 53, 891-904. https://doi.org/10.1007/s11858-021-01228-3
21. Gonzales, N. A. (1998). A blueprint for problem posing. School Science and Mathematics, 98, 448-453. http://dx.doi.org/10.1111/j.1949-8594.1998.tb17437.x
22. Guo, M., Leung, F. K., ve Hu, X. (2020). Affective determinants of mathematical problem posing: The case of Chinese Miao students. Educational Studies in Mathematics, 105(3), 367-387. https://doi.org/10.1007/s10649-020-09972-1
23. Hiebert, J., ve Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. İçinde J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (s. 1-27). Lawrence Erlbaum Associates.
24. Işıksal, M. (2006). A study on pre-service elementary mathematics teachers’ subject matter knowledge and pedagogical content knowledge regarding the multiplication and division of fractions (Yayımlanmamış doktora tezi). Middle East Technical University.
25. Leung, I. K., ve Carbone, R. E. (2013). Pre-service teachers’ knowledge about fraction division reflected through problem posing. The Mathematics Educator, 14(1-2), 80-92.
26. Lortie-Forgues, H., Tian, J., ve Siegler, R. S. (2015). Why is learning fraction and decimal arithmetic so difficult? Developmental Review, 38, 201–221. https://doi.org/10.1016/j.dr.2015.07.008
27. Ma, L. (2010). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Routledge.
28. Mayer, R. E. (1998). Cognitive, metacognitive, and motivational aspects of problem solving. Instructional Science, 26, 49–63. https://doi.org/10.1023/A:1003088013286
29. Milli Eğitim Bakanlığı [MEB]. (2009). İlköğretim matematik dersi 6-8. sınıflar öğretim programı. https://ttkb.meb.gov.tr
30. Milli Eğitim Bakanlığı [MEB]. (2013). Ortaokul matematik dersi (5, 6, 7 ve 8. sınıflar) öğretim programı. https://ttkb.meb.gov.tr
31. Milli Eğitim Bakanlığı [MEB]. (2018). Matematik dersi öğretim programı (ilkokul ve ortaokul 1, 2, 3, 4, 5, 6, 7 ve 8. sınıflar). http://mufredat.meb.gov.tr
32. Milli Eğitim Bakanlığı [MEB] (2024). Türkiye Yüzyılı Maarif Modeli: Ortaokul Matematik dersi öğretim programı ( 5-8. sınıflar). http://mufredat.meb.gov.tr
33. Mishra, S., ve Iyer, S. (2015). An exploration of problem posing-based activities as an assessment tool and as an instructional strategy. Research And Practice İn Technology Enhanced Learning, 10, 1-19. https://doi.org/10.1007/s41039-015-0006-0
34. Mutlu, E. (2023). Ortaokul matematik öğretmeni adaylarının kesirlerle bölmeye yönelik problem oluşturma ve çözme becerilerinin gelişimi. MANAS Sosyal Araştırmalar Dergisi, 12(3), 927-943. https://doi.org/10.33206/mjss.1245927
35. NCTM [National Council of Teachers of Mathematics]. Principles and standards for school mathematics. Reston: NCTM.
36. National Curriculum of Korea. (2015). The national curriculum of mathematics (Ministry of Education Notification No. 2015-74, Book 8). National Curriculum of Korea.
37. Newton, K. J. (2008). An extensive analysis of preservice elementary teachers’ knowledge of fractions. American Educational Research Journal, 45(4), 1080-1110. https://doi.org/10.3102/0002831208320851
38. Nillas, L. (2003). Division of fractions: Preservice teachers’ understanding and use of problem-solving strategies. The Mathematics Educator, 7(2), 96-113.
39. Olanoff, D., Lo, J. J., ve Tobias, J. (2014). Mathematical content knowledge for teaching elementary mathematics: A focus on fractions. The Mathematics Enthusiast, 11(2), 267-310. https://doi.org/10.54870/1551-3440.1304
40. Rittle-Johnson, B., ve Schneider, M. (2015). Developing conceptual and procedural knowledge of mathematics. İçinde R. C. Kadosh ve A. Dowker (Eds.), Oxford handbook of numerical cognition (pp. 1102–1118). Oxford University Press.
41. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. İçinde D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (ss. 334–370). Macmillan.
42. Schulz, A. (2024). Assessing student teachers’ procedural fluency and strategic competence in operating and mathematizing with natural and rational numbers. Journal of Mathematics Teacher Education, 27(6), 981-1008. https://doi.org/10.1007/s10857-023-09590-7
43. Sharp, J., ve Adams, B. (2002). Children's constructions of knowledge for fraction division after solving realistic problems. The Journal of Educational Research, 95(6), 333-347. https://doi.org/10.1080/00220670209596608
44. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.
45. Siegler, R. S., ve Lortie-Forgues, H. (2015). Conceptual knowledge of fraction arithmetic. Journal of Educational Psychology, 107(3), 909-918. https://doi.org/10.1037/edu0000025
46. Silber, S., ve Cai, J. (2021). Exploring underprepared undergraduate students’ mathematical problem posing. ZDM–Mathematics Education, 53(4), 877-889.https://doi.org/10.1007/s11858-021-01272-z
47. Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19-28.
48. Silver, E. A., ve Cai, J. (2005). Assessing students’ mathematical problem posing. Teaching Children Mathematics, 12(3), 129–135. https://doi.org/10.5951/TCM.12.3.0129
49. Singer, F. M., Ellerton, N., ve Cai, J. (2013). Problem-posing research in mathematics education: New questions and directions. Educational Studies in Mathematics, 83(1), 1–7. https://doi.org/10.1007/s10649-013-9478-2
50. Son, J. W., ve Crespo, S. (2009). Prospective teachers’ reasoning and response to a student’s non-traditional strategy when dividing fractions. Journal of Mathematics Teacher Education, 12, 235-261. https://doi.org/10.1007/s10857-009-9112-5
51. Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404–411.
52. Stoyanova, E., ve Ellerton, N. F. (1996). A framework for research into students' problem posing in school mathematics. Technology in Mathematics Education, 518-525.
53. Tichá, M., ve Hošpesová, A. (2013). Developing teachers’ subject didactic competence through problem posing. Educational Studies in Mathematics, 83(1), 133–143. https://doi.org/10.1007/s10649-012-9455-1
54. Tirosh, D. (2000). Enhancing prospective teachers' knowledge of children's conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31(1), 5-25. https://doi.org/10.2307/749817
55. Toluk-Uçar, Z. (2009). Developing pre-service teachers’ understanding of fractions through problem posing. Teaching and Teacher Education, 25(1), 166-175. https://doi.org/10.1016/j.tate.2008.08.003
56. Yao, Y., Hwang, S., ve Cai, J. (2021). Preservice teachers’ mathematical understanding exhibited in problem posing and problem solving. ZDM–Mathematics Education, 53(4), 937-949. https://doi.org/10.1007/s11858-021-01277-8
57. Yıldırım, A., ve Şimşek, H. (2016). Sosyal bilimlerde nitel araştırma yöntemleri (11. bs.). Seçkin Yayınları.
58. Zeitz, P. (2006). The art and craft of problem solving (2nd ed.). John Wiley ve Sons.
59. Zembat, İ. Ö. (2007). Working on the same problem-Concepts; with the usual subjects, pre-service elementary teachers. İlköğretim Online, 6(2), 305-312.
60. Zhang, H., ve Cai, J. (2021). Teaching mathematics through problem posing: insights from an analysis of teaching cases. ZDM–Mathematics Education, 53(4), 961-973. https://doi.org/10.1007/s11858-021-01260-3