Fraction Multiplication and Division Word Problems Posed by Different Years of Pre-Service Elementary Mathematics Teachers

Abstract

It is important for pre-service teachers to know the conceptual difficulties they have experienced regarding the concepts of multiplication and division in fractions and problem posing is a way to learn these conceptual difficulties. Problem posing is a synthetic activity that fundamentally has multiple answers. The purpose of this study is to analyze the multiplication and division of fractions problems posed by pre-service elementary mathematics teachers and to investigate how the problems posed change according to the year of study the pre-service teachers are in. The study employed developmental research methods. A total of 213 pre-service teachers enrolled in different years of the Elementary Mathematics Teaching program at a state university in Turkey took part in the study. The “Problem Posing Test” was used as the data collecting tool. In this test, there are 3 multiplication and 3 division operations. The data were analyzed using qualitative descriptive analysis. The findings suggest that, regardless of the year, pre-service teachers had more conceptual difficulties in problem posing about the division of fractions than in problem posing about the multiplication of fractions.

Keywords

• Pre-service mathematics teachers,fractions,problem posing,multiplication,division

Kaynakça

1. Abramovich, S. (2014). Revisiting mathematical problem solving and posing in the digital era: toward pedagogically sound uses of modern technology. International Journal of Mathematical Education in Science and Technology, 45(7), 1034-1052.
2. Abu-Elwan, R. (2002). Effectiveness of problem posing strategies on prospective mathematics teachers’ problem solving performance. Journal of Science and Mathematics Education in S.E. Asia, 25(1), 56-69.
3. Akay, H., Soybas, D. & Argun, Z. (2006). Problem posing experiences and using open-ended questions in mathematics teaching. Kastamonu University Kastamonu Education Journal, 14(1), 129-146.
4. Armstrong, B.E. & Bezuk, N. (1995). Multiplication and division of fractions: The search for meaning. In J. Sowder & B.P. Schappelle (Eds), Providing a foundation for teaching mathematics in the middle grades (pp.85-119). Albany, NY:State University of New York Press.
5. Ball, D. L. (1990). Pre-service elementary and secondary teachers’ understanding of division. Journal for Research in Mathematics Education, 21(2), 132-144.
6. Ball, D.L. & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.), Multiple Perspectives on Teaching and Learning (pp. 83-104). Westport CT: Ablex Publishing.
7. Ball, D.L., Lubienski, S. & Mewborn, D. S. (2001). Research on teaching mathematics: The unsolved problem of teachers’ mathematical knowledge. In V. Richardson (Ed.), Handbook of Research on Teaching (pp. 433-456). Washington, DC: American Educational Research Association.
8. Ball, D. L., Thames M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407.

9. Behr, M. J., Harel, G., Post, T. R. & Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 296–333). New York: Macmillan.
10. Behr, M. J., Lesh, R., Post, T. & Silver, E. A. (1983). Rational number concepts. In R. Lesh, & M. Landau (Eds.), Acquisitions of mathematics concepts and processes (pp. 92–126). New York: Academic Press.
11. Cankoy, O. & Darbaz, S. (2010). Effect of a problem posing based problem solving instruction on understanding problem. Hacettepe University Journal of Education, 38, 11-24.
12. Chapman, O. (2012). Prospective elementary school teachers’ ways of making sense of mathematical problem posing. PNA, 6(4), 135–146.
13. Chen, L., Dooren, W.V., Chen, Q. & Verschaffel, L. (2011). An investigation on Chinese teachers’ realistic problem posing and problem solving ability and beliefs. International Journal of Science and Mathematics Education, 9, 919-948.
14. Crespo, S. (2003). Learning to pose mathematical problems: Exploring changes in preservice teachers’ practices. Educational Studies in Mathematics, 52, 243–270.
15. Cakan, M. (2004). Assessment-evaluation applications of teachers and their qualification levels: Primary and secondary education. Ankara University Journal of Educational Sciences, 37, 99–114.
16. Ervin, H.K. (2017). Fraction multiplication and division models: A practitioner reference paper. International Journal of Research in Education and Science (IJRES), 3(1), 258-279.
17. Hudson, P. (2013). Desirable attributes and practices for mentees: Mentor teachers’ expectations. European Journal of Educational Research, 2(3), 107-119.
18. Isik, C. (2011). Conceptual analysis of multiplication and division problems in fractions posed by preservice elementary mathematics teachers. Hacettepe University Journal of Education, 41, 231-243.
19. Isik, C., Isik, A. & Kar, T. (2011). Analysis of the problems related to verbal and visual representations posed by pre-service mathematics teachers. Pamukkale University Journal of Education, 30(2), 39-49.
20. Julie, H. (2017). The elementary school teachers’ ability in adding and subtracting fraction, and interpreting and computing multiplication and division fraction. International Journal of Science and Applied Science: Conference Series, 1(1), 55-63.
21. Kilic, C. (2013a). Determining the performances of pre-service primary school teachers in problem posing situations. Educational Sciences: Theory & Practice, 13(2), 1195-1211.
22. Kilic, C. (2013b). Pre-service primary teachers’ free problem-posing performances in the context of fractions: An example from Turkey. The Asia-Pacific Education Researcher, 22(4), 677–686.
23. Kilic, C. (2015). Analyzing pre-service primary teachers’ fraction knowledge structures through problem posing. Eurasia Journal of Mathematics, Science & Technology Education, 11(6), 1603-1619.
24. Kieren, T. E. (1976). On the mathematical, cognitive and instructional foundations of rational numbers. In R. A. Lesh (Ed.), Number and measurement: Papers from a research workshop (pp. 101–144). Columbus: ERIC/SMEAC.
25. Kojima, K., Miwa, K. & Matsui, T. (2009). Study on support of learning from examples in problem posing as a production task. In. S.C. Kong, H. Ogata, H.C. Arnseth, C.K.K. Chan, T. Hirashima, F. Klett, J.H.M. Lee, C.C. Liu, C.K. Looi, M. Milrad, A. Mitrovic, K. Nakabayashi, S.L. Wong & S.J.H. Yang (eds.). Proceedings of the 17th International Conference on Computers in Education [CDROM]. Hong Kong: Asia-Pacific Society for Computers in Education.
26. Korkmaz, E. & Gur, H. (2006). Determining of prospective teachers’ problem posing skills. Journal of Balikesir University Institute of Science and Technology, 8(1), 64-74.
27. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and The United States. Erlbaum: Mahwah, NJ.
28. Mack, N. K. (1998). Building a foundation for understanding the multiplication of fractions. Teaching Children Mathematics, 5(1), 34-38.
29. Menard, S. (2008). Handbook of longitudinal research: Design, measurement and analysis. Academic Press, First Edition, London, UK.
30. Miller, S. A. (1998). Developmental research methods (2nd Edition). New Jersey, USA.
31. MEB - Milli Egitim Bakanligi [MOE - Ministry of Education], (2005). Ilkogretim Matematik (6, 7 ve 8. Siniflar) Dersi Ogretim Programi [Elementary School Mathematics Curriculum (6, 7 and 8. Grades)], Ankara.
32. MEB - Milli Egitim Bakanligi [MOE - Ministry of Education],  (2013). Ortaokul Matematik (5, 6, 7 ve 8. Siniflar) Dersi Ogretim Programi [Middle School Mathematics Curriculum (5, 6, 7 and 8. Grades)], Ankara.
33. MEB - Milli Egitim Bakanligi [MOE - Ministry of Education],  (2017). Matematik Dersi Ogretim Programi (1, 2, 3, 4, 5, 6, 7 ve 8. Siniflar) [Mathematics Curriculum (1, 2, 3, 4, 5, 6, 7 and 8. Grades)], Ankara.
34. Pagni, D. (1998). Giving meaning to multiplication and division of fractions. Ohio Journal of School Mathematics, 54(4), 11-13.
35. Romberg, T. A. & Carpenter, T. P. (1986). Research on teaching and learning mathematics: Two disciplines of scientific inquiry. In M. C. Witrock (Ed.), Handbook of Research on Teaching (pp. 850–873). New York: Macmillan.
36. Shulman, L.S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57, 1-22.
37. Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19-28.
38. Stoyanova, E. (2003). Extending students' understanding of mathematics via problem-posing. Australian Mathematics Teacher, 59(2), 32–40.
39. Tirosh, D. (2000). Enhancing pre-service teachers’ knowledge of children’s conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31(1), 5-25.
40. Tirosh, D. & Graeber, A. (1991). The effect of problem type and common misconceptions of preservice elementary teachers’ thinking about division. School Science and Mathematics, 91, 157-163.
41. Toluk, Z. (2002). Conceptual schemes used by elementary school students for relating division and rational numbers. Bogazici University Journal of Education, 19(2), 81-103.
42. Toluk-Ucar, Z. (2009). Developing pre-service teachers understanding of fractions through problem posing. Teaching and Teacher Education, 25, 166–175.
43. Turhan, B. & Guven, M. (2014). The effect of mathematics instruction with problem posing approach on problem solving success, problem posing ability and views towards mathematics. Cukurova University Faculty of Education Journal, 43(2), 217-234.
44. Utley, J. & Redmond, A. (2008). Prospective elementary teachers attitudes towards and knowledge of the division of fractions. Paper presented at the annual meeting of the Research Council on Mathematics Learning, Oklahoma City, OK.
45. Whitin, D. J. (2004). Building a mathematical community through problem posing. In R. N. Rubenstein (Ed.), Perspectives on the teaching of mathematics: Sixty-sixth yearbook (pp. 129–140). Reston, VA: National Council of Teachers of Mathematics.
46. Xie, J., & Masingila, J. O. (2017). Examining interactions between problem posing and problem solving with prospective primary teachers: A case of using fractions. Educational Studies in Mathematics, DOI 10.1007/s10649-017-9760-9.
47. Yim, J. (2010). Children’s strategies for division by fractions in the context of the area of a rectangle. Educational Studies in Mathematics, 73, 105–120.