Investigation of Middle School Students’ Solution Strategies in Solving Proportional and Non-proportional Problems

Abstract

The purpose of this study was to investigate middle school students’ solution strategies in solving different types of proportional (i.e., missing value, numerical comparison and qualitative reasoning problems) and non-proportional problems and to compare if differences existed between sixth and eighth grades students’ solution strategies. Data were collected from 101 sixth grade (n=44) and eighth grade (n=57) students from three different public middle schools. The students were asked to solve ten open-ended items that included seven proportional problems and three non-proportional problems. Descriptive data analysis methods were used to analyze data. The results revealed that the students’ solution strategies differed based on problem type and grade level. The eighth grade students used cross-multiplication as a leading strategy whereas the sixth grade students used factor of change strategy. Moreover, the results showed that students commonly used incorrect proportional strategies to solve non-proportional problems.

Keywords

• Middle school students,solution strategies,proportional problems,ratio and proportion,proportional reasoning

Kaynakça

1. Atabaş, Ş. (2014). An examination of fifth and sixth grade students’ proportional reasoning (Unpublished master’s thesis). Boğaziçi University, Graduate School of Social Sciences, İstanbul.
2. Atabaş, Ş., & Öner, D. (2016). An examination of Turkish middle school students’ proportional reasoning. Boğaziçi Üniversitesi Eğitim Dergisi, 33(1), 63-85.
3. Avcu, R., & Avcu, S. (2010). 6th grade students’ use of different strategies in solving ratio and proportion problems. Procedia-Social and Behavioral Sciences, 9, 1277-1281.
4. Ayan, R., & Işıksal-Bostan, M. (2018). Middle school students’ reasoning in nonlinear proportional problems in geometry. International Journal of Science and Mathematics Education, 16(3), 503-518.
5. Ayan, R., & Işıksal-Bostan, M. (2019). Middle school students’ proportional reasoning in real life contexts in the domain of geometry and measurement. International Journal of Mathematical Education in Science and Technology, 50(1), 65-81.
6. Baroody, A. J., & Coslick, R. T. (1998). Fostering children's mathematical power: An investigative approach to K-8 mathematics instruction. New York: Lawrence Erlbaum.
7. Behr, M., Lesh, R., Post, T., & Silver E. (1983). Rational number concepts. In R. Lesh, & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 91-125). New York: Academic Press.
8. Ben-Chaim, D., Fey, J. T., Fitzgerald, W. M., Benedetto, C., & Miller, J. (1998). Proportional reasoning among 7th grade students with different curricular experiences. Educational Studies in Mathematics, 36, 247–273.

9. Ben-Chaim, D., Keret, Y., & Ilany, B. S. (2012). Ratio and proportion: Research and teaching in mathematics teachers’ education (pre- and in-service mathematics teachers of elementary and middle school classes). Rotterdam, The Netherlands: Sense Publishers.
10. Borg, W. R., & Gall, M. D. (1989). Educational research. An introduction (5th ed.). White Plains, New York: Longman.
11. Cramer, K., Post, T., & Behr, M. (1989). Interpreting proportional relationships. Mathematics Teacher, 82(6), 445-452.
12. Cramer, K., & Post, T. (1993). Connecting research to teaching proportional reasoning. Mathematics Teacher, 86(5), 404-407.
13. Cramer, K., Post, T., & Currier, S. (1993). Learning and teaching ratio and proportion: Research implications. In D. Owens (Ed.), Research ideas for the classroom (pp. 159-178). New York: Macmillan Publishing Company.
14. Duatepe, A., Akkuş-Çıkla O., & Kayhan, M. (2005). Orantısal akıl yürütme gerektiren sorularda öğrencilerin kullandıkları çözüm stratejilerinin soru türlerine göre değişiminin incelenmesi. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 23, 73-82.
15. Gall, M. D., Gall, J. P., & Borg, W. R. (2007), Educational research: An introduction (8th ed.). Boston: Pearson.
16. Hart, K. (1988). Ratio and proportion. In J. Hiebert, & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 198-219). Reston, VA: National Council of Teachers of Mathematics.
17. Heller, P., Ahlgren, A., Post, T., Behr, M., & Lesh, R. (1989). Proportional reasoning: The effect of two context variables, rate type and problem setting. Journal for Research in Science Teaching, 26(1), 205-220.
18. Heller, P., Post, T., Behr, M., & Lesh, R. (1990). Qualitative and numerical reasoning about fractions and rates by seventh and eighth grade students. Journal for Research in Mathematics Education, 21(5), 388-402.
19. Hillen, A. F. (2005). Examining preservice secondary mathematics teachers’ ability to reason proportionally prior to and upon completion of a practice-based mathematics methods course focused on proportional reasoning (Unpublished doctoral dissertation). University of Pittsburgh, Johnstown, the USA.
20. Kaput, J. J., & West, M. M. (1994). Missing-value proportional reasoning problems: Factors affecting informal reasoning patterns. In G. Harel, & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 235-287). Albany: State University of New York Press.
21. Karplus, R., Pulos, S., & Stage, E. K. (1983). Early adolescents' proportional reasoning on 'rate' problems. Educational Studies in Mathematics, 14(3), 219-233.
22. Kayhan, M. (2005). 6. ve 7. sınıf öğrencilerinin oran-orantı konusuna yönelik çözüm stratejilerinin; sınıf düzeyine, cinsiyete ve soru tipine göre değişiminin incelenmesi (Unpublished master’s thesis). Hacettepe University, Graduate School of Science and Engineering, Ankara.
23. Lamon, S. J. (1995). Ratio and proportion: Elementary didactical phenomenology. In J. T. Sowder, & B. P. Schapplle (Eds.), Providing a foundation for teaching mathematics in the middle grades (pp. 167-183). Albany, NY: State University of New York Press.
24. Lamon, S. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–667). Charlotte, NC: Information Age Publishing.
25. Lamon, S. (2012). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (3rd ed.). Mahwah, NJ: Erlbaum.
26. Lesh, R., Post, T., & Behr, M. (1988). Proportional reasoning. In J. Hiebert, & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 93-118). Reston, VA: National Council of Teachers of Mathematics.
27. Lo, J. J., & Watanabe, T. (1997). Developing ratio and proportion schemes: A story of a fifth grader. Journal for Research in Mathematics Education, 28, 216-236.
28. Ministry of National Education [MNE]. (2009). İlköğretim matematik dersi öğretim programı 6-8. sınıflar: Öğretim programı ve klavuzu [Elementary mathematics curriculum and guide for grade levels 6 to 8]. Ankara, Turkey: Author.
29. Ministry of National Education [MNE]. (2013). Ortaokul matematik dersi öğretim programı 5-8. sınıflar: Öğretim programı ve klavuzu [Middle school mathematics curriculum and guide for grade levels 5 to 8]. Ankara, Turkey: Author.
30. Ministry of National Education [MNE]. (2018). Matematik dersi öğretim programı ilkokul ve ortaokul 1, 2, 3, 4, 5, 6, 7 ve 8. sınıflar: Öğretim programı ve klavuzu [Mathematics curriculum and guide for grade levels 1, 2, 3, 4, 5, 6, 7 and 8]. Ankara, Turkey: Author.
31. Nassaji, H. (2015). Qualitative and descriptive research: Data types versus data analysis. Language Teaching Research, 19(2), 129–132.
32. Özgün-Koca, S. A., & Kayhan-Altay, M. (2009). An investigation of proportional reasoning skills of middle school students. Investigations in Mathematics Learning, 2(1), 26-48.
33. Pelen, M. S., & Artut, P. D. (2016). Seventh grade students’ problem solving success rates on proportional reasoning problems. International Journal of Research in Education and Science (IJRES), 2(1), 30-34.
34. Pişkin-Tunç, M. (2016). Pre-service middle school mathematics teachers’ proportional reasoning before and after a practice-based instructional module (Unpublished doctoral dissertation). Middle East Technical University, Graduate School of Social Sciences, Ankara.
35. Post, T., Behr, M., & Lesh, R. (1988). Proportionality and the development of pre-algebra understandings. In A. Coxford, & A. Shulte (Eds.), The idea of algebra K-12 (pp. 78-90). Reston, VA: National Council of Teachers of Mathematics.
36. Post, T., Cramer, K., Behr, M., Lesh, R., & Harel, G. (1993). Curriculum implications of research on the learning, teaching, and assessing of rational number concepts: An integration of research. In T. Carpenter, E. Fennema, & T. Romberg (Eds.), Rational numbers: An integration of research (pp. 327-358). Hillsdale, NJ: Lawrence Erlbaum Associates.
37. Singh, P. (2000). Understanding the concepts of proportion and ratio constructed by two grade six students. Educational Studies in Mathematics, 43, 271-292.
38. Sowder, J., Armstrong, B., Lamon, S., Simon, M., Sowder, L., & Thompson, A. (1998). Educating teachers to teach multiplicative structures in the middle grades. Journal of Mathematics Teacher Education, 1, 127-155.
39. Sowder, J. T., Philipp, R. A., Armstrong, B. E., & Schappelle, B. P. (1998). Middle-grades teachers' mathematical knowledge and its relationship to instruction: A research monograph. Albany, NY: State University of New York Press.
40. Toluk-Uçar, Z., & Bozkuş, F. (2016). İlkokul ve ortaokul öğrencilerinin orantısal durumları orantısal olmayan durumlardan ayırt edebilme becerileri. Journal of Kirsehir Education Faculty, 17(3), 281-299.
41. Toluk-Uçar, Z., & Bozkuş, F. (2018). Elementary school students’ and prospective teachers’ proportional reasoning skills. International Journal for Mathematics Teaching and Learning, 19(2), 205-222.
42. Van de Walle, J., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics methods: Teaching developmentally (7th ed.). New York: Allyn and Bacon.
43. Van Dooren, W., De Bock, D., Depaepe, F., Janssens, D., & Verschaffel, L. (2003). The illusion of linearity: Expanding the evidence towards probabilistic reasoning. Educational Studies in Mathematics, 53, 113–138.
44. Van Dooren, W., De Bock, D., Hessels, A., Janssens, D., & Verschaffel, L. (2005). Not everything is proportional: Effects of age and problem type on propensities for overgeneralization. Cognition and Instruction, 23, 57–86.
45. Yıldırım, A., & Şimşek, H. (2013). Sosyal bilimlerde nitel araştırma yöntemleri (Genişletilmiş 9. baskı). Ankara: Seçkin Yayıncılık.