Pre-service Teachers’ Imaginary Creative Approaches to Address Students’ Erroneous Understanding of Algebraic Expressions

Abstract

This study examines the creative responses of pre-service mathematics teachers in their lesson plays designed to address sixth-grade students’ misconceptions about algebraic expressions. This research employs a qualitative descriptive research design, involving 78 third-year students enrolled in an elementary mathematics education program. Using lesson plays, the pre-service teachers developed hypothetical lessons that demonstrated how dialogues between teachers and students could unfold in a classroom. The research revealed that the pre-service teachers exhibited pedagogical and mathematical flexibility in addressing students’ misconceptions in algebraic expressions. While the participants did not display mathematical and pedagogical originality, they were able to create a variety of hypothetical instructional settings. This study highlights the potential of lesson plays as an effective tool to examine pre-service teachers’ creativity and explores various pedagogical approaches in their hypothetical instruction. The findings suggest that teacher education programs should include more opportunities for pre-service teachers to develop their creativity using lesson plays and for preparing them to effectively and originally address students’ misconceptions about algebraic expressions.

Keywords

• Algebra
• Creativity
• Lesson play
• Pre-service matematics teachers

References

1. Abaté, C. J., & Cantone, K. A. (2005). An evolutionary approach to mathematics education: enhancing learning through contextual modification. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 15(2), 157-176. https://doi.org/10.1080/10511970508984115
2. Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations. Learning and Instruction, 16(3), 183–198. https://doi.org/10.1016/j.learninstruc.2006.03.001
3. Ball, D., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407. https://doi.org/10.1177/0022487108324554
4. Bolden, D. S., Harries, A. V., & Newton, D. P. (2010). Pre-service primary teachers’ conceptions of creativity in mathematics. Educational Studies in Mathematics, 73(2), 143–157. http://dx.doi.org/10.1007/s10649-009-9207-z
5. Bush, S. B., & Karp, K. S. (2013). Prerequisite algebra skills and associated misconceptions of middle grade students: A review. The Journal of Mathematical Behavior, 32(3), 613-632. https://doi/10.1016/j.jmathb.2013.07.002
6. Chamberlin, S. A., & Moon, S. M. (2005). Model-eliciting activities as a tool to develop and identify creatively gifted mathematicians. The Journal of Secondary Gifted Education, 17(1), 37–47. https://doi.org/10.4219/jsge-2005-393
7. Chang, J. M. (2011). A practical approach to inquiry-based learning in linear algebra. International Journal of Mathematical Education in Science and Technology, 42(2), 245–259. https://doi.org/10.1080/0020739X.2010.519795
8. Chapman, O. (2013). Investigating teachers’ knowledge for teaching mathematics. Journal of Mathematics Teacher Education, 16(4), 237–243. http://dx.doi.org/10.1007/s10857-013-9247-2

9. Creswell, J. W., & Poth, C. N. (2018). Qualitative inquiry and research design: Choosing among five approaches. Sage Publications.
10. Daro, P., Mosher, F. A., & Corcoran, T. B. (2011). Learning trajectories in mathematics: A foundation for standards, curriculum, assessment, and instruction. CPRE Research Reports. Retrieved from https://repository.upenn.edu/cpre_researchreports/60
11. Even, R., Tirosh, D., & Robinson, N. (1993). Connectedness in teaching equivalent algebraic expressions: Novice versus expert teachers. Mathematics Education Research Journal, 5, 50–59. https://doi.org/10.1007/BF03217254
12. Gabina, S. (2019). The effects of using manipulatives in teaching and learning of algebraic expression on senior high school (SHS) one students’ achievements in Wa municipality. Journal of Educational Development and Practice, 3(3), 83–106. https://doi.org/10.47963/jedp.v3i.951
13. Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative research. Aldine.
14. Haines, C., & Crouch, R. (2001). Recognising constructs within mathematical modelling. Teaching Mathematics and its Applications, 26(3), 132-138. https://doi.org/10.1093/teamat/20.3.129
15. Hallagan, J. E. (2006). The case of bruce: A teacher’s model of his students’ algebraic thinking about equivalent expressions. Mathematics Education Research Journal, 18, 103–123. https://doi.org/10.1007/BF03217431
16. Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students' learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371–404). Information Age Publishing.
17. Jao, L. (2013). From sailing ships to subtraction symbols: Multiple representations to support abstraction. International Journal for Mathematics Teaching and Learning, 33, 1–15. Retrieved from www.cimt.org.uk/journal/jao.pdf
18. Kaput, J. J. (1999). Teaching and learning a new algebra. In E. Fennema, & T. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 133–155). Routledge.
19. Kaput, J. J. (2008). What is algebra? What is algebraic reasoning? In J. J. Kaput, D. W. Carraher,ü & M. L. Blanton (Eds.), Algebra in the early grades (pp. 5–17). Lawrence Erlbaum Associates.
20. Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 707–762). Information Age Publishing.
21. Kieran, C. (2020). Algebra teaching and learning. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 27-32). Springer.
22. Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. National Academies Press.
23. Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A., & Stephens, A. C. (2005). Middle school students’ understanding of core algebraic concepts: Equivalence & variable. ZDM, 37(1), 68–76. https://doi.org/10.1007/BF02655899
24. Lambert, V. A., & Lambert, C. E. (2012). Qualitative descriptive research: An acceptable design. Pacific Rim International Journal of Nursing Research, 16(4), 255-256.
25. Lee, L., & Freiman, V. (2006). Developing algebraic thinking through pattern exploration. Mathematics Teaching in the Middle School, 11(9), 428–433. https://doi.org/10.5951/MTMS.11.9.0428
26. Leikin, R. (2009). Exploring mathematical creativity using multiple solution tasks. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 129–135). Rotterdam: Sense Publishers.
27. Leikin, R. (2013). Evaluating mathematical creativity: The interplay between multiplicity and insight. Psychological Test and Assessment Modeling, 55(4), 385–400.
28. Leikin, R., & Pitta-Pantazi, D. (2013). Creativity and mathematics education: the state of the art. ZDM, 45(2), 159–166. https://doi.org/10.1007/s11858-012-0459-1
29. Lesh, R., & Zawojewski, J. (2007). Problem solving and modeling. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 763–804). Information Age Publishing.
30. Lev Zamir, H., & Leikin, R. (2011). Creative mathematics teaching in the eye of the beholder: focusing on teachers' conceptions. Research in Mathematics Education, 13(1), 17-32. https://doi.org/10.1080/14794802.2011.550715
31. Levenson, E. (2011). Exploring collective mathematical creativity in elementary school. Journal of Creative Behavior, 45(3), 215–234. http://dx.doi.org/10.1002/j.2162-6057.2011.tb01428.x
32. Liljedahl, P. (2016). Building thinking classrooms: Conditions for problem-solving. In P. Felmer, J. Kilpatrick, & E. Pekhonen (Eds.), Posing and solving mathematical problems: Advances and new perspectives (pp. 361–386). Springer.
33. MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation: 11–15. In G. C. Leder & H. Forgasz (Eds.), Stepping stones for the 21st century: Australasian mathematics education research (pp. 63–81). Brill.
34. Ministry of Education (2018). Mathematics course (Grades 5, 6, 7 and 8) Curriculum. Ankara.
35. Mokhtar, M. Z., Tarmizi, R. A., Ayub, M. & Tarmizi, M. A. A. (2010). Enhancing calculus learning engineering students through problem-based learning. WSEAS Transactions on Advances in Engineering Education, 7(8), 255–264.
36. Moschkovich, J. N. (1999). Understanding the needs of Latino students in reform-oriented mathematics classrooms. Changing the faces of mathematics: Perspectives on Latinos, 4, 5–12.
37. Nathan, M. J., & Koedinger, K. R. (2000). An investigation of teachers’ beliefs of students’ algebra development. Cognition and Instruction, 18(2), 209–237. https://doi/10.1207/S1532690XCI1802_03
38. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. NCTM.
39. O’Connor, C., & Joffe, H. (2020). Intercoder Reliability in Qualitative Research: Debates and Practical Guidelines. International Journal of Qualitative Methods, 19, 1-13. https://doi.org/10.1177/1609406919899220
40. Patton, M. Q. (2001). Qualitative research and evaluation methods (2nd Ed.). Thousand Oaks, CA: Sage Publications.
41. Radford, L. (2006). Algebraic thinking and the generalization of patterns: A semiotic perspective. In S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 2–21). PME-NA.
42. Saldaña, J. (2015). The coding manual for qualitative researchers. Sage.
43. Schoenfeld, A. H. (2014). What makes for powerful classrooms, and how can we support teachers in creating them? A story of research and practice, productively intertwined. Educational Researcher, 43(8), 404–412. https://doi.org/10.3102/0013189X14554450
44. Shure, V., & Rösken-Winter, B. (2022). Developing and validating a scriptwriting task in the context of student difficulties with fraction multiplication and division. Research in Mathematics Education, 24(3), 267-290. https://doi.org/10.1080/14794802.2021.1988689
45. Shure, V., Rösken-Winter, B., & Lehmann, M. (2022). How pre-service primary teachers support academic literacy in mathematics in a scriptwriting task encompassing fraction multiplication and division. The Journal of Mathematical Behavior, 65, 100916. https://doi.org/10.1016/j.jmathb.2021.100916
46. Silver, E. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM, 29, 75–80. https://doi.org/10.1007/s11858-997-0003-x
47. Sriraman, B. (2009). The characteristics of mathematical creativity. ZDM, 41, 13–27. https://doi.org/10.1007/s11858-008-0114-z
48. Stephens, A. C. (2005). Developing students’ understandings of variable. Mathematics Teaching in the Middle School, 11(2), 96–100. http://dx.doi.org/10.5951/MTMS.11.2.0096
49. Tirosh, D., Even, R., & Robinson, N. (1998). Simplifying algebraic expressions: Teacher awareness and teaching approaches. Educational Studies in Mathematics, 35, 51–64. https://doi.org/10.1023/A:1003011913153
50. Torrance, E. P. (1974). The Torrance Tests of Creative Thinking. Personnel Press.
51. Trouche, L., Drijvers, P. (2010). Handheld technology for mathematics education: flashback into the future. ZDM Mathematics Education 42, 667–681. https://doi.org/10.1007/s11858-010-0269-2
52. Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. F. Coxford (Ed.), The ideas of algebra, K-12 (pp. 8–19). National Council of Teachers of Mathematics.
53. Warren, E. (2003). The role of arithmetic structure in the transition from arithmetic to algebra. Mathematics Education Research Journal, 15(2), 122–137. https://doi.org/10.1007/BF03217374
54. Weinberg, A. D., Stephens, A. C., McNeil, N. M., Krill, D. E., Knuth, E. J., & Alibali, M. W. (2004). Students initial and developing conceptions of variable. Paper presented at the annual meeting fo the American Educational Research Association San Diego, CA,. April.
55. Zazkis, R. (2017). Lesson play tasks as a creative venture for teachers and teacher educators. ZDM, 49, 95–105. https://doi.org/10.1007/s11858-016-0808-6
56. Zazkis, R., & Leikin, R. (2010). Advanced mathematical knowledge in teaching practice: Perceptions of secondary mathematics teachers. Mathematical Thinking and Learning, 12(4), 263–281. https://doi.org/10.1080/10986061003786349
57. Zazkis, R., Liljedahl, P., & Sinclair, N. (2009). Lesson plays: Planning teaching versus teaching planning. For the learning of mathematics, 29(1), 40-47.
58. Zazkis, R., Sinclair, N., & Liljedahl, P. (2013). Lesson play in mathematics education: A tool for research and professional development. Springer.
59. Zazkis, R., & Zazkis, D. (2014). Script writing in the mathematics classroom: Imaginary conversations on the structure of numbers. Research in Mathematics Education, 16(1), 54-70. https://doi.org/10.1080/14794802.2013.876157