Elementary students’ functional thinking: From recursive to correspondence

Abstract

This study aims to identify elementary students’ functional thinking processes in solving pattern problems. Previous studies showed that elementary students' functional thinking still often experience errors in solving pattern problems. The study of the functional thinking process in solving pattern problems is a fundamental key as a solution to find out the strengths and weaknesses of elementary school students, so that they are better prepared in generalizing relationships, representing and analyzing function behavior in advanced algebra classes. This study used a descriptive qualitative approach with a case study method. Participants of study was sixty-five elementary students who had not yet received generalization patterns material. The instruments were tasks and interview guidelines. Based on the task results, students who had correct answers were chosen using purposive sampling to be given an in-depth interview. The finding indicated that elementary students are able to think functionally in different ways. Students’ functional thinking begins with recursive thinking in the pre-finding formula in the entry stage. Students find the formula by corresponding thinking in the attack stage. Finally, students use the formula to get inverse in the review stage.

Keywords

• Fuctional thinking
• Recursive
• Correspondence
• Mathematics education

References

1. Amir, M. F., Mufarikhah, I. A., Wahyuni, A., Nasrun, & Rudyanto, H. E. (2019). Developing ‘Fort Defending’ Game as a Learning Design for Mathematical Literacy Integrated to Primary School Curriculum in Indonesia. Elementary Education Online, 18(3), 1081–1092. https://doi.org/10.17051/ilkonline.2019.610145
2. Amit, M., & Neria, D. (2008). ‘“ Rising to the challenge ”’: using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students. ZDM Mathematics Education, 40(2), 111–129. https://doi.org/10.1007/s11858-007-0069-5
3. Blanton, M., Brizuela, B., Gardiner, A., Sawrey, K., & Newman-Owens, A. (2017). A progression in first-grade children’s thinking about variable and variable notation in functional relationships. Educational Studies in Mathematics, Online First. https://doi.org/doi:10.1007/s10649-016- 9745-0
4. Blanton, M. L. (2008). Algebra and the elementary classroom. Transforming thinking, transforming practice. Portsmounth, NH: Heinemann.
5. Blanton, M. L., & Kaput, J. J. (2004). Elementary Grades Students ’ Capacity for Functional Thinking. In Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 135–142).
6. Blanton, M. L., & Kaput, J. J. (2005). Characterizing a Classroom Practice That Promotes Algebraic Reasoning. Journal for Research in Mathematics Education, 36(5), 412–446.
7. Blanton, M. L., Levi, L., Crites, T., & Dougherty, B. J. (2011). Developing essential understandings of algebraic thinking, Grades 3-5. Reston, VA: The National Council of Teachers of Mathematics.
8. Bush, S. B., & Karp, K. S. (2013). Prerequisite algebra skills and associated misconceptions of middle grade students : A review. Journal of Mathematical Behavior, 32(3), 613–632. https://doi.org/10.1016/j.jmathb.2013.07.002

9. Cohen, L., Manion, L., & Morrison, K. (2000). Research methods in education (5th editio). London: Routledge.
10. Confrey, J., & Smith, E. (1991). A framework for functions: Prototypes, multiple representations, and transformations. In R. Underhill & C. Brown (Eds.), Proceedings of the thirteenth annual meeting of the north American chapter of the international group for the psychology of mathematics education (pp. 57–63). Blacksburg, VA: Virginia Polytechnic Institute & State University.
11. Creswell, J. W. (2012). Educational research: Planning, conducting, and evaluating quantitative and qualitative research. Educational Research (Vol. 4). https://doi.org/10.1017/CBO9781107415324.004
12. Eisenmann, P. (2009). A contribution to the development of functional thinking of pupils and students. The Teaching of Mathematics, (23), 73–81. Ellis, A. B. (2007). Connections Between Generalizing and Justifying : Students ’ Reasoning with Linear Relationships. Journal for Research in Mathematics Education, 38(3), 194–229.
13. Fraenkel, J. R., Wallen, N. E., & Hyun, H. H. (2012). How to Design and Evaluate Research in Education. New York: McGraw-Hill.
14. Huntley, M. A., Marcus, R., Kahan, J., & Miller, J. L. (2007). Investigating high-school students ’ reasoning strategies when they solve linear equations. Journal of Mathematical Behavior, 26(2), 115–139. https://doi.org/10.1016/j.jmathb.2007.05.005
15. Jupri, A., & Drijvers, P. (2014). Difficulties in initial algebra learning in Indonesia. Mathematics Education Research Journal, 26(4), 683–710. https://doi.org/10.1007/s13394-013-0097-0
16. Kaiser, G., & Willander, T. (2005). Development of mathematical literacy: Results of an empirical study. Teaching Mathematics and Its Applications, 24(2–3), 48–60. https://doi.org/10.1093/teamat/hri016
17. Kemendikbud. Peraturan Menteri Pendidikan dan Kebudayaan Republik Indonesia Nomor 21 Tahun 2016, Pub. L. No. 21 (2016). Indonesia.
18. Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. In F. K. Lester Jr (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (p. 707). Charlotte, NC: New Age Publishing; Reston, VA: National Council of Teachers of Mathematics.
19. Knuth, E. J. (2000). Student Understanding of the Cartesian Connection : An Exploratory Study. Journal for Research in Mathematics Education, 31(4), 500–507. https://doi.org/DOI: 10.2307/749655
20. Lannin, J. K. (2005). Generalization and Justification : The Challenge of Introducing Algebraic Reasoning Through Patterning Activities. Mathematical Thinking and Learning, 7(3), 231–258. https://doi.org/DOI: 10.1207/s15327833mtl0703_3
21. Lichti, M. (2018). How to Foster Functional Thinking in Learning Environments Using Computer-Based Simulations or Real Materials. Journal for STEM Education Research, 1(1–2), 148–172.
22. Lichti, M., & Roth, J. (2019). Functional thinking: A three-dimensional construct? Journal Für Mathematik-Didaktik, 40(2), 169–195. https://doi.org/10.1007/s13138-019-00141-3
23. Mason, J., Stacey, K., & Burton, L. (2010). Thinking Mathematically. Edinburgh: Pearson.
24. Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis: an Expanded Sourcebook. London: SAGE Publications Inc.
25. NCTM. (2000). Principless and Standards for School Mathematics. Reston: NCTM.
26. Smith, E. (2008). Representational thinking as a framework for introducing functions in the elementary curriculum. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 95–132). London & Newyork: Lawrence Erlbaum/Taylor & Francis Group & NCTM.
27. Stacey, K. (1989). Finding and Using Patterns in Liniar generalising Problems. Educational Studies in Mathematics, 20(2), 147–164. https://doi.org/10.1007/bf00579460
28. Stacey, K. (2011). The PISA View of Mathematical Literacy in Indonesia. Journal on Mathematics Education, 2(2), 95–126. https://doi.org/10.22342/jme.2.2.746.95-126
29. Stephens, A. C., Fonger, N., Strachota, S., Isler, I., Blanton, M. L., Knuth, E., & Gardiner, A. M. (2017). A learning progression for elementary students’ functional thinking a learning progression for elementary students’ functional. Mathematical Thinking and Learning, 19(3), 143–166. https://doi.org/10.1080/10986065.2017.1328636
30. Subanji, & Nusantara, T. (2013). Karakterisasi Kesalahan Berpikir Siswa dalam Mengonstruksi Konsep Matematika. Jurnal Ilmu Pendidikan, 33(2), 208–217. Tanıslı, D. (2011). Functional thinking ways in relation to linear function tables of elementary school students. The Journal of Mathematical Behavior, 30, 206–223. https://doi.org/10.1016/j.jmathb.2011.08.001
31. Warren, E. A., Cooper, T. J., & Lamb, J. T. (2006). Investigating functional thinking in the elementary classroom : Foundations of early algebraic reasoning. Journal of Mathematical Behavior, 25, 208–223. https://doi.org/10.1016/j.jmathb.2006.09.006
32. Warren, E., & Cooper, T. O. M. (2005). Introducing Functional Thinking in Year 2 : a case study of early algebra teaching. Contemporary Issues in Early Childhood, 6(2), 150–162.
33. Wijaya, A., Heuvel-panhuizen, M. Van Den, Doorman, M., & Robitzch, A. (2014). Difficulties in solving context-based PISA mathematics tasks : An analysis of students ’ errors. The Mathematics Enthusiast, 11(3), 555–584. https://doi.org/10.1139/t02-118
34. Wilkie, K. J. (2015). Learning to teach upper primary school algebra : changes to teachers ’ mathematical knowledge for teaching functional thinking. Mathematics Education Research Journal, 28(2), 245–275. https://doi.org/10.1007/s13394-015-0151-1
35. Wilkie, K. J., & Clarke, D. M. (2016). Developing students’ functional thinking in algebra through different visualisations of a growing pattern’ s structure. Mathematics Education Research Journal, 28(2), 223–243. https://doi.org/10.1007/s13394-015-0146-y