Examining Algebraic Habits of the Mind through a Problem Solving: Elementary School Example
top of page
Asian Institute of Research, Journal Publication, Journal Academics, Education Journal, Asian Institute
Asian Institute of Research, Journal Publication, Journal Academics, Education Journal, Asian Institute

Education Quarterly Reviews

ISSN 2621-5799

asia institute of research, journal of education, education journal, education quarterly reviews, education publication, education call for papers
asia institute of research, journal of education, education journal, education quarterly reviews, education publication, education call for papers
asia institute of research, journal of education, education journal, education quarterly reviews, education publication, education call for papers
asia institute of research, journal of education, education journal, education quarterly reviews, education publication, education call for papers
crossref
doi
open access

Published: 15 December 2022

Examining Algebraic Habits of the Mind through a Problem Solving: Elementary School Example

Bilge Yilmaz Aslan, Begüm Özmusul

Gaziantep University, Turkey

asia institute of research, journal of education, education journal, education quarterly reviews, education publication, education call for papers
pdf download

Download Full-Text Pdf

doi

10.31014/aior.1993.05.04.642

Pages: 542-556

Keywords: Algebraic Thinking, Algebraic Habits of Mind, Elementary Students, Problem Solving

Abstract

In this study, it is aimed to determine the algebraic thinking habits of two eighth grade school students through the answers they gave in the process of solving mathematical problems. The algebraic habits of mind (ZCA) theoretical framework developed by Driscoll (1999) was used to reveal these thinking habits. The research design of this study is a case study and the participants consist of two eighth grade students. The data were analyzed using Driscoll (1999)'s ZCA framework, which is algebraic habits of mind. Descriptive analysis was used in the analysis of the data. When we look at the findings obtained from the research; It is seen that both students can do describing a rule and justifying a rule in the solutions of the problems. In addition, it is seen that computational shortcuts, equivalent expressions and symbolic expressions come to the fore in the solutions of students' problems. On the other hand, the habit of undoing in solving problems was not encountered very rarely in both students. In the light of the findings obtained, the reasons for the existing and non-existent algebraic habits of mind are discussed. As a result of this discussion, it is thought that it is effective to include guide questions to create and develop algebraic thinking habits in students in classroom teaching practices of teachers.

References

  1. Bilgiç, E. & Argün, Z., (2018). Examining middle school mathematics teacher candidates’ algebraic habits of mind in the context of problem solving. International e-Journal of Educational Studies (IEJES), 2(4), 64-80. https://doi.org/10.31458/iejes.426052

  2. Blanton, M., & Kaput, J. (2011). Functional thinking as a route into algebra in the elementary grades. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives. Advances in mathematics education (pp. 5–23). Berlin Heidelberg, Germany: Springer-Verlag. https://doi.org/10.1007/BF02655895.

  3. Booker, G., & Windsor, W. (2010). Developing algebraic thinking: Using problem-solving to build from number and geometry in the primary school to the ideas that underpin algebra in high school and beyond. Procedia-Social and Behavioral Sciences, 8, 411-419. https://doi.org/10.1016/j.sbspro.2010.12.057

  4. Cai, J., Morris, A., Hohensee, C., Hwang, S., Robison, V., Cirillo, M., Kramer, S. L., Hiebert, J., & Bakker, A. (2020). Maximizing the quality of learning opportunities for every student. Journal for Research in Mathematics Education, 51(1), 12-25. https://doi.org/10.5951/jresematheduc.2019.0005

  5. Creswell, J. W. (2007). Qualitative inquiry and research design: Choosing among five approaches. Thousand Oaks, CA: Sage.

  6. Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers, grades 6-10.Heinemann, 361 Hanover Street, Portsmouth, NH 03801-3912.

  7. Driscoll, M., Zawojeski, J., Humez, A., Nikula, J., Goldsmith, L., & Hammerman, J. (2003). The fostering algebraic thinking toolkit: A guide for staff development.

  8. Eroğlu, D., & Tanışlı, D. (2017). Integration of algebraic habits of mind into the classroom practice. Elementary Education Online, 16(2), 566-583. https://doi.org/10.17051/ilkonline.2017.304717

  9. Evans, C. W., Leija, A. J., & Falkner, T. R. (2001). Math links: Teaching the NCTM 2000 standards through children's literature. Libraries Unlimited.

  10. Hart K.M., Brown M.L., Kuchermann D.E., Kerslach D., Ruddock G. & Mccartney M., (1998). Children's understanding of mathematics: 11-16, General Editor K.M. Hart, The CSMS Mathematics Team.

  11. Hawker, S. ve Cowley, C. (1997). Oxford dictionary and thesaurus. Oxford: Oxford University.

  12. Herbert, K., & Brown, R. H. (1997). Patterns as tools for algebraic reasoning. Teaching Children Mathematics, 3(6), 340-345.

  13. Herbst, P., & Chazan, D. (2011). Research on practical rationality: Studying the justification of actions in mathematics teaching. The Mathematics Enthusiast, 8(3), 405-462. https://doi.org/10.54870/1551-3440.1225

  14. Hout-Wolters, B. V. (2000). Assessing active self-directed learning. In New learning (pp. 83-99). Springer, Dordrecht.

  15. Kaf, Y. (2007). The effect of model use in mathematics on 6th grade students' algebra achievement (Matematikte model kullanımının 6. sınıf öğrencilerinin cebir erişilerine etkisi [Unpublished master’s thesis]). Hacettepe University, Ankara, Turkey.

  16. Kaput, J. (2008). What is algebra? What is algebraic reasoning? In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades(pp. 5-17). New York, NY: Lawrence Erlbaum.

  17. Kaput, J. J. (1998). Transforming algebra from an engine of inequity to an engine of mathematical power by ‘‘algebrafying’’ the K-12 curriculum. In S. Fennel (Ed.), The nature and role of algebra in the K-14 curriculum: Proceedings of the national symposium (pp. 25–26). Washington, DC: National Research Council, National Academy Press.

  18. Kaput, J. J., Blanton, M. L., ve Moreno, L. (2008). Algebra from a symbolization point of view. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 19-55). New York: Lawrence Erlbaum.

  19. Kieran, C., & Chalouh, L. (1993). Prealgebra: The transition from arithmetic to algebra. Research ideas for the classroom: Middle grades mathematics,179-198.

  20. Kieran, C., Pang, J., Schifter, D., & Ng, S. F. (2016). Early algebra: Research into its nature, its learning, its teaching. Springer Nature.

  21. Lee, F. L. (2002, December). Diagnosing students' algebra errors on the web. In International Conference on Computers in Education, 2002. Proceedings. (pp. 578-579). IEEE.

  22. Lew, H. C. (2004). Developing algebraic thinking in early grades: Case study of Korean elementary school mathematics. The Mathematics Educator, 8(1), 88-106.

  23. Lins, R., Rojano, T., Bell, A. & Sutherland, R. (2001). Approaches to algebra. In R. Sutherland, T. Rojano, A. Bell & R. Lins (Eds), Perspectives on school algebra (pp. 1–11). Dordrecht, NL: Kluwer Academic Publishers.

  24. Magiera, M. T., Moyer, J. C., & van den Kieboom, L. A. (2017). K-8 pre-service teachers’ algebraic thinking: Exploring the habit of mind building rules to represent functions. Mathematics Teacher Education and Development (MTED), 19(2), 25-50.

  25. Magiera, M. T., Van den Kieboom, L. A., & Moyer, J. C. (2013). An exploratory study of pre-service middle school teachers’ knowledge of algebraic thinking. Educational Studies in Mathematics, 84(1), 93-113. https://doi.org/10.1007/s10649-013-9472-8

  26. Mason, I. G. (2008). An evaluation of substrate degradation patterns in the composting process. Part 2: Temperature-corrected profiles. Waste management, 28(10), 1751-1765. https://doi.org/10.1016/j.wasman.2007.06.019

  27. Mason, J. (1987). What do symbols represent? in C. Janvier (ed.), Problems of representation in the teaching and learning of mathematics, Lawrence Erlbaum Associates, Hillsdale, NJ, pp. 73-81. ISBN 0-89859-802-8

  28. Mason, J. (1989). Mathematical abstraction as the result of a delicate shift of attention. For the Learning of Mathematics, 9(2), 2-8.

  29. Max, B., & Welder, R. M. (2020). Mathematics teacher educators’ addressing the common core standards for mathematical practice in content courses for prospective elementary teachers: A focus on critiquing the reasoning of others. The Mathematics Enthusiast, 17(2), 843-881. https://doi.org/10.54870/1551-3440.1505

  30. Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis: an expanded sourcebook (2. Edition) CA: Sage. Thousand Oaks.

  31. Montague, M., Applegate, B., & Marquard, K. (1993). Cognitive strategy instruction and mathematical problem-solving performance of students with learning disabilities. Learning Disabilities Research & Practice.

  32. Özdemir, M. İ. (2019). A scale development study on the evaluation of mathematics teaching in the context of operational and conceptual knowledge(Matematik öğretiminin işlemsel ve kavramsal bilgi bağlamında değerlendirilmesine ilişkin bir ölçek geliştirme çalışması [Unpublished master’s thesis]). Gaziantep University, Gaziantep, Turkey.

  33. Polotskaia, E., Fellus, O. O., Cavalcante, A., & Savard, A. (2022). Students’ problem solving and transitioning from numerical to relational thinking. Canadian Journal of Science, Mathematics and Technology Education, 22(2), 341-364. https://doi.org/10.1007/s42330-022-00218-1

  34. Polya, G. (1973). How to solve it 2nd. New Jersey: Princeton University.

  35. Pourdavood, R., McCarthy, K., & McCafferty, T. (2020). The impact of mental computation on children's mathematical communication, problem solving, reasoning, and algebraic thinking. Athens journal of Education, 7(3), 241-253. https://doi.org/10.30958/aje.7-3-1

  36. Radford, L., & Sabena, C. (2015). The question of method in a Vygotskian semiotic approach. In A. Bikner–Ahsbahs, C. Knipping, & N. C. Presmeg (Eds.). Approaches to Qualitative Research in Mathematics Education (pp. 157–182). Springer Netherlands.

  37. Rosenzweig, C., Krawec, J., & Montague, M. (2011). Metacognitive strategy use of eighth-grade students with and without learning disabilities during mathematical problem solving: A think-aloud analysis. Journal of learning disabilities, 44(6), 508-520. https://doi.org/10.1177/0022219410378445

  38. Schliemann, A. D., Carraher, D. W., & Brizuela, B. M. (2007). Bringing out the algebraic character of arithmetic: From children's ideas to classroom practice. Hillsdale, NJ: Erlbaum.

  39. Schoenfeld, A. (1987). What’s all the fuss about metacognition? Schoenfeld, A.H. (ed.), Cognitive Science and Mathematics Education, 189-215. Lawrence Erbaum.

  40. Sezer, N., & Altun, M (2020). 6. Sınıf öğrencilerinin zihnin cebirsel alışkanlıklarının geliştirilmesi üzerine bir çalışma. Uludağ Üniversitesi Eğitim Fakültesi Dergisi, 33(2), 446-476. https://doi.org/10.19171/uefad.568737

  41. Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C., & Strawhun, B. T. F. (2005). Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. The Journal of Mathematical Behavior, 24(3-4), 287-301. https://doi.org/10.1016/j.jmathb.2005.09.009

  42. Simon, M. A., & Blume, G. W. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. The Journal of Mathematical Behavior, 15(1), 3-31.

  43. Swafford, J. O., & Langrall, C. W. (2000). Grade 6 students' pre instructional use of equations to describe and represent problem situations. Journal for Research in Mathematics Education, 31, 89-112. https://doi.org/10.2307/749821

  44. Veenman, M. V., & Spaans, M. A. (2005). Relation between intellectual and metacognitive skills: Age and task differences. Learning and individual differences, 15(2), 159-176. https://doi.org/10.1016/j.lindif.2004.12.001

  45. Vygotsky, L. S. (1997). Research method. In R. W. Rieber (Ed.), The collected works of L.S.Vygotsky Vol. 4 (pp. 27–65). New York: Plenum Press.

bottom of page