The Role of Cognitive Factors in Year Five Pupils' Algebraic Thinking: A Structural Equation Modelling Analysis

Piriya Somasundram

doi:10.29333/ejmste/9612

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Piriya Somasundram ¹ ^*

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¹ School of Mathematical Sciences, Sunway University, MALAYSIA^* Corresponding Author

Abstract

Algebraic thinking in children can bridge the cognitive gap between arithmetic and algebra. This quantitative study aimed to develop and test a cognitive model that examines the cognitive factors influencing algebraic thinking among Year Five pupils. A total of 720 Year Five pupils from randomly selected national schools in Malaysia participated in this study. Two mathematics instruments were used to evaluate the participants’ number, symbol, operation, and pattern senses and their algebraic thinking respectively. Data was analysed using a structural equation modelling and Partial Least Squares regression. The results indicated that the proposed cognitive factors influence algebraic thinking significantly, with the most influential factor being symbol sense followed by pattern sense, number sense, and operation sense. The finding implies that educators should consider implementing activities related to these cognitive factors when teaching mathematics to enhance the pupils’ transition from arithmetic to algebra.

Keywords

Algebraic thinking
Arithmetic generalisation
functional thinking
pattern generalisation

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, Volume 17, Issue 1, January 2021, Article No: em1935

https://doi.org/10.29333/ejmste/9612

Publication date: 20 Jan 2021

Article Views: 2086

Article Downloads: 1612

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Disclosure

Declaration of Conflict of Interest: No conflict of interest is declared by author(s).

Data sharing statement: Data supporting the findings and conclusions are available upon request from the corresponding author(s).

References

Alibali, M. W., Knuth, E. J., Hattikudur, S., McNeil, N. M., & Stephens, A. C. (2007). A longitudinal examination of middle school students’ understanding of the equal sign and equivalent equations. Mathematical Thinking and Learning, 9, 221-247. https://doi.org/10.1080/10986060701360902
Anderson, J. C., & Gerbing, D. W. (1988). Structural equation modeling in practice: A review and recommended two-step approach. Psychological Bulletin, 103(3), 411-423. https://doi.org/10.1037/0033-2909.103.3.411
Anderson, J. R. (1983). The architecture of cognition. Psychology Press.
Arcavi, A. (1994). Symbol sense: Informal sense-making in formal mathematics. For the Learning of Mathematics, 14(3), 24-35.
Ardiansari, L., & Wahyudin. (2019). Operation sense in algebra of junior high school students through an understanding of distributive law. Journal of Physics: Conference Series, 1521, 032003. https://doi.org/10.1088/1742-6596/1521/3/032003
Ayhan. K., & Nilufer. Y.K., (2017). Relational thinking: The bridge between arithmetic and algebra. International Electronic Journal of Elementary Education, 10(1), 131-145. https://doi.org/10.26822/iejee.2017131893
Blanton, M. L., & Kaput, J. J. (2003). Developing elementary teachers’: “Algebra eyes and ears”. Teaching Children Mathematics, 10(2), 70-77. https://doi.org/10.5951/TCM.10.2.0070
Blanton, M. L., & Kaput, J. J. (2004). Elementary grades students’ capacity for functional thinking. In M. J. Hoines, & A. B. Fuglestad (Eds), Proceedings of the 28th International Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 135-142). Bergen University College.
Blanton, M. L., & Kaput, J. J. (2008). Building district capacity for teacher development in algebraic reasoning. In J. J Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 361-388). Taylor & Francis Group. https://doi.org/10.4324/9781315097435-17
Brizuela, B., & Schliemann, A. (2004). Ten-year-old students solving linear equations. For the Learning of Mathematics, 24(2), 33-40.
Byrd, C. E., McNeil, N. M., Chesney, D. L., & Matthews, P. G. (2015). A specific misconception of the equal sign acts as a barrier to children’s learning of early algebra. Learning and Individual Differences, 38, 61-67. https://doi.org/10.1016/j.lindif.2015.01.001
Cai, J., & Moyer, J. (2008). Developing algebraic thinking in earlier grades: some insights from international comparative studies. In National Council of Teachers of Mathematics (pp. 169-193). NCTM.
Carpenter, T. P., & Levi, L. (2000). Developing conceptions of algebraic reasoning in the primary grades (Report no. 002). Wisconsin Center for Education Research.
Carraher, D. W., & Schliemann, A. D. (2007). Early algebra and algebraic reasoning. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 669-705). Information Age Publishing.
Carraher, D. W., Schliemann, A. D., & Schwartz, J. L. (2008). Early algebra is not the same as algebra early. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 235-272). Taylor & Francis Group. https://doi.org/10.4324/9781315097435-12
Carraher, D. W., Schliemann, A. D., Brizuela, B. M., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics Education, 37(2), 87-115.
Chan, Y. L., Maun, R., & Krishnan, G. (2018). Mathematics Year 5. Alias, A. R., Hamdan, N.A. (Eds.). Dewan Bahasa dan Pustaka.
Childs, K. M. (1995). An investigation of the role of patterns in developing algebraic thinking (UMI No. 9615789) (Doctoral dissertation), Texas A&M University.
Chrysostomou, M., Pitta-Pantazi, D., Tsingi, C., Cleanthous, E., & Christou, C. (2013). Examining number sense and algebraic reasoning through cognitive styles. Educational Studies in Mathematics, 83, 205-223. https://doi.org.10.1007/s10649-012-9448-0
Cohen, J. (1988). Statistical power analysis for the behavioral sciences. Lawrence Erlbaum.
Diamantopoulos, A., Riefler, P., & Roth, K. P. (2008). Advancing formative measurement models. Journal of Business Research, 61(12), 1203-1218. https://doi.org/10.1016/j.jbusres.2008.01.009
Driscoll, M., & Moyer, J. (2001). Using students’ work as a lens on algebraic thinking. Mathematics Teaching in the Middle School, 6(5), 282-286. https://doi.org/10.5951/MTMS.6.5.0282
Ferrini-Mundy, J., Lappan, G., & Phillips, E. (1997). Experiences with patterning. Teaching Children Mathematics, 3(6), 282-288. https://doi.org/10.5951/TCM.3.6.0282
Gozde. A., & Dilek, T. (2017). An analysis of middle school mathematics textbooks from the perspective of fostering algebraic thinking through generalization. Educational Sciences: Theory & Practice, 17(6), 2001-2030. https://doi.org/10.12738/estp.2017.6.0506
Hair, J. F., Hult, G. T., Ringle, C. M., & Sarstedt, M. (2014). A primer on partial least squares structural equation modeling (PLS-SEM). SAGE Publications, Inc.
Haldar, L. C. (2014). Students’ understandings of arithmetic generalizations (UMI No. 3640454) (Doctoral dissertation), University of California, Berkeley.
Henseler, J., Ringle, C. M., & Sinkovics, R. R. (2009). The use of partial least squares path modeling in international marketing. New Challenges to International Marketing Advances in International Marketing, 20, 277-319. https://doi.org/10.1108/S1474-7979(2009)0000020014
Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics, 27(1), 59-78. https://doi.org/10.1007/BF01284528
Houssart, J., & Evens, H. (2003). Approaching algebra through sequence problems: Exploring children’s strategies. Research in Mathematics Education, 5(1), 197-214. https://doi.org/10.1080/14794800008520122
Hsu, C. Y., Yang, D. C., & Li, F. M. (2001). The design of “The fifth and sixth grade number sense rating scale”. Chinese Journal of Science Education, 9(4), 351-374.
Jacobs, V. R., Franke, M. L., Carpenter, T. P., Levi, L., & Battey, D. (2007). Professional development focused on children’s algebraic reasoning in elementary school. Journal for Research in Mathematics Education, 38(3), 258-288. https://doi.org/10.2307/30034868
Jurdak, M. E., & El Mouhayar, R. R. (2014). Trends in the development of student level of reasoning in pattern generalization tasks across grade level. Educational Studies in Mathematics, 85, 75-92. https://doi.org/10.1007/s10649-013-9494-2
Kaput, J. 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). Taylor and Francis Group. https://doi.org/10.4324/9781315097435-2
Kaput, J., Carraher, D. W., & Blanton, M. L. (Eds.), Algebra in the early grades (pp. 361-388). Taylor and Francis Group.
Kieran, C. (1996). The changing face of school algebra. In C. Alsina, J. Alvarez, B. Hodgson, C. Laborde, & A. Pérez (Eds.), 8th International congress on mathematical education: selected lectures (pp. 271-290). S.A.E.M. Thales.
Kieran, C., Pang, J., Schifter, D., & Ng, S. F. (2016). Early algebra: Research into its nature, its learning, its teaching. Springer International Publishing. https://doi.org/10.1007/978-3-319-32258-2
Kim, J., Kim, J., DesJardins, S. L., & McCall, B. P. (2015). Completing Algebra II in high school: does it increase college access and success? The Journal of Higher Education, 86(4), 628-662. https://doi.org/10.1353/jhe.2015.0018
Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A., & Stephens, A. C. (2011). Middle school students’ understanding of core algebraic concepts: Equivalence & variable. In J. Cai, & E. Knuth (Eds.), Early algebraization (pp. 259-276). Springer. https://doi.org/10.1007/978-3-642-17735-4_15
Lannin, J. K., Barker, D. D., & Townsend, B. E. (2006). Recursive and explicit rules: How can we build student algebraic understanding? Journal of Mathematical Behavior, 25, 299-317. https://doi.org/10.1016/j.jmathb.2006.11.004
Mason, J. (2008). Making use of children’s power to produce algebraic thinking. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 57-94). Taylor and Francis Group. https://doi.org/10.4324/9781315097435-4
McIntosh, A., Reys, B. J., & Reys, R. E. (1992). A proposed framework for examining basic number sense. For the Learning of Mathematics, 12(3), 2-8.
Miller, G. A. (1956). The magical number seven, plus or minus two: Some limits on our capacity for processing information. Psychological Review, 63, 81-97. https://doi.org/10.1037/h0043158
Ralston, N. C. (2013). The development and validation of a diagnostic assessment of algebraic thinking skills for students in the elementary grades (UMI No. 3588844) (Doctoral dissertation), University of Washington.
Rittle-Johnson, B., Matthews, P. G., Taylor, R. S., & McEldoon, K. L. (2011). Assessing knowledge of mathematical equivalence: A construct-modeling approach. Journal of Educational Psychology, 103(1), 85-104. https://doi.org.10.1037/a0021334
Schubert, C., Gfeller, M., and Donohue, C. (2013). Using Group Explorer in teaching abstract algebra. International Journal of Mathematical Education in Science, 44(3), 37-41. https://doi.org/10.1080/0020739X.2012.729680
Slavit, D. (1999). The role of operation sense in transitions from arithmetic to algebraic thought. Educational Studies in Mathematics, 37(3), 251-274. https://doi.org/10.1023/A:1003602322232
Smith, E. (2008). Representational thinking as a framework for introducing functions in the elementary curriculum. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 133-163). Taylor and Francis Group. https://doi.org/10.4324/9781315097435-6
Stacey, K. (1989). Finding and using patterns in linear generalising problems. Educational Studies in Mathematics, 20(2), 147-164. https://doi.org/10.1007/BF00579460
Swafford, J. O., & Langrall, C. W. (2000). Grade 6 students’ preinstructional use of equations to describe and represent problem situations. Journal for Research in Mathematics Education, 31(1), 89-112. https://doi.org/10.2307/749821
Swangrojn, P. (2003). Solving algebra word problems: Solution strategies Thai students used and potential connections with teachers’ instructional strategies (UMI No. 3103575) (Doctoral dissertation), Oregon State University.
Van Amerom, B. A. (2002). Reinvention of early algebra (Doctoral dissertation, Utrecht University). https://dspace.library.uu.nl/handle/1874/874
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
Whitley, J. E. (2019). The unfolding of third graders’ conceptualization of a function: A collective case study (UMI No. 27549067) (Doctoral dissertation), Kansas State University.
Yang, D. C. (2003). Teaching and learning number sense—An intervention study of fifth grade students in Taiwan. International Journal of Science and Mathematics Education, 1, 115-134. https://doi.org/10.1023/A:1026164808929
Yang, D. C., & Jan H. J. (2019). The study of primary school teachers’ performance on number sense. International Journal of Information and Education Technology, 9(5), 342-349. https://doi.org/10.18178/ijiet.2019.9.5.1224
Yang, D. C., Hsu, C. J., & Huang, M. C. (2004). A study of teaching and learning number sense for sixth grade students in Taiwan. International Journal of Science and Mathematics Education, 2(3), 407-430. https://doi.org/10.1007/s10763-004-6486-9

How to cite this article

APA

Somasundram, P. (2021). The Role of Cognitive Factors in Year Five Pupils' Algebraic Thinking: A Structural Equation Modelling Analysis. Eurasia Journal of Mathematics, Science and Technology Education, 17(1), em1935. https://doi.org/10.29333/ejmste/9612

Vancouver

Somasundram P. The Role of Cognitive Factors in Year Five Pupils' Algebraic Thinking: A Structural Equation Modelling Analysis. EURASIA J Math Sci Tech Ed. 2021;17(1):em1935. https://doi.org/10.29333/ejmste/9612

AMA

Somasundram P. The Role of Cognitive Factors in Year Five Pupils' Algebraic Thinking: A Structural Equation Modelling Analysis. EURASIA J Math Sci Tech Ed. 2021;17(1), em1935. https://doi.org/10.29333/ejmste/9612

Chicago

Somasundram, Piriya. "The Role of Cognitive Factors in Year Five Pupils' Algebraic Thinking: A Structural Equation Modelling Analysis". Eurasia Journal of Mathematics, Science and Technology Education 2021 17 no. 1 (2021): em1935. https://doi.org/10.29333/ejmste/9612

Harvard

Somasundram, P. (2021). The Role of Cognitive Factors in Year Five Pupils' Algebraic Thinking: A Structural Equation Modelling Analysis. Eurasia Journal of Mathematics, Science and Technology Education, 17(1), em1935. https://doi.org/10.29333/ejmste/9612

MLA

Somasundram, Piriya "The Role of Cognitive Factors in Year Five Pupils' Algebraic Thinking: A Structural Equation Modelling Analysis". Eurasia Journal of Mathematics, Science and Technology Education, vol. 17, no. 1, 2021, em1935. https://doi.org/10.29333/ejmste/9612