Enhancing functional thinking: Identifying the prior schemas of seventh grade students in generalization of two-variable figural patterns

Afkhami, R.; Asghary, N.; Medghalchi, A.

doi:10.22061/jte.2019.4844.2127

Document Type : Original Research Paper

Authors

¹ Department of Mathematics, Faculty of sciences, Islamic Azad University, Central Tehran Branch, Tehran, Iran

² Department of Mathematics, Faculty of Mathematical sciences, Kharazmi University, Tehran, Iran

https://doi.org/10.22061/jte.2019.4844.2127

Abstract

Background and Objectives: The figural patterns have a unique capacity to enhance functional thinking. The patterns generalization in school mathematics is considered as a way to promote functional thinking. Variable is one of the concepts in patterns generalization. Paying attention to figural patterns provides an opportunity for students to understand the meaning of variable and how to use it. Reference is also a central concept in patterns generalization. The number of variables is one of the characteristics that has been proposed in the pattern generalization tasks, but all the research has been related to one variable, linear and quadratic patterns. The aim of this study was to identifying the prior schemas in generalization of two-variable figural patterns. As regard to the concept of two variables, understanding three-dimensional space is a prerequisite for understanding and generalizing two-variable patterns. In these patterns, instead of one independent variable, there are two independent variables that change simultaneously and affect the dependent variable. Understanding these patterns requires the development of the R2 space scheme to R3 space, which is not a cognitively complex step and does not require the reconstruction of the existing scheme.
Methods: The present research is part of a broad research which is done using quantitative-qualitative (mixed) research method. The research framework is APOS theory and based on the use of ACE (Activities, Class discussions and Exercises) teaching cycles. This research was conducted in three steps. In the first step, initial genetic decomposition for generalization of two-variable figural patterns was designed using the background, self-concept analysis and researchers’ experiences. It includes the prior schemas for generalization. In the second step, from the total 493 students of Malekan city (in East Azerbaijan) as the statistical population of research, a sample of 220, 7th grade students were selected based on the Cochran formula for determination of sample size. Then, a test that includes 7 tasks was designed based on APOS framework. The validity of the test was confirmed by three experts in mathematics education and four experienced teachers. Internal consistency of questions was estimated with Cronbach’s alpha and reported to be 0.69. Students responded the test at 90 minutes. The third step of research began with 19 students, with permission from the education and training office of Malekan, and school principals and parents of students. This step is done in three cycles.
Findings: Using the analysis of students' responses to this test based on the APOS framework and doing three cycles of the research were conducted with the teaching method of Activity-Class Discussion-Exercise (ACE) with 19 students; genetic decomposition was finalized in this way, and defects of students in reference schema, R³schema and variables schema as prior schemas in generalization of two-variable figural patterns were identified and encoded. Most of students had a good understanding of working with two variables. However in the context of generalization of two-variable figural pattern revealed many difficulties at the naming of variables, and using independent and dependent variables in proper positionConclusion: Identifying the mental constructs of students in generalizing patterns eases better teaching and learning.
Conclusion: By identifying the mental structures of students in generalizing patterns, the path of teaching and learning will be smoother.

Keywords

20.1001.1.20080441.1399.14.3.18.0

Main Subjects

Modern Educational Approaches

COPYRIGHTS
©2020 The author(s). This is an open access article distributed under the terms of the Creative Commons Attribution (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, as long as the original authors and source are cited. No permission is required from the authors or the publishers.

References

[1] Mason J. Expressing generality and roots of algebra. In: Bdnarz N, Kieran C, Lee L (Eds) Approches to algebra:perspective for research and teaching. Dordrecht: Kluwer Academic;1996. p.65-86.

[2] Blanton ML , Kaput JJ. Functional thinking as a route into algebra in the elementary grades. ZDM- International Reviews on Mathematicl Education. 2011; 37(1):34-42.

[3] Markworth K. Growing and growing: promoting functional thinking with geometric growing patterns [dissertation]. University of North Carolina, ChapelHill; 2010.

[4] Asghari N. Developing a model to enhance elementary teachers, ability to foster functional thinking and algebraic resoning in elementary students. Jornal of Theory and Practice in Curriculum. 2014; 2(3): 141-162. Persian

[5] Rivera F. Teaching and learning patterns in school mathematics: Psychological and pedagogical considerations. Switzeland: Springer Science & Business Media; 2013.

[6] Huntzinger EM. Exploring generalization through pictorial growth patterns. In Greenes C E & Rubenstein R (Eds.), Algebra and algebraic thinking in school mathematics Reston, VA: NCTM; 2008. p. 279-293.

[7] National Council of Teachers of Mathematics. Principles and standards for school mathematics. The National Council of Teachers of Mathematics, Inc., Reston: USA; 2000.

[8] English LD, Warren EA. Introducing the variable through pattern exploration. In Moses B (Ed.), Algebraic thinking, Grades K-12, Reston, VA: NCTM; 1999.

[9] Rivera F, Becker JR. Abductions in pattern generalization, In Woo JH, Lew HC, Park KS and Seo DY (Eds.). Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education. Seoul: PME; 2007: p. 97-104

[10] National Center for TIMSS and PIRLS :International Studies. A brief report of Most Important Results of TIMSS and PIRLS 2011 and Comparison with Iranian Students'Performance in the Past Peroids. Research Center of Education, Tehran: Iran; 2013. Persian.

[11] Hashemi N, Abu MS , Kashefi H, Rahimi KH. Generalization in the learning of mathematics. The 2nd seminar on quality and affordable education. Malazia. 2013.

[12] Chua BL, Hoyles C. Modalites of rules and generalizing strategies of year 8 students for a quadratic pattern. In Nicol C, Liljedahl P, Oesterle S and Allan D. (Eds.) Proceedings of the Joint Meeting of PME 38 and PME-NA 36. Vancouver, Canada: PME; 2014. p. 305-312.

[13] Samson DA. The Heuristic significance of enacted visualisation. [dissertation]. Rhodes University, Grahamstown, South Africa;2011.

[14] Shojaie K. The investigation of ability of the nine grade students’ generalization and justification in numeric and geometric number patterns. [master’s thesis]. Shahid Rajaee Teacher Training University, Tehran; 2013. Persian.

[15] Lannin JK, Barker DD, Townsend BE. Algebraic generalization strategies: factors influencing student strategy selection. Mathematics Education Research Journal. 2006;18 (3): 3-28.

[16] Samson DA. An analysis of the influence of question design on pupils’ approaches to number pattern generalisation tasks. [master’s thesis]. Rhodes University, Grahamstown, South Africa; 2007.

[17] Chua B. L.Features of generalising task: Help or hurdle to expressing generality? Australian Mathematics Teacher. 2009; 65(2): 18 – 24.

[18] Chua BL, Hoyles C. The effect of different pattern form on secondray two students’ ability to generalize. In Tso,T. Y. (Ed.). Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education. Taipei, Taiwan: PME; 2012. p.155-162.

[19] Rivera F. Visual templates in pattern generalization activity. Educational Studies in Mathematics. 2010;73(3): 297-328.

[20] Kaput J, Carraher DW, Blanton ML. (Eds.). Algebra in the early grades. Mahwah.(NJ): Lawrence Erlbaum Associates; 2007.

[21] Smith E. Representational thinking as a framework for introducing functions in the elementary curriculum. Algebra in the Elementary Grade; 2008. P.133-163.

[22] Wilkie k , Clarke D. Developing students' functional thinking in algebra through different visualisation of a growing pattern's structure. In J. Anderson, M. Cavanagh & A. Prescott (Eds.). Curriculum in focus: Research guided practice, Proceedings of the 37^th annual conference of the Mathematics Education Research Group of Australasia; 2014. p. 637–644.

[23] Sutarto, Toto N, Subanji S. Local conjecturing process in the solving of pattern generalization problem. Educational Research and Reviews. 2016;11 (8): 732-742.

[24] Sutarto, Nusantara T, Subanji S, Hastuti ID. Global conjecturing process in pattern generalization problem.Journal of Physics: Conference Series.1008 012060; 2018.

[25] Arnon I, Cottrill J, Dubinsky E, Oktaç A, RoaFuentes S, Trigueros M, Weller K. APOS Theory: A framework for research and curriculum development in the mathematics education. New York: Springer-Verlag; 2014.

[26] Dubinsky E. Reflective abstraction in advanced mathematical thinking. In Tall D (Ed.), Advanced mathematical thinking. The Netherlands: Kluwer Academic Publishers; 1991; 95-126.

[27] Zohrevand SH. Students’ misconceptions about concept of variable in algebra [master’s thesis]. Shahid Rajaee Teacher Training University: Tehran. 2013.Persian.

[28] Álvarez I, Gómez-Chacón I, Ursini S. Understanding the algebraic variable: comparative study of Mexican and Spanish students. Eurasia Journal of Mathematics, Science & Technology Education; 2015; 11(6): 1507-1529.

[29] Peirce CS. Collected papers of Charles SandersPeirce. InC.Hartshorne&P.Weiss (Eds.). Cambridge, MA: Harvard University Press: 1985; 1-6.

[30] Pedemonte B. Argumentation and algebraic proof. ZDM Mathematics Education; 2008; 40: 385–400.

[31] Reid D. Forms and uses of abduction. Paper presented to Working Group 4: Proof and argumentation. Third Annual Conference of the European Society for Research in Mathematics Education. Bellaria, Italy; 2008.

[32] Harel G, Tall D. The general, the abstract, and the generic in advanced mathematics. For the Learning of Mathematic. 1989; 11 (1): 38-42.

[33] Asiala M, Brown A, DeVries D, Dubinsky E, Mathews D, &Thomas KA framework for research and curriculum development in undergraduate mathematics education. In Research in Collegiate mathematics education II. Providence, RI: American Mathematical Society. CBMS issues in mathematics education; 1996; 6: 1–32.

[34] Khosroshahi GhL. Algebraization for Pre-schoolers with a focus on Associativity.[ dissertation]. shahid beheshti university,Tehran; 2016. Persian.

[35] Li X. Cognitive analysis of student’errors and misconceptions in variables, euations and functions.[dissertation]. A&M University, Texas ; 2006.

[36] Asghary N, Shahvarani A, Medgalchi A. R. Significant Process of Change for Elementary Teachers to Foster Functional Thinking. Mathematics Education. Bulletin- BOLEMA, Brazil; 2013.