Authors retain the copyright without restrictions for their published content in this journal. HSSR is a SHERPA ROMEO Green Journal.
This is an open-access article distributed under the terms of
STAGES IN PARTIAL FUNCTIONAL THINKING IN THE FORM OF LINEAR FUNCTIONS: APOS THEORY
Corresponding Author(s) : Suci Yuniati
Humanities & Social Sciences Reviews,
Vol. 8 No. 3 (2020): May
Purpose of the study: The purpose of this study is to describe students' partial functional thinking processes in solving mathematical problems based on APOS Theory. The problem in this study was formulated into the question, what are the stages of students' partial functional thinking in solving mathematical problems based on APOS Theory?.
Methodology: This study was conducted by with 44 students from the Department of Mathematics Education. The subjects of this study were asked to solve mathematical problems developed from (Wilkie, 2014). Then some of them were interviewed to learn their functional thinking processes. The subjects’ partial functional thinking processes were analyzed using APOS theory.
Main Findings: The results showed that, based on APOS theory, the students’ partial functional thinking consisted of several stages: 1) identifying the problem, 2) organizing the data, 3) determining the recursive patterns, 4) determining the covariational relationships, 5) generalizing the relationships between variations in quantities (correspondence), and 6) re-checking the generalization results. In this case, the students generalized the relationships between variations in the form of functions done partially using the arithmetic formula .
Applications of this study: The findings of this study can help teachers understand the stages in students' thinking processes in solving problems about functions and the difficulty faced by the students in understanding the functions.
Novelty/Originality of this study: The researchers identified stages in students' partial functional thinking in solving mathematical problems in the form of functions based on APOS Theory.
Download CitationEndnote/Zotero/Mendeley (RIS)
Allday, R. A. (2017). Functional Thinking for Managing Challenging Behavior. Intervention in School and Clinic. 1–7. https://doi.org/10.1177/1053451217712972 DOI: https://doi.org/10.1177/1053451217712972
Ana Stephens, Fonger, N., Maria Blanton, & Eric Knuth. (2016). Functional Thinking Learning Progression Stephens, Fonger, Blanton, Knuth. Annual Meeting of the American Educational Research Association.
Blanton, M., Brizuela, B. M., Gardiner, A. M., Sawrey, K., Newman-owens, A., Blanton, M., Gardiner, A. M., & Newman-owens, A. (2016). A Learning Trajectory in 6-Year-Olds’ Thinking About Generalizing Functional Relationships. Journal for Research in Mathematics Education. 46(5), 511–558. https://doi.org/10.5951/jresematheduc.46.5.0511 DOI: https://doi.org/10.5951/jresematheduc.46.5.0511
Blanton, M. L., & Kaput, J. J. (2004). Elementary Grades Students’ Capacity For Functional Thinking. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education. 2. 135-142.
Blanton, M. L., & Kaput, J. J. (2005). Helping Elementary Teachers Build Mathematical Generality into Curriculum and Instruction. ZDM.37(1). https://doi.org/10.1007/BF02655895 DOI: https://doi.org/10.1007/BF02655895
Blanton, M., Stephens, A., Knuth, E., Gardiner, A. M., Isler, I., Kim, J., Blanton, M., Knuth, E., Gardiner, A. M., & Kim, J. (2015). The Development of Children ’ s Algebraic Thinking : The Impact of a Comprehensive Early Algebra Intervention in Third Grade.Journal for Research in Mathematics Education. 46(1), 39-87. https://doi.org/10.5951/jresematheduc.46.1.0039 DOI: https://doi.org/10.5951/jresematheduc.46.1.0039
Brizuela, Bárbara M. (2015). Children’s Use of Variables and Variables Notation to Represent Their Algebraic Ideas. Mathematical Thinking and Learning, 1–30. https://doi.org/10.1080/10986065.2015.981939 DOI: https://doi.org/10.1080/10986065.2015.981939
Canadas, M. C., & Castro, E. (2007). A Proposal of Categorisation for Analysing Inductive Reasoning. PNA, 1(2), 67–78.
Cañadas, M. C., Deulofeu, J., Figueiras, L., Reid, D. A., & Yevdokimov, O. (2007). The Conjecturing Process: Perspectives in Theory and Implications in Practice. Journal of Teaching and Learning, 5(1), 55–72. https://doi.org/10.22329/jtl.v5i1.82 DOI: https://doi.org/10.22329/jtl.v5i1.82
Doorman, M., Drijvers, P., Gravemeijer, K., Boon, P., & Reed, H. (2012). Tool use and Development of the Function Concept : from Repeated Calculations to Functional Thinking. International Journal of Science and Mathematics Education. 1243–1267. https://doi.org/10.1007/s10763-012-9329-0 DOI: https://doi.org/10.1007/s10763-012-9329-0
Dubinsky, E. (2001). Using a Theory of Learning in College. TaLUM, 12, 10–15. https://doi.org/10.11120/msor.2001.01020010 DOI: https://doi.org/10.11120/msor.2001.01020010
Dubinsky, E., & Michael A. McDonald. (2008). APOS: A Constructivist Theory of Learning in Undergraduate Mathematics Education Research. Animal Genetics, 39(5), 561–563.
Dubinsky, E., Weller, K., McDonald, M. A., & Brown, A. (2005). Some Historical Issues and Paradoxes Regarding .
the Concept of Infinity: An APOS Analysis: Part 2. Educational Studies in Mathematics, 60(2), 253–266. https://doi.org/10.1007/s10649-005-0473-0 DOI: https://doi.org/10.1007/s10649-005-0473-0
Markworth, K. A. (2010). Growing and Growing: Promoting Functional Thinking With Geometric Growing Patterns.A Dissertation Submitted to the Faculty of the University of North Carolina at Chapel Hill in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the School of Education. Chapel Hill.
Mceldoon, K. L., and Rittle-Johnson. (2010). Assessing Elementary Students Functional Thinking Skills.The Case of Function Tables.
Muir, T. &Livy, S. (2015). Two of Everything Developing Functional Thinking in Primary Grades Through Children’s Literature. In APMC. Vol. 20 (1).
NCTM. (2000). Principles and Standards for School Mathematics. Reston: The National Council of Teachers of Mathematics.
Pinto, E., &Cañadas, M. C. (2012). Functional Thinking and Generalisation in Third Year of Primary School. Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education, At Institute of Education, Dublin City University. 1–8.
Polya. (1973). How to Solve it. 2nd ed. Princeton: Princeton University Press. ISBN 0-691-08097-6.
Reid, D. A., & Jniversitv, A. (2002). Conjectures and Refutations In Grade 5 Mathematics. Journal for Research in Mathematics Education, 33 (1), 5-29. https://doi.org/10.2307/749867 DOI: https://doi.org/10.2307/749867
Stephens, A., Blanton, M., Strachota, S., Knuth, E., & Gardiner, A. (2017). The Interplay between Students’ Understandings of Proportional and Functional Relationships. Proceedings of the 39th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Indianapolis, IN Hoosier Association of Mathematics Teacher Educators. 251–258.
Stephens, A. C., Blanton, M. L., Knuth, E. J., Marum, T., & Gardiner, A. M. (2011). From Recursive Pattern To Correspondence Rule: Developing Students’ Abilities To Engage in Functional Thinking. Proceedings of the 34th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Kalamazoo, MI: Western Michigan University.
Stephens, A. C., Fonger, N., Strachota, S., Isler, I., Blanton, M., Knuth, E., & Murphy Gardiner, A. (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 DOI: https://doi.org/10.1080/10986065.2017.1328636
Sutarto, Nusantara, T., Subanji, Sisworo. (2016). Local Conjecturing Process in the Solving of Pattern Generalization Problem. Educational Research and Review, 11(8), 732–742.
Tanişli, D. (2011). Functional Thinking Ways in Relation to Linear Function Tables of Elementary School Students. Journal of Mathematical Behavior, 30(3), 206–223. https://doi.org/10.1016/j.jmathb.2011.08.001 DOI: https://doi.org/10.1016/j.jmathb.2011.08.001
Warren, E. A.,and Cooper. (2012). Exploring Young Students Functional Thinking. Proceeding of the 35th Conference of the International Group for the Psychology of Mathematics Education. 4. 75–84.
Warren, E. A., Cooper, T. J., & Lamb, J. T. (2006). Investigating Functional Thinking in the Elementary Classroom : Foundations of Early Algebraic Reasoning. Mathematical Behavior. 25, 208–223. https://doi.org/10.1016/j.jmathb.2006.09.006 DOI: https://doi.org/10.1016/j.jmathb.2006.09.006
Warren, E., &Cooper, T. O. M. (2005). Introducing Functional Thinking in Year 2: a Case Study of Early Algebra Teaching. In Contemporary Issues in Early Childhood. 6(2), 150–162. https://doi.org/10.2304/ciec.2005.6.2.5 DOI: https://doi.org/10.2304/ciec.2005.6.2.5
Wilkie, K. J. (2004). Learning to like Algebra through Looking. APMC. 19(4).
Wilkie, K. J. (2014). Upper Primary School Teachers’ Mathematical Knowledge for Teaching Functional Thinking in Algebra.Journal of Mathematics Teacher Education. 17(5). https://doi.org/10.1007/s10857-013-9251-6 DOI: https://doi.org/10.1007/s10857-013-9251-6
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 DOI: https://doi.org/10.1007/s13394-015-0151-1
Wilkie, K. J., &Clarke, D. M. (2015). Developing Students’ Functional Thinking in Algebra Through Different Visualisations of a Growing Pattern’s Structure. Mathematics Education Research Journal, 223–243. https://doi.org/10.1007/s13394-015-0146-y DOI: https://doi.org/10.1007/s13394-015-0146-y
Yuniati, S., Nusantara, T., Subanji, & Made Sulandra, I. (2019). The use of Multiple Representation in Functional Thinking. International Journal of Recent Technology and Engineering, 8(1C2), 672–678.
Yuniati, S. Nusantara, T. Subanji,Sulandra, I. M.(2018). The Process of Discovering Student’s Conjecture in Algebra Problem Solving.International Journal of Insights for Mathematics Teaching. 1(1), 35-43.