عنوان مقاله [English]
The figural patterns have a unique capacity to enhance functional thinking. The aim of this study is to identifying the prior schemas in generalization of two-variable figural patterns .The data collection method was quantitative and qualitative. The statistical population of the study were consisted of 493 students from Malekan township (East Azarbayjan) and 220 students were selected as sample. Using the self-concept analysis, background and experiences of researcher, initial genetic decomposition was developed. Then, a test of 7 task (46 question) was designed based on Action-Object-Process- Schema (APOS) framework. The validity of the test was confirmed by three experts in mathematics education and four experienced teachers. Internal coordination of questions was confirmed with Cronbach's alpha of 0.68. 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 volunteer students, genetic decomposition was finalized and defects of student in reference schema , R3 schema and variable schema as prior schemas in generalization of two-variable figural patterns were identified and encoded. Identifying the mental constructs of students in generalizing patterns, will make it easier to better teaching and learning.
 Asghari, N. (2014). Developing a model to enhance elementary teachers, ability to foster functional thinking and algebraic resoning in elementary students. Jornal of the theory and practice in curriculum, 2(3), 141-162.[in Persian].
 Huntzinger, E.M. (2008). Exploring generalization through pictorial growth patterns. In C. E. Greenes & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics (pp. 279-293). Reston, VA: NCTM.
 Rivera, F. & Becker, J. R. (2007 ). Abductions in pattern generalization .In Woo, J. H., Lew, H. C., Park, K. S. & Seo, D. Y. (Eds.). Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, 4, 97-104. Seoul: PME.
 National Center for TIMSS and PIRLS International Studies. (2013). 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. [in Persian].
 Chua, B.L, Hoyles, C. (2014). Modalites of rules and generalizing strategies of year 8 students for a quadratic pattern. In Nicol, C., Liljedahl, P., Oesterle, S., & Allan, D. (Eds.) Proceedings of the Joint Meeting of PME 38 and PME-NA 36 (pp. 305-312). Vancouver, Canada: PME.
 Shojaie, k. (2013). The investigation of ability of the nine grade students’ generalization and justification in numeric and geometric number patterns.( pablished master thesis). Shahid Rajaee Teacher Training University, Tehran. [InPersian].
 Samson, D.A. (2007). An Analysis of the Influence of Question Design on Pupils’ Approaches to Number Pattern Generalisation Tasks. (published Master thessis). Rhodes University, Grahamstown, South Africa.
 Chua, B.L, Hoyles, C. (2012). 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 (pp. 155-162). Taipei, Taiwan: PME.
 Wilkie, k. & Clarke, D. (2014). 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 37th annual conference of the Mathematics Education Research Group of Australasia, (pp.637–644).
 Arnon, I.Cottrill, J., Dubinsky, E., Oktaç, A., RoaFuentes, S., Trigueros, M., & Weller, K. (2014). APOS Theory: A framework for research and curriculum development in the mathematics education. New York: Springer-Verlag.
 Zohrevand, Sh. (2013). Students’ Misconceptions about Concept of Variable in Algebra (unpublished master thesis). Shahid Rajaee Teacher Training University: Tehran.[In Persian].
 Álvarez, I. Gómez-Chacón, I. Ursini, S. (2015). Understanding the algebraic variable: comparative study of Mexican and Spanish students. Eurasia Journal of Mathematics, Science & Technology Education, 11(6), 1507-1529.
 Reid, D. (2003). 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.
 Khosroshahi, Gh.L. (2016). Algebraization for Pre-schoolers with a focus on Associativity. (Unpublished Ph.D. dissertation). shahid beheshti university,Tehran. [in Persian].