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

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.

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.

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, R3 schema 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 position

Identifying the mental constructs of students in generalizing patterns eases better teaching and learning.

By identifying the mental structures of students in generalizing patterns, the path of teaching and learning will be smoother.