From student difficulties to design of a hypothetical learning trajectory: The case of system of linear equations in two variables

This research aims to design a hypothetical learning trajectory for junior high school students in the case of systems of linear equations in two variables. The hypothetical learning trajectory is primarily designed according to the Realistic Mathematics Education (RME) theory to address students' difficulties. To do this, we carried out a two-stage qualitative case study. First, we investigated difficulties of 19 junior high school students (age 13-14 year-olds) when dealing with solving systems of equations in two variables and their application through an individual written test and interviews. Second, based on the first stage and the RME theory, we designed a hypothetical learning trajectory for the case of systems of linear equations in two variables. The results revealed that the main difficulties of the participants include, among others, transforming word problems into an appropriate mathematical model in the form of systems of linear equations of two variables, and making calculation or algebraic manipulation errors. Also, a hypothetical learning trajectory was designed, which consists of a learning goal, activities of the learning and teaching process, and predictions of student thinking and learning processes. In particular, we propose the use of the model method for overcoming difficulties in dealing with word problems. We conclude that the investigation of student difficulties provides fruitful information in the design process of hypothetical learning trajectory.