Students’ Awareness of Algebraic Structure in Linear Equations

Abstract

This thesis investigates the teaching and learning of linear equations, with particular focus on how different ways of perceiving an equation and its components may influence students’ equation solving, especially when decimal or negative numbers are involved. The dissertation also more broadly examines how students visually attend to various aspects of equations, and how instruction can direct students’ attention toward aspects that are relevant to algebraic structure. Much of the previous research on the teaching and learning of linear equations has focused either on students’ conceptual challenges (such as understanding the concept of equality) or on difficulties related to equation solving. However, algebraic thinking involves perceiving and attending to equations differently than arithmetic, and little is known about how this way of perceiving can be supported through teaching. Therefore, the six papers included in this thesis examine how students perceive equations (Paper I–III), how their awareness of algebraic structure changes (Paper IV), and what types of tasks can support this learning (Paper V–VI). Together, the studies draw on data collected from four lower-secondary and six upper-secondary schools. The research has largely followed a mixed-methods approach with a practice-oriented focus, using interventions to explore how teaching can be designed to address the identified challenges. Variation theory serves as the theoretical foundation, conceptualising learning as a change in awareness, and patterns of variation have guided the instructional designs. Overall, the results indicate that students need support in recognising equations that share the same algebraic structure, since they are sometimes influenced by irrelevant aspects such as the types of numbers involved. To support this shift in attention, teaching activities involving the building and sorting of equations have been developed and investigated. The thesis also makes a theoretical contribution to research on algebra structure sense by emphasising that awareness of algebraic structure involves both discerning critical aspects of expressions and disregarding irrelevant aspects. Based on these findings, teachers can support students in perceiving linear equations in more powerful ways – by revealing their algebraic structure – thereby creating conditions for more efficient equation solving.