Encountering algebraic letters, expressions and equations: A study of small group discussions in a Grade 6 classroom

Abstract

Introductory algebra has a pivotal role for pupils’ continued learning in algebra. The aim of this licentiate thesis is to contribute to knowledge about how pupils appropriate introductory algebra and the kind of challenges they encounter. In this thesis, the term introductory algebra is used to refer to the introduction of formal algebra at compulsory school level – algebraic letters, algebraic expressions and equations. The work is based on two research articles: What’s there in an n? Investigating contextual resources in small group discussions concerning an algebraic expression and Moving in and out of contexts in collaborative reasoning about equations. The studies are positioned in a socio-cultural tradition, which implies a focus on pupils’ collaborative meaning making. A dialogical approach was applied when analyzing the pupils’ communication. Two case studies have been conducted in class-rooms, both consisting of video recorded small group discussions between 12-year-old pupils working with algebraic tasks. The first study shows how pupils tried to interpret the algebraic letter n and provide an answer formulated as an algebraic expression. The results show that the pupils used a rich variety of contextual resources, both mathematical and non-mathematical, when trying to understand the role of the n. In addition, the meaning of the linguistic convention “expressed in n” was a barrier for the pupils. In the second study the pupils used their experiences of manipulatives (boxes and beans, which had been used during prior lessons), as a resource when solving a task formulated as an equation expressed in a word problem. The study shows that the manipulatives supported the pupils in working out the equation, but did not help them to solve the task. Three general conclusions can be drawn from the empirical studies. Firstly, the interpretations of an algebraic letter can be dynamic and the nature of the meaning making may shift quickly depending on the contextual resources invoked, indicating that an interpretation is not a static, acquired piece of knowledge, but more like a network of associations. Secondly, mathematical conventions may work as obstacles to the pupils’ understanding. This indicates that learning mathematics is about learning a specific communicative genre in addition to learning about mathematical objects and relationships. Thirdly, the studies show that a critical part of appropriating introductory algebra is being aware of ‘what is the example’ and ‘what is general’ in different activities. A conclusion is that although pupils are able to mobilize resources that are helpful for managing specific cases, additional problems may arise when they try to comprehend fundamental algebraic principles.