A Hierarchy of Strategies for Solving Linear Equations

This paper describes an investigation of students’ strategies for solving linear equations. The assessment techniques used for investigating algebra paralleled those used for investigating number in the New Zealand Numeracy Development Projects (NDPs). In this study of 621 Year 7 to Year 10 students oral interviews were used to investigate the strategies that students used to solve equations. Rasch analysis was used to investigate item difficulty and student ability, and then the strategies associated with each question were examined. The data suggest that there is a hierarchy of sophistication of strategies. Many students were unable to solve a lot of the equations as they were restricted to less sophisticated strategies. The most sophisticated strategy of solving equations by performing transformations was understood by very few students. A new curriculum for New Zealand schools was introduced in 2007 (Ministry of Education, 2007). The learning area of Mathematics and Statistics is now divided into three strands rather than the previous six, with Number and Algebra being one strand. The implementation of the curriculum in Number up to Year 10 of schooling is guided by the Numeracy Development Projects (NDPs), which provide a framework for children’s development in number. The achievement objectives of the new curriculum are grouped to reflect the structure of the Number Framework (Ministry of Education, 2003), which details the number strategies that students use and the number knowledge required for these strategies. At the lower levels of the new curriculum the Number and Algebra achievement objectives are divided up into number strategies, number knowledge, equations and expressions, and patterns and relationships. The integration of number and algebra into one strand follows debate within the mathematics education community in New Zealand and within international research (see for example Carraher & Schiemann (2007) and Kieran (1992)) as to what constitutes algebraic thinking. The NDPs have been successful at raising the achievement of New Zealand children in the strand of Number (G. Thomas & Tagg, 2007) and various initiatives are currently underway to extend the projects into early algebra. This study takes a similar approach to the NDPs by specifically examining students’ strategies for solving linear equations through the use of oral interviews. Children’s Strategies Students have struggled with introductory algebra for a long time (Cockcroft, 1982) and teachers have little to guide them in designing programmes of learning. Little is known about the strategies that students use to solve equations or how these strategies are related to conceptual development. A useful summary of strategies used by students is, however, provided by Kieran (1992), who describes the use of known basic facts, counting techniques, guess and check, cover-up, working backwards and formal operations. The difficulties that students experience, related to their use of algebraic strategies, are well documented. Because much arithmetic in schools is presented as a computation ready to complete, e.g. , and because pressing the equals button on a calculator performs a calculation on whatever has been entered, children understand equals as meaning compute now rather than is equivalent to (Booker, 1987; Booth, 1988). Linchevski (1995)