Differences in Swedish and Norwegian pre-service teachers’ explanations of solutions of linear equations

Abstract

Solving linear equations is a cornerstone in the learning of algebra. There are two main strategies for solving a linear equation, ‘swap sides swap signs’ (SSSS) and ‘do the same to both sides’ (DSBS). While SSSS can often be more efficient for solving equations, DSBS has been shown to better promote the learning of algebra. Thus, the preference of SSSS or DSBS might depend on the purpose of solving equations. Since both approaches are common, mathematics teachers, and thus also pre-service teachers (PSTs), must be familiar with both SSSS and DSBS. This study draws on data from 161 Swedish and 146 Norwegian PSTs. They were given a correct but short and unannotated solution to the linear equation x + 5 = 4x − 1. The PSTs were invited to explain the provided solution for a fictive friend. Of the Norwegian PSTs, 2/3 explained the additive steps in the solution by SSSS, while only 1/3 of the Swedish PSTs applied SSSS. Consequently, DSBS was more frequent among the Swedish PSTs regarding the additive steps. However, in the final, multiplicative step, 3/4 of the Norwegian PSTs invoked DSBS. On the contrary, among the Swedish PSTs, the proportion applying DSBS for the multiplicative step decreased, and it was common to provide an incomplete explanation of the final operation. We also analysed how mathematics textbooks for secondary school presented how to solve linear equations. In Sweden, all textbooks utilised DSBS through the whole solution for all years in secondary school. This also applied for Norwegian textbooks for the first two years of lower secondary school. However, in last year of lower secondary school, they changed their approach and promoted an SSSS strategy in additive steps, while DSBS was still suggested for multiplicative steps. This might explain the differences between the two countries regarding the PSTs’ preferences of solution strategies. We suggest that these results can be useful for teacher education, since increased awareness of PSTs’ pre-knowledge is beneficial to support their development of teaching linear equations.