Danish preservice teachers’ solving of linear equations: A challenge for teacher education

Solving linear equations, which requires a relational understanding of the
equals sign, predicts later mathematical learning. The Danish national curriculum, despite expectations of equation-solving competence, privileges flexible approaches to mathematical problem solving above the learning of standard algorithms, goals reflected in the regulations governing teacher education. This paper presents an evaluation of the linear equations-related competence of 50 first year preservice teachers. Undertaken at the start of their programme, the assessment comprised four tasks, each designed to offer the possibility of shortcut solutions – solutions that involve fewer steps and are
executed faster than standard algorithms. The first two included only integer
coefficients, while the final two incorporated a mixture of integers, fractions or decimals.
Success rates on the equations with integer coefficients were relatively high at 83% and 76% respectively, while the rates for those with mixed coefficients were significantly lower at 48% and 54% respectively. Procedurally, most solutions involved the application of a standard algorithm (56%), beginning with the expansion of brackets, before strategies based on, in order of popularity, a swap the side swap the sign principle, a do the same to both sides principle, or both. Shortcut opportunities were acknowledged more frequently on the mixed coefficient equations than on the integer, highlighting the appeal of standard algorithms for solving seemingly familiar equations and the need for reflection on those that appeared unfamiliar. Failures, despite most solutions exhibiting a relational understanding of the equals sign, were typically due to inadequate understandings of the distributive law, the arithmetic of signed numbers, transposition of terms, or division of two integers when the divisor was larger than the dividend or, if smaller, not a factor of it. Overall, the results highlight the extent to which Danish preservice teachers are ill-prepared for a programme that assumes a base level of competence.