Mental Calculation Strategies as a ‘Missing Link’ between Arithmetic and Algebra – Insights into the 'Auxiliary Task' and its Role in the ‘Cognitive Gap’

Abstract

Mental calculation strategies are important arithmetical operations in primary school mathematics lessons, but a pro­ceptual understanding of highly relational strategies such as the ‘Auxiliary Task’ might also play an important role in the emer­gence of pre-algebraic thinking processes as prior insights from this study show. Following upon these indications, n = 18 learners’ proceptual understanding of the mental calculation strategy ‘Auxiliary Task’ is analysed in this article with a Design-based Research study utilizing interpretative analyses of learning processes. The analyses show that the fostering of a pro­cep­tual under­standing of the ‘Auxiliary Task’ might lead to a) pre-algebraic generalizations of the numbers and a creative ‘ana­ly­tical noticing’ and b) specific ‘Grundvorstellungen’ – thus multi-facetted mental models – might emer­ge. Especially two ‘Grundvorstellungen’ seem to be of high relevance in such a pre-algebraic thinking of the ‘Auxiliary Task’: The understanding of mathematical structures behind the ‘Auxiliary Task’, which is the ‘compen­sation strategy’, as well as a notion of numbers-as-indeterminate. The latter is an important aspect in the transition from primary to secondary school since it precedes an understanding of variables-as-indeterminate, ‘bridging’ an arithmetic and algebraic understanding of numbers.