Pre-algebraic aspects in arithmetic strategies – The generalization and conceptual understanding of the ‘Auxiliary Task’

Abstract

In the last decades, a broad international reform approach was visible in support of mental calculation strategies: Instead of being solely ‘transition strategies’ for learning the standard algorithms, the understanding of numerical relations is essential for mental calculation strategies, making them highly important for a viable understanding of arithmetics. Yet, mental calculation strategies are not only important for understanding arithmetics, but highly relational strategies such as the ‘Auxiliary Task’ might have an important role in the emergence of a pre-algebraic understanding of numerical relations. In this qualitative study from Germany, 4<sup>th</sup> and 5<sup>th</sup> grade learners’ (n=18) processes of interpreting the ‘Auxiliary Task’ are examined by conducting linguistic and epistemological analyzes of their conceptual understanding of the ‘Auxiliary Task’ utilizing a design-based research framework. Insights are given into specific, language-related forms of pre-algebraic generalizations of the ‘Auxiliary Task’ as well as into developmental processes within the designed learning-environment.