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Subtraction can be understood by two basic models-taking away (ta) and determining the difference (dd)-and by its inverse relation to addition. Epistemological analyses and empirical examples show that the two models are not relevant only in single-digit arithmetic. As curricula should be developed in a longitudinal perspective on mathematics learning processes, the article highlights some exemplary steps in which the inverse relation is discussed in light of the two models, namely mental subtraction, the standard algorithms for subtraction, negative numbers and manipulations for solving algebraic equations. For each step, the article presents educational considerations for fostering a flexible use of the two models as well as of the inverse relation between subtraction and addition. In each section, a mathedidactical analysis is conducted, empirical results from literature as well as from our own case studies are presented and consequences for teaching are sketched. Two models of subtractionMany adults as well as school children understand subtraction solely as taking away. In this paper, we shall show the importance of the second model of subtraction (determining the difference) and the relevance of the inverse relation between addition and subtraction by adopting a longitudinal perspective. 1 Beforehand, some remarks are necessary with respect to the notions that we use. 1 Note that we do not present an empirical longitudinal study where we followed a set of students or a programme over a longer period of time. We adopt a longitudinal perspective for conducting a mathedidactical analysis (van den Heuvel-Panhuizen and Treffers, 2009).C. Selter (*) : S. Prediger : M. Nührenbörger : S. Hußmann

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