Taking away and determining the difference—a longitudinal perspective on two models of subtraction and the inverse relation to addition

Abstract

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 mathe-didactical analysis is conducted, empirical results from literature as well as from our own case studies are presented and consequences for teaching are sketched.

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Notes

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 mathe-didactical analysis (van den Heuvel-Panhuizen and Treffers, 2009).

2. For the sake of clarity, we did not include the problem 3 + 4 here as well. For the same reason, we restrict ourselves to these different syntactic structures and do not primarily deal with numerical or semantic structures (Carpenter and Moser, 1982; De Corte and Verschaffel, 1987; Fuson, 1992), which would have made an analysis even more complex.

3. Both students attended a qualitative field study conducted by one of the authors.

4. This distinction is made, although quite some children might proceed entirely mechanically, using basic facts without any understanding. However, at least in Germany, it is a goal of mathematics teaching that children also understand the algorithm they apply.

5. Note, that this method—at least as it is taught in Austria and Germany—does not use decomposition, where the amount “borrowed” from a given order of the minuend is added to the corresponding order of the subtrahend using the equivalence between the differences (x − 1) − y and x − (y + 1), like Fiori and Zuccheri (2005, pp. 324–325) claim. Instead, the law $$x-y = \left( {x + a} \right)-\left( {y + a} \right)$$ is used, namely by adding ten ones to the minuend and compensating this by adding one ten to the subtrahend.

6. The data are taken from a qualitative field study by one of the authors (Hußmann, 2010, p. 1).

7. And analogically the inverse relation between multiplication and division and their models, cf. Prediger (2008).

8. Analogically, the inverse relation of multiplication and division is the foundation for the second transposing rule: $$A \times B = C\, \Leftrightarrow C\, \div B = A\,({\hbox{and}}\,{\hbox{analogically}}\,A\, \times \,B = C \Leftrightarrow C \div A = B).$$

9. And accordingly: $$A{ } = { }B \Leftrightarrow A{ } \times { }C = B \times C$$

10. The balance model only applies for positive numbers and values of the variable, not for equations like $${4}x + {15} = - {3}$$ (Malle, 1993, p. 224).

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Selter, C., Prediger, S., Nührenbörger, M. et al. Taking away and determining the difference—a longitudinal perspective on two models of subtraction and the inverse relation to addition. Educ Stud Math 79, 389–408 (2012). https://doi.org/10.1007/s10649-011-9305-6