Greet Peters

Solving subtraction problems flexibly by means of indirect addition

Background. Subtraction problems of the type a2 b 1⁄4 ? can be flexibly solved by various strategies, including the indirect addition (IA) strategy ‘howmuch do I have to add to b to get a?’ Although rational task analyses indicate that IA is highly efficient especially on subtractions with small differences between the integers, little research has been done on the frequency and efficiency of this strategy on different types of subtractions.

Subtraction by Addition Strategy Use in Children of Varying Mathematical Achievement Level: A Choice/No-Choice Study

We investigated the use of the subtraction by addition strategy, an important mental calculation strategy in children with different levels of mathematics achievement. In doing so we relied on Siegler’s cognitive psychological model of strategy change (Lemaire & Siegler, 1995), which defines strategy competencies in terms of four parameters (strategy repertoire, distribution, efficiency and selection), and the choice/no-choice method (Siegler & Lemaire, 1997), which is essentially characterized by offering items in two types of conditions (choice and no-choice). Participants were 63 11-12-year-olds with varied mathematics achievement levels. They solved multi-digit subtraction problems in the number domain up to 1,000 in one choice condition (choice between direct subtraction or subtraction by addition on each item) and two no-choice conditions (obligatory use of either direct subtraction or subtraction by addition on all items). We distinguished between two types of subtraction problems: problems with a small versus large difference between minuend and subtrahend. Although mathematics instruction only focused on applying direct subtraction, most children reported using subtraction by addition in the choice condition. Subtraction by addition was applied frequently and efficiently, particularly on small-difference problems. Children flexibly fitted their strategy choices to both numerical item characteristics and individual strategy speed characteristics. There were no differences in strategy use between the different mathematical achievement groups. These findings add to our theoretical understanding of children’s strategy acquisition and challenge current mathematics instruction practices that focus on direct subtraction for children of all levels of mathematics achievement.

Indirect Addition: Theoretical, Methodological and Educational Considerations

The development of fundamentally important arithmetic principles related to the four basic operations, and of arithmetic strategies that are based on these principles, is an intriguing and important element of psychological, mathematical and math educational research. As far as addition and subtraction are concerned, we have, for instance, the following principles: (a) the commutativity principle, which says that the order of the addends is irrelevant to their sum (a + b = b + a); (b) the principle prescribing that if nothing is added to or removed from a collection its cardinal value remains unchanged (a + 0 = 0; a 0 = a); (c) the principle that adding an amount to a collection can be undone by subtracting the same amount and vice versa (a + b b = a or a b + b = a); and (d) the principle that if a + b = c, then c-b = a or c a = b. Previous theorizing and research shows that understanding these principles plays an important role in children’s construction of the additive composition of number and in additive reasoning. Moreover, the implicit or explicit application of these principles can also considerably facilitate people’s arithmetic performance by eliminating computational effort and increasing solution efficiency (Baroody, Torbeyns, & Verschaffel, 2009). For example, the first principle underlies the well-known computation shortcut for solving additions starting with the smaller given number (like 2 + 9 or 4 + 58), that consists of reversing the order of operands and adding the smaller addend to the larger one. The fourth principle underlies the computation shortcut for solving subtractions involving a small difference between the two integers (like 11 9 or 61 59), by determining how much has to be added to the smaller integer to make the larger one. Whereas the first three abovementioned principles and their accompanying computational shortcut strategies have already received a great amount of research attention (Verschaffel, Greer, & De Corte, 2007), the fourth principle has not. In this contribution, we will present a series of closely related studies in the domain of elementary subtraction that we have done so far on this fourth principle and its accompanying computational shortcut, namely indirect addition (IA). We will use the term direct subtraction (DS) for the more common straightforward strategy for doing subtraction whereby the smaller number is directly taken away from the smaller one.