Commentary 2: knowledge integration in mathematics learning: the case of inversion

Abstract

In their most common form, inversion problems have the form of a+b−b0?. The correct answer in this case is a. It can be derived without computation because addition and subtraction of the same number cancel each other out. This solution method, that is, finding the answer without computation, is called the inversion-shortcut strategy. Many researchers use inversion problems, related principles, and related problem types to investigate understanding by the learners of the inverse relation between mathematical operations. In the following, I use the terms “inversion” and “area of inversion” to refer to the principles, strategies, problem types etc. that are related to the concept of inversion. Researchers frequently emphasize the importance of learning about inversion. However, they rarely explain why inversion is important, even though this is hardly self-evident. There are at least three reasons to doubt the importance of learning about inversion. First, empirical studies show that it is possible to be good at arithmetic problem solving without having a good understanding of inversion and vice versa (Bryant, Christie, & Rendu, 1999; Gilmore & Papadatou-Pastou, 2009). To my knowledge, there is not a single experiment that shows a direct positive causal effect of an understanding of inversion on arithmetic or other mathematical skills (cf. Schneider & Stern, 2009). Second, inversion problems rarely occur in everyday life. The reader of this article might try to remember when he or she last encountered a problem of the form a+b−b (with arbitrary numbers for a and b) in their life. Most people have troubles coming up with even just one or two examples. Third, learning about the inversion shortcut might be an inefficient use of learning time. Based on the existing studies, it can be estimated that using the inversion shortcut instead of left-to-right computation saves roughly one second, at least if relatively small and easy numbers are involved. Consequently, if a teacher used only 90 min on inversion instruction, a learner would have to solve 90×6005,400 inversion problems by using the shortcut strategy before he gained more time by using the shortcut than he invested by learning about inversion. Educ Stud Math DOI 10.1007/s10649-011-9373-7