Integers: History, textbook approaches, and children's productive mathematical intuitions

Introduction The introduction of the notion of signed numbers is a critical point in children’s learning of mathematics. As with fractions, the advent of integers widens the domain of numbers in children’s mathematical worlds and, as such, presents notorious difficulties. On the one hand, some of the challenges that children experience in coming to grips with the notion of negative numbers parallel those that mathematicians encountered historically. On the other hand, the experience of most children today as they learn about integers is quite distinct from that of mathematicians of old. After all, at the time Diophantus was denying the possibility of negative solutions to linear equations—a position that was perfectly reasonable on the basis of his conceptions of number—he had no teacher or textbook telling him otherwise (Gallardo, 2002; Henley, 1999). We looked to the history of integers to guide our review of current textbook approaches to integer instruction. The juxtaposition of these two bodies of evidence raises questions about the utility of prevalent textbook approaches. At the same time, we have seen children in interviews engage productively with ideas concerning positive and negative numbers prior to formal instruction. These findings show potentially fruitful alternative directions for textbook instruction concerning integers.