Children’s understanding of the relationship between addition and subtraction

Abstract

In learning mathematics, children must master fundamental logical relationships, including the inverse relationship between addition and subtraction. At the start of elementary school, children lack generalized understanding of this relationship in the context of exact arithmetic problems: they fail to judge, for example, that 12 + 9 − 9 yields 12. Here, we investigate whether preschool children's approximate number knowledge nevertheless supports understanding of this relationship. Five-yearold children were more accurate on approximate large-number arithmetic problems that involved an inverse transformation than those that did not, when problems were presented in either non-symbolic or symbolic form. In contrast they showed no advantage for problems involving an inverse transformation when exact arithmetic was involved. Prior to formal schooling, children therefore show generalized understanding of at least one logical principle of arithmetic. The teaching of mathematics may be enhanced by building on this understanding.To a large degree, mathematics is the discovery and use of general, abstract principles that make hard problems easy. The inverse relationship between addition and subtraction is a case in point. Problems of the form x + y − z = ? are intractable for those who lack knowledge of specific arithmetic facts (e.g., what is x + y?), and they require two successive calculations for those who possess the relevant knowledge. In contrast, problems of the form x + y − y = ? can immediately be solved, without arithmetic fact knowledge or calculation, by anyone who understands the logical relationship between addition and subtraction. The present research explores the origins of this understanding in children on the threshold of formal instruction in arithmetic.Previous research suggests that children's understanding of this relationship develops over many years of instruction in elementary mathematics. Children who have received arithmetic instruction perform more accurately on inverse problems of the form x + y − y than on matched problems of the form x + y − z (e.g. Bisanz & LeFevre, 1990;Bryant, Christie & Rendu, 1999;Gilmore & Bryant, 2006, Gilmore 2006Rasmussen, Ho & Bisanz, 2003;Siegler & Stern, 1998;Stern 1992), but they appear to learn about this principle in a piecemeal fashion. For example, children may recognize that subtracting 4 cancels the operation of adding 4, but they fail to recognize inversion as a general principle that can be applied to all numbers (Bisanz & LeFevre, 1990). Furthermore, these studies all involved children who were already receiving formal instruction in arithmetic, and thus the roots of this understanding are unclear. Publisher's Disclaimer: This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final citable form. Please note that ...

Children’s understanding of addition and subtraction concepts

“…Studies on preschoolers have shown that formal instruction is not necessary to understand the inversion concept, although not all children demonstrate understanding (e.g., Baroody & Lai, 2007;Gilmore & Spelke, 2008;Klein & Bisanz, 2000;Sherman & Bisanz, 2007). A number of studies have also examined school-aged children's understanding of the concept (e.g., Bryant, Christie, & Rendu, 1999;Gilmore, 2006;Gilmore & Bryant, 2006;Robinson et al, 2006;Siegler & Stern, 1998;Stern, 1992).…”

Patterns of Individual Differences in Conceptual Understanding and Arithmetical Skill: A Meta-Analysis

“…Recent evidence also suggests that a system of approximate non-symbolic representation of number may play a role in children's developing understanding of quantitative inversion. Five-year-old children can identify and use inverse relationships for large numbers with approximate symbolic or nonsymbolic (dot array) quantities before they can do so for exact symbolic quantities (Gilmore & Spelke, 2008).…”

Preschoolers’ nonsymbolic arithmetic with large sets: Is addition more accurate than subtraction?

“…Although several other studies demonstrate that children between 4 and 9 years of age perform better on inversion problems such as a + b À b than on control problems such as a + b À c, these have involved children who were already receiving formal instruction in arithmetic (Gilmore & Bryant, 2008;Rasmussen, Ho, & Bisanz, 2003). More recently, 5½-year-old preschoolers were shown to perform better on inverse problems than on control problems on approximate nonsymbolic tasks with large sets (Gilmore & Spelke, 2008), like those used in the current study, as well as on symbolic tasks with large sets. Together with the current results, this pattern suggests that children may gradually come to understand the conceptual nature of inversion with large sets between 3½ and 5½ years of age.…”

“…First, it contributes to scientific understanding about the nature of the number sense that humans share with other animals before humans' ability to learn symbolic mathematics sets them apart. Second, it contributes to the literature on the development of knowledge about the mathematical principle of inversion-that addition and subtraction are inversely related and, thus, cancel each other out (e.g., Gilmore & Spelke, 2008). Finally, the answer to this question has educational implications.…”