Measure 27 — MATH 2

Number Sense: Math Operations

Child shows increasing ability to add and subtract small quantities of objects

The ability to reason about arithmetic operations and to solve simple addition and subtraction problems with objects develops gradually during infancy and early childhood. Knowledge about adding and subtracting builds on the early experiences children have with manipulating groups of objects to change amounts and on their later interest in manipulating and thinking about number. Young children’s earliest notions of adding and subtracting tend to be inexact (adding makes more; subtracting makes less), except with very small numbers. Very young children can begin to calculate exact sums and differences with one, two, or three objects. As they gain experience and knowledge across the preschool years, children can learn to apply their number skills, including counting, to adding and subtracting larger numbers.

Young infants begin to construct ideas about objects and quantity as they attend to and interact with objects in their environment. In their play, older infants put objects together into groups and change the groups by adding and removing objects. For example, an infant might put several rings on a stacking cone and then remove and replace them, one at a time, over and over again. These early manipulating activities provide repeated experiences in seeing how amount and number are changed by additions and subtractions.

Toddlers are increasingly aware of and interested in number and amount. Quantity is an important feature of their world, and they actively and purposefully manipulate amounts of things. For example, a child might gather up as many train cars as he can hold in order to make a long train, or a child might ask for another cracker so she can have one for each hand. Toddlers understand addition and subtraction to the extent that they know that adding objects increases the number in a group and subtracting objects decreases the number. They add and take away objects to make more or fewer and may begin to talk about number and quantity. For example, a child might say, “All gone. I get more,” as she takes several more raisins from a serving bowl after eating those on her plate. Experiences manipulating very small numbers of objects (up to three) may be particularly important to children, since the process readily allows them to see how the number of objects has changed.

Young preschoolers are typically able to calculate simple additions and subtractions with two or three objects. They generally do this without counting. For example, if an adult shows the child two pennies in her hand, then covers the pennies and adds one more, most young preschoolers can say that she is now holding three pennies. Some young preschoolers begin to solve slightly larger number problems with four or more objects by counting, especially if they are encouraged to do so. Importantly, even when young children are able to count, they may not apply these skills spontaneously to solving problems.

Children’s arithmetic abilities undergo considerable development at around four or five years of age. Many older preschoolers are learning to solve larger addition and subtraction problems with five to ten objects, usually by counting. Even when they are not accurate, children’s solutions tend to be in the correct direction of the operation (adding results in a larger number; subtracting results in a smaller number). Accuracy is greatly improved when children learn to systematically count the objects to get the result. Older preschoolers may also begin to solve simple, everyday verbal problems involving imaginary objects, although these are more difficult than similar problems with real objects, particularly if they involve more than two or three elements. With experience, older preschoolers can readily solve problems in which only one object is added or subtracted. Adding or subtracting one is easier than the same operation with two or more, since it only requires going up one or going down one in the counting sequence. Accuracy on verbal problems improves, however, when older preschoolers can begin to use a strategy, such as representing the problem with fingers or objects and counting to get the solution. These abilities continue to develop in the early years of elementary school.