Training in mathematics has conceptual and procedural components. Before the fourth grade, teachers concentrate on the procedural aspects of addition, subtraction, multiplication and division. After children learn that addition and multiplication involve acts of combining objects, while subtraction and division involve acts of separating objects, they master the four operations by memorizing and repeatedly reciting problems in which at least two of the three numbers in the operation are whole numbers from one to nine. These problems are called "fast math facts."

#### Addition

Children learn to quickly calculate sums up to five =in kindergarten, sums up to 10 in first grade, and sums up to 20 in second grade. Consequently, children should be able to add any two whole numbers from one to nine by the end of the second grade. Children initially do these calculations by counting and combining objects. They also use techniques such as "counting up" in which they start with a large number and count up the smaller number. For example, to add three to eight, a child starts with eight and counts up three numbers. Another useful technique is decomposing a large number into smaller ones. For example, to add five and eight, a child might add 5 + 4 + 4 = 13.

#### Subtraction

Children learn to quickly subtract single digits from five in kindergarten, subtract single digits from 10 in first grade, and subtract single digits from 20 in second grade. They learn to subtract by reversing the steps involved in addition. Objects that were put onto a pile during addition are taken away during subtraction. Children also learn how to check their subtraction of numbers such as, 7 - 4 = 3, by adding 3 + 4 = 7.

#### Multiplication

By the end of the third grade, children should be able to multiply in their heads any two whole numbers from one to nine. Third-graders also learn to quickly multiply multiples of 10 (10, 20 … 90) by whole numbers from one to nine. Children develop their understanding of multiplication by working with equal-size groups of objects. For example, they might learn that 5 x 3 = 15, by counting out three groups of five objects, combining the groups into one pile, and then counting the number of objects in the combined pile.

#### Division

By the end of the third grade, children should be able to quickly divide a whole number from one to nine into a two-digit number where the result also is a whole number from one to nine. Children develop their understanding of division by reversing the steps involved in multiplication. For example, they might learn that 15 / 3 = 5, by distributing the 15 objects one at a time into three groups, and then counting the number of objects in each of the smaller groups. Children also should learn how to check their division of numbers, such as 15 / 3 = 5, by multiplying 5 x 3 = 15.

#### Online Games

Children can improve their ability to do math problems quickly by playing online games that require a particular skill. Many such games are available, and some of them let children choose the operation to practice and their grade level (see References).