Young children learn through a process of constructing their own knowledge of the world around them through interaction with people and things. In math, this means that developing an understanding of concrete operations is much more important than simply memorizing facts.1 For the purposes of this article, I want to concentrate on giving you some practical ideas and tips for working with children in the area of math up through about age 9 or 10.

I personally don’t encourage starting textbook work in math until about grade five. Up to that time, I think children need a lot of hands-on experiences in math-related concepts and the building of math vocabulary.

In the primary grades, children typically learn about measurement, time, and money. These can all be best done with real objects. Measurement can be learned while cooking, baking, gardening (measure between the rows and between plants), and in the workshop. Time can best be learned with a real watch. Money can best be learned through having a small allowance that is not tied to performance, plus a little chance to earn some extra money for working (extra chores) around the house. When children have a little money, they need to learn early how to save, tithe, and budget. Of course, at this stage parents need to walk a fine line between expecting too much and expecting nothing at all.

Even young children can begin to develop a sense of numbers through things like setting the table for four people and then adding two more place settings when Grandma and Grandpa come to call. Egg cartons can be used for learning sorting skills, perhaps in connection with science collections — black rocks, brown rocks, broken seashells, whole seashells, etc.

As the children get a little older, somewhere between kindergarten and second-grade level, they can begin to learn adding, subtracting, and even multiplying and dividing, by using concrete materials. My favorite manipulatives are Cheerios, M&Ms, and Popsicle sticks.

It is best to teach children by using groups of operations. For example, start with six Cheerios. As much as possible, let the students discover things rather than just showing them everything right away. For example, you might ask, “How many different ways can you combine (or divide up) these Cheerios into different groups?”

Take a group of operations, like 6 plus 2 equals 8, 2 plus 6 equals 8, 8 minus 2 equals 6, and 8 minus 6 equals 2. After they have had some time to play with concepts themselves, show them how you can rearrange the Cheerios to demonstrate these facts.

For multiplication and division, either draw chalk circles on the ground to create groups, or make circles out of string. Then show them that 2 times 3 means “two groups of three,” and if you just rearrange them, the same number of Cheerios can be “three groups of two” ... and you’ve just taught them the commutative property of multiplication! (Just don’t call it that yet!) Two times 3 gives the same answer as 3 times 2! Then show them that the same group of Cheerios can be rearranged to show the corresponding division problems ... 6 divided into three groups gives you 2 Cheerios per group; 6 divided into two groups gives you 3 Cheerios per group.

To teach carrying and borrowing, take a large piece of paper (we used end rolls from the newspaper office). Make three large columns for ones, tens, and hundreds places. Then use Popsicle sticks to teach the concepts. Be sure to have some rubber bands handy. Take a problem like 34 plus 57. First put three bundles of ten in the tens columns and four individual Popsicle sticks in the ones column ... and you have 34 Popsicle sticks. Then tell them to put 7 more in the ones column. At this point there will be more than nine in the ones column, so point that out and tell them that’s too many and they have to “get rid of some of them ... how?”