Imagine That

Physical fractions

Laura Mann

The main purpose of the Muppet character Count von Count is educating very young children on the simple mathematical concept of counting. When most kids first learn to count, they begin by counting objects they see in the physical world.

Perhaps you have pointed at a group of objects and asked your preschooler: How many? How many blocks; how many bananas; how many birds? The Count from Sesame Street helps children learn with this technique: He will count anything, anywhere, and anytime. The Count models how a child can look around and see how numbers function in the world around us.

After kids learn how to count, however, parents and teachers usually begin instructing kids in conceptual mathematics: Six minus three is presented using the symbols 6 – 3, rather than six objects with three taken away.

The switch from physical mathematics to conceptual mathematics is necessary. When learning higher forms of math, the thinking goes, kids must have an idea of how numbers behave that goes beyond counting blocks.

From concrete to concept

This move from the physical to the conceptual may be happening a little too soon, however. A study published last year by the Journal of Education and Human Development suggests that learning conceptual mathematics using physical objects is a useful tool beyond the initial counting, and can help older elementary-age children learn more advanced multiplication, fractions, and division.

In this study, researchers separated fourth graders, ages 9 and 10, into two groups. The first group was given small square tiles; the second was given pictures of the tiles, and scratch paper to write on.

This move from physical math to conceptual math may be happening a little too soon.

All the kids were taught fractions using these tools. Simple questions, such as “what is ¼ of 8?,” were posed to the kids before and after the instruction. In the post-tests, the fourth graders who were given tiles scored higher than kids given scratch paper and pictures.

Manipulating the physical objects helped kids understand grouping, which is a fundamental piece of knowledge in completing math problems that involve fractions. The kids with the tiles set up their eight tiles, grouped them into four equal groups, then counted the tiles in each group. Not only did the kids in the tile group get the right answers, they understood the problems better than the kids with just scratch paper and drawings.

Counting with fingers, kind of

Parents can take advantage of these findings by using physical manipulatives to help older elementary kids complete their math homework with a better grasp of how the problems work. If your child is having trouble understanding multiplication, division, and fractions, try introducing objects — blocks, Legos, popsicle sticks, toothpicks — into study time.

For example, if the question is 3×8, give your child a box of toothpicks, ask them to count out eight toothpicks, then a second group of eight, then a third group. Have your child count out how many toothpicks there are in all, which should lead to the correct answer.

If the question asks for ¼ of 24, count out 24 toothpicks, and have your child group them into four equal groups. Ask your child to count out the number of toothpicks in each group, making sure there are equal numbers of toothpicks in each of the four groups.

Using physical manipulatives isn’t the quickest way to understanding math, but it helps kids understand how numbers work in the physical world and connect this to the conceptual mathematics of numbers and symbols.

We know that creativity is fostered in group environments, where others’ ideas help us look at a problem in new ways. Physical manipulators like tiles or toothpicks, accompanied by guidance from a parent or teacher, has the same basic effect. The child is presented with a totally new way of looking at the same problem, which allows their brain to look at new, more effective ways of reaching a solution.

Count von Count would be very pleased. Ah … Ah … Ah! Imagine that!

Laura Mann graduated in Emerging Media and Communications from the University of Texas, and writes for The Dallas Observer. She also provides media assistance to the MediaWise Movement.

© Michael Mann, 2013, all rights retained. Printed by permission of the author.

Tags: , , , , , , , ,