*Note: a shorter version of this post will be published in the September 2009 issue of the Southwest Homeschool Network newsletter*

“My son has trouble with division,” a young mom told me once. “I think it’s because he hasn’t memorized his multiplication facts.”

She explained that her child had figured out his own method for getting the right answers to multiplication problems. He just kept adding the multiplied number mentally until he had added it enough times for a correct answer. His multiplication method was slow, but it gave him right answers. Division had him stumped though. He couldn’t figure out the problems.

Although it may not appear that way, this boy’s trouble with division was the same problem that 5-year-old Elias had with addition the day I asked him, “How many places should we set for lunch today?”

First Elias counted himself and me. Then we talked about the other people who would be eating lunch with us – my husband (who was working in the garage), Grandma (who lived in a mobile home on the back of our lot), and Daniel (who was asleep in the loft). This talking wasn’t enough. Elias still couldn’t figure out how many places to set. If all five people had been there in the room, he could have easily figured out the answer by counting them. But since he couldn’t see the people, he couldn’t count them.

I tried to help him by showing him how to count people in his head, using my fingers to represent each person : “You (thumb), me (index finger), Dennis (middle finger), Grandma (ring finger), Daniel (pinkie) – one-two-three-four-five – see?”

His face went completely blank. Obviously, to Elias, a finger did not represent a person. He could not count people by counting fingers.

This is a developmental characteristic.

Adults use three ways of thinking about math: the manipulative mode, the mental mode, and the abstract mode. They can switch back and forth, using the abstract mode, for example, to figure using only symbols ($50 – $22.48 = $27.52), or the manipulative mode to do the same problem by counting correct change into a customer’s hand. Young children like Elias, on the other hand, can think only in the manipulative mode. They have to see and touch objects in order to understand math concepts like adding and subtracting.

As children develop they progress into the mental mode. When Elias grew older, he was able to do math problems like this mentally – to count me and himself, then mentally image the other three people and count them, too. But at that time, my attempt to represent each person with one finger, and then count the fingers, required more mental imaging than Elias was capable of yet. We solved the problem by drawing a picture of each person. When Elias counted the pictures, he knew how many places to set. Pictures of objects help children make the transition from the manipulative to the mental mode of thinking.

This process was a slow way to get my lunch table set, but a good way to teach Elias essential math. Practicing addition with pictures and objects helps children develop a strong concept of what addition *is.* Hands-on math is the foundation on which all other kinds of mathematical understanding is built. Lots of hands-on math experiences prepare children to grow into the next two stages of thinking development.

When children are hurried too quickly through this manipulative thinking stage, they feel anxious and uncertain. “Failure (to teach children in the manipulative mode) is probably the greatest single cause of children’s arithmetic difficulties,” Beechick says. “It is why people grow up with Arithmetic Anxiety.”

Until children understand with their eyes and hands what addition is all about, there is no point in having them memorize addition facts. Meaningless memorization falls out of people’s brains. Children must develop a strong mental image of a particular math process, like addition or multiplication, before memorizing math facts has enough meaning to make the facts stick in their heads. Thus, the little boy who had trouble with division probably could not memorize multiplication tables because he lacked a strong mental image of what multiplication (and division) mean.

A stack of index cards and some counters, like pennies or beads, could help him understand with some practice that 3 x 4 means that you have three groups with 4 things in each group. You take 3 cards, place four counters on each card, and count up the total. Making a group of four three times gives you 12. Children can draw pictures to demonstrate problems, too. For example, 3 x 4 could be drawn as 3 trees with 4 apples on each tree.

When children can demonstrate easily that they understand what multiplication means, then you can show them what division means. The problem 12 divided by 3 means that you have 12 things in a group, and then you separate those twelve things into three different, equal groups. So to demonstrate 12 divided by 3, count out 12 counters, lay out 3 cards, place one counter on each card, and then keep going round and round until all 12 counters are on cards, and each card has the same number of counters.

“When we say that a child doesn’t understand something (in math), we usually mean that he is not able to image it in his head,” said Ruth Beechick in *The Three R’s.* “The cure for that is to provide more manipulative experience.”

Using math manipulatives helps children to develop and strengthen the mental images they need for understanding math concepts. It also helps them to become able to do mental math (which they must learn before being able to do abstract math).

“Try showing something one way and a second way and a third way” said Beechick. “Wait awhile and teach it again next month. After sufficient manipulative experience, the child eventually will image the troublesome process in his head. He will understand it.”

© Becky Cerling Powers 2009 Reprint with attribution only.

Related posts/articles on Story Power: How to Teach Games to Kids: https://beckycerlingpowers.wordpress.com/articles/how-to-teach-games-to-kids/

Related links: children’s mental development

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