Math with manipulatives: Preschool number activities designed to foster your child’s number sense

© 2008 -2013 Gwen Dewar, Ph.D., all rights reserved

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Never mind the talking toys and fancy video games. These preschool number activities require only a dose of imagination and a few household supplies.

Discoveries in cognitive psychology and neuroscience suggest that preschool number activities should address more than verbal counting.

Young children need to develop an intuitive feeling for numerosity—-the “how many-ness” associated with specific numbers.

These activities are designed to help kids sharpen their “number sense” and provide them with opportunities to put several math concepts into practice, including

• the notion of relative magnitudes

• the one-to-one principle of numerosity (two sets are equal if the items in each set can be matched one-to-one with no items left over)

• the one-to-one principle of counting (each item to be counted is counted once and only once)

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• the stable order principle (number words must be recited in the same order)

• the principle of increasing magnitudes (the later number words refer to greater numerosities)

• the cardinal principle (the last word counted represents the numerosity of the set)

Most of these preschool number activities rely on a set of cards and a set of tokens.

Here’s what you need to get started.

Making a set of cards and manipulatives

The cards will be used in two ways—as displays of dots for kids to count, and as templates for kids to cover with tokens.

Preparation

For this group of preschool number activities, you’ll need

• 10 or more tokens (each over 1.25” in diameter to avoid a choking hazard)

• 10 or more sheets of heavy-stock paper or large index cards

• Felt-tip pen

• Optional: a set of small stickers

Finding tokens that aren't distracting or hazardous

A variety of objects can be used for tokens, but keep in mind: Kids can get distracted if your tokens are too interesting, so it's best to avoid the fancy plastic frogs and fully-embellished coins (Petersen and McNeil 2012).

Also, you need to be conscious of choking hazards for kids under 3. According to the U.S. Consumer Product Safety Commission, a ball-shaped object is unsafe if it is smaller than a 1.75” diameter golf ball. Other objects are unsafe if they can fit inside a tube with a diameter of 1.25” inches. I’ve used plastic poker chips. You can also use something safe and edible, like “O”-shaped cereal pieces.

Creating the cards

Each card will be marked by an Arabic numeral and corresponding number of dots. Make the dots with a felt tip marker. Alternatively, you can use stickers to make your dots. The dots should be spaced far enough apart for your child to place a token over each dot. The larger your tokens, the larger your cards will need to be.

Make at least one card for each number between 1 and 10. In addition, make multiple cards for the same number—each card bearing dots arranged in different configurations. For example, one “three” card might show three dots arranged in a triangular configuration. Another might show the dots arranged in a line. Still another might show the dots that appear to have been placed randomly. Whatever your configuration, leave enough space between dots for your child to place a token over each dot.

Preschool number activities: Mix and match

One you have your cards and tokens, you can play any of the preschool number activities below. As you play, keep in mind the points raised in my guide to preschool math lessons:

• Start small. It’s important to adjust the game to your child’s attention span and developmental level. For beginners, this means counting tasks that focus on very small numbers (up to 3 or 4).

• Keep it fun. If it’s not playful and fun, it’s time to stop.

• Be patient. It takes kids about a year to learn how the counting system works.

The basic game: One-to-one matching

Place a card, face up, before your child. Then ask your child to place the correct number of tokens on the card—one token over each dot.

After the child has finished the task, replace the card and tokens and start again with a new card.

Once your child has got the hang of this, you can modify the game by helping your child count each token as he puts it in place.

The Tea Party: Relative magnitudes

Choose two cards that display a different number of dots, taking care that the cards differ by a ratio of at least 2:1. For instance, try 1 vs. 2, 2 vs. 4, and 2 vs. 5. You can also try larger numbers, like 6 vs. 12.

Then set each card down in front of a toy creature / doll / teddy bear, and show your child how to cover the dots with a token. When I’ve played this game, I used poker chips and called them cookies. But you could also use edible tokens, like pieces of cereal.

After your child has covered each dot with a token, ask him

“Which (creature) has more (cookies / treats)?”

After he answers you, you can count each “tray” of treats to check the answer. But I’d skip this step if you are working with larger numbers (like 6 vs. 12) that are beyond your child’s current grasp. You don’t want to make this game feel like a tedious exercise.

As your child becomes better at this game, you can try somewhat smaller ratios (like 5 vs. 9).

Bigger and bigger: Increasing magnitudes

Instead of playing with the tokens, have your child place the cards side-by-side in correct numeric sequence.

For beginners, try this with very small numbers (1, 2, 3) and with numbers that vary by a large degree (e.g., 1, 3, 6, 12).

Sharing at the tea party: The one-to-one principle

I’ve stolen this one directly from experiments done by Brian Butterworth and his colleagues (2008). Choose three toy creatures as party attendees and have your child set the table—providing one and only plate, cup, and spoon to each toy. Then give your child a set of “cookies” (tokens or real edibles) and ask her to share these among the party guests so they each receive the same amount. Make it simple by giving your child 6 or 9 tokens so that none will be left over.

As always, go at your child’s pace and quit if it isn’t fun.

If your child makes a mistake and gives one creature too many tokens, you can play the part of another creature and complain that it isn’t fair.

You can also play the part of tea party host and deliberately make a mistake. Ask for your child’s help? Did someone get too many tokens? Or not enough? Have your child fix it.

Once your child gets the hang of things, try providing him with one token too many and discuss what to do about this "leftover."

One solution is to divide the remainder into three equal bits. But your child may come up with other, non-mathematical solutions, like eating the extra bit himself.

Matching patterns: Counting and numerosity

Play the basic game as described above, but instead of having your child place the tokens directly over the dots, have your child place the tokens alongside the card. Ask your child to arrange his tokens in the same pattern that is illustrated on the card. And count!

Matching patterns: Conservation of number

For this game, use cards bearing dots only-—no numerals. To play, place two cards—-each bearing the same number of dots, but arranged in different patterns—-side by side.

Ask your child to recreate each pattern using his tokens. When she’s done, help her count the number of tokens in each pattern. The patterns look different, but they use the same number of dots/tokens.

Spotting the goof: The one-to-one and cardinal principles

Here’s another activity swiped from the experimental literature.

In one study, researchers asked preschoolers to watch—-and help-—a rather incompetent puppet count a set of objects (Gelman et al 1986). The puppet would occasionally violate the one-to-one principle by double-counting (e.g., “one, two, three, three, four…). He also sometimes skipped an object or repeated the wrong cardinal value.

Kids ranging in age from 3 to 5 were pretty good at detecting these violations. So your child might have fun correcting your own “goof up” puppet or toy at home. Have the puppet count the number of tokens in a set, and, sometimes, make mistakes. If your child doesn’t notice the error, you can correct the puppet yourself.

But either way, ask your child to explain what went wrong. Experimenters working with 4- and 5-year olds found that kids didn’t make conceptual progress unless they were asked to explain either their own or the experimenter's reasoning (Muldoon et al 2007).

The cookie maker: Making predictions about changes to a set

Even before kids master counting, they can learn about the concepts of addition and subtraction. Here are some research-inspired preschool number activities that ask kids to make predictions about addition and subtraction

For these games, have a puppet or toy “bake cookies” (a set of tokens). Ask your child to count the cookies (helping if necessary) and then have the puppet bake one more cookie and add it to the set.

Are there more cookies or fewer cookies now? Ask your child to predict how many cookies are left. Then count again to check the answer.

Try the same thing with subtraction by having the puppet eat a cookie.

Don’t expect answers that are precise and correct. But you may find that your child is good at getting the gist. When researchers asked 3-, 4- and 5-year olds to perform similar tasks, they found that 90% of the predictions were in the right direction (Zur and Gelman 2004).

The Big Race: Increasing magnitudes and the number line

As your child begins to master the first few number words, you can also try these research-tested preschool number activities for teaching kids about the number line.

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References: Preschool number activities

Butterworth B, Reeve R, and Lloyd D. 2008. Numerical thought with and without words: Evidence from indigenous Australian children. Proceedings of the National Academy of Sciences 105(35): 13179-13184.

Gelman R, Meck E, and Merkin S. 1986. Young children’s numerical competence. Cognitive Development 1(1): 1-29.

Muldoon KP, Lewis C, Francis B. 2007. Using cardinality to compare quantities: the role of social-cognitive conflict in early numeracy. Developmental Psychology 10(5):694-711.

Petersen LA and McNeil NM. 2012. Effects of Perceptually Rich Manipulatives on Preschoolers' Counting Performance: Established Knowledge Counts. Child Dev. 2012 Dec 13. doi: 10.1111/cdev.12028. [Epub ahead of print]

Zur O and Gelman R. 2004. Young children can add and subtract by predicting and checking. Early childhood Research Quarterly 19: 121-137.

Content last modified 3/13