# Scavenge for Quantity

The beginning of the college semester has kept me extra busy for a couple weeks, but I’m back. And I have a game this time that can be adapted for a very wide variety of ages – a scavenger hunt!

One way to help your child to internalize mathematical thinking is to open their eyes to the quantities around them. As parents, we help provide our children ways to see the world from an early age. We name objects, identify colors, describe what we’re doing. A child doesn’t necessarily see the color blue on their own – they see blue because adults point blue out to them (listen to this totally fascinating Radiolab podcast about the color blue). We point out big and little, noisy and quiet, dark and bright – we help show them which characteristics are worth noticing.

Quantity is also a characteristic. So many things in our world come in quantities, but since kids don’t tune in naturally to the characteristic “how many,” adults can help by pointing it out. I’m not talking about learning to count – I’m talking about learning to see quantity as an attribute, something we can recognize, describe, and use to characterize parts of our world, just like color. Just as Monkey’s t-shirt and the playground equipment in the picture below, although entirely different in other ways, share the attribute “gray,” the flower petals and the dots on the dice below the photograph share the attribute “five.”

A scavenger hunt for quantities can help your child become accustomed to seeing quantity as an attribute. There are all sorts of ways to do a quantity scavenger hunt. Here are three:

1. Open Scavenge: Pick a room in the house and work together to find as many quantities as possible. You might choose the kitchen and find that there are two salt/pepper shakers, six chairs around the table, four drawers beneath the counter, three slats on the back of the chair, two towels on the front of the oven, and so on and so on. This can be played one-on-one with a single child or cooperatively with two or more children. Some children may enjoy labeling quantities with sticky notes or a label maker.
2. Number Challenge: Give your child (or children) a paper labeled with the numbers 1-12, or a set of sticky notes labeled 1-12. Then roam throughout the house and try to find a quantity for every number. With multiple players, everyone can have their own set, the rule being that no two people can choose the same set of objects for their number. Because some numbers will be much harder than others (how many things come in sets of 11?*), you may wish to eliminate hard numbers, set a time limit, or simply play for fun with the expectation that you might not find everything.
3. Quantity Match: Can be played with a parent and child, or with two children. Have one player find a quantity (four shelves, for example). The other player then has to find something different, but with the same quantity (four pillows on the couch!). Then switch places. Repeating quantities is okay (and even repeating sets of objects for very young children).

Again, this is really easy to adapt, and it’s easy to involve multiple members of the family – including younger and older children. Have fun with it!

And as a side note, it’s never to early to start helping your child notice quantity. With very young children, you can point out any quantity but especially two – there are all sorts of twos in a baby or toddler’s life! Two ears, two hands, two feet… This morning as I was changing 16-month-old Monkey out of his pajamas, I counted “one…” as I pulled one foot out of the footie, and as I reached for his second foot he responded with, “dooh!” A few more repetitions convinced me that it wasn’t just coincidence – he’s picked up on the “one…two!” count he’s been hearing from me, and I wasn’t even really thinking about it. Way to make your math mama proud, Monkey!

* Look! 11 slats on the shoe rack!

# How to Count: A Guide for Grownups

One of the things I love about my job is that I get to look in-depth at mathematics concepts that appear basic, but are surprisingly complex. Understanding the complexity of the skills our kids are learning can really help us as parents to appreciate what they are capable of. Knowing what we’re watching can also change how we interact with our children.

Take counting. Counting is second-nature to us grownups and we probably can’t even remember when it wasn’t. But it takes children several years of practice and play to become really, truly proficient. Think about all that a child needs to be able to do just to count a set of objects:

• Learn all the counting words (“one, two, three, four, …”).
• Remember the correct order for all the counting words.
• Make sure every single thing in the set gets counted. No skipping an object!
• Make sure everything gets counted just once. No double counting!
• Know that the last number we say tells us how many there are.

That’s a lot of stuff! No wonder it takes several years to get it right.

Counting Before You Know How

Not long ago I recorded my nephew counting cookie dough blobs on a cookie sheet. The video quality isn’t fantastic, but I love this clip because his “mistakes” so nicely illustrate the skills needed to count correctly. I also love this clip for thinking about what he does know—what he gets right even when it looks like he gets most things wrong.

Notice that my nephew uses the correct counting sequence through ten, but then skips straight to 14 and 16. Notice how he points with great abandon, touching the first three cookies and then jumping all the way to the back corner, skipping many and touching several of them repeatedly.  And notice how when I ask my nephew how many there are he does not say 16, even though that’s the number he ended on. Instead he says, “We should have seven!”

What he doesn’t know (yet)… He doesn’t know all the counting words he needs, or the order of the higher numbers. He doesn’t know that everything needs to be counted, with no double-counting. He doesn’t know how to keep track of what he has counted. He doesn’t know that the last number you say tells you how many there are. He probably doesn’t have a good sense of what “how many” even means!

But what he does know… He knows the “number rhyme” through ten, and he knows that it keeps going, even if he doesn’t know exactly how. He knows that when you count, you’re supposed to point at things as you count him. Eventually he’ll realize that the purpose of pointing is to help him keep track, but it’s okay that all he knows now is that you point.He knows that when someone asks him, “How many are there?” he is supposed to say a number.

See? He knows stuff! He just doesn’t know all the stuff.

Getting It Right by Getting It Wrong

That’s another thing I love about this clip. It doesn’t just demonstrate a transitional stage in learning all the building blocks for counting. It demonstrates this awesome characteristic that little kids have, this ability to learn to get something right by getting it wrong over and over and over again.

My nephew really doesn’t understand the concept of “how many” at this point, but as he engages in this adult activity called counting (because it’s so fun to do!), he’ll gradually start to notice that adults touch every object just once. He’ll gradually start to remember the number words in order up to higher and higher quantities. He’ll gradually notice that there’s a visible difference between sets where the last number he says is “5” and sets where the last number he says is “7” and by so doing will begin to understand that the number words he is saying actually represent something. He’ll begin to develop an understanding of quantity, which is really the purpose of learning to count.

What do you have to do as a parent to encourage this? Not much—you don’t have to tell a 3-year-old: “The last number you say is how many there are.” You don’t have to drill or isolate the skills. All you have to do is model and encourage.

MODEL: Model good counting: Say the numbers, point at each thing you are counting and, when you are finished, say how many there are (“1, 2, 3, 4, 5. There are 5 crackers.”).

ENCOURAGE: Count everything! Count toes, eyes, food, people. Read counting books. Count things in non-counting books. There are so many counting opportunities that you really don’t have to worry about how many you’re missing. But every time you see and use a counting opportunity will make it that much easier for you, and your child, to catch the next one.

# Replicate an Experiment! Kids and Algebra (ages 4 – 6)

## Question: Are young children able to reason algebraically?

This experiment comes from a recent study published by researchers at Johns Hopkins University.

Materials
1. A set of about 20-25 each of four different kinds of small objects (buttons, beads, fish crackers, cereal, pennies, etc.)
2. Two identical, opaque cups (paper cups work well)
3. A stuffed animal with a name

Setup
Separate the objects into piles somewhere that will be accessible to you, but out of your child’s view. (You may want to lay out the objects on your table, then hide them behind a book.) For three of the objects, you will want 12 in the first pile (those 12 will go in the “magic cup”), and anywhere from 5 to 9 in the second pile. For the fourth object (I used chocolate chips), you will want 12 in one pile and 4 in the other.

How to Conduct the Experiment
Tell your child you are going to play a number game with the stuffed animal (ours is Puppy). Tell your child that Puppy (or whatever the animal is named) has a magic cup, and show them one of the cups. Tell them that any time you put the cup on a pile of something, it adds the same number of that object to the pile, no matter what the object is. Say, “Let’s try and see what happens.”

Out of your child’s sight, place 12 of the first object (I used pennies) into the magic cup Place the remaining pennies in front of Puppy. Point at the pennies and say, “See Puppy’s pennies?” Then bring the magic cup out, without showing the child the pennies inside. Tilt it upside down over the pile of 5 pennies, then lift it to reveal the (now larger) pile of pennies. Say, “It worked! Shall we try again?”

Remove the pennies and do the same thing with the next two objects (I used cherries and Cheerios). The key is to move fairly rapidly, without rushing. You want to discourage counting, and you want to make sure that your child never actually sees the objects in the magic cup until they have been added to the pile in front of the animal.

Now that you’ve laid the groundwork, you’re going to see if your child can tell how many objects the magic cup creates. Discretely place your fourth objects (chocolate chips in this case) into the two cups, 12 in one and 4 in the other. Bring both cups out and tell your child that you found two cups and you’re not sure which is Puppy’s magic cup. Turn both cups over onto the table and lift them to reveal the 12 objects and the 4 objects. Ask your child to tell you which is Puppy’s cup. You should realize that, at this point, your child has never seen the contents of the magic cup—only the starting amount and the final amount. Will she now be able to tell which of the two is the amount that was added to each pile?

What is the purpose of this experiment?
The researchers who conducted this experiment already knew that children are born with number sense that does not need to be taught. They wanted to know if that same number sense extends to algebraic reasoning.

The task in this experiment may seem simple, but it’s not. There’s a big difference between being able to think about a situation where the unknown quantity is the result of an operation, such as 5 + 12 = ?, and being able to think about a situation where the unknown quantity is within the operation, such as 5 + ? = 17. Identifying the change amount (5 + ? = 17) rather than the result amount (5 + 12 = ?) is something that older kids often struggle with when they first begin algebra.

In this experiment, the children were too young to solve an algebraic situation symbolically. When given equations like 5 + ? = 17, and asked to choose between the numbers 4 and 12, they just guessed. But when the magic cup story presented children with the exact same situation in a non-symbolic form, the children were suddenly successful.

This experiment reveals that your child’s brain is already wired to do simple algebra! Knowing what our children are capable of can give us confidence as parents who want our children to achieve their potential.

What if my kid failed the experiment? Does it mean my child will struggle with algebra?
Of course not! For one, it’s hard to perform an experiment perfectly, and you only had one go at it. And maybe your child was tired or hungry or distracted. Maybe they would have gotten the right cup 4 out of 5 times (better than chance), and you happened to see the one they got wrong. More important than the outcome of your particular experiment is the opportunity you gave your child to think algebraically just by doing the experiment with her. Now give them more opportunities!

Reference: Kibbe, M. M. and Feigenson, L. (2014). Young children ‘solve for x’ using the Approximate Number System. Developmental Science. doi: 10.1111/desc.12177

# Four Super Simple Counting Games

As your child is learning to count, it’s great to give them lots of opportunities. To you, it may appear that they’re learning an important mathematical skill, but to them it’s a game, and it’s fun! Here are some simple ways you can spice up the “counting game” with new games that are easy to play almost anywhere, and help build a young child’s mathematical brainpower.

Hidden Objects: When your child has counted a number of objects, hide them by covering them with your hand, a blanket, a bowl, or a book, or simply turn around with your child so you can no longer see. Ask, “how many are there?” and see if they give you the same number.

Why? Part of learning to count is recognizing that the number you say is how many there are, and that objects don’t have to be seen to have a count. Seeing whether your child can tell that the number of hidden objects is the same as the number they counted gives you insight into what they understand about counting.

Variation: After your child has counted a set of objects, mix the objects up, push them closer together, or spread them farther apart, and ask how many there are. Early on, a child will recount the set. With counting experience, though, they will learn that moving objects around doesn’t change how many there are.

Guess the Number: Ask your child, “How many [crackers, birds, toys] do you think there are?” Then count together to see how close the guess was.

How many shoes do you think there are in the picture?

Why? Build number sense! It’s horribly difficult to tell the difference between, say, 12 and 13 at a glance. But a collection of 7 things looks very different from a collection of 15 things. Guessing before counting can help children to get a sense of the relative size of different quantities.

Variation: Give a child two options. For instance, if there are 6 swings at the park, ask, “Do you think there are 6 swings, or do you think there are 10 swings?”

Now How Many? If your child is counting objects that you can manipulate (beans, toys, spoons, etc.), take one away or put one in after your child has counted and ask, “Now how many are there?”

Why? This will help your child learn how to utilize the number sequence for simple addition and subtraction. Initially, children will need to re-count the set of objects, and that’s okay! Be sure to remind them how many there were before, and how many there are now. Eventually, your child may realize that all they have to do when you put one in is to say the next counting word, and all they have to do when you take one away is to say the previous counting word.

Variation: If a child can play “Now How Many?” without recounting, begin removing or adding two objects at a time, or even three.

Backward and Forward: Count steps as you’re walking. Stop on a certain number of steps and walk backwards, counting down. Make a game of it, counting up for walking forward and down for walking backward. When they get the hang of it, let your child lead.

Why? Being able to count backward is an important skill for early arithmetic. It also helps solidify a child’s familiarity with the number sequence.

Variation: Count when you walk up or down stairs with your child. If you’re walking up or down an unfamiliar flight of stairs this could also be combined with “Guess the Number.” Guess how many stairs there are, and then count (up or down) to see if you’re right.