# Sharing Brownies

The other night at dinner as we all debriefed our days, I mentioned that I’d given my college students a problem about sharing brownies. “I should give the problem to you and see how you would solve it,” I mused to my kids, and my husband immediately said, “Let’s make brownies and solve the problem in real life!”

So we did. We threw together a pan of brownies from a mix we had sitting in our cupboard, and when they had cooled I cut and plated three square brownies for each kid. Then I handed them each a plate and a dinner knife and told them to figure out how to give every member of our 4-person family the same amount of brownie.

We like to ask sharing problems about food (“If there are 8 pancakes, how many can everyone have?”), because our kids, like all kids, are highly motivated by food and by fairness. But this is a challenging problem, without a clear, immediate solution.

Still, my kindergartener dove right in, deftly cutting all three brownies in half. She then paused for a moment with the two extra pieces before cutting them in half too and stacking a quarter brownie on top of each half-brownie portion. “Everyone gets a half and a piece,” she said when I asked her about her solution.

(Surprisingly, very few of my college-age students come up with this particular method initially, although once they have seen it they tend to prefer it. This semester one student commented on how surprised he was that a kindergartener would come up with this method right away when he, a math education major, didn’t think of it on his own.)

My 7-year-old stared and stared and stared at the brownies and I could see the gears turning in his head. Finally, several minutes after my daughter had confidently offered up her half and a piece, he said, “Okay, I think this will work,” and embarked upon a complicated cutting exercise that I would call “split the brownies into smaller and smaller pieces and hope it will all work out eventually.” First he cut one brownie into thirds, cut another brownie into fourths, cut a fourth in half, and put the half-quarter together with a full quarter to make another “third”. He then cut the last brownie into fourths, and proceeded to cut and re-cut any odd pieces out until he felt confident that he had a workable number of pieces.

He explained that everyone got a third and a quarter and a half quarter and a “small quarter” (half of half of a quarter). But when he actually distributed the pieces onto each of our plates, there were a few extra bits left behind. “Hmm, I don’t know if that really worked,” he said, and then shrugged and popped the extra pieces into his own mouth.

It was obviously interesting and fun to watch how my kids approached this problem on the very same day I watched my college students approach the same problem. But it was also interesting to listen to the informal language they used to talk about their solutions. They both already had some language for talking about fractions, and they both ran up against limits. The kindergartener could talk about halves, but once she got to quarters they became “pieces”. Nevertheless, with the motivation of actually getting to eat the brownies at the end, she was remarkably accurate at splitting the brownies into equal parts. The second grader could talk about halves and thirds and quarters, but then began talking about half-quarters and “smaller quarters” when he got down to eighths and sixteenths. And he made an interesting approximation of 1/3 by combining 1/8 and 1/4. This was not precisely equal to 1/3, but it was certainly close enough to feel fair.

My college students like to ask me questions like: “When would you start teaching this to students?” My answer is often: “Much earlier than you’d think!” When can kids start understanding fractions? Much earlier than you’d think! When can kids begin making sense of probability? Much earlier than you’d think! When can you give kids multiplication problems? Much earlier than you’d think! When can kids understand that a square is a type of rectangle? Much earlier than you’d think!

This particular problem involves both division and fractions, and while I had no idea what my 5- and 7-year-old children would do with it, I knew they would be able to do something. Kids have great ideas, and they have great ideas much earlier than you’d think!

# 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!

# Taking Turns: From Quizzing to Playing

A lot of doing math with young kids involves asking questions. Sometimes the hard part is knowing what questions to ask and when to ask them, and I hope this blog can provide you with plenty of ideas.

But sometimes I find that the real challenge is taking these questions and turning them into a conversation. I mean, asking questions is a great start to igniting conversation. “What shape is that?” is a lot more likely to give your child reason to interact with you than, “Look! There’s a triangle!” once your child is old enough to meaningfully respond. (Or even before – I ask Monkey questions all the time, and even though the response is often jabber, he’s still learning patterns of conversation, it it feels like I’m now talking with him, not just at him.)

But once you ask a question and get an answer, how to you keep the conversation going? How do you keep it feeling like you’re playing rather than quizzing? How do you keep it interesting and engaging?

One very simple and very versatile tool is to take turns asking questions. This is a fantastic tool because it works with a wide variety of ages, with any mathematical idea (and even with non-mathematical conversations!).

Going back to shapes, imagine three different conversational outcomes of a parent on a walk with their child.

Conversation 1:

Parent: [points at Yield sign] Look! There’s a triangle!

Child: [looks]

Conversation 2:

Parent: [points at Yield sign] What shape is that?

Child: Triangle!

Parent: Good job!* [points at wheel on a car] What shape is this?

Child: Circle!

Parent: How about….that! [points at window on a house]

Child: Square!

etc.

Conversation 3:

Parent: [points at Yield sign] What shape is that?

Child: Triangle!

Parent: Good job! Now you ask me one.

Child: [points at stop sign] What’s that!

Parent: That’s an octagon. Can you say that?

Child: Octagon.

Parent: Good! My turn?

Child: Yeah!

etc.

So what’s better about outcome 3? For one, the child is much more empowered in this conversation. The child can choose the focus of conversation. The child can communicate to the parent more easily what they are interested in or curious about. They might maintain interest longer, because the conversation is a game rather than a quiz. And thinking about what kinds of questions fit in the game involves a much higher level of thinking than just answering the questions that you choose.

Now, kids are totally unpredictable, which means that sometimes this (like any tool) might work fantastically…and sometimes it might not work out so well. Kids don’t always play the game you choose.  And sometimes a kid won’t play by the rules (but sometimes their rules can take you to interesting places). And sometimes a kid will ask you a really hard question, and when this happens it’s not a fail – it’s a total success! What if your child were to ask you about the shape of a tree? What would you say? What if they ask what a million times a gajillion is? Or what number comes before zero? These are great conversation starters, and even if you don’t know an answer, you have an opportunity to model thinking and curiosity and how adults go about figuring out something they don’t know.

What other tools do you use to engage your kids in mathematical conversation?

* P.S. A great question to ask at times like these, although it’s not the purpose of this post, is “How do you know?” Sometimes this might be beyond your child, but go ahead and ask it anyway. See what happens!

# Measure! Early Measurement in Everyday Interactions

A child’s very early math experience is made up in large part of learning how to quantify, or to use numbers to describe characteristics of their world. Usually we think of this skill in terms of counting. But another form of quantifying is measurement. Whereas counting answers the question “How many?”, measuring assigns a number to attributes such as height, weight, length, and temperature.

Measurement allows kids access to all sorts of interesting questions about their environment. It’s also useful—we use measurement in chores, cooking, art, sports, sewing, building, etc. And measurement provides fertile ground for starting to think about fractions: what, for instance, do we call a measurement that’s somewhere between 4 inches and 5 inches?

Formal measurement skills (such as using a ruler) develop out of informal measurement experiences. Here ere are some easy, natural ways to introduce early measurement ideas in conversations and play with your toddler or preschooler.

Comparison Questions

Ask your child, “Which is bigger, ___ or ___?” This is a simple question with infinite variations, and easily tailored to your child’s ability and interests.

• Use different measurement words for different attributes. “Which is bigger?” is appropriate for some comparisons, but you might also use, “Which is taller?” or “Which is longer?” or “Which is more full?” Use comparisons questions to compare length, height, weight, volume, width, area, or even coldness or loudness.
• Don’t shy away from silly questions! “Which is bigger, the real car or the toy car?” “Which is heavier, the piano or the pencil?” These silly questions can be very fun for a young child, and still get them to think about, notice, and talk about measurable attributes.
• When your child is ready, make comparisons of objects that have very similar measurements. Your child might just guess, and that’s okay. When they begin to show interest in actually knowing, you have a great gateway to introducing formal measurement. If it’s not clear whether the pink cup or the glass tumbler has more water in it, pull out the measuring cups and find out!

Find Something Bigger/Smaller

This is a game that can be played in the home, outdoors, at the grocery store, in the car. It is better for preschoolers than toddlers, but older children can join in. Start with an object, and then ask your child to find something bigger. Proceed from there, taking turns finding bigger and bigger objects.

• You can also, of course, find smaller and smaller objects – just be sure to start with something very big.
• You can also use different measurable attributes – find something longer, or heavier. The term “bigger” is actually vague, because big could mean lots of things. Is a bookshelf bigger than a couch or smaller? It depends on what you’re measuring! But don’t avoid the term “bigger” – using the word can give you insight into how your child is thinking about “big”, and can open up conversations that lead to noticing other size attributes.

Fill It Up

We usually measure things with tools, like rulers, measuring cups, or scales. But informal measuring techniques can help children understand what formal tools are actually doing. Most informal techniques involve “filling up”:

• Fill different-size containers (cups, bowls, tupperware, etc.) with “scoops” of water or Cheerios or flour (whatever you’re willing to clean up!). Use a single measuring cup (it doesn’t matter what size) to scoop with. Notice that some containers take lots of scoops and some take very few.
• Draw outlines of shapes on paper and fill up the outline with objects of the same size, like small blocks, cotton balls, dry pasta. Again, notice that some shapes take more and some take fewer. Or you can fill the same shape with different-sized objects – goldfish crackers first, and then wheat squares. Notice that the size of the objects relates to how many fit.
• Line silverware up end-to-end and see how many pieces of silverware it takes to go from one end to the other, or all the way around. Then use silverware to measure other lengths as well. You can also use pens, crayons, pretzel sticks – anything long and thin and approximately the same length.

As with all activities on this blog, the most important thing is to follow your child’s interest, and to have fun! Your child naturally wants to learn about his/her world, and you as a parent are there to give them more tools to do so.

# 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.