Measure! Early Measurement in Everyday Interactions

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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!

Cold feet

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.

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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.

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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.

How to Count: A Guide for Grownups

Toddler Counting

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.

That’s Math!

The word “math” has a lot of baggage behind it. I find this unfortunate, but I also totally get it. A lot of people have experienced math that was hard, or boring, or didn’t make much sense. A lot of people watched as other students seemed to “get it” effortlessly, while they struggled along. Some people didn’t have great teachers; some people had great teachers and wondered why they still couldn’t understand.

And while there are still plenty of people who love math, or at least don’t dislike it, there are so, so many people for whom the word “math” rings lots of scary bells. But my experience is that when those people are parents, they really, truly want their children to feel differently.

One possible solution is for a parent to think, “I didn’t like math, and so I’ll sneak math in without telling them. That way they won’t be scared off by knowing that they’re doing math.” I think of this as the “hidden vegetables” approach to doing mathematics, akin to adding squash to macaroni and cheese or sneaking zucchini into a chocolate cake. It might get kids to eat their squash and zucchini, but it won’t help them learn to love it. (But seriously, give these recipes a try. Yum!)

So I recommend a completely different approach. Instead of hiding math, showcase it! Let kids know they’re doing math! I don’t mean you should surround every math experience with bells and whistles and confetti, because some kids are going to get suspicious. A lot of young children will find natural pleasure in learning about numbers, identifying shapes, measuring, counting, adding— just as they naturally want to learn language. Build on that natural interest and tell them exactly what they’re doing—they’re doing math! Say, “Do you know what we just did? We did math!” Or even just, “That’s math!” Labeling mathematics for what it is while they are having positive experiences with it will help children develop an early positive association with the word math.

I like this blog to be positive, to tell you what you can or should do rather than what you shouldn’t do. But there is one thing that I warn parents against without hesitation, and that is putting your own fear or dislike of mathematics on display for your children.

Don’t tell your kids, “I’m bad at math,” even if you think you are. Don’t tell your kids, “Math isn’t much fun, but you’ve got to do it,” even though math sometimes does require hard work. That’s true of everything, but somehow the “not fun” label and the “hard” label snap on to mathematics like magnets.

Rather than telling your kids that you aren’t good at mathematics, show them that you are willing to learn this stuff right along with them. Empower your children with a belief that mathematics (like everything) is not a natural ability, but something that they can access through persistence and effort. I believe that hearing that message, early and often, will do far more for your child than any mathematics instruction you give them.

Book Review: Zero by Kathryn Otoshi

zero

It seems to me that children’s math books fall somewhere along a continuum. On one side of the continuum the books practically shout, “I am a MATH book! I will teach you something!” On the other side, the math is so subtle that you might not even notice it’s there.

I’m much more partial to the subtler side of the continuum, where the author weaves math into the fabric of the story, rather than fitting a story onto the math. It’s not that I think kids will only swallow the math if we sneak it into their literary diet. There are some fantastic, mathiest-of-math books that are engaging and fun and fantastic for children. But sometimes when the mathematics doesn’t jump out and hit you on the head there can be a lot more room for conversation. You just need to know what to look for.

One of my students recently lent me her copy of Zero, by Kathryn Otoshi. I had neither seen nor heard of this book, but I fell in love with it from the very first page. This book hits my sweet spot—simple, gorgeous illustrations; an engaging, child-friendly story; solid mathematics; and plenty of conversation fodder. It’s fantastic. I’m going to give it to myself for Christmas.

In this book, there are two big mathematical ideas:

1. Zero is a different kind of number.
I could write a whole blog post about this idea. (I probably will.) But here’s the short of it: young children don’t initially think of zero as a number because zero is not “part of the count.” When we count, we typically start with “one”. And we never ask our children to count a set of no objects. They’d probably think we were joking if we pointed at an empty tabletop and said, “Count the M&Ms!”

2. With only ten symbols, we are able to write numbers as big as we want to.
This is an under-recognized beauty of our number system. Many historical number systems (like Roman numerals) can only represent quantities up to a certain size before running out of symbols. But our modern Arabic numerals have infinite possibilities for combining just ten symbols into any number we can imagine.

So what can you do with this book (besides read it)? Talk about the math! Here are just a few ideas to get you started, but let your child lead the way and see where the conversation takes you.

  • Ask your child, “Is zero a number?” and see what they say. Probe their thinking. You can use the context of the book (but you don’t have to). Why would zero feel left out? What’s different about zero?
  • Have your child write the biggest number they can think of, and the smallest number they can think of. Together, see if you can think of bigger and bigger numbers (and for older children, smaller and smaller). It’s a game, but a game that can help your child think about how writing numbers “works.”
  • Talk about how the numbers in the book can join themselves together. What kinds of numbers could two numbers make? Four numbers?
  • And you don’t have to stop at math. Talk about friendships, fitting in, being yourself, working together.

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

objects

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.

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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?”

pennies

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.

cherrios

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?

Final

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

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.

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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.

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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?”

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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.

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About this Blog

I knew I loved to write long before I knew I loved mathematics. As a child I was a voracious reader, and dreamed of creating the same magic with my own pen.

Instead I became a math teacher.

I kept up a personal blog on the side, and my writing background served me well when I decided to earn a masters degree in math education, and then doctorate. Then followed a new career, and marriage, and a baby, and in the last few years my writing has mostly fallen to the wayside. Still, in the back of my mind I had a feeling that writing could still fit in my life, that there was some way I could merge my professional passions and my personal life in a way that would be useful and interesting to a wider audience than just my friends and family.

One night, when our son was still very young, I shared some of my thoughts about mathematics and parenting with my husband, Brian. I felt it was important for our son to get positive messages about mathematics from both of his parents, and for both of us to engage with him in mathematical thinking.

Brian, a trained opera singer who hadn’t taken a formal math class since his first semester of college, agreed that this was important, but expressed some concern. “What would talking mathematics with Monkey even look like?” he asked me. “I don’t think I would know where to begin.” Honestly, I didn’t have a good answer. Talking math felt intuitive to me, but that was because I was totally immersed in the world of mathematics learning. My husband, like so many other parents, is not.

Not long after that conversation I had a chance to give a 20-minute presentation to a group of mothers of young children. I focused my presentation on mathematical literacy for toddler-aged children, and when I rehearsed the Power Point with my husband he said, “That. That’s exactly what I was looking for.” The presentation was very well received. Several mothers told me that they very much wanted their children to have a good experience with mathematics, maybe a better one than they themselves had, but they knew very little about what they could do as parents, and that my presentation had given them a way to think about it. They felt excited to begin talking math with their kids.

That was the moment this blog was born.

There are many, many early literacy resources for parents. We know to talk to our children from the moment they are born, and to read to them. In fact, giving your children a strong foundation in literacy is one of the best things you can do to provide them with a strong foundation for mathematics. But some very literate adults can nevertheless remember struggling with mathematics, and I have found that many parents are looking for more. They want to actively help their children to feel excited about mathematics, and able, and empowered. These parents also want to feel empowered themselves when it comes to helping their children learn, understand, and enjoy mathematics. I created The Kids’ Quadrant to be a resource for these parents. For you!

The Kids’ Quadrant lies exactly at the intersection of my personal and professional life. I have a PhD in math education and a familiarity with and access to research on teaching mathematics to children. I teach, coordinate, and write curriculum for mathematics classes for future elementary teachers, which means I spend my career working to impress hearts and minds with an understanding of how children learn, perceive, and sometimes struggle with mathematics. And with my one-year-old son now at the cusp of language, I am deeply, personally invested in creating a natural and positive environment for him to explore the world of numbers and shapes and patterns.

I have much to share and much to learn, and this blog is my journey as well.

toddlers doing math