Math versus Literacy?

June 4, 2015June 4, 2015 / Amy / Leave a comment

One possible concern about focusing on mathematics at an early age is that too much focus on mathematics could take time away from learning crucial language and literacy skills. There’s an incredible body of research on the importance of early literacy, and no parent, caregiver, or educator would want to detract from a child’s literacy and language development.

But recently, as I’ve been digging in to the research on early mathematics learning, I came across the intriguing finding: Early mathematics skills may be a better predictor of later reading achievement than early reading skills. For example, a large study of the effects of various school-entry skills on later achievement showed that “early math skills have the greatest predictive power, followed by reading and then attention skills” for both boys and girls, and for children from high and low socioeconomic backgrounds (1). Another study on the effects of a high quality, intensive preschool math curriculum on children’s later language and literacy abilities showed that children who were taught from the math curriculum performed as well as the control group on some skills, and better on most skills (2).

These studies aren’t alone. The evidence is not perfect and doesn’t yet address why the link between early math and later literacy might exist. But I find the idea that strong, early math exposure could also boost a child’s language and literacy development to be fascinating.

And, honestly, it’s not all that surprising to me. Talking with your child about numbers or shapes or measurement is still talking to your child. Asking your child how they thought about a simple addition problem gives them opportunity to articulate their thought processes. Making sense of the world through quantities and spatial reasoning is still making sense of the world. Bringing math talk and math play into a child’s world, in ways that are fun and challenging and build on their natural curiosity, provides them with even more and broader contexts for making use of language and interpreting symbols and recalling facts and ideas from memory and linking ideas.

References

(1) G. J. Duncan et al., School readiness and later achievement. Developmental Psychology 43, 1428 (2007).

(2) J. Sarama, A. Lange, D. H. Clements, C. B. Wolfe, The impacts of an early mathematics curriculum on emerging literacy and language. Early Childhood Research Quarterly 27, 489 (2012).

Taking Turns: From Quizzing to Playing

August 12, 2014 / Amy / 2 Comments

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!

Naming Shapes

July 28, 2014 / Amy / Leave a comment

As Monkey slowly learns to talk, my husband and I find it fascinating to observe when and where he uses his “words.” On a hike in the spring he was “woof-woof”-ing at deer, squirrels – any animal we encountered. Now, a couple months later, he seems much better, though not perfect, at differentiating between dogs and not-dogs. I have every bit of confidence that eventually, as he encounters and names more and more animals, he will understand the difference between dog and not-dog as well as any adult.

For a child to learn what a dog is, they need two kinds of experiences:

1. Experience observing things that are like dogs, but are not, and giving them names – some things with four legs and fur are dogs, but some are squirrels, some are deer, some are horses…

2. Experience observing lots of kinds of dogs and hearing them called “dog.” We own a big German Shepherd mix, but in our family and in our neighborhood, Monkey has played with big dogs and little dogs, grown dogs and puppies. He is developing a large context for the word “dog.”

This is exactly the way kids learn about shapes. But children’s experiences with shape are often much more limited than their experiences with animals. Take a look at all the real-world examples of “triangle” in the pictures below:

These are all great examples of triangles to point out to your toddler or preschooler as they learn what it means for something to be a triangle. The problem is that, as varied as the contexts might be, these triangles are all the same! They’re all equilateral triangles (all sides and angles are the same) – what we might call the prototypical triangle, or the kind of shape we think about or draw automatically in response to the term “triangle.” We all have prototypes for words and concepts – particular images that come to mind when we hear the word. But when the prototype is all a child encounters (or even most of a what a child encounters), they miss out on the wide variety of objects that are considered triangles, and they lose some opportunity to identify what a triangle really is – not a “shape that looks like this,” but a shape made up of three straight sides.

Everything in the picture below is a triangle, but many children will fail to identify at least some of them as triangles because they don’t “look right.” They’re turned the wrong way, they’re too skinny, they’re upside down, they’re funny-shaped.

Non-prototypical triangles like these are harder to find in real life. Not impossible, just harder. So when you give your child the opportunity to identify, reason with, and talk about a wide variety of types of shapes, they have a huge leg up when it comes to learning geometry later on in school settings.

What can parents do to help their kids gain a broader experience with shapes?

Toddler: You are probably already pointing out and naming simple shapes to your child. You’ll likely focus mostly on prototypical rectangles, squares, triangles, but be on the lookout for non-prototypical shapes as well – long skinny rectangles, squares standing on a corner, triangles with sides that are all different lengths. Children’s books with nice, solid, colorful illustrations can be a great place for finding a variety of shapes.

Preschool: Have your child identify shapes in real life, and shapes that you draw yourself. Play sorting games (look for an upcoming blog post). Have your child draw shapes. Most importantly, when your child identifies a shape, ask, “How do you know?” Do they say something is a triangle because it is pointy? Has three sides? Looks like one? The correct answer is less important at this point than getting them to articulate what they are noticing.

Early Grades: Give names to less standard shapes: octagons, trapezoids, rhombuses, kites. Notice shapes that have more than one name – a square is also a kind of rectangle (and a rhombus and a kite!). Keep asking the question, “How do you know?” and challenge your child. If something is a triangle because “it’s pointy,” find something that’s pointy and not a triangle to help them focus on what really makes it a triangle.

Measure! Early Measurement in Everyday Interactions

July 21, 2014 / Amy / Leave a comment

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.

How to Count: A Guide for Grownups

July 11, 2014July 18, 2014 / Amy / 1 Comment

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!

July 3, 2014 / Amy / Leave a comment

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

June 27, 2014July 7, 2014 / Amy / 2 Comments

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.

Four Super Simple Counting Games

June 13, 2014July 10, 2014 / Amy / 2 Comments

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.

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