The Math

Naming Shapes

/ 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:

Triangles

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.

triangles 2

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.

 

unusual 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.

How to Count: A Guide for Grownups

/ Amy / 1 Comment

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.