If You Give a Kid a Calculator, Part II

January 15, 2021January 19, 2021 / Amy / 2 Comments

I have a confession: I gave my oldest a calculator when he was four. But I did not give my second child a calculator when she was four. In fact, she is now almost 6 and it was only when her older brother found the calculator the other day that it occurred to me that I should give her a chance to play with it, too.

This is partly a first child/second child matter. I didn’t think to get her a calculator because I had already thought to give a child of mine a calculator, and my brain had checked the “done” box on that one.

But it’s also because my oldest has always shown a very obvious interest in numbers and patterns and is therefore very easy to give a calculator to, or a math problem, or a math game. My youngest is…not. At least, not always. My son has already been identified by his teachers as a “math person”, but my daughter is very different from her older brother and I worry that she will be identified just as early as someone who is not a math person. I am especially aware of this because I see how easy it is for me, a parent who believes deep within my soul that every child is a math person, to nevertheless give different math opportunities to my older child (who is practically begging me to do math with him) than to my younger child (who would rather just do her own thing, thank you very much).

“Do you want to play a math game with me?” I ask my older child, and he says, “Yes!” and drops whatever he is doing, and we play the math game.

“Do you want to do play a math game with me?” I ask my younger child, and she doesn’t even look up, but just smiles knowingly into the pages of her sticker book, confident that I cannot make her play anything she does not want to play, and we don’t play the math game.

The problem is that if a child is particularly interested in or receptive to mathematics activities early on, they will likely get more opportunities to do math, to do more interesting and open-ended math, and to experience more positive feelings around mathematics. If a child tends to show less interest in math over other activities at an early age, or takes longer to learn some of the skills we think of as foundational, that child will probably be given fewer opportunities, more basic skills practice, and a less interesting mathematics experience overall.

I can’t (and shouldn’t) force my daughter to do math with me. But I’ve also learned that with the right approach she’s really receptive to doing math and talking about math and exploring what it means to do math. She actually loves to play math games when she’s in the mood (Tiny Polka Dot is our current favorite), and she solves problems in interesting ways. Doing math with her just looks different than doing math with my son, and it requires me to be more a little more conscious about looking for opportunities and helping her build a positive view of mathematics and her own ability to do math.

So here’s my advice (and it’s advice for myself, too):

Give your kid a calculator. Give one to the kid you know will love playing with it, and give one to the kid you’re not so sure will love it. You may be surprised to see what they do.

Show interest in how your child thinks about math. Show as much interest in the child who is using the commutative property to solve a complex multiplication problem, as you do in the child who is counting on their fingers, because all children’s thinking is interesting when you really stop and listen.

Find math activities that encourage creativity and multiple approaches. These types of activities allow any kid to shine by sharing their unique ideas. Right now my son is enjoying the game Proof!, which gives us a chance to combine numbers in interesting ways. My daughter loves the activity “Foot Parade” (we’ve also created our own version called “Alien Parade” where there’s no limit on the number of feet you can create). And both my kids love the book How Many? by Christopher Danielson, which is an open-ended and interactive counting book that we never get tired of looking at.

My own children may or may not decide that math is their thing, but as a parent I hope to give them the gift of knowing that it can be if they want it to, and to share ideas with other grownups who want to do the same thing for the children in their lives.

Sharing Brownies

January 13, 2021January 13, 2021 / Amy / 2 Comments

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!

Finger Counting

February 14, 2020February 14, 2020 / Amy / 1 Comment

How do you count on your fingers?

Here’s how I do it:

I think it’s a pretty normal way to count on fingers, and when I asked a group of adults to count on their fingers, most of them did either this, or started with their thumb.

But then I asked my 6-year-old how he counts on his fingers and this is what he showed me:

I had to have him do it for me a few times because it was so strange! It starts off normal enough, but then when he moves from two to three, instead of just holding up the next finger, he puts down his first finger and holds up a completely different set of fingers!

I have lots and lots of thoughts about this, but for now I’ll just leave this here as an example of the weird and wonderful things that happen when you ask kids to show you how they think.

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).

Seeing Improper Fractions

September 23, 2014 / Amy / 2 Comments

Today I want to put in a good word for improper fractions. You know, those fractions where the numerator is larger than the denominator, like 5/3 or 289/18.

We don’t usually tell kids about improper fractions until kind of late in their fraction learning trajectory, after they’re comfortable with “normal” fractions like 1/2 and 3/4 and 2/3 and 5/8. So children naturally get used to thinking that a fraction is like a partially filled pie. 3/4, for example, means we have a pie with 4 pieces but there are only 3 left. And then once they’re really good at this idea, we spring 5/4 on them, and the kids think, “Huh? If there are only 4 pieces in the pie to begin with, how is it possible to have 5?”

Even adults can have a hard time with improper fractions.

The conceptual root of this problem (if you’ll allow me to go into math educator mode for a moment) is that when kids see only fractions less than one, they start to think that 3/4 means “3 out of 4 things” which is not quite right, because 5/4 is a totally legitimate fraction but “5 out of 4 things” doesn’t make any sense. A better way to think about 3/4 is as three 1/4’s, where it takes four 1/4’s to make a whole. 5/4 then means five 1/4’s, where it takes four 1/4’s to make a whole. When we think of a 1/4 as a unit, we can have as many of them as we want.

So moving back to parent mode, when we give our child chances to see improper fractions, like 5/4, in real life, and when we do this early in their fraction learning trajectory, we’re not only making improper fractions themselves easier, we’re helping them develop a strong and solid understanding of what a fraction is in the first place.

Here are a few ideas for seeing improper fractions in everyday situations:

Graham Crackers

Graham crackers are great for introducing fractions to young learners because they break naturally into halves and into fourths, and because those halves and fourths are an identifiable unit. A graham cracker square can be called a half. A small graham cracker rectangle can be called a fourth. Four small rectangles make a whole cracker – that’s why they’re fourths. And it’s not at all inconceivable that you could have 5 or 6 or 7 or more small rectangles: 5/4, 6/4, 7/4, and so on.

Measuring Cups

Measuring cups are also great for thinking of fractions as units. It takes three 1/3-cup measuring cups to fill up a 1-cup measuring cup; that’s why it’s called 1/3. 1/2-cup measures and 1/4-cup measures are similar. You could experiment and see how many 1/3-cup measures would fill a glass measuring cup up to the 2 cup line. That’s six thirds (6/3)! How many thirds would it take to fill it all the way to the top, above the line? 7 thirds? 8?*

Pizza

Or quesadillas, pies, mini bread loaves… The important thing is that a) you can cut the food into equal-sized servings, and b) you have more than one whole (whole pizza, whole quesadilla, whole pie, whole loaf). If a child can identify a piece of pizza as 1/8 of a pizza, and can count pieces as eights (one pieces is 1/8, three pieces are 3/8, etc.), they can also tell how many eights 10 pieces would be, or 15, or how many eights there are in 2 pizzas.

Ruler Measurements

It’s common to use fractions in measurements – a quarter inch, a half centimeter. We have to have a way of naming measurements that are in between whole number measurements. If you’re working on measurement with your child, you’re probably using mixed numbers (e.g., 2 1/2 inches) rather than improper fractions. But go ahead and try to make the leap. If something measures 2 1/2 inches, ask: “How many half inches is that?” or “How many quarter inches is it?” This is more for older children – this is more challenging than pizza where you can see and hold and count an eighth. But it never hurts to ask something that’s beyond the child, and come back to it later if you find it’s beyond the child’s current capacity.

* As an aside, a colorful set of plastic measuring cups and spoons (maybe even one with a 1/8-cup measure) is a great Christmas or birthday gift for a child. It’s not terribly expensive, and opens up opens up all sorts of opportunities for experiences, creativity, and one-on-one time with parents or older siblings and (bonus!) they have their very own tool for thinking about and reasoning with fractions in a completely natural context.

Scavenge for Quantity

September 5, 2014September 5, 2014 / Amy / Leave a comment

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:

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.
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.
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!

Reading Share: 5 Ways To Help Your Kid Not Stink At Math (NY Times)

August 4, 2014August 6, 2014 / Amy / Leave a comment

Recently the New York Times magazine published an article on the recent history of math education in the U.S., written by Elizabeth Green. It’s a fantastic read for anyone who is at all invested in the education of children (which, of course, includes all parents!). I highly recommend it, and I’m looking forward to reading her new book, Building a Better Teacher.

But the article, while very readable, is also quite long. As a parent, and a parent who loves to read, I still may not have made it through if it wasn’t exactly up my alley. So if you’re looking for something shorter and directly relevant to how you interact with your kids around math, I’d like to point you to Elizabeth Green’s accompanying blog post on the New York Times parenting blog, Motherlode: 5 Ways To Help Your Kid Not Stink At Math. It contains excellent advice, and it’s solid – these are things that really work, no matter what math curriculum your child might be using.

While all 5 of her main points are good, my favorite is the first:

1. Listen to What’s Going Wrong

Teaching children math requires first figuring out what they don’t understand. Instead of getting to the heart of a misunderstanding, we are far more likely to tell children something like, “No, that’s not right, try it this way instead.” The better response to a wrong answer begins with asking the child to explain her thinking.

You’ll hear this from me over and over again, but the very best thing you can do to help your child, whether you’re encouraging a preschooler’s interest in numbers, or helping a high schooler with their math homework, is to make an effort to find out what they’re thinking.

Click over and take a look. And if you have other resources (books, articles – anything!) that have helped you as a parent to help your child with math, please share!

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.

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