# Seeing Improper Fractions

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

The beginning of the college semester has kept me extra busy for a couple weeks, but I’m back. And I have a game this time that can be adapted for a very wide variety of ages – a scavenger hunt!

One way to help your child to internalize mathematical thinking is to open their eyes to the quantities around them. As parents, we help provide our children ways to see the world from an early age. We name objects, identify colors, describe what we’re doing. A child doesn’t necessarily see the color blue on their own – they see blue because adults point blue out to them (listen to this totally fascinating Radiolab podcast about the color blue). We point out big and little, noisy and quiet, dark and bright – we help show them which characteristics are worth noticing.

Quantity is also a characteristic. So many things in our world come in quantities, but since kids don’t tune in naturally to the characteristic “how many,” adults can help by pointing it out. I’m not talking about learning to count – I’m talking about learning to see quantity as an attribute, something we can recognize, describe, and use to characterize parts of our world, just like color. Just as Monkey’s t-shirt and the playground equipment in the picture below, although entirely different in other ways, share the attribute “gray,” the flower petals and the dots on the dice below the photograph share the attribute “five.”

A scavenger hunt for quantities can help your child become accustomed to seeing quantity as an attribute. There are all sorts of ways to do a quantity scavenger hunt. Here are three:

1. Open Scavenge: Pick a room in the house and work together to find as many quantities as possible. You might choose the kitchen and find that there are two salt/pepper shakers, six chairs around the table, four drawers beneath the counter, three slats on the back of the chair, two towels on the front of the oven, and so on and so on. This can be played one-on-one with a single child or cooperatively with two or more children. Some children may enjoy labeling quantities with sticky notes or a label maker.
2. Number Challenge: Give your child (or children) a paper labeled with the numbers 1-12, or a set of sticky notes labeled 1-12. Then roam throughout the house and try to find a quantity for every number. With multiple players, everyone can have their own set, the rule being that no two people can choose the same set of objects for their number. Because some numbers will be much harder than others (how many things come in sets of 11?*), you may wish to eliminate hard numbers, set a time limit, or simply play for fun with the expectation that you might not find everything.
3. Quantity Match: Can be played with a parent and child, or with two children. Have one player find a quantity (four shelves, for example). The other player then has to find something different, but with the same quantity (four pillows on the couch!). Then switch places. Repeating quantities is okay (and even repeating sets of objects for very young children).

Again, this is really easy to adapt, and it’s easy to involve multiple members of the family – including younger and older children. Have fun with it!

And as a side note, it’s never to early to start helping your child notice quantity. With very young children, you can point out any quantity but especially two – there are all sorts of twos in a baby or toddler’s life! Two ears, two hands, two feet… This morning as I was changing 16-month-old Monkey out of his pajamas, I counted “one…” as I pulled one foot out of the footie, and as I reached for his second foot he responded with, “dooh!” A few more repetitions convinced me that it wasn’t just coincidence – he’s picked up on the “one…two!” count he’s been hearing from me, and I wasn’t even really thinking about it. Way to make your math mama proud, Monkey!

* Look! 11 slats on the shoe rack!

# Book Review, and a Message about Math Messages

Yesterday I passed by a bookstore display of The Numberlys by William Joyce and Christina Ellis,* and my children’s math radar buzzed – was this some hot new math picture book that should be on my radar? So I stopped, and I read the book, cover to cover (it’s not long). And it turns out that it’s not a math book…but the math teacher/math lover/math advocate in me had a strong negative reaction to the message about math that it nevertheless sends.

I hate to write a negative review of The Numberlys, because it has so many charming qualities – beautiful illustrations, a cute and quirky setting, a message about creativity and nonconformity and cooperation that doesn’t (to me) feel tired or trite. It’s cute, it’s different, it’s fun.

But my problem with the book, and I think it’s a relevant one whether or not this particular picture book has crossed your parent path, is with the subtle message it sends about numbers and, therefore, mathematics.

Here’s the plot summary, directly from Amazon:

Once upon a time there was no alphabet, only numbers…

Life was…fine. Orderly. Dull as gray paint. Very…numberly. But our five jaunty heroes weren’t willing to accept that this was all there could be. They knew there had to be more.

So they broke out hard hats and welders, hammers and glue guns, and they started knocking some numbers together. Removing a piece here. Adding a piece there. At first, it was awful. But the five kept at it, and soon it was…artful! One letter after another emerged, until there were twenty-six. Twenty-six letters—and they were beautiful. All colorful, shiny, and new. Exactly what our heroes didn’t even know they were missing.

And when the letters entered the world, something truly wondrous began to happen…Pizza! Jelly beans! Color! Books!

The message? Numbers are gray, dull, orderly but boring, and definitely not colorful and creative. That’s what letters are for!

Do you see the problem with this message? It’s a message that I doubt Joyce and Ellis intended to send, and it’s a message easily overlooked. But it’s a message gets repeated over and over to children. Math is boring. Words are fun. Math is rigid and structured and words are beautiful and creative. I am particularly attuned to these messages because I was a child bursting with imagination (like most children!) and I got the message – math had no place for me and my creativity. It wasn’t until college that the message was disrupted and I discovered I could find just as much beauty and creativity (just of a different kind) in the world of numbers.

As parents, we can help combat that message by recognizing it, and by countering it. Have you heard of the praise-to-criticism ratio? This originally comes (as far as I can tell) from a study at my alma mater, the University of Michigan, examining low- and high-functioning business teams. They found that the praise-to-criticism ratio in high functioning teams was almost 6 to 1. That is, every criticism was balanced out by six compliments. I’ve heard this idea extended to marriages, sibling relationships, and parent-child relationships as well, and I like to think (with, admittedly, no solid research to back myself up) that a similar idea might apply to math messaging: for every (subtle or not-so-subtle) negative message a child gets about mathematics, they should hear six positive messages.

And so with that in mind, here are six positive messages I’d like to send out in response to the unfortunate negative message in The Numberlys.

Math-terpieces: The Art of Problem Solving by Greg Tang (get your art and math fix at the same time!)

Anno’s Mysterious Multiplying Jar by Masaichiro and Mitsumasa Anno

How Much is a Million? by David M. Schwartz (my husband still remembers this one from Reading Rainbow)

Math Curse by Jon Scieszka and Lane Smith

G is for Googol: A Math Alphabet Book by David M. Schwartz

The Boy Who Loved Math: The Improbable Life of Paul Erdos by Deborah Heiligman and LeUyen Pham

What are some of your favorite math-positive books for kids?

* Apparently this book began life as an iPad app, but I am not familiar with the app, and \$5.99 was more than I was willing to pay to check it out. If any of you are familiar with the app, I’d be curious to hear how it compares with the book.

# Taking Turns: From Quizzing to Playing

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

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

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

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

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

Conversation 1:

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

Child: [looks]

Conversation 2:

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

Child: Triangle!

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

Child: Circle!

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

Child: Square!

etc.

Conversation 3:

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

Child: Triangle!

Parent: Good job! Now you ask me one.

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

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

Child: Octagon.

Parent: Good! My turn?

Child: Yeah!

etc.

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

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

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

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

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

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

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.

# 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.