What do you love about math?

I asked this question in our Math Ed 101 class this week, a class for students beginning or considering the math education major. As I expected, several students eagerly raised their hands.

The first student said, “Maybe I’m not supposed to say this, but I actually love mindless exercises. I like following a process and getting an answer, over and over. It’s very satisfying.”

Another student said, “I love the certainty of math. I love knowing that there’s a right answer, and that there’s a way to get there.”

And so it continued, with all the answers in more or less the same vein.

I ask this question when new majors come into my office for advisement, too, and I have heard what many, many prospective math teachers love about mathematics.

Here’s what no one says:

  • I love how there are so many ways you can solve a math problem.
  • I love how creative mathematics is.
  • I love it exploring a problem when I don’t even know where to start.
  • I love how math helps you see connections between ideas that seem totally unrelated on the surface.
  • I love using math to try to understand complex, real-world problems.

One of the things I love about my work is that I get to take my prospective teachers’ love of mathematics and expand it, from a math that is simple and certain, to a math that is complex and surprising and connected and beautiful.

But on Wednesday when I listened to this particular group of students, I wondered what would happen if future math teachers came to us already loving the complexity and surprise and connectedness and beauty of mathematics, instead of loving math because of the exercises. Are those students even out there? And if they are, why aren’t they the ones choosing a math education major?

This is an incomplete thought, but I’m learning to be okay with pressing publish on an incomplete thought.

The power and potential of an incomplete idea. That’s one of the things I’ve learned to love about mathematics.

Black Magic Math

Yesterday I asked my elementary education majors to solve this division problem in as many ways as they could.

A teacher has some crayons. There are 24 crayons in each box. If there are 192 crayons altogether, how many boxes of crayons does the teacher have?

During the sharing out portion, one student said, “I changed it from 192 divided by 24 to 200 divided by 25, because that was easier, and got 8. And that was the right answer. I don’t know why it worked. It’s pretty much black magic.”

Once I had a student who performed black magic on her math all. the. time. She would do something like this: “First I divided 192 by 2 to get 96, and then I divided 96 by 3 to get 32. Then I multiplied the 2 and the 3 and got 6. Then I divided the 24 crayons by 6 and got 4. And then I multiplied that by 2 since I divided by 2 at the beginning, and I got 8, which was the answer. But I have no idea how it worked!”

And while the other students would puzzle over it, I would stare at the board trying to figure out whether it actually did work, because when I looked at it from one angle it looked like a random string of operations, but when I looked at it from another angle it seemed like maybe there was some sense to be made of it all.

I love solutions like these, because sometimes it’s the former, and sometimes it’s the latter. Usually in my classes “black magic math” happens because the student already knows the destination (8 boxes, in this case) and simply performs operations until they land there. That’s what happened in the second solution.* But sometimes behind those operations there turns out to be an underlying mathematical structure that’s completely reasonable. This is true of the first.** The fun is figuring out which it is. The challenge is bringing my students along for the ride without losing them.

What “black magic math” have you encountered, inside the class or out?

* But I think it only takes one change, in the last step, for this black magic solution becomes mathematically justifiable. I’m curious what other readers might think.

** The crayon boxes can help you see the structure in the first solution. 8 boxes of 24 crayons is 192 crayons. If we add a crayon to each box, there are now 8 more crayons (200 altogether) and 25 in each box. The number of boxes remains unchanged. No magic required.

“I’m not very good at counting.”

This is what my almost-5-year-old daughter said to me on the way to preschool this morning when I asked how many kids were in her preschool class. She was thinking about how many lollipops she might need to bring on her birthday in two days, and I actually know how many kids are in her preschool class but was curious about what she would say.

What she said was, “I don’t know. I’m not very good at counting.”

“Hm,” I said, because I couldn’t just let that statement stand. “How high can you count?”

“Let me tell you!” she said excitedely, and proceeded:

“One, two, three, four, five, six, seven, eight, nine, ten, twelve, thirteen, fourteen, sixteen, seventeen, eighteen, nineteen, seventeen, eighteen, nineteen, twenty, twenty-one, twenty-two, twenty-three, twenty-four, twenty-five, twenty-six, twenty-seven, twenty-eight, twenty-nine, twenty-ten…” Here she trailed off.

“Thirty,” I prompted, and she was off again:

“Thirty-one, thirty-two, thirty-three, thirty-four, thirty-five, thirty-six, thirty-seven, thirty-eight, thirty-nine, twenty!”

“Forty?”

“Forty!”

I won’t try to unpack what my daughter meant when she said she can’t count very well (she probably meant she has never actually counted how many children are in her preschool class, and what preschooler has?). But I can tell you that Anya is actually a fairly competent counter. She’s also an enthusiastic counter, so I don’t feel too worried about this one statement in this one context.

But what she’s not good at is the teens. She’s pretty terrible at the teens, actually. Yet with minimal prompting, she was great at 20 through 40.

When we talk about numeration systems in my History of Mathematics class I give them a list of the English counting words from one to one hundred, and the first thing students notice is that the words for eleven through nineteen are weird. “Sixty-seven” at least suggests the idea of “six tens and seven” but “twelve” is just another word in a sequence of unrelated words.

This means that once children learn the number words from one through ten, they still have to conquer eleven through nineteen by brute force. To learn the numbers from 20 on kids can make use of patterns, but before that it’s pretty much pure memorization.

Ultimately, though, patterns are way more important for understanding mathematics than memorization. So when my preschool-age daughter gets held up on the memorization part, but moves on with little hesitation to the patterns of the twenties and thirties, I feel pretty good about her counting abilities.

Finger Counting

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?

numbers and lettersOne 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).

Feeding, Fishing, Prompting, Probing

number crackersOur 2-year-old, Monkey, loves numbers, and my husband and I are doing what we can to encourage his enthusiasm. For example, a few weeks back I bought a big container of number and letter cookies at Target. When we give him a cookie, we have him try to identify the letter or number on the cookie. He loves it, and so it’s both a game and a teaching moment. We’ve been surprised and delighted at how well he recognizes numbers, but he’s two and far from perfect. This means that my husband and I have to think about how to respond when he’s wrong.

There’s a subtle but real difference between the two conversations below. In the first, I give the answer away. In the second, I give Monkey a chance to come up with the answer himself. I’m communicating very different expectations and providing different learning opportunities.

Conversation 1

Mom (holding up a 6 cookie): What number is this?

Monkey: Nine!

Mom: No. That’s close. It’s a six! What number is it?

Monkey: Six!

Conversation 2

Mom (holding up a 5 cookie): What number is this?

Monkey: Two!

Mom: Not two. What number is it?

Monkey: Five!

Whether we’re helping with homework or just engaging in conversation, the way we respond to our children in interactions about mathematics can make a difference in their opportunities to learn and their attitude toward math. Our questions can feed or fish, prompt or probe, and thereby subtly or even dramatically affect the nature of our interaction.

Fishing

Feeding is when we give our child the answer outright. This can be appropriate with very young children. If Monkey can’t identify a 6, telling him that the cookie he sees is a 6 helps him make a connection between the written numeral and the word. As he makes more of those connections, he’ll gradually become able to identify 6 on his own. However, once a child is older or more fluent in a skill, feeding is less productive because it removes the necessity for thinking.

Fishing is when we have an answer or solution path in mind and we’re trying to get our child to land on it. It’s a game of “guess what the grownup wants me to say,” and it’s not very conducive to learning. But it can also sometimes be hard to recognize when we’re fishing. In the second conversation with Monkey, I knew he could identify a 5 and thought I was giving him an opportunity to think harder about what he was seeing. But it could very well be that he was throwing out number words to see which one landed and earned him the cookie.

Prompting is when we ask our children questions to help lead them to the correct answer or down a solution path we know they are capable of following. In teaching we use the term “scaffolding,” which is when, with a little bit of help, a child can do something that they are not capable of doing on their own. With good prompting, the child is doing most of the thinking and getting just the help they need. Less effective prompting can keep a child from connecting the dots and truly understanding:

Parent: If everyone in our family eats two pieces of pizza, how many pieces of pizza do we need?

Child: I don’t know.

Parent: How many people are in our family?

Child: 4.

Parent: What’s 4 x 2?

Child: Umm…

Parent: What’s 2 + 2?

Child: 4.

Parent: Plus 2 is…?

Child: 6.

Parent: Plus 2 is…?

Child: 8.

Parent: How many pieces of pizza do we need?

Child: 8.

In this conversation, the child could get every answer right without every seeing the logic of the solution method. Contrast the conversation above with the one below, in which prompting is used to help the child follow their own inner logic.

Parent: If everyone in our family eats two pieces of pizza, how many pieces of pizza do we need?

Child:  I don’t know.

Parent: How could you figure it out?

Child:  Umm…

Parent: How about four pieces. Is that enough?

Child:  No. Then only two people could have pizza.

Parent: So how many more do we need?

Child:  Um, four?

Parent: Why?

Child: Because we have two more people.

Parent: So how much pizza do we need?

Child:  Eight pieces.

Probing is when we are asking our child questions not to get them to an answer, but to get at their thinking. Probing is often the most effective type of questioning a parent can do, because it encourages a child to be aware of their thought processes and to have confidence in their own reasoning. And probing without prompting is really hard to do when your child gets an answer wrong! When a child answers a question wrong, we want to get them to the right answer. But sometimes all a child needs is to think aloud in order to see what they’re doing wrong, and sometimes as a parent we need to know how they’re thinking in order to best help them. Both of these require the parent to just give children space and opportunity to reason on their own.

A note: Young children are not very good at explaining their thinking, but they should be given the opportunity anyway. As they grow older, they will become more sophisticated at explaining their thinking because they have had the expectation laid out, and have had opportunity to practice.

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.

graham cracker

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?*

measuring cups

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.

pizza

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.

ruler

 

* 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.”

IMG_1689

flower dice

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!

scavenge

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!

IMG_0503