# The Real Math

There’s a phrase my college students often use that used to drive me crazy. The phrase is “do the math.” When my students use this phrase they always, without fail, mean, “do the algorithm,” or “follow the procedure,” as in, “I solved the problem with a picture, and then I did the math to make sure I was right.” The implication, of course, is that other solution methods are not doing math, that you’re only actually doing math if you’re doing what you learned in school.

A couple days ago I had an interesting interaction with my 7-year-old. Bear with me for a moment. These are related thoughts. He had taken it upon himself to count up the number of dots on the step stool in our bathroom.

He proudly showed me how he counted the number of dots in a row (13) and the number of rows (9), and then broke the 9 apart into 5 and 4, used skip-counting to multiply 13 by 5, and then by 4, and then added everything together to get 117. It was wonderful. He came up with it all himself, and it made my math teacher heart happy. And to top it off, once he finished showing me his method he wondered if he could think about it differently, and he broke apart the 13 into 10 and 3 and solved it anew and got the same answer. I was just melting with delight.

But that’s not what I want to write about. What I want to write about is the thing he said next, which was: “Now I’m going to figure it out for real.”

Wait, what? He just did figure it out for real! Twice!

My son ran into the bathroom and grabbed the step stool and brought it in and set it on the chair (not the table, thank goodness) and proceeded to count all the dots one by one by one by one. 117 is a lot of dots to count, with a lot of room for error, but it wasn’t until nearly the end that he did indeed make an error. I watched as he counted the dots in the 11th column and then brought his finger back down to the 11th column and counted them all over again, and as a result his final count was 126, not 117.

“Huh,” he said. “I guess there are 126 dots.” He was completely unfazed by this. Never mind that he had already solved the problem and gotten 117, and then solved it in a different way and gotten 117 again. To him, the count was obviously the more trustworthy method.

Of course I intervened. I pointed out that he got 117 twice, and shouldn’t his count match up with his earlier solutions? (He didn’t buy it.) I suggested that he count again, and I helped him keep track of the dots this time, and he found it very exciting when they turned up 117. In fact, he ran to tell his dad (who had already seen his first solution):

“Daddy! I found the proof that there were 117 dots!”

“Oh, really?” his dad asked, with interest. “How did you find the proof?”

“I counted!”

Of course I immediately wrote this all down.

Okay, so now to connect the dots (ha ha). This experience with my son reminded me of an experience with a college student just a few weeks ago, who came to me with her solution to a fraction division problem, one that she had solved beautifully with picture. “But then I did the math,” she told me (there it is, did the math), “and I got a different answer. I just don’t understand what I did wrong in the picture.”

As it turned out, she hadn’t done anything wrong in the picture. She’d just made a simplification error in the algorithm, when she did “the math”, and she never caught it because when the answers turned out to be different, she immediately assumed her picture was wrong, and that was what she checked, over and over again. It never occurred to her to double check her algorithm.

It was just like my son, who didn’t even consider that counting could lead him astray. The bug in relying on the old reliable method is that it’s much harder to see when it fails.

Like I said, the phrase “doing the math” used to drive me crazy. And it used to drive me crazy that when my students solve a problem in a new, much more conceptually meaningful way, they always want to check themselves by “doing the math”. But it doesn’t drive me crazy anymore (or at least not as much as it used to). Because for my students, “doing the math” has meant doing a procedure or an algorithm for years and years and years, and as long as they’re following the steps it’s been pretty reliable. New methods, even if they make sense, even if they make more sense than the algorithm ever did, are still new, and there’s still a level of distrust that they’ll actually work. Or maybe not distrust so much as surprise that it really does work. When they check themselves with “the math” it helps them build their trust in the new way. It’s confirmation.

That’s exactly what my son was doing, too. He knows how to count, but multiplying? That’s strange, uncharged, exciting new territory for him. Multiplication is still a little bit like magic, and counting is real, but it’s the counting that helps him pull back the curtain on multiplication magic and come to see that his reasoning, if not yet completely trustworthy, is just as real.

# Ice cream! Get your ice cream!

A couple weeks ago I was browsing the kids’ educational selection at my University bookstore and I wondered, “What would I put in a kids’ math workbook?” Something more interesting than “count the birds and trace the number,” something I would enjoy talking about with my kids, something involving some creative mathematical thinking.

It was just a thought, but then suddenly all the schools were closing and my Pre-K daughter and 1st grade son were faced with two weeks outside of school, and suddenly I saw an opportunity to play around with that little idea I’d had.

So yesterday I made my first attempt, using one of my favorite exploratory math topics – permutations! Then I brought them home to my very eager test subjects.

When I did it with my 1st grade son it was so fun! He loved it, and it was fascinating for me to see and hear him think through the permutations of ice cream cones. We had some great mathematical discussion, and he was very engaged, and had some surprisingly sophisticated thinking.

My 5-year-old, just as I predicted, basically turned it into an art project. I think there was still at least a little bit of mathematical value in the activity, but I had some thoughts afterward about adapting it to be more Pre-K friendly. I’m still not quite sure I’m there, and would love some feedback.

I’m happy to share the files and would love to know how your kids respond if you use them. I’d also love feedback on how to adapt them. The “Ice Cream” file could probably be used with 1st through 3rd graders (and maybe even older), and the PreK version is, of course, intended for Pre-K.

What would you like to see in a “workbook” geared toward creative mathematical thinking for young children?

# 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?

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).

# Feeding, Fishing, Prompting, Probing

Our 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.

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.