Notice and Wonder

I had a small math interaction with my kindergartener yesterday while we were doing some much-needed toy decluttering. We have a collection of toy cars that she plays with occasionally, but not often enough that it makes sense to keep all of them, so I told her she could keep ten. She pulled out a handful of favorites and counted: “One, two, three.” Then she added a couple more to the pile and counted them all again: “One, two, three, four, five.” Then two more: “One, two, three, four, five, six, seven.”

“Wait,” I said, seeing an opportunity. “You have seven already, and you get to keep ten. How many more can you choose?”

“I wish you hadn’t of asked me that!” she said, with a big sigh of 5-year-old exasperation. But she took up the problem. She began holding fingers up one at a time while quietly counting them, but stopped after a moment and just held up all the fingers on one hand and two fingers on the other. Then she looked at her hands and said, “Three!”

“Great!” I said. “Go ahead and choose three more cars that you want to keep.” 

“Do you know what I did?” she asked, before she started picking up the cars. “I looked at my fingers and I saw that I had three fingers down and I knew that’s what I needed.”

This was a brief interaction…but there was so much in that interaction for me to see! 

If you spend any time in the math education world these days, you’ll probably come across the questions What do you notice? and What do you wonder? The questions, when used well, can turn a mathematical activity from a teacher-driven activity to a student-driven activity, by stirring up and amplifying students’ inherent curiosity.

But these terms, notice and wonder, are also helpful in thinking about how we as parents or caregivers interact with children around their mathematical thinking. Instead of trying too hard to direct their mathematical learning, we often do more good by developing our own curiosity about their mathematical thinking, and our ability to simply notice what is happening in a given interaction.

Here are some of the things I noticed as I watched my daughter solve this problem: 

I noticed that she was smiling when she said, “I wish you hadn’t of asked me that!” Her big performative sigh told me that she knew I was asking her to do something she considered harder than simply counting the cars over and over until she reached ten. But this is a child who will not do something that she doesn’t want or have to do, and the smile told me she was up for the challenge.

I noticed that she didn’t count out all seven fingers. She started to, but quickly realized that she could just hold up five and two. This is a new development. She’s been counting on her fingers enough now that she is beginning to know what seven looks and feels like. Seven is more than just a count that ends at seven. It is also a relationship: a five and a two. 

I noticed that she also didn’t count the three or the ten. Once she had seven fingers up, she could see everything she needed to solve the problem: seven fingers up, three down, and ten in all. This is another relationship: seven fingers up and three down forms a ten. But also, she didn’t have to model all the actions of the story. She could look at a static image of seven and three and ten and relate it to a story with a beginning (cars she’s already chosen), middle (cars she still needs to choose), and end (all the cars she gets to keep).

I noticed that she explained her thinking unprompted. I almost always ask my kids, “How did you think about that?” and my husband does this, too. It’s just something we do in our household. So it wasn’t the explaining that was noteworthy here, but rather the fact that when I didn’t ask her (because my adult brain was already moving back the goal of decluttering the play space), she launched into an explanation anyway. It’s no longer just a parent-driven part of a math interaction–she now expects and wants to share her thinking.

One of the best things we can do to lay the groundwork for a love of mathematics is to notice what our children are thinking and seeing and doing (not just what we want them to think or see or do), to wonder with them, and to truly experience the wonder of watching them. When we approach our mathematical interactions with children with genuine curiosity, we see more, and children see that we see them.

If You Give a Kid a Calculator, Part I: Playgrounds

Our family is lucky to live in a small city with great playgrounds. There are at least ten playgrounds in less than 20 square miles, which means that no matter where you live within the city, you’re likely within walking distance of one of them. They’re good playgrounds, too. Each is a little unique, most of them have been updated within the past few years, and all of them are surrounded by a good amount of open, grassy space for running around. As a parent of young children, this is one of the things I like best about where I live.

I wasn’t initially thinking about playgrounds when I first sat down to write this particular blog post. I went through several drafts before I got here. What I wanted to write about initially (and still do) was calculators, and how you should give your kid a calculator, earlier than you think you should give your kid a calculator, just to see what they do with it.

I got this idea from another parent, who had found that a calculator was the perfect toy to keep her kindergartener occupied during hour-long church services. So when my oldest was four, I went out and bought a cheap, solar-powered calculator and handed it to him just to see what he would do with it, and he loved it. A couple weeks ago at the start of the holiday break we were cleaning the house from top to bottom and came across the dusty calculator, languishing underneath a piece of furniture, and my now-7-year-old cried out, “My calculator! I love this!” and immediately jumped back into playing with it. 

Calculators sometimes get a bad rap, partly because most of us adults grew up in math classrooms where we were discouraged from using calculators except in carefully monitored situations. Then we all grew into a world where we basically have a calculator within arms reach 24 hours a day, 7 days a week, and it’s a tool we all use, but that we’ve nevertheless been conditioned to think of as sort of a cheat.

But to my child, the calculator is not a cheat or a shortcut, but a playground. No one has told him how it should or shouldn’t be used, and he can play with it and learn things about numbers and relationships that go well beyond his current numerical reasoning ability. He doesn’t play with it constantly, but each time he rediscovers it, his calculator play has changed a little to reflect his growing mathematical knowledge. His favorite calculator game when he was four was to enter in a randomish string of digits, like 827518271, and ask me to name the number. But now at age 7 he’s doing things like noticing that 8 ✕ 10 and 10 ✕ 8 give the same answer, wondering what other operations this works for, and stumbling across negative numbers when he tries it out on subtraction. A lot of this exploration is certainly stuff he might be able to do without a calculator, but the calculator removes the tedium of hand calculation and allows him to jump straight to the big questions.

I’ve been thinking a lot about playgrounds in my own classes of college students. Years ago when I was a beginning teacher, I would create “explorations” that were really just carefully guided activities where I had a series of pre-planned discoveries I intended my students to make. Unfortunately, I often found that my students could answer every question correctly without ever truly discovering the thing I’d been excited for them to discover. I found that my students were missing out on the wonder and delight of mathematics when I laboriously pre-arranged what they were going to wonder about and delight in.

I don’t think it’s wrong to have a sense of where you want students to end up, and while I’m constantly learning, I am nevertheless a much better lesson writer now than I was twenty years ago. But more and more I’ve been thinking about what it would mean to create and make room for true mathematical playgrounds in my instruction, spaces there’s the bare minimum structure and students can pursue the ideas that spark their curiosity. This is a hard vision for me to realize in practice. College students (most adults, really) often struggle with completely open exploration, and with knowing how to access the curiosity needed to explore. But kids come at this curiosity quite naturally, and don’t feel any compulsion to make their play look productive to an adult.

I wrote a guest post at Everywhere Math a little while ago where I talked about following your child’s agenda rather than your own, and the best mathematical playgrounds give the child agency to follow their own agenda. That’s what I’ve loved about handing my son a calculator.

What makes a tool or a toy or a book or an app or a game a playground? I don’t know that I have a full answer, but here are some questions I thought of:

  • Is there more than one way to use it? 
  • Do kids have a say in how to use it?
  • Can you create your own rules?
  • When adults join in, are they playing with the child (as opposed to directing the child’s play)? 
  • Is it possible to imagine uses for which it was never intended?

If the answer to these questions is yes, then it might be a playground, and the very last step is to put it into a child’s hand and see what they do with it.

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.

Please ignore the dust. We haven’t washed it in awhile and none of the camera filters I used could hide it.

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!”



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.