Interesting Mathematics

My 8-year-old was talking about doing some math this summer. “But interesting math,” he added.

“What’s something that’s not interesting math?” I asked.

“You know, like, 1 + 1. That’s not interesting.”

Well, he’d thrown down the gauntlet. So I got a piece of paper and wrote: 1 + 1 = 10. “True statement,” I told him, and then I introduced him to binary and he was engrossed in translating numbers to and from binary for the next half hour. 

Binary was the first thing that came to my mind, but binary isn’t the only way to make 1 + 1 interesting, and I have many more ideas where that came from.

What Counts as Counting?

On Wednesday we hiked to Corona Arch, just outside of Moab, Utah. The trail cuts across red rocks, too sturdy for human feet to cut a clear path, so much of the trail is marked with faded blue paint strokes every few yards. As we followed along, my 6-year-old said, “I’m counting the blue marks, but without numbers.”

“That’s impossible,” her older brother retorted. “You can’t count without numbers.”

“Yes I can,” she insisted.

“Are you noticing each mark?” I asked.

She agreed that this was what she was doing.

“But that’s not counting,” my son pressed.

“It’s like the ‘fish, fish, fish’ sketch from Sesame Street,” my husband said, and he was exactly right. In the “123 Count With Me” Sesame Street sketch (brilliantly commented on by Steven Strogatz in the New York Times), a furry monster tries to communicate a certain number of fish to Ernie as “fish, fish, fish, fish, fish, fish,” whereupon Ernie explains that a better way to communicate the quantity is with the number “six.” 

So when my daughter said she was “counting without numbers’, she was fish-fish-fishing the blue marks – noting each one, without assigning a number word. I knew what she was doing. But the dispute was about what to call what she was doing. Certainly she was engaging in a component of the counting process. I teach my college students that successful counting involves a one-to-one correspondence between words and counted objects, a consistent ordering of counting words, and a recognition that the last number said is how many there are. On the surface, she was just doing the first of these.

But I want to go deeper here, because the next thing that happened was that my son started wondering about how to say a number in binary, something that had come up the other day. In base ten, we group numbers by tens. As we count, we begin with “1, 2, 3, 4, 5, 6, 7, 8, 9”, but then instead of inventing a new number for one more than 9, we write 10, which means 1 ten and 0 ones. This means we can count as high as we want without continuing to invent new numbers to place in the sequence.

We don’t have to use ten, though. Various languages and writing systems have used 5, or 20, or even 60 for grouping. Binary uses 2. 1 is followed by 10, which doesn’t mean ten but rather 1 two and 0 ones. This is followed by 11 (1 two and 1 one), then 100 (1 group of two twos, or 1 four), and so on. If you count in binary, it’s still counting. 

But juxtaposing the counting without numbers discussion with the binary discussion made me wonder: what if we go still further and just count by ones? What if we called 1 thing “one”, 2 things “one one”, 3 things “one one one”, 4 things “one one one one”, and so on? Is this base one? And wouldn’t this still be counting in a sense? It’s like a tally. If I keep track of points in a game with a series of hash marks, I’m still counting points, aren’t I? If Ernie can ask, “How many fish?” and the monster can respond, “fish, fish, fish, fish, fish, fish”, and Ernie can provide him with the correct number of fish, haven’t the fish been counted? 

Granted, if you asked my daughter how many blue marks there were, she would not have been able to respond “mark, mark, mark…” with the correct number of marks. I don’t think what she was doing could actually be defined as counting. Her “counting without numbers” was not actually counting. And once I started asking my family if what she was doing was counting in base one, it became clear that I was the only one still interested in pursuing this line of thought. 

Still, I did find it intriguing that my daughter called what she was doing “counting without numbers.” Counting doesn’t stop being interesting once kids move out of preschool.

Everything is Math?

Not long ago my kindergartner was sitting at the counter finishing her breakfast toast when she pointed to two water bottles sitting in front of her. “Mama, is this math?” she asked. I tried to figure out what she was asking about. Were there number markings on one of the water bottles? I didn’t see any.

“Is what math?” I asked. 

“This,” she said, grabbing the yellow water bottle and moving it back and forth.

I stood up and walked over to see. She was looking intently at the water in the yellow bottle as it passed back and forth in front of the green one. As she did so, light shone through and turned the water in the yellow bottle green. 

“Oh,” I said, “how it changes the color?”

“Yes,” she confirmed. “Is that math?”

In retrospect I should have turned the question back to her: “Do you think it’s math?” or “Tell me why you think it might be math.” The question “What is mathematics?” is a really interesting and complicated question–we spend a whole semester talking about it in my college-level History and Philosophy of Mathematics course!

But sometimes daily life is distracting and I don’t think to ask the good questions until later. This time I just said, “Actually, what that looks like to me is science. Science and math do have a lot in common, though.” 

My second grader was listening in and he was not satisfied with my response. “But it kind of is math,” he piped up. “Because really, everything is math.”

I paused. “You’re right,” I said to him, and to my daughter. “There’s math in everything.” Because, even though I think it’s a bit more complicated than “everything is math”, so many of my college-age students tell me they never felt math was applicable to them, that math seemed unrelated to the questions that interested them. So I should want my kids to see math wherever they want to look for it, and be willing to look for it where they want to see it.

A Problem Interrupted (and thoughts on the “right” way to respond)

Yesterday I showed this picture to my kids and asked whether they thought there were more black ants or more red ants. Both of them immediately wanted to start counting the ants, but I stopped them because I was interested in how they thought about it without counting. 

My kindergarten-age daughter initially thought there were more red ants, because the arrangement looked “longer”, and indeed when she counted how many ants there were in a row, she found that the rows of red ants were longer and was satisfied that there were, indeed, more red ants. But when (with a little prompting) she counted the columns, she seemed surprised to count a longer line of black ants. She seemed more delighted than puzzled at this outcome, and was perfectly content to leave the question unresolved.

My 2nd-grade son decided pretty quickly that there were the same amount. He had seen his sister count the rows and columns, and in response to the puzzle of the different outcomes, he said, “You have to figure out how many are inside.” He then proceeded to count all the ants around the perimeter, and found 24 in both cases. “So since there’s the same amount all around,” he told me, “you know there will be the same amount inside.”

And then it was bedtime, and so I left it at that, with my daughter not knowing the answer and feeling interested but not overly invested in the question, and my son feeling quite confident in his incorrect, but very reasonable, solution. 

The fact that the conversation was interrupted gave me more time than usual to think about what happens next. Do I…

  • Have them count the number of ants to see that there are actually 49 black ants and 48 red ants?
  • Explore similar problems, with different numbers of ants?
  • Manipulate the original pictures to help them  visually see the difference between black ants and red ants?
  • See if they can create configurations of ants that push up against their initial theories?

When confronted with a “how would you respond?” question, my college students often ask, “But what is the right way to respond?” or “What is the best way to respond?” This is a hard question to answer! There are often some good ways and some not so good ways that you could respond (and sometimes a “good” way can backfire, and occasionally a “not so good” way can turn up something surprising). I don’t know that you can know the right way to respond, and sometimes you’ll experiment and it won’t go so well and then there’s always a next time.

I haven’t decided yet what direction to take (I’ve gotten lots of great ideas from folks on Twitter), but while whatever I do may or may not be “right”, I do have some general principles. I want to work with the thinking they’ve already given me rather than impose my own thinking on them. I want to spark their curiosity. I want to be flexible enough to go a different direction if that’s where their interest takes them. And I want whatever I do to happen in an environment of love and care.

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?