Representations and Manipulatives and Tools and Things

One of the things I think about is the relationship between teaching math and the physical stuff that goes along with teaching math. This relationship gets distorted sometimes.

For a while, the elementary world got all kinds of swept up in manipulatives. All lessons became “hands-on” because somehow “hands-on” led to “minds-on.” Deborah Ball’s classic Magical Hopes article does the best job I know exposing the flaws in this stance. If you haven’t read it, by all means, click that link and read it right this second.

People sometimes get hung up on tech in the same way. Recently, I had the chance to share some outstanding work Kristin Gray got from her students when she asked them to take out their notebooks and write down what they were wondering about doubling and halving. They’d been working on 14 x 25 = 7 x 50. Check out these conjectures:

I mean, so great.

Both times I shared this work, people oohed and aahed, and then asked the same question: “Could you use tech to do this? Maybe a google doc?”

I have to confess, I don’t understand this question. With paper and a pencil, students were able to shift back and forth between words and numbers effortlessly, much faster than 11-year-old kids can type. If they’d wanted to make a quick sketch or doodle (perhaps an area model, in this situation), they could. They didn’t have to lose their train of thought while hunting through their device’s symbols for ÷ (an obelus, for fellow #wordnerds). The only apps I know that allow students to think and write so freely are apps that turn tablets into $800 notebooks by letting you write on a screen with a stylus.

I kept wondering, what’s the value added there? What’s the rationale for adding tech? What can it do for you that cheapo paper notebooks can’t?

That’s the question I ask myself about tools, in general, whether they require charging or storage in a plastic tote. What will they do for the mathematical teaching and learning here? Sometimes, the answer is not much. Other times, A LOT.

This year, I did something new in my school. My principal and I made it a priority for me to work with our paraprofessionals. These colleagues are overworked, underpaid, undertrained, and almost never supported to go to PD. Yet they’re responsible for educating about 20% of our students–the neediest 20%. So, this past year, my principal worked out a schedule where, one week per month, I had all the paras in two different shifts on Tuesday and Thursday afternoons. Over the year, we did a book study of the second edition of Children’s Mathematics. I chose it because it’s the foundation of everything in elementary math. The book and its embedded videos teach educators how students think about the operations, and how that thinking progresses. They also teach educators to listen closely–and quietly–to students while they think through problems. Given that the paras had been jumping in too quickly to tell or spoonfeed or direct kids, I wanted them to start learning how to tolerate wait time, how to trust that their students have mathematical ideas, and how to listen as a core part of teaching mathematics.

[Update, June 30 2017. I received a note from one of my colleagues–a proud paraprofessional–today. It was a hard note to read, but I am so grateful that she wrote. She taught me a ton. I am leaving the blog intact so you can see what I wrote, and what I learned. Let’s take a look at that paragraph again, but I’ll fix it, and then make changes throughout:]

This year, I did something new in my school. My principal and I made it a priority for me to work with our special education team, including certified teachers, therapists, and paraprofessionals. These colleagues are overworked and underpaid, and there are many demands on their time because they frequently have trainings about specific disabilities and student needs, as well as IDEA compliance, not to mention the meetings required for IEPs. undertrained, and almost never supported to go to PD. Yet they’re These colleagues are responsible for educating about 20% of our students–the neediest 20%–but both the literacy coach and I get less time with them than we do with the classroom teachers. So, this past year, my principal worked out a schedule where, one week per month, I had all the paras the full special education team in two different shifts on Tuesday and Thursday afternoons. Over the year, we did a book study of the second edition of Children’s Mathematics. I chose it because it’s the foundation of everything in elementary math. The book and its embedded videos teach educators how students think about the operations, and how that thinking progresses. They also teach educators to listen closely–and quietly–to students while they think through problems. Given that I’d observed that some of the paras had been jumping in too quickly to tell or spoonfeed or direct kids, I wanted them to start learning how to tolerate work on their wait time (which we all always need to work on). I’d spent a few years working with the classroom teachers, reading the same book (Children’s Mathematics), encouraging them how to trust that their students have mathematical ideas, and  how helping them learn to listen as a core part of teaching mathematics. I wanted the special education teachers, therapists, and mainstream coaches to have similar opportunities to work on their listening.

I also knew that digging into students’ mathematical ideas would allow me to get the parasspecial education team digging into the mathematics itself. One para loves math and I knew she’d be game, but many were reluctant and some were downright hostile to the idea of this year-long focus on mathematics. I had my work cut out for me last fall.

Fast forward to June, and I gotta tell you, I loved my time with the paras special education team. I think it’s some of the most important work I’ve done in the building and I can’t believe it took me so long to get there. We built a safe space and strong relationships and, most of the time, they got more and more willing to try new ideas, wonder about why things worked, and make sense for themselves. I hope they understand that I’m sharing this story partly to encourage my fellow math coaches to think about how they can support their special education colleagues, partly because it helps me make my point about tools, and partly because I’m so proud of the growth I saw over the year, and want to share it. I hope I’ve done the story justice, and I hope, if I haven’t, they’ll tell me. Nothing matters more than the trust and safety we’ve built together.

One observation I’d made over the year was that several parascolleagues openly despised both the area model and the related partial products strategy for multiplication. They didn’t understand why anyone would do that, and were resistant to multiplying any other way than the standard algorithm. As our year drew to a close, I wanted to devote time to multiplication of double-digit numbers and see if I could get anywhere with this animosity toward these two essential multiplication strategies. I knew that if I just drew or recorded the strategies with equations on chart paper, I would lose them. I’d learned that lesson the hard way, and wanted to avoid the shut down I’d caused before. So I needed something. I needed a tool that would unsettle our typical patterns.

In this case, I reached for graph paper. I handed them each a rectangle of graph paper and asked them, “How many squares are there?” Note, I did not use the word “multiply.”

Nobody shut down. Everybody got to work, and I got a great range of strategies, including the ones above. (The graph paper was 14 x 21, if that helps.)

Man I love a variety of strategies. It’s just the best. Now I had a whole range of decisions to make about where to go from here. (If you want a thoughtful discussion about that decision-making process, you need to read Intentional Talk. It’s had a huge impact on me.) There were a number of things I could do, and a number of competing goals in my head. A few of them:

  • I needed to explicitly connect what they did with the graph paper to multiplication.
  • I needed to get them more comfortable with representations of what they did with the graph paper, both in pictures and numbers. Optimally, they’d be the ones recording, not me.
  • I needed to expose some rich mathematics by digging down into one of these, or by drawing connections among a few of them. Which ones?
  • I needed to take this opportunity to highlight the fresh mathematical thinking from some parascolleagues who have had negative histories in math, who started out the year reluctant but dove into this problem bravely, and who still needed support to see themselves “as math people.”
  • I needed them to explain their strategies to one another so they could put words to their own thinking, and listen to and try to follow their peer’s thinking. (This was an ongoing goal all year and we’d made a lot of progress.)

So here’s what we did. Each person explained their strategy. While they did, I asked for volunteers to come up and record on paper what their colleague had done on our chart paper.

I stayed quiet while the person with the marker recorded, and then naturally turned to their colleagues and said, “Is this what you did?” or “What was your equation again?” We were getting somewhere.

In each group, someone had surprised me with their strategy. In the first group, J said she looked at her rectangle and thought, “Way too many to count.” So she folded her rectangle in half:

She thought, “Still too big,” and folded it in half again:

She looked at that and thought, “I can count that and then multiply by four.” The thing is, one of the factors was odd, so quartering led to a fractional result. She didn’t bat an eyelash:

She had a 7 x 10 rectangle, which yielded 70 squares. She then combined 6 half-squares to make 3, and had 1/2 left over. Each quarter had 73 1/2 squares. To figure out the total number of squares, she pinched together the 4 half-squares into 2 whole squares, multiplied 73 x 4 to get 292, and added them together to find 294 squares. She told us that the folding grew out of her comfort with sewing, and she was completely in command of her strategy.

I did the recording on J’s work. Not beautiful, but everyone agreed that what I drew matched what she did.

In the next section, L also began by folding her rectangle in half, but the other way:

Then she groused at me, “Oh, you made it not work out evenly!” A moment later, she said, “That’s OK, I love thirds.” Look how pretty:


A perfect square!

When she unfolded it, she had this:

Now we see one power of a tool. If I had explicitly asked them to solve the multiplication problem 21 x 14, I would have had almost all identical column multiplication solutions, which aren’t ripe for rich discussions. But this little graph paper rectangle yielded a wide range of approaches, including two strategies that made beautiful sense, visually, but had almost no chance of emerging if we’d only worked with numbers:

21 x 14 = 4 x (10.5 x 7) and

21 x 14 = 6 x (7 x 7)

Not only that, both J and L were in the spotlight for innovative math, when both J and L have historically not been so keen on the subject. They were the experts on these strategies, teaching me and the rest of their curious peers. If I could have bottled that moment and given it to them for safe keeping, I would have. It was a highlight of my year.

That brought us to the end of Tuesday. I knew I’d start with these on Thursday. Because J and L were in different sections, they hadn’t seen each other’s solutions, although word spread quickly throughout the staff and I heard them comparing notes after school (yeah baby). We began Thursday by marveling over the two strategies, comparing and contrasting the difference it made to want to avoid fractions or not be bothered by them. I then focused us on L’s strategy, written numerically. When we looked at the piece of paper, we could all see that there were six 7 x 7 squares. But when I wrote the equation:

21 x 14 = 6 x (7 x 7)

there was a lot of wondering about it. They all agreed that, if they’d been working with numbers only, they never ever would have transformed 21 x 14 into 6 x (7 x 7). They wondered, where did those numbers come from? Especially that 6? How could a 6 come out of 21 x 14? Eventually, there was recognition that 7 was a common factor of both 21 and 14. That insight led us to write:

21 x 14 = (3 x 7) x (2 x 7).

The 6 was in there somewhere, starting to become more visible, but this is a place where nobody was sure about the rules they’d learned once. What could you do with that 3 and 2? Add or multiply? Five or 6? We went back to the paper:

Do you see a 2 x 3 array there? A 2 x 3 array of 7 x 7 squares? Six 7 x 7 squares? Holy smokes, there it is. They saw where the 6 came from.

We played a little with the quartering strategy in the same way:

21 x 14 = (10.5 x 2) x ( 7 x 2) = (10.5 x 7) x 4

We wrote them up, talked some more about the associative property and what happens when you break factors up by multiplication. Now, to be clear, I am not saying everyone in the group would be able to recreate this logical flow independently yet. But I am saying everyone in the group was following along. They didn’t shut down at the formal math vocabulary, at the symbolic representation, at the diagram.

And that’s why I was glad to have a tool. In this case, the tool made it possible for everyone to access the mathematics here. It helped me gather a variety of solutions so we could make connections among them. It made tangible what had been abstract. It allowed my colleagues to bridge from something intuitive to something a little out of reach. And it made us talk with each other about the mathematics more, not less.

I’d call that added value.

I don’t get fussed over whether tools are high tech or low tech. I love and use them all. But I do take care to use them thoughtfully, not for the sake of using tools or tech, but for the sake of the mathematical learning and conversation they’ll allow me to engineer.

In early drafts of my book, there was a chapter called Mathematicians Use Tools. I was planning to get into all of this stuff. I cut it for the sake of length–it was already a huge book–and I thought tools had been written about a lot elsewhere. They have. I decided, instead, to showcase thoughtful use of tools throughout the book, which wasn’t hard because effective lessons often involved the strategic use of tools. Probably the right call.

There are times I regret not taking the deep dive into tools, though. I see so much tech-for-the-sake-of-tech, tools-for-the-sake-of-tools. I also see teachers still afraid to use tools for fear of mess or noise or lack of control or time or organization. I’d love to explore when and how and why we reach for them–or don’t.

Maybe this is the space.

Disrupting the Usual Rhythm

Update 12-9-15 Since pressing publish, I’ve thought of much better titles for this blog. Take your pick:

The Rhythm Method


Mathematicus Interruptus

I returned last night from a thought-provoking PCMI weekend workshop. A+ professional development, friends. Seriously, if you have a chance, go. Huge thanks to Tina Cardone for organizing it.

Each day, we alternated doing lots of math with facilitator Brian Hopkins and reflecting on our practice with facilitators Cal Armstrong and Jennifer Outz. Plus we socialized, shared ideas, and learned from PCMI alumni. It was a great mix and there are many things I could blog about, but I want to focus on some of Brian’s choices that I found surprising in the best way possible. He really has me reflecting on my practice.

Spoiler alert: I’m going to introduce some math problems – problems we thought about with other teachers for long stretches of time. After each problem, I’m going to have to give away some of the mathematics to make the point I want to make. If you want a chance to think through the math yourself, pause after each picture. Go play. Then come back. If you keep reading, you’ve been warned. You’re missing out on some of the fun.

Assurance: I’m going to be using some math vocabulary, ideas, and representations that were new to me and I spent many hours exploring this weekend. They’re still tentative for me. If you don’t take the time to figure out the math and you don’t know it already, that stuff won’t make sense as you read. Don’t worry about it for these purposes. I’m not trying to recreate PCMI here. I’m trying to make a larger point that you can totally get even if you skim over the math. So skim away, especially in this first bit. Resume close reading after the circular pennies picture.

The first morning, we started with a penny game. (Thanks, Heather, for tweeting these out!)

We tested what would happen if we had different numbers of pennies or were allowed more moves. We found that multiples of 3 were very important if we were allowed to move 1 or 2 pennies. Multiples of 4 were very important if we were allowed to move 1, 2, or 3 pennies. Brian encouraged us to think about these ideas in terms of modular arithmetic, which we’d been told was going to be a focus of this weekend. My hunch is they chose this topic knowing it was outside of most of our comfort zones. Perfect. It certainly was new to me. I went in having no idea what 2mod3 meant. By the end of the morning, I think lots of us were becoming more comfortable with mod notation and thinking. Brian closed the session with a good laugh over Survivor’s 21 Flags game.

After a reflection session and lunch, Brian passed out this page of primitive Pythagorean Triples. He asked what we noticed and wondered, and encouraged us to play with mods the way we had in the morning.

So, what do you notice and wonder? What patterns do you see? What conjectures can you make?

We had all kinds of observations about the ones that were one apart or two apart, different ways to generate subclasses of triples, and what was going on with the evens and odds. Super fun. After the group share out, our group started looking at other group’s conjectures that each triple had a multiple of 3, a multiple of 4, and a multiple of 5. This got us going again with mods, and the squares of mods.

Scan 3

When we stopped, my group was trying out mod8 to see if that helped. I’m still not sure why, but it was fun.

When we arrived the second morning, Brian had a new challenge for us:

My partner and I tried and tried to explain this game using mods.

We failed.

The pair next to us was exploring a different technique. They found that the player who forced their opponent to face a symmetric situation won every time, as long as they kept resetting the symmetry. I really encourage you to try this one. It’s lovely and satisfying.

Brian pulled us together and asked what we found. Consensus was that mods were unhelpful, and that symmetry was the more powerful idea here.

This was the most exciting moment of the weekend for me, pedagogically. The way I thought about it was that Brian disrupted the predictable, pitter-pat routine of math class. Up until this moment, all signs pointed toward mods. I assumed each new activity would build on what we’d done before in this familiar story arc. We’d deepen, add nuance, try new aspects and applications. I did not expect to have Pennies II be totally different than Pennies I, and not involve the handy new tool we’d just learned.

I started scribbling in my notes. When do we ever do this? When do we teach a new tool and then introduce a similar task where this tool is not helpful? When do we teach kids the limits of the tool right from the beginning?

What I see in schools is we cue kids to know what tool to use. If we’re two weeks into a unit on fractions and we give them a story problem, the kids figure fractions are involved. If the name of the chapter is “Multiplying Two-Digit Numbers” and it’s written on the bottom of the worksheet, the kids are going to assume they should multiply some 2-digit numbers. If we’ve written an objective about linear equations on the board, kids figure the answer is going to involve linear equations. If my new tool is the hammer that divides fractions, I’m going to use that hammer until my teacher tells me it’s time to switch hammers.

At PCMI, Brian disrupted that process for me. He gave us a new tool, and then gave us a problem where it didn’t apply. Blew my mind. Next up, though, was Pennies III, which he referred to as 11 plus 1, and later told us was called Kayles. It has the same rules as Pennies II, but the pennies start out in this arrangement.


At first, I thought, “Oh, we’re back in the rhythm. I bet this game will give us some amazing synthesis of mods and symmetry. I get it. I see his plan.” Because that’s how math classes normally work, right? I’ve cracked that code.

My table worked and worked and worked at this problem and couldn’t come up with any good rules using mods or symmetry, or anything else for that matter. When we got back together, Brian told us this problem has been around forever and wasn’t analyzed well until computers were invented. It turns out there are no elegant solutions. It’s just a messy, case-by-case thing. There were some groans in the room. Not from me. I was delighted because he’d disrupted the rhythm again. He’d surprised me.

We headed downstairs to set up for the Josephus problem. The rules he played in this version were that Josephus was going to count by 3s and kill whomever he landed on. He was also going to be in the circle. So where should he stand? Which position will be the last one standing?

Brian’s shorthand for this problem was “Duck, Duck, Die.” He began killing us off, round and round. With 29 people, player 26 survived. With 20 people, player 13 survived. What was going on here? I heard participants say, “What does this have to do with the pennies?”

Over lunch, participants tried out different numbers and Brian synthesized the data.


What do you notice? What do you wonder?

We had lots of questions and observations. Fun. But then we got to the meaty question of can you predict the solution for any number? What do you think the answer was? What do you think I thought the answer was?

If he went by typical math class rhythm, the answer would have been some marvelous synthesis of all we’d done. It was the afternoon on the last day. So could he tie together modular arithmetic and symmetry and something wise from Kayles and Josephus? Would he wrap it up in a neat, satisfying bow?

By this point, I had come to expect surprises from Brian, so I really didn’t know what was going to happen. It turns out this problem has to be solved recursively. He had a clean, recursive approach to do it, but people were unsatisfied and wanted a closed formula. He gave us this “big ugly thing.”


I don’t know what it means either. I do know we all cracked up when he revealed the constant. It turns out you have to solve the problem recursively to use the big ugly thing anyway, because that’s how you can calculate κ to enough digits.

Brian talked to us about how not everything in mathematics is solvable in a nice, neat formula. Sometimes this is how it works out.

I can’t tell you how satisfying I found it that there wasn’t some “satisfying” synthesis in the end. I mean, I love making connections. I love thinking relationally. But not everything connects nicely. In math class, we often create an artificial story arc where one thing leads to the next and then the next in this nice flow. We leave out the part of the story where mathematicians struggled for hundreds of years between these ideas we’ve connected in 45 minutes. We leave out the part of the story where mathematicians aren’t told what tool to use when they attempt a problem. They might try this or that. Maybe mods are helpful? Nope. Symmetry? Nope. Should I try to graph something here? Maybe that will be illuminating?

What I’m thinking about most, though, is that we teach in this familiar rhythm – this unit is factoring and every question will be solved using factoring – almost all year long. And then we’re frustrated when kids don’t “transfer” what we’ve taught them in novel situations. “We covered that!” we yell, when we see they didn’t get the factoring question right on the big state test because they didn’t recognize it as a situation where factoring would be helpful. We shouldn’t be at all surprised. We’ve been cuing them all along. The tool of the week is tables, or skip counting, or measurement. Attention everyone, the new tool of the week is negative numbers, or symmetry, or finding the intercepts. Without the cuing, where are they?

Some of this is for good reasons. It’s important to dig into ideas with depth and connect them. I’m not saying to jump around willy nilly. But I wonder if we can learn something from Brian? Next time you teach a tool, how about giving students a chance to figure out when it doesn’t help as soon as they’ve started figuring out when it does? That would give them a fighting chance of deciding when it might be helpful in novel situations. Brian’s way to help us think about the usefulness and limitations of our new gadget was to give us seemingly related problems that were actually quite different, mathematically.

I’d love to put our heads together and think about content examples in the comments. Pick a concept. How could we use Brian’s technique?

I keep thinking about a snippet of an Ira Glass talk from 1998 that has always stayed with me. He was talking about the shortcomings of the rhythm of a typical National Public Radio news story, which led him to create This American Life:

And there was something dull about the rhythm, to me as a radio producer, where every story was set up so there was a little bit of script and then you’d hear a quote, and some script and then some quote. And radio, you know, functions a lot like music, even though it’s speech. It had this very predictable rhythm.

And we never get to know any of the characters, enough to feel anything or empathize in any way or to be amused or to feel angry or to be surprised.

This weekend, I felt things. By disrupting the typical rhythm of math class, Brian gave me a chance to feel amused and confused, surprised and delighted.

It’s really worth thinking about how we can do that for our students.

Elementary Teachers as Math Learners

One of the central messages of my upcoming book is that elementary teachers have an incredibly hard cycle to break. Almost all of us were taught math very badly. We memorized procedures. We learned rules. We circled keywords. We did it the teacher’s way. Now, we’re all grown up and are expected to teach math for understanding. Yippee!

Except we don’t understand.

In my coaching and writing, I spend a lot of time creating safe spaces for elementary teachers to engage with the powerful and fascinating math we teach, so we can build deep, connected, conceptual understanding. It’s my life’s work to help teachers learn together and alone, from each other and with our students.

But what happens when it’s me? What happens when I don’t understand?

This question came up for me on Twitter today. Kristin asked:

Screen Shot 2014-09-06 at 5.31.19 PM

I looked at her question, and thought, “Of means multiply. Why? Why does ‘of’ mean multiply in fractions?” I learned that rule as well as anybody else, but, like Kristin’s students, I am struggling to make the connection. As a fourth-grade teacher, I’ve never had to teach this content, so haven’t yet dug into it enough to re-learn the math meaningfully. Here we were on Twitter, for all the world to see, and she was asking for advice from me, as if I had expertise. I thought for a minute, and then put my money where my mouth is.

Screen Shot 2014-09-06 at 5.34.40 PM

That was a hard tweet to write, for a few seconds. Then I saw Kristin’s reply:

Screen Shot 2014-09-06 at 5.40.54 PM

And we were off! So began one of my favorite Twitter conversations to date, in which several incredible teachers chimed in, especially Brian Stockus and Zak Champagne. Each of us freely admitted the concepts that were hard for us. We made sense together. We laughed. I had paper next to me, and was sketching and working out ideas and models as we talked. I emerged from today’s conversation a whole lot wiser about multiplying and dividing fractions. Feel free to read the storified version. It’s really good.

When I coach teachers about learning content more deeply, I always talk about how much bravery it takes to admit we don’t understand. Today, it felt great to be brave myself.


Brian blogged about “of” after this conversation.

Kristin blogged about our online professional learning community.

And the twitter conversation continues: This a meaty topic. We’re not done!