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.


I had a great time at the California Math Council (South) conference in Palm Springs a few weeks ago. I went to lots of super talks, but the two that have stayed with me the most were Ruth Parker’s and Megan Franke’s. I’m still mulling over both, and want to start by posting about Megan’s talk because it has made an immediate impact on my practice.

Megan discussed the relationship between children’s counting and children’s problem solving. She made a compelling argument against viewing them sequentially, or thinking that one is a prerequisite to the other, and instead talked about how they can develop in an intertwined, mutually reinforcing way. She argued that children can use what they know about counting to think about problem solving. And she argued strongly that children’s partial understandings about counting are incredibly valuable. Amen to that, sister. The CGI researchers have been leaders in focusing us on what students do know, rather than taking a deficit approach, and it was so gratifying to hear Megan make this argument in forceful terms, in person. I’m a total fan.

Megan showed several videos of students counting collections of objects (rocks, teddy bears, etc.) with partial understanding. For example:

  • perhaps they organized what they counted and had one-to-one correspondence, but didn’t have cardinality, so they didn’t know the last number they said represented the quantity of the group. Or,
  • perhaps they didn’t have one-to-one correspondence and didn’t count accurately, but knew their last number signified the total. Or,
  • they had a lot of things going for them, but didn’t know the number sequence in the tweens (because they’re a nightmare and make no sense). And so on.

Megan showed a video in which a girl was counting a group of teddy bears (I think 15). She did pretty well in the lower numbers, but got lost as the numbers got bigger. The questioner then asked the student, “What if all the green bears walked away? How many bears would be left?” The girl giggled at the thought, collected up all the green bears, shoved them across the table, and counted the remaining 9 bears accurately.

My jaw dropped.

Megan made a powerful case that we can springboard off counting collections into problem solving, even if the counting is partial. She argued that students are already invested and engaged in the collection, so we might as well convert the opportunity. Some of her reasons:


I sat at my table thinking about how much work I’ve asked teachers to do. (If you don’t know what the counting collections routine looks like, take a peek at Stephanie’s kindergarten in this video:)

Teachers have gathered all these little objects, bagged them up, collected muffin tins and cups and plates, created representation sheets, taught the routine of counting collections. And yet, after the kids count and represent their collection, we just clean up.

It’s like we’re leaving players on base at the end of an inning. We’ve done all that work to get the hits and load the bases, but then we don’t bring them home. We don’t make full use of the opportunity we have designed. It was suddenly all so plain.

I tweeted about this idea of bouncing right from counting collections into problem solving, and my friend and colleague Debbie Nichols got the idea right away. She didn’t even wait for a counting collection. She started springboarding off an image-based number talk.

I visited yesterday, and Debbie’s K-1 kids were counting these cupcakes. They had all sorts of beautiful ways to count them.


  • “I see two sixes, one on each side.”
  • “I see four in each row, and there are three rows.”
  • “I counted by ones. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.”
  • “I counted by twos. 2, 4, 6, 8 10, 12.”
  • “I counted by fours. 4, 8, 12.”
Counting by twos, of course.

Students stood up and marked up the image on the smartboard, showing how they counted. They played with strategies to keep track and record. It was all great.

This is normally where we would have stopped, proud of ourselves for a worthy counting exploration. Not this time.

Debbie asked, “If we were going to write a story problem about this picture, what might we write?”

Arms went up right away. A student suggested, “Somebody had 12 cupcakes and took away 10. How many are left?”

Another student said, “Six cupcakes have raspberries. Two have rulers. Two have apples. Two have papers. How many are there altogether?”

Another student said, “I have 12 cupcakes. Evelyn gave me 10 more. Now I have 22.”

Debbie asked students to go get their notebooks and write a story. Solving it was optional.

Remember, this was a K-1. So, some kids who are very young K’s drew a picture and then talked about cupcakes:

Ned has one cupcake. How many Neds have cupcakes?

Most kids were able to write a story problem and read it to us. And most of those kids wanted to solve it and were able to do so successfully:

Davin had 12 cupcakes. Arik brought 12 more. How many does Davin have now?

One precocious student made the context a little silly so he could work with the kinds of numbers he likes. He explained to me how he found the answer of 450 cupcakes: “1000-500 would be 500, so another 50 more would leave 450”:

David had 1000 cupcakes. Aric took 550. How many does David have now?

But my favorite conversation grew out of this piece of work:

Bella had 5 cupcakes. Morgan gave her 35 more. Now how many does Bella have?

I asked her how she’d figure out how many cupcakes she had now. She said, “I’d think 5 in my mind, and then come over to these cupcakes and start counting 6, 7, 8, 9…” A few minutes later, she told me, “It’s 40!”

After admiring her careful work, I said, “If I were going to count this, I would have thought 35 in my mind, and then come over to these cupcakes and counted 36, 37, 38, 39, 40.”

Now it was her turn to drop her jaw. She was so excited that we both found the same answer. I asked her if I could do that–can I switch the numbers around like that? I could literally see the gears turning and the wheels spinning. She is off to the races, starting to abstract and decontextualize and play with numbers.

I especially loved Debbie’s work because kids got to pose problems. We are way, way too stingy with opportunities for kids to pose problems. Most kids think math class is a place where the teacher asks questions or the book asks questions, and kids answer them. If we want students to understand that math is a way to ask and answer your own questions, we need to give them some chances to do the asking.

I hasten to add, during morning meeting, these same children were asking questions about infinity. “What’s half of infinity?” “What’s the biggest number?” “Is infinity a number?” Students who have their thinking honored–who are used to generating questions in math–will ask specific math-problem-type questions, but also large, important, relational, lofty questions. Problem-posing and question-asking in mathematics has a wide range. Kids need practice asking at all the different grain sizes.

Circling back to springboarding, today’s lesson drove home what Megan was saying. These kids had already spent 15 minutes studying these cupcakes, counting the cupcakes, listening to their classmates count the cupcakes. They were already invested, had thought about how they were organized, and were certain they were starting with 12. It hardly took any nudging at all to bounce them from their counting investigation to a problem-solving one. In fact, the two investigations were seamless.

My friends at different grade levels, can this idea transfer? When do you do loads of work, get students invested in a context, and then walk away too soon? I’d love to know.

In Debbie’s room yesterday, I kept thinking how lovely it was for students to see sensemaking as integral to counting and problem-solving, right from the start. How lovely it was for Deb to double the bangs for her bucks with this scenario. How lovely for her students to go deeper into a context that they could already visualize and understand.

Springboarding. I’m a fan.

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.

Comparisons: A Little Bit More Older

Ask any elementary school teacher, and she or he will tell you that comparison problems are much harder for most kids than operations with other actions. For example, fourth-grade-teacher Jennifer Clerkin Muhammad asked her students to draw a picture of this problem from Investigations:

Darlene picked 7 apples. Juan picked 4 times as many apples. How many apples did Juan pick?

Her students are pros at representations and skillful multipliers, but we saw a lot of this:

apples unsure

Kids who were used to thinking of multiplication as either groups, arrays, or repeated addition had some productive, furrowed-brow time ahead of them. How can you represent comparisons? How is this multiplication? How do you draw the 4?

Then they started to get somewhere.

apples groups

apples array

With each representation they discussed, Jen asked the excellent question:

“Where do we see the 4 times as many in this representation?”

Seriously. Jot that one down. It’s a keeper.

Jen and her class played with a series of problems involving multiplicative comparison, (height, quantity, distance, etc.). For each problem, she asked student to create a representation. And then she’d choose a few to discuss, asking, “Where is the ______ times as ______ in this representation?” Lightbulbs were going off everywhere as students broadened and deepened their understanding of multiplication.

Teachers often see the same struggle with comparison situations that can be thought of as subtraction or missing addend problems, like:

Marta has 4 stuffed animals. Kenny has 9 stuffed animals. How many more stuffed animals does Kenny have than Marta?

If kids have been taught “subtract means take away” or “minus means count backwards,” then this problem doesn’t fit the mold. And if kids have been taught keyword strategies, they’re likely to hear “more,” pluck those numbers, and add them together.

Knowing this as I do, I jump on opportunities to explore comparison problems with my own kids. I want them to have a fuller sense of the operations from the beginning, rather than developing too-narrow definitions that have to be broadened later. So, subtract doesn’t mean take away in our house. They’re not synonyms. I actually don’t worry about defining the operations at all: we just play around with lots of different types of problems, and I trust that my kids can make sense of them.

Yesterday, I was talking with Daphne about a neighbor visiting tomorrow. Our neighbor has a little girl, whom I’ll call Katrina. She’s about 2. We also have more awesome girls who live across the street. Let’s call them Lily, 15, and Gloria, 13. Gloria helps out once a week, and my girls idolize her. Here’s how the conversation kicked off:

Me: “If Katrina’s mother comes over to help me on Sunday, could you play with Katrina for a bit so her mom and I can work?”

Daphne: “Yes!!! We love playing with her. And she loves us. She cries when she has to leave us. We’re her best friends.”

Me: “I think Katrina feels the way about you and Maya that you and Maya feel about Lily and Gloria. It’s fun to hang out with older kids. You learn a little about what’s coming for you.”

Daphne: “Yeah. But for Katrina, Lily and Gloria would be TOO old. Like, she’d think they were grown-ups because they’re SO much older than she is. We’re older than Katrina, but we’re less older than Lily and Gloria are.”

So there’s my opportunity. She was comparing already. With a little nudge, I could encourage her to quantify and mathematize this situation. I thought I’d start with the simplest numbers so she could think about meaning first.

Me: “How much older are you than Katrina?”

Daphne: “I think she’s 2. So…”

There was a pretty good pause.

Daphne: “Are we counting on here?”

That’s paydirt.

Me: “What do you think?”

Daphne: “I think so. Because I want to know how many MORE. So, 2. 3, 4, 5, 6. I’m 4 years older than she is. Yeah. That makes sense.”

Me: “OK. So how many years older is Maya than Katrina?”

Daphne: “6.”

Me: “You didn’t count on that time. What did you do?”

Daphne: “Well, I knew that I was 4 years older than Katrina, and Maya is 2 years older than me, so 4, 5, 6. Because Maya is OLDER than me, I knew they had to be farther.”

Me: “What do you mean farther?”

Daphne: “Like, it had to be more. I was starting with 4, and then it had to be more farther apart.”

Me: “Oh! So you were trying to figure out whether to go 2 more or 2 fewer?”

Daphne: “Yeah. And it had to be 2 more, because Maya’s more older than Katrina than I am. I mean, I’m older than Katrina, and Maya’s older than me, so she’s more older than Katrina than I am. It had to be 2 MORE.”

Hot damn!

Daphne’s life might have been easier right here if she had some tools she doesn’t have yet. I’m thinking about number lines, symbols, and a shared mathematical vocabulary including words like distance or difference. Those tools might help her keep track and facilitate communicating her thinking to somebody else. But I love that we were able to have this conversation without those tools, while driving in the car where I couldn’t even see her gestures. Her thinking is there, and clear. The tools will come as a relief later on, when the problems are more complicated.

We kept going.

Me: “OK. So you’re 4 years older than Katrina and Maya’s 6 years older than Katrina. I wonder how that compares to Gloria? Is she about the same amount older than you two as you two are older than Katrina?”

Daphne: “Well, she’s 13. This is gonna be hard.”

Long pause.

Daphne: “She’s 7 years older than me.”

Me: “How’d you figure that out?”

Daphne: “I went 6. 7, 8, 9, 10, 11, 12, 13.” She held up her fingers to show me.

Me: “So how much older is Gloria than Maya?”

Daphne: “Umm…so Maya’s 8…umm…5. She’s 5 years older than Maya.”

Me: “How do you know?”

Daphne: “Because Maya’s 2 years older than me, so it’s 2 closer.”

This is the kind of problem where kids lose track of their thinking easily. Do we add or subtract that 2? In the last problem, she’d added. This time she subtracted. And she knew why!

When I glanced at Daphne in the rearview mirror, she had this awesome look on her face. She was looking out the window, but not seeing what was outside because she was so deep in thought. I wish I could have peeked inside her mind, and seen what she was seeing. Because, to use Jen’s question, where do we see the 2 years older? How did she see it? There was no time to ask, because she was onto pulling it all together.

Daphne: “So, Gloria’s a little bit more older than us than we are older than Katrina, because Gloria is 5 and 7 years older, and we’re 4 and 6 years older. But it’s pretty close.”

We pulled into the school parking lot on that one.

In our family, we count and manipulate objects all the time. Counting more abstract units like years, though, gives me a chance to open the kids’ thinking a little bit. In this conversation, Daphne was both counting and comparing years. Tough stuff, but oh so good.

Making Sense

Last week, between work with grade-level bands, I had a complete K-6 staff for about an hour. I wanted to tackle “story problems,” so I started by showing them How Old Is the Shepherd, by Robert Kaplinsky:

Seriously, take the 3:07 to watch it. You’ll see that 75% of 8th graders presented with the nonsensical problem, “There are 125 sheep and 5 dogs in the flock. How old is the shepherd?” answered with a number.

I had given my close friend and colleague, Debbie Nichols, a sneak peek at the video. She often helps me think things through, and is one of the teachers I am profiling heavily in my book. She decided to give the same problem to her 1st and 2nd graders to see what would happen. We both expected the younger kids to do better on the nonsense problem, mostly because they haven’t been trained to pluck-numbers-and-do-something-with-them the way the older kids have. In our experience, younger kids are great sense-makers.

Debbie changed the problem to read, “There are 25 sheep and 5 dogs in the flock. How old is the shepherd?” Partway through, she wondered about vocabulary, and added a second question, “There are 25 kids and 5 dogs in the classroom. How old is the painter?” Debbie interviewed her students individually, recorded it on video, and sent me her notes with the subject header WOW:

Response to sheep version Response to painter version Grade
Is this times or add? I have no idea. I do not get it. 33? 2
37? I thought, my mom is 37, so he might be 37. I was going to try to count to the highest number and then count on with the other number. I was gonna try to but then I changed my mind. Or I could just do that number and then count on. 1
7? I don’t really know, because I’m guessing. What’s a shepherd? 2
A shepherd? 5. I just, I don’t really know. I’m just really good at it and I like animals. 1
11. I’m trying to hold numbers in my head and I just counted. 20 years old? 1
I know 25 + 5 = 30. 30 2
25. You said 25 sheep. 25, because you said 25 kids. 1
30, because I added them up. 30 2
25. I was just thinking. I was counting. 30? 32 1
Shrugged shoulders. 30? 32? 7. I thought of the question. 2
8. I was counting. 8. I counted by 9s. 1
8. Is that correct? I thought it was a little boy so I chose 8. 20. I thought of another number. 1
5. I just got it out of my head. 20. It just came out of my head. 1
What are flocks? What’s a shepherd? 29? 59? 69? I got it out of my brain and my brain is made of pink worms. 1

After individual questioning, Debbie brought the kids to the rug and asked the whole class, “There are 25 sheep and 5 dogs in the classroom. How old is the teacher?” Students quickly fell in line behind some opinionated students, and came to consensus that the teacher was 30 because 25 + 5 = 30.

Wow indeed. What’s especially striking to me is how many students admitted they didn’t understand the problem, and still gave a numeric answer anyway. Some of the nonsense answers even sound kind of right–like they are imitating the way a math answer should sound. “The answer is 8. I counted by 9s.” (As much as I love the brain made of pink worms, the counting by 9s answer is the most fascinating one to me.)

The following day, Debbie posted, “There are 4 kids and 3 chickens in the room. How old is Mrs. McCabe?” (Mrs. McCabe is another teacher in the building.) Students wrote their answers on the chart paper. Take a peek:

photo (16)

43, 7, 7, 40, 64, 70, 2, 2, 10, 9, 30, 36, 100, 34, 7, 44.

Nobody wrote, “I can’t tell because this problem makes no sense.”

43 and 34 come from students taking the 4 children and 3 chickens and using them as digits in a new number.

7 came from students adding the 4 children and the 3 chickens.

I suspect 70 comes from a student adding the 4 and 3, then deciding 7 was too young, so they made it 70.

40, 64, 30, 36, and 44 may come from students disregarding the information in the scenario, and just making a reasonable guess about Mrs. McCabe’s age (56). The 40 and the 30 might be similar to the 70, in that the numbers 4 and 3 were in the problem, and they were clearly too young, so 40 and 30 sounded reasonable. (In this group of answers, if students were thinking about reasonable ages for teachers, that’s something to build on. My next step would be to ask them for evidence in the question.)

2, 10, 9, and 100 remain mysteries to me.

I can see in the anchor chart that Debbie took out her purple marker and tried to help students recognize the lack of relationship between the number of chickens, the number of children, and Mrs. McCabe’s age. She had them try to come up with a question that would make sense from the chickens and children, like how many legs are there?

After reading Debbie’s notes, I wondered what my own children would do with this problem. I asked my 5-year old daughter, (who asked to be called D), “There were 4 children and 3 chickens in a room. How old is the teacher?”


“Where’d the 9 come from?”

“No, wait, that’s not enough. 90.”

“Where’d the 9 come from and where’d the 90 come from?”

“Well, I added the 3 and the 4, and that made 9. But that’s not enough. Like, Maya is 7, and that’s 2 years less than 9, and she’s not old enough to be a teacher! So I made the 9 a 90.”

“Can you show me how you added 3 and 4?”

She counted on her fingers.

“Oops. It’s not 9. It’s 7. So she’s 70.”

I was quiet for a while, thinking about a question.

“We have 2 dogs and 1 fish, right?”


“So, we have 2 dogs and 1 fish in our house. How old is D?”

She laughed uproariously. “Mommy, that doesn’t make sense! The dogs don’t have anything to do with how old I am! 2 and 1 is 3, but I’m 5!”

“Really? OK. Let me try again, then. We have 2 dogs and 1 fish in our house. You are 5 years old. Let’s say we bought another fish. How old is D now?”

She was hysterical at this point. When she calmed down, I said, “OK, let’s go back to this question again. There were 4 children and 3 chickens in a room. How old is the teacher?”


I recognize that these nonsense word problems are contrived. I think they’re revealing, though. In particular, I think they show some problematic beliefs our students have about doing math:

  1. All math problems have to be answered with numbers.
  2. All math problems can be answered.
  3. It’s normal for math not to make sense.

Where do these beliefs come from?

In the last week, as I’ve been mulling all this over, I’ve been revisiting some of the books I have about the intersection of reading comprehension and math, like Comprehending Math by Arthur Hyde, Mathwise by Arthur and Pamela Hyde, and From Reading to Math by Maggie Siena. This quote of Siena’s about the foundations of reading jumped out at me:

“Children must…expect the things they read to mean something and expect to be satisfied by that meaning” (17).

Do we teach the same expectation in math? Or do we teach students to answer every problem with a number, guessing if they must, and it’s OK if it doesn’t make sense?

I’m still scratching my head over where this message comes from with my kids and Debbie’s students. D has grown up during the writing of my book, when I am hyper-aware of the math messages I am sending and hearing. Debbie’s students had a great year of kindergarten with a teacher who emphasizes making sense, and now they’re with Debbie, who teaches math for understanding. And yet, 100% of them answered the nonsense question with a number.

As a fan of CGI, I know children are naturally sense-makers. But I also know that reading mathematical problems is a special kind of reading, and students need instruction in it. Historically, teachers have used two different types of instruction for reading word problems:

  1. Teach students to “decode” math problems with “keywords,” like “in all means add.”
  2. Teach students to recognize unnecessary information, red herrings, and traps that “they put in the problems to trick you.”

On keywords, some of us have been having fun over on Twitter, creating a list of problems that show why it’s a doomed strategy. For example, Tommy buys 3 bags of avocados. There are 4 avocados in each bag. How many avocados did Tommy buy in all? Hmm. I thought in all meant add?

As for focusing on the traps “they” are putting in the problem, I am no fan of this strategy either. What sort of message are we sending kids with this teaching? That there are rooms full of nasty adults, rubbing their hands together, trying to set traps that catch nice little children taking math tests? Ahem. Though there may be some truth to that image, I refuse to cede mathematics to the standardized-test and curriculum writers who write crappy, trappy math problems. I want to snatch math back, and teach students to see the beauty and usefulness of math around them, and to enjoy the journey through a perplexing, puzzling problem. So red herrings and tricks be damned!

What do we do instead? How do we teach students to read math problems for understanding in a way that will yield empowered students who expect to make sense? I’m looking for resources on this question, so please pass them along in the comments. In the meantime, let me share three of my favorite approaches. They all have something in common, which is that they are all strategies to make it impossible for students to leap right to answering the question. All three approaches force students to slow down and make sense, first.

1. The Math Forum at Drexel University is a fantastic group of people who are all about teaching students to make sense of math. One of the strategies they have been promoting is Notice and Wonder, where teachers share a scenario without a question, and ask students what they notice and wonder. You can read about it in Max Ray‘s Powerful Problem Solving, by following Annie Fetter, or at a whole bunch of sites here, here, here, here, here, here, and here.

2. Brian Stockus wrote a great blog called “Numberless Word Problems,” in which he described a co-worker removing the numbers from a word problem. Again, this strategy eliminates the option of racing to an answer, and introduces students to the idea that we can do quite a bit of mathematical thinking about quantities without knowing what they are, which Kate Nowak framed as the the rich idea at the heart of algebra:

Screen Shot 2014-10-18 at 4.08.39 PM3. In Mathwise, Art and Pamela Hyde wrote, “Getting students to slow down and think about a problem is not always easy, especially if they are used to calculating answers quickly to one-step translation problems. We have found that students can be encouraged to think through their assumptions with an intriguing type of problem called “Fermi questions'” (66). Fermi Questions are mathematical questions where answers seem impossible, but we can get close by making some assumptions and then approximating:

  • How many piano tuners live in Chicago?
  • How many kids could fit in the gym with no furniture inside?
  • How many hairs are on your head?

From a teaching point of view, Fermi Questions can be fantastic for helping kids realize they are making assumptions and connections and using their prior knowledge in mathematics.

All three of these strategies–Notice and Wonder, Numberless Word Problems, and Fermi Questions–force students to slow down and make sense of the situation before worrying about the answer.

I’m hoping to learn more about high quality instructional strategies for math teachers that are rooted in what we know about teaching reading comprehension. Annie Fetter presented on this idea at NCTM in New Orleans, and I think it’s an idea with long, strong legs. The connection between making sense in literacy and math is something I talked about in the workshop last week, and it seemed to resonate with Shawna Coppola, a wonderful literacy specialist. I loved her notes:


Making sense is the thread that ties everything together, in every content area. Right? If our students arrive having already internalized the message that making sense isn’t part of math, or that math doesn’t make sense, or that word problems are just a bunch of numbers hidden in words and traps, we have our work cut out for us. Time for some intentional, creative, inquiry-based teaching that empowers students to make sense.