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.

Counting Circles Variant: Tens and Ones

Last week I was hanging out in a kindergarten during Counting Collections. It was amazing and beautiful. Afterwards, the kindergarten teacher, Becky Wright, and I were talking about the challenge of switching from counting by tens to counting by ones. These three students will give you a sense of what happens at the transition.

“10, 20, 30, 40, 50, 60, 70, 71, 72, 73, 74.”
“10, 20, 30, 40, 50, 51, 52, 53. But these 3 each have 3.”
“10, 20, 30, 40, 50, 60, 70, 100.”

We’ve all been thinking about what might help students get more comfortable switching back and forth between counting by tens and counting by ones. Today, Becky and I were talking with Debbie Nichols, who teaches 1st and 2nd grade. Together, we landed on the idea of passing out 10s and 1s – connected sticks of ten cubes and single cubes, base 10 rods and units, etc. – and then having a counting circle.

In kindergarten in the late fall/winter, Becky would have the kids holding tens positioned at the beginning of the circle. As kids counted around, adding what they have, they’d keep a running total of the cubes. So a count with 20 kids might sound like “10, 20, 30, 40, 50, 60, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83” if the first 7 kids were holding 10s and the rest had 1s.

What about having one of the kids with tens switch places with one of the kids with ones?  Now the count might be, “10, 20, 30, 40, 41, 51, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 81, 82, 83.” Ooh!

Or, what about going around the other way, starting with the ones and ending with the tens? “1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 23, 33, 43, 53, 63, 73, 83.” Hard!

With 1st and 2nd graders, Debbie wanted to pass out tens and ones at random. Her kids came back from recess and we gave it a go right away:


As kids counted around, Debbie kept track by pointing on the hundreds chart. After one count around, we found we had 77 cubes.


The kids asked if we could count around the other way – counterclockwise instead of clockwise – and we had a great discussion about whether we would still land on 77 or not. I suggested Deb keep track of the running total by tracing its path on the hundreds chart, using a different color for each count:


We tried clockwise and counterclockwise. We tried rearranging the kids by switching every third spot. For our last count, everyone stood up and traded cubes with somebody else.

What do you notice? What do you wonder?

Some of the kids’ claims:

“It will always be 77 unless we add some cubes or take some away.”

“It doesn’t matter what order we add in. It will always work out the same!”

There’s so much potential here.

Debbie’s planning to do the same thing with dimes and pennies on a different day. And, of course, we could give older kids multiples of ten and/or multiples of one.

After doing it just four times, we noticed an increasing smoothness for some of the kids. They were noticing that they’d either move over or down on the hundreds chart.

I’m excited to see what other versions we might come up with. Have an idea? Please put it in the comments!

And let me know if you’ve read about this idea somewhere else. Happy to cite it if we reinvented the wheel.


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.

The Structure of Popovers

We spent a lovely week on Mount Desert Island this summer, playing in Acadia and eating lots of popovers, which are apparently kind of a thing in that part of Maine. When we returned home, the kids asked if we could make some popovers ourselves. Happily, I dug out an old favorite, this Maida Heatter recipe, which says it makes ten popovers. We use a standard twelve-muffin tin, so I asked my five-year old daughter, Daphne, to butter all the way around the outside and leave the middle ones empty. When she was done, I asked, “How many popovers are we making?”

IMAG3021Daphne looked at the tin for a minute, and then started counting from the left-hand side, one column at a time. She looked at the first column and said “Three” without counting, which means she is now able to visually recognize, or “subitize,” three objects as a group. Then she moved her finger to the second column, counted the top cup, skipped the middle one, and counted the bottom one. “Four, five.” She kept going: “Six, seven. Eight, nine, ten.”

I found her count really interesting because I expected her to count all the way around the outside, following her butter tracks. Instead, she looked at the layout of the muffin pan, and used the grid. This is an exciting development! She is looking for some structure, rather than plunging right into counting without a plan. I’ve watched this thinking of hers grow over this summer, starting when a friend asked her, “Do you think you could count all those boats?”


Daphne took one look and answered, “If they were in a line I could.”

Daphne is feeling a need for organization, and for developing strategies to keep track. As numbers get bigger and problems get more complex, she is looking under the numbers for structures like lines, columns, patterns, and groups. These structures are essential for the important mathematics to come, like place value and operations.

Counting doesn’t get the respect it deserves. When I watch kids count lots of things, I see them develop that need for organization. After a child tries to count a pile of action figures, sneezes on figure 31, and has to go back to the beginning because she’s lost track, she starts to think about strategies that might work out better next time. Before long, students are naturally counting out groups of 10, 20, 50, or 100, like the students in these videos: Counting Collections with Kindergarteners and Counting Collections with Third Graders. (This Teaching Children Mathematics article by Julie Schwerdtfeger and Angela Chan has a nice description of the value of counting collections in the classroom.)

In our home, we count all the time. We count anything and everything, from popover tins to ponytail holders to toys on the floor the kids haven’t picked up. After all that real and varied counting, Daphne is starting to count strategically. Once we made the batter, I said, “It looks like we have a lot. Maybe we’ll be able to fill the whole pan. How many popovers would we make if we filled these empty ones also?” I pointed to the two empty cups in the center of the muffin tin.

She thought for a moment, then sub-vocalized, “Ten. Eleven, twelve.” She turned to me and said, “Twelve!”

I asked, “Where did the ten come from?”

“When I counted it before.”

“You didn’t need to start back at the beginning and count them all?”

“No, Mommy. I remembered the ten. I know that’s ten. I just counted from there. Ten…then eleven, twelve.”

Her older sister, Maya, came barreling into the kitchen and said, “Daphne! You’re counting on! I heard you!” She explained the term to Daphne, who was grinning broadly.

I asked, “How did you learn to count on, Daphne? That’s the first time I’ve ever heard you do it!”

She said, “Well, I just counted a lot of times, and then I started remembering the first number. When I counted it again, it hadn’t changed.”

When I coach teachers, I sometimes hear primary teachers instruct kids to count on instead of count all. “I don’t want to hear anybody go back to one. Keep track of where you were and count on.” I hear intermediate teachers say, “I don’t want to see anybody using repeated addition. You need to multiply.” Or I’ll hear, “I don’t want to see any dots. Make gridlines. No more dots.”

I think about this kind of talk a lot. On the one hand, I’m glad when teachers have a sense of the thinking that’s on the horizon for the student, and it’s sometimes appropriate to nudge students toward more efficient techniques. On the other hand, I’ve never seen this kind of nudge work. What if, instead of disparaging kids’ strategies and telling students they have to abandon their trusted techniques, we think about creating the need for more organization, better keeping track, more efficient strategies? For example, if kids insist they can multiply by repeated addition, what happens if we give them some bigger numbers to multiply, and then keep quiet? The children will eventually become frustrated–exasperated even–with the laborious calculations. Now there’s an opening! Now they feel the need for an improved strategy. Now they are looking for a different, more appropriate structure than the one they were overtaxing. And now they own all kinds of good math thinking, which we get to honor as theirs. If we’ve created a classroom climate where students are respected as mathematical thinkers, they’ll be perfectly comfortable saying, “This problem was a total nightmare and I’m sure I made some mistakes doing it this way! I think I need to figure out a new way to do it.”

Daphne believed there were ten buttered muffin cups because she’d counted them herself. She believes in her new strategy of counting on because she’s verified her counts enough times to trust them. She owns her thinking. I see no skipping over these phases. If students enter school without this kind of experience, we need to provide opportunities for them to build this foundation in our classrooms. Otherwise, the student who is counting on because his teacher taught him a procedure, (whether he is convinced it works or not), is the same student who finds an answer of 6,080 instead of 680, and doesn’t bat an eyelash.

The recipe, by the way, made 12 perfectly imperfect eggy popovers, which we ate with butter and jam. Enjoy!