My stellar week of NCSM/NCTM fun ended with my family flying to San Francisco for a weeklong holiday. We started out at the Embarcadero, and ended up in Ghirardelli Square. We were sitting on a bench, eating our hot fudge sundaes, when my husband, Sam, started talking about the sale on big bags of Squares® inside.
He said he’d worked out the math, and big bags came out to about three Squares® per dollar. He was thinking about picking up a bag to take back to his office. I thought about grabbing one for Stenhouse, and one for my school. Conversation moved on. Eventually we all fell quiet.
Several minutes later, I was people-watching when my first grader, Daphne, asked, “Is it possible to break ten into three equal pieces?”
I asked why she was asking. (I do this often.)
“This is a question I have a lot. Like, today, when I wanted to buy my new doll, Olivia, it was some amount of money for three of them, and I wondered how much it would be for one of them, because I only wanted one.”
I was a little slow on the uptake. Long conference week. I sat there, blinking, for a minute. And then I said, “Oh! You’re asking about the chocolate, aren’t you?”
“Yeah. Because Daddy said they’re three for a dollar. What if you only wanted to buy one of them? How much would that be?”
“I see. But you didn’t ask if you could break a dollar into three equal parts. You asked if you could break ten into three equal parts. Where’d the ten come from?”
“Well, you know how there are ten 10s in 100?”
“I do.”
“It’s basically the same thing. Like, 6 + 4 = 10, and 60 + 40 = 100.”
I asked where the 100 came from.
“The 100 is because there are 100 pennies in a dollar. So if I figure it out for 10, then I’ll be able to figure it out for 100, because it’s basically the same thing.”
This is where I stopped her for a second so we could high-five. I mean, holy use of mathematical structure, Batman.
Daphne went on, “What I really need are ten things. Oh, rocks! Perfect!”
Daphne worked with the rocks for a long time. She was thinking hard. I kept her sister quiet, which is the challenge in moments like this.
Daphne finally said, “I don’t think I can make three equal pieces with rocks, because I can’t break this last rock apart. It works out to 3 + 3 + 4 or 3 + 3 + 3 + 1. I need a piece of paper to show you.” She grabbed the envelope I had in my bag.
“See? It doesn’t work because I can’t break this rock! I can’t cut rocks into thirds.”
I asked, “What if they were crackers or cookies instead?”
“Then I could break up the last piece into three equal pieces. Then they’d each have three whole ones and one-third.”
“What do you mean, one-third?”
“Well, a third is one of three equal pieces.”
“What if you have three pieces, but they’re different sizes?”
“Then they’re not thirds. They’re just three chunks. I learned this from listening to you and Maya talk, by the way.”
Hmm. +1 for older sisters.
I asked, “So what do you think they do if they want to sell one piece of chocolate? How much should it be?”
She said, “With money, I should be able to to break it up. I can make change for the dollar. So I have 100 cents. So, 30 + 30 + 30 is 90, and that leaves 10 more…”
At this moment, she leapt up. “Wait! It’s the same thing again! It’s going to go on forever! With 100, it was 30 + 30 + 30 with 10 left. With 10, it’s 3 + 3 + 3 with 1 left. I can split up that 1 into three pieces, but there’s going to be a piece left. That one extra piece MAKES IT GO ON FOREVER! There’s always going to be an extra piece! Three, three, three!”
I happened to have a conference schwag calculator in my bag, and she got to see 100 ÷ 3.
“So how much should one piece of chocolate cost?”
“One-third of a dollar.”
“How much is that?”
“About 33 cents. If they charge 33¢, they get pretty close to a dollar.”
“How close?”
“Well, three chocolates would be 33 + 33 + 33, so 30 + 30 + 30 is 90, and 3 + 3 + 3 is 9, so that’s 99¢.”
I just spent a week thinking about the teaching and learning of mathematics with all kinds of amazing people. So much of the conversation is about how we can create the conditions so students do what Daphne did here:
She noticed math in the world around her, and wondered about it.
She posed an original (to her) mathematical question.
She used structure to think about that question (in this case, the structure of place value).
She used the strategy of solving a simpler problem.
She looked for patterns and regularity.
She stuck with her problem when it was hard for her.
She used tools, representations, and models.
She decontextualized and recontextualized the problem.
She reasoned and justified.
I mean, she was all over the SMPs, right? And naturally, too. I wasn’t pushing any math at that moment, believe me. I was sitting there like a lump, very tired and very happy to be with my family. But I’ve done things at other times. Namely, I’ve made it clear to Daphne that math belongs to her. That her ideas are valuable. That I’m interested in them. That math involves asking questions. That she can figure things out for herself. That she owns the results of her investigations. That math is all around her. That she is a mathematician.
It’s paying off. Now, if only she can hold on to all of that.
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:
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.
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.
Maya (7) brought home a folder full of completed math worksheets yesterday, which put me in a funk. First, there was the bugs problem.
I couldn’t decide which part of this problem bothered me most. Was it the ridiculous premise? I mean, come on. Was it that the problem was, yet again, a multiple-choice question? Was it the stubborn insistence on drawing bugs with the wrong number of legs? Was it that students have no room to work, but the publisher took plenty of room for cutesie drawings? Harumph.
As awful as the bugs problem was, the page that really upset me was this one:
Sigh. A few issues:
The strategies are named wrong, and made to look more complicated than they are.
There are better and other ways to solve this problem.
Students are left with 2″ to do math on this entire page, which is barely enough room to do the standard algorithm. There’s no chance they’ll try out one of the other strategies with no space to work.
Once again, we’re turning useful, general computation strategies into prescriptive algorithms. Breaking the numbers up by place value and adding the partial sums becomes: Step 1) Add hundreds…
I’m not the first math teacher to notice that prescriptive worksheets are a problem. Kamii, the CGI group, and others have all written about it. Yesterday, though, I was upset as a parent because Maya has stellar, flexible mental math skills, and her instinct to think is being undermined by this curriculum. I asked her why she’d opted for the standard algorithm on all the problems, and she said, “Those other strategies are too confusing.” I covered up her solution to 597 + 122 and asked her to solve it mentally.
“Well, I’d give 3 to the 597 to make it 600. Then 600 + 122 is 722. I’d take the 3 back, so it’s 719.”
I did the same thing for 209 + 376:
“200 + 300 is 500. 500 + 70 is 570. 9 + 6 is 15. 570 + 15 is 585.”
I pointed to the top of the page and said, “You just used this strategy. You broke the numbers up into place value parts, and then added each part together, starting with the biggest part.”
Her jaw dropped.
And my mind clicked. She has made NO connection between the mental math strategies she uses with fluency and all this junk on the worksheets. The reason? She’s never been given the chance to record her own thinking at school.
I think I’ve decided what one of my bigger problems with this curriculum is: they never use blank paper. They never write 209 + 376 at the top of a big piece of paper and let kids have at it. The kids never get a chance to wrestle with keeping track of their thinking or figure out organizational strategies. All math problems are either on worksheets or educational technology. The kids just don’t write enough.
So now I know what to do with Maya at home. We’re going to spend some time with blank paper, where she has to work out how to write down what she does in her head. She needs to make mistakes, lose track, not be able to follow her own thinking, and then ultimately figure out ways that make sense. She needs to be able to write down her thinking so that she and her mathematical community can follow it. I’m on it.
Two hours later, after dinner, Daphne (5) got us started. Our dining room chairs have decorative nailheads, and the kids are forever running their fingers over them and counting them. Daphne said, “Someday, I’m going to get out a math journal and count all these nailheads and write it down so I know how many there are.” Before I knew it, she was off! Someday turned out to be right then. Both kids got in the game.
Daphne was incredibly excited to count AND write down her results. Check them out:
If you want to understand her notation, take a peek at the short videos:
The power of blank paper, baby.
Perhaps inspired by all of this discussion and my venting, Maya asked if she could get a piece of blank paper when she did her homework, which is truly a counter-cultural act with this curriculum. “Of course!” I nearly sang.
She created this number line and used it to solve the final problem.
When I asked her about it, I pointed out that she didn’t just answer the question by saying, “The red one.” She wrote about the problem more generally: “I used a number line and found that anything less than 350 would fit and 270 is less so red paint is less than 350!”
She said, “Well, when I wrote it myself I thought about it more.”
This morning, as I was nearing the top of a curvy highway entrance ramp, I broke into a skid. I don’t mean a little fishtail. I mean a full-on skid where I desperately wanted to be on the entrance ramp but was actually sliding across the highway into traffic, at speed.
I grew up in a snowy climate, and my dad taught me how to handle skids in an empty mall parking lot when I was 16. (Thanks, Dad.) I’m now 42, and today was the first time I’ve had a car skid at speed because, well, I know better. Everything about this morning was surprising and wrong. I was not driving too fast and I took the turn well. The car in front of me was going faster and handled it fine. I drive a Subaru Outback, for crying out loud!
I recovered control while careening back and forth between traffic and the ten foot snow bank on the guardrail. Thankfully, all the other drivers were paying attention and got out of the way. We didn’t make contact with anything and emerged unscathed. Maya was so engrossed in Harry Potter and the Chamber of Secrets that she barely looked up from her book. Daphne took it all in stride. But I was shaken.
I wanted to sit and think, but the highway is no safe place for that, so I started driving again. As I drove, I was running through all the poor traction I’ve had at stop signs lately, the slide I had in a parking lot last week, the way the brakes haven’t been responding well in slush, and now this terrifying skid. In case you’re in a newsless cave, we’ve had a lot of snow in New England this winter, and the car just hasn’t been handling the way I expect it to. I dropped the kids off and headed to the mechanic.
Six hours later, I left with four new tires and new rear brakes. The mechanic said he could barely get the car to stop as he entered the garage. Leaving, I felt like I was driving a new car. That’s the Outback I remember! Fist pump!
I picked the kids up and told them about my day. Here’s where we get to the mathy part. As a math teacher and parent, I am attuned to everyday opportunities to talk about math with my kids. I notice them, seek them out, and take advantage of them, aspiring to what Christopher Danielson described in his perfect TMWYK post, “A Tale of Two Conversations.” I knew I could open a mathematical conversation here. But which one?
One option was to get into measurement. How often do you get to talk about 5/32 of an inch of tread?
I could also get into the math of risk. My kids are afraid of the wrong things, just like everybody else, even though the most dangerous thing we do is drive. Perhaps I could have introduced a little statistical perspective. Then again, the last thing I wanted to do this morning was scare my kids further.
Speed, rates, and distance are always a choice when talking about cars and driving.
The mechanic talked with me at length about tire tread design, which took us to rotational symmetry. If you’re curious, google words like directional and asymmetrical tire tread. It’s really interesting, I swear!
There were lots of opportunities to play with larger numbers. The tires are supposed to last 50,000 miles. We bought them when the car had 72,000 miles, and it now has 96,000. Hmm. That’s very appealing.
In the end, I guided us toward costs and money for a very specific reason. My old friend Ron Lieber has written a terrific new book called The Opposite of Spoiled: Raising Kids Who Are Grounded, Generous, and Smart About Money. I’m a few chapters in and loving it. Ron has convinced me that we need to do a much better job of raising financially literate kids. He’s encouraging adults to break down the taboos surrounding money, and engage in meaningful conversations about decisions, debt, needs, and wants. He points out that kids are aware of the thorny ethical, political, personal, and societal issues surrounding money from really early ages, and we might as well talk about them openly rather than hide them. I feel like a dope for not seeing it this way before.
The truth is, while I’ve always taken opportunities to talk about small money with kids because it gets me to good math talks, I haven’t done as good of a job talking about big money. Four new tires and new rear brakes is a big money moment that gave me a chance to talk about math and money at the same time. Here’s how that part of the conversation went:
Tracy: “So how much do you think all of that cost?”
Maya (7): “I think about $1,000.”
Tracy: “Wow! That’s really close! Let’s see. At first, he told me each tire would be $156.”
Maya: “So…that’s $624 just for the tires.”
Tracy: “How’d you do that?”
Maya: “I did the hundreds first. 100 and 100 is 200, and then I doubled that to get 400. 50 and 50 is 100, and so I doubled that to get another 200, which got me to 600. And then 6 and 6 is 12, and 12 and 12 is 24. So that’s $624. Wait. What do you mean, ‘At first?'”
Tracy: “Well, remember how we figured out that the tires should have lasted for another 26,000 miles? He had sold me those tires, and he felt bad they didn’t hold up well, so he gave me a discount on the new tires. He charged me $146 for each tire instead.”
As a math teacher, I set it up this way on purpose. I was curious if Maya would solve $146 x 4 to find the new total, or use her solution from the prior problem and solve some version of ($156 x 4) – ($10 x 4). At the same time, I was mindful that Daphne (5) didn’t have a way to access this conversation, and was looking for a chance to bring her in.
Maya: “So, he charged $40 less.”
Tracy: “Right, because he charged $10 less for each tire. Hey Daphne, can you count by tens for me?”
Daphne: “10, 20, 30, 40. Forty dollars.”
As she counted, I held up my fingers to keep track, and then pointed each finger toward a tire in turn, trying to help her associate $10 with each tire. She counted them by tens again, pointing at the tires herself. Pleased with the mental math the kids were doing, I decided to turn to the bigger issues around money, starting with a question.
Tracy: “Where do you think the $1,000 comes from?”
Daphne: “From the bank.”
Tracy: “How did it get in the bank?”
Daphne: “When you and Daddy work, you put the money you make in the bank.”
She made such a great connection to the breakfast conversation we’d had about allowance. As per Ron’s policies, we’d been talking about categories of give, spend, and save. I was able to add nuance to their ideas about saving by introducing the idea of a short-term cash reserve for unexpected expenses. We don’t know when the roof will leak or the dishwasher will go on the fritz or someone will get a cavity or relatives will fall ill and we’ll need to travel suddenly. We just know that these things do happen, and we need to have money ready to pay for them right away.
But part of talking about money is talking about the hard truths too, which is why I found myself going on:
“I feel really grateful today. I feel grateful that we can pay this $1,000, which we absolutely needed to pay for safety, and we still have enough money for heat and health care and food. Families who are having a harder time right now have to choose between those things, even though they need them all. $1,000 is a lot of money. Yes, we work hard and we save up for costs like this, but we’re also really lucky, and I am grateful.”
So began a conversation that lasted the rest of the way home. I’d opened the doors, and the questions came pouring through. They’re still coming, and we had a long talk after dinner tonight about how sometimes society is like a game that’s not played fair. Privilege, race, class, advantage. It’s all coming up. So good.
I feel grateful again. I started my day grateful that I got the car stopped. I end it grateful that I have a new way to engage in meaningful conversations with my children.
Justin Lanier shared this bit of loveliness on Twitter the other day:
I love everything about it. I printed out a couple of copies this morning and made them available at breakfast. My kids are 5 and 7. Maya, the 7-year old, has read A Fly on the Ceilinga time or two, but has otherwise not yet been introduced to coordinate grids, graphing, slope, parallel and perpendicular lines, etc. Daphne, the 5-year old, is in the same boat, except she hasn’t read A Fly on the Ceiling. I was curious what would happen.
After studying it for a bit, Maya went straight for the rulers. She spent some time measuring the axes, trying to figure out how far apart the numbers were. She was surprised they weren’t an even centimeter or inch apart. She figured out their distance, then compared the x-axis to the y-axis for a while. For those of us familiar with coordinate grids, the axes quickly recede into the background so we can see the information laid on top. For someone new to coordinate grids, they carried lots of information.
“Mommy, I see these faint lines–like graph paper–that line up with the numbers. I think that helps you see where you are. It’s like the tape lines in A Fly on the Ceiling.”
After a while, she seemed content-for-now about the axes, and moved on to looking at the two lines. She started measuring the distance between them.
I got up to make my oatmeal, and noticed someone else in the family was using a different approach.
When I asked Sam what he’d been doing, he told me he was thinking about ways to see if the lines were parallel. Maya overheard him, and said, “Oh! I don’t think they’re parallel. She started showing us with her hands how the lines were ever so slightly tipped toward each other. She said, “They’ll meet someday, way, way over here.” I asked her how she knew. Back to work she went.
Meanwhile, Daphne got in on the action with her own ruler.
And I noticed this sketch in the newspaper. Sam wasn’t done.
This problem is so lovely because it’s tantalizing, and open, and full of possibilities. It reminded me of this tweet that went by yesterday.
This problem has plenty of space inside it to learn. What do you think Daphne learned? What’s she thinking about? Maya kept wanting to measure the distance between the lines, and we ended up playing around with that idea some. She isn’t there yet, but she’s starting to see the helpfulness of right angles. She’s really into rulers right now, and she spent quite a bit of time figuring out how to write down the lengths she found. She told me one was 3 1/4″ and the other was 3 1/8.”
I asked her which was longer.
“3 1/4.”
I said, “Wait a minute! Eight is more than 4! That can’t be right!”
She said, “The 8 tells me that the inch is cut into 8 pieces, so each piece is smaller because there are more of them. Actually, 1/8 is half the size of 1/4.”
Happy dance. Of course we spent a little more time playing with that idea. Then she said, “I want to write down what I think. It asks, ‘What do you think,’ and I want to write about that. I think I want to write about whether the lines are parallel or not.”
What a lovely change from her curriculum at school, which is nothing but bubbles, boxes, and blanks to fill in.
Maya (7) asked me about this sign in the coffee shop drive-thru the other day:
And so, we had a lovely conversation about clearance. Once she understood the notation, she asked, “Does that mean if your truck is 9’5″, you could drive underneath?”
That’s just gold.
I said, “Well, it depends how confident you are in the two measurements. Are you sure the clearance is exactly 9’6″? Are you sure your truck is exactly 9’5″?”
She thought about it a while, and decided that was too close for comfort. We brainstormed a nice list of sources of error while we waited in line:
What if the tires weren’t as inflated when you measured, and then you filled them up with air?
What if the person measuring made a mistake?
What if the tape measure was a little bent?
What if they didn’t get to the very top of the truck, like an antenna or exhaust pipe sticks out or something?
What if there’s ice on the road, so your truck sits a few inches higher? (There was ice, in fact. Happy Thanksgiving.)
What if they’ve paved the parking lot since they made the sign, so now the road is higher?
What if whatever is overhead is curved, and the 9’6″ is the highest part, even though it should be the lowest?
What if it was really almost 9’6″, so they just went to the next closest number to make it even, but it’s really a little under. And what if the truck is really a little over 9’5″, but they went to the next closest number to make it even too? It could be a lot less than an inch!
That last one was Maya’s, and it’s my favorite, probably because I’ve been spending time with the (unfortunately out of print) 1986 NCTM handbook Estimation and Mental Computation lately. It’s a worthwhile read about an important and neglected topic, and I hope an update is in the works this (ahem) century. Peter Hilton and Jean Pederson caught my eye with this quote in their article, “Approximation as an Arithmetic Process”:
It is an extraordinary triumph of the human intellect that the same number system may be used for both counting and measurement. Nevertheless, we argue that the arithmetic of counting is not at all the same as the arithmetic of measurement; and that the failure to distinguish between these two arithmetics (which one might call discrete and continuous arithmetic) is responsible for much of our students’ confusion and misunderstanding with respect to decimal arithmetic and the arithmetic of our rational number system (57-58).
That’s a doozy, and it’s been rattling around in my head for a while. The authors point out that, in counting, 3.00 = 3, but in measurement, 3.00m ≠ 3m. Do we make a big enough deal about this distinction in class? I’m thinking no, not by a long shot.
Hilton and Pederson make the case that all measurements are approximations, and we should teach them that way. This is a really interesting idea. What would happen if, from the outset, we were honest about the inherent uncertainty and decisions made in every measurement? What if we stopped referring to the “actual” measurement, and instead talked about the closest approximations we could get, or a range of reasonable measurements?
Thus there are situations in which we round off in view of the accuracy we NEED and situations in which we round off in view of the accuracy we can genuinely GET—and, of course, situations in which both kinds of consideration apply. The technique of rounding off is a familiar one, but we emphasize that the technique, easy in itself, is based on the nontrivial, nonalgorithmic art of deciding the appropriate level of accuracy (58).
Teaching the “art of deciding the appropriate level of accuracy” is a far cry from a worksheet with the directions, “round off to the nearest hundredths place.” The art of deciding involves the idea of authorship, which runs headlong into the myth of math as objective truth. The art of deciding means there is always an interval of reasonable answers to a measurement problem, not just one “right” answer printed in the back of the book. The art of deciding is empowering; the opposite of regurgitation.
As a student, I loathed the dreary lessons on significant figures, uncertainty, and measurement error. They were full of hierarchical rules and tedium and a complete absence of intuition and sense-making. Booorrriiiing. As an adult, I’m baffled that we somehow make boring this element of mathematics that involves a little bit of mystery, the explicit acknowledgment that sometimes we can’t know no matter how hard we try, and an honest admission that we make decisions when doing math. How do we kill it? We mask the interesting parts because we, the teachers, are the ones practicing the art of deciding. We are the ones who choose whether to measure to the nearest 1/2 inch or millimeter or gram. We leave the trivial implementation of our decisions to our students, but we keep the decisions for ourselves. Not good.
So, I left the deciding to Maya. Would you drive your truck underneath if it was 9’5″? She said no. I asked, what if your truck was 9’3″? She said no.
“I don’t think I’d be comfortable going under unless my truck was under 9′. Then I’d be confident I wouldn’t hit.”
She’d decided that 6″ is a reasonable margin of error, so she wouldn’t end up like these guys.
You might decide on a different amount of wiggle room. That’s cool too. Deciding is personal. It’s an art.
Appendix:
If you end up wanting to pursue any of this with kids, here are a few things I found helpful:
This is a video the girls and I made when they were 4 and 6, in response to 1st and 2nd graders who thought math was only found in worksheets and classrooms. I discuss this clip in my book, and needed a permanent home for the video, so here it is!
In case you’re wondering, Daphne named that bald baby doll Rapunzel on purpose. She has a well developed sense of irony, that kid.
Mathematically, my favorite part is Maya’s exploration of the leaf, because she asks and answers her own question.
My 5-year old daughter, Daphne, was looking out the car window when she laid this question on me:
“I’ve been wondering something. If there’s a whole of a tree, and half of a tree next to it, what do you call it? A whole half?”
At the next light, I turned on my phone and recorded our conversation. What follows is a word-for-word transcript. It’s one of my favorite math discussions ever, because I learned so much about her thinking.
“First of all, tell me about the half of a tree. How do you know it’s a half?”
“It’s a stump.”
“It’s a stump?”
“No. I mixed up the word. It’s a trunk.”
“It’s a trunk. OK. So tell me how you know something is half again. What does half mean?”
“Half means they are even pieces and you cut one of the even pieces up and you leave the other one, then they’re even, and that’s half. The one that’s down and the one that’s up without the other one is half.”
“OK. I have a whole lot of questions. Are you ready? This is the first one. Tell me what you mean by even.”
“Even? They’re both the same amount of tall.”
“They’re both the same amount of tall.”
“Yeah.”
“Whoa.” (Long pause.) “You’re talking about trees. What about wide? Do they have to be the same amount of wide also, or just the same amount of tall?”
“Just the same amount of tall.”
“Just the same amount of tall. OK. So here’s my next question about halves. How many pieces do you have?”
“Only two!”
“Only two. How come?”
“Because you cut one off and leave one on.”
“And that’s halves?”
“Yeah.”
“What if you cut it into three pieces that were the same amount of tall?”
“That would be quarters.”
“How come?”
“Because they just are!”
“What’s a quarter?”
“A little bit of something.”
“A little bit of something. What if I cut it into, like, 20 pieces of the same amount of tall?”
“That would be like…inches.”
“Inches! OK. So, it goes halves, the next smallest is quarters, and then the next smallest is inches?”
“Yeah.”
“What if I cut it into, like 1,000 pieces?”
“It would be just specks.”
“Specks…Hmm. And you said it doesn’t matter, like, I’m looking at that tree over there, and it’s fatter at the bottom–the trunk is fatter at the bottom and skinnier at the top–but the top has lots of leaves and the bottom doesn’t. Does any of that matter, or it’s just the same amount of tall?”
“Where?”
“That tree with the orange leaves.”
(It wasn’t this tree, but it was a sugar maple like this one.)
Long pause.
“That one has too many branches.”
“OK. So what kind of tree does it work for?”
“Trees…like that one!” (Pointing at a sapling.)
“OK, so what is it about that tree?”
“It’s smaller and it doesn’t have very many branches.”
“Hmm. What sorts of things can we cut in half?”
“Bread…flags…” She trailed off, and we were quiet for a while. I decided to revisit the thirds and quarters idea a little more.
“Can we think about cookies for a second? We made cookies yesterday. If you and Maya and I shared a cookie and we all had the same amount of cookie, how much cookie would we each have?”
“A quarter.”
“How come?”
“Because that’s a little bit of sugar and it’s all the same.”
“How many quarters does it take to make a whole cookie?”
“It matters how big the cookie is.”
“Tell me about that.”
“It would be a lot of pieces if it was a huge cookie, but just a little bit of pieces if it was small.”
“So, if I broke it into pieces so you got one piece, and Maya got one piece, and I got one piece of a small cookie, and then I took a big cookie, and I broke it so you got one piece, and Maya got one piece, and I got one piece of the big cookie, it would take different numbers of pieces, you’re saying?”
“Yes. Because they’re bigger.”
“So if it’s a bigger cookie that the three of us share evenly, we have more pieces?”
“Yeah.”
We kept going for a while longer, but you get the gist.
I love this conversation. I love it partly because it’s evidence that kids are mathematical thinkers who come to us full of ideas. Some of the ideas hold up to scrutiny and some don’t. Some are misconceptions we’ll need to address, and some are deep conceptual understandings and experiences we’ll want to build on.
What does Daphne know? We can divide wholes into parts. We can keep dividing into lots of pieces, and the different pieces have different names. We can use fractions to share. And, most importantly, we can look out the window and see mathematics, and wonder, and have conversations about our ideas.
What does Daphne not understand? Units, although I adore her sequence of halves, quarters, inches, and specks. She has a lot of fraction ideas intertwined. For example, even though she thinks we can have halves of big things, like trees, and halves of smaller things, like flags, she thinks a quarter is a size, and it’s small. There’s a lot of room to explore the distinction between the number of pieces and the size of the pieces. She has developing ideas about measurement and the idea of identifying a dimension: the “same amount of tall” goes in my math Hall of Fame. Volume didn’t occur to her here. That’s OK.
In case you’re wondering, I didn’t “address” any misconceptions, and I’m not worried about any of them. My goal here was not to have Daphne identify fractions with accuracy, or learn any definitions, or figure out how to measure something as complex as a tree, or answer my questions with an expected answer. My goal was to join her wondering, listen to her ideas, help her clarify her thinking here and there, and gain a deeper sense of how she’s seeing the world. I look forward to revisiting fractions and measurements with her many times, and to bearing witness as she makes sense of these ideas. It’s my privilege, as a parent and a teacher. See, math is fascinating. And Daphne is fascinating. Together? It’s the best.
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:
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?”
“9.”
“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?”
“Yeah.”
“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?”
“OH!”
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:
All math problems have to be answered with numbers.
All math problems can be answered.
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:
Teach students to “decode” math problems with “keywords,” like “in all means add.”
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:
3. 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.
This past week, my older daughter shared with me that something “horrible” had happened in math class. She didn’t understand the worksheet (it was a terrible worksheet, but that’s another post), so she asked about it. She said:
“Everyone was staring at me, and I heard some people whispering, and I even heard one person say, ‘I thought Maya was so good at math. What happened?'”
And then she started sobbing.
“I don’t ever want to ask a question in math again! It was horrible!”
I’ve been playing with my kids and math from the get-go. Maya has had more opportunities to practice math, more exposure to different aspects of math, and more support for thinking mathematically than most kids–especially girls–in our country. So, she started school with a leg up. The students in her class have decided she is good at math, and given her the kind of status that Ilana Horn writes about so well.
At the same time, my husband and I intentionally give our kids lots of growth-mindset messages. We emphasize how good productive struggle, confusion, and hard work are. We ask probing questions, encourage our kids to go deeper, support risk-taking, model not-knowing, and emphasize effort.
That wasn’t enough.
My daughter’s teacher is fairly new, but she clearly encourages risk taking and effort, and wants to have a classroom where all kids are valued. She loops, and I helped out in math once a week all last year, so I’ve been in there classroom enough to see how she supports kids.
That wasn’t enough either.
Parents’ messages and teacher’s messages were no match for peer status.
One of the major challenges teachers face is to deconstruct pervasive cultural messages about math and construct new, authentic, positive ones instead. This is incredibly hard work, and a major focus of my upcoming book. I actually had my daughter curl up with my laptop the other night so she could read a chapter about creating a mathematically productive classroom climate. It starts this way:
“I like math, but I’m not very good at it.”
“I have a hard time in math. Sometimes I get stuck and I don’t know what to do.”
“I’m not very good at math. It takes me a long time.”
“When I have to do math fast, like on a test or something, I have trouble.”
“Davon always gets the answer first. He’s like a math genius.”
“Math’s not really my thing. No one in my family can do math.”
“I suck at math. It takes me forever to do my homework.”
During my travels through dozens of math classes, I’ve been gathering examples of student talk about math. Sometimes I interview students, but most of the time I overhear them making these off-handed, revealing comments to each other. When I look through the long list of statements I’ve gathered, there is an unmistakable pattern. Students have a firmly established, deep-rooted, working definition of what it means to be “good at math,” and it goes something like this:
Being good at math means you answer the teacher’s questions fast, right, easily.
At this point in the text, my daughter looked up and said, “I think that’s what the kids in my class think too.”
I asked, “Do you think that’s what it means to be good at math?”
“No. Doing math well takes time. And you make a lot of mistakes. And it can be really challenging. And it’s not just about answering questions in school. It’s about seeing math everywhere, and wondering about it, and trying to figure it out.”
My messages had been sinking in after all. We spent some time that night talking about questions, and how important they are in learning math, and how my husband and I expect that she ask questions. We also rehearsed what she could say or do if kids reacted the same way the next time she had a question. (I wasn’t there, so I don’t know how many kids actually stared or whispered, but her perception is what matters here anyway.) She seemed to feel better by the end.
They had a test the next day. I asked about it.
She said, “This time, I told myself before the test, the only thing that matters is that I’m learning.”
Atta girl. Now we just need to change the culture so her peers get that message too.
