A Brief Ode to Blank Paper

Maya (7) brought home a folder full of completed math worksheets yesterday, which put me in a funk. First, there was the bugs problem.

IMAG3939-1I 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:

IMAG3936Sigh. 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:

IMAG3950-1If 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.

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She created this number line and used it to solve the final problem.

IMAG3951When 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.”

Precisely my point.

The Up Sides of a Near Miss

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 MoneyI’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.

A Problem with the Space Inside it to Learn

Justin Lanier shared this bit of loveliness on Twitter the other day:

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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 Ceiling a 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.

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I got up to make my oatmeal, and noticed someone else in the family was using a different approach.

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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.
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Meanwhile, Daphne got in on the action with her own ruler.

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And I noticed this sketch in the newspaper. Sam wasn’t done.

IMAG3854This problem is so lovely because it’s tantalizing, and open, and full of possibilities. It reminded me of this tweet that went by yesterday.

Screen Shot 2015-02-01 at 12.48.48 PMThis problem has plenty of space inside it to learn. What do you think Daphne learned? What’s she thinking about?
IMAG3858Maya 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.

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