The title of this blog post is the last line to a beautiful, short film called GÖMBÖC. For a film where almost nothing happens, it’s compelling watching. And wonderfully, it captures the simple, profound, mathematical joy of thinking really, really hard about a beautiful problem.
Back in August, an email with the subject “Help, my daughter hates math!” showed up in the Math for Love inbox. To quote that email:
Here’s our situation: We are entering our fourth year of home schooling and my 10 y/o daughter has been utterly unable to memorize basic math facts. She does seem to grasp new concepts quickly (though with minimal retention), but the arithmetic isn’t sticking however much we practice. (This spring, for example, we practiced x3 every day for an entire month and every day it was like she was seeing them for the first time.) Last year I decided to put her back a year with the curriculum. My thinking was that the repetition and slower pace would fill in gaps and strengthen the foundation, while also giving her time to mature and hopefully the math facts would ‘click’. Didn’t happen. She’s going into 5th grade and last night it took her 42 minutes to do a 100 problem subtraction worksheet. She, of course, hated every minute of it.
I am at my wit’s end and am looking for any assistance/advice/resources you might be able to offer.
To this mom, we sent the following advice:
You’ve got an interesting problem on your hands, and it may be a bit tricky to handle it. If I were you, I’d pull back from the kind of practice that doesn’t seem to be clicking. It’s possible (though unlikely) that she has a kind of numerical dyslexia. I think it’s more likely that the work doesn’t feel motivated for her, or the arithmetic was introduced too early, before she was developmentally ready, and she’s suffering the continuing results of an initial negative experience, or series of experiences. The fact that she hates every minute of her math homework may have some effect on her ability to do it (and vice versa).
If it’s the case that she doesn’t have an initial understanding of math as something worth doing for its own sake—and let me know what her initial experiences with math were (did she ever like it?)—then you need to decouple the arithmetic from the stress and incoherence (for her) of the practice. There’s an art to this, as well as a science. If she’s a kid who likes playing games, I’d see if I could find games that she liked playing that helped her practice these facts. And I’d start super simple, with no worries about whether she’s at the level she’s “supposed” to be at. Start with war. There’s a million variations on where to go from there. Other games to check out are nim, casino, cribbage, shut the box… even classics like monopoly will work fine. Anything that involves arithmetic gives you a starting point. The trick is, don’t get on her case during the games to emphasize the arithmetic… just play with her, and let her see that she needs to do the arithmetic to play well (or play quickly, depending on the game). If you pay attention to exactly what she’s responding well to and struggling with, you can tweak whatever you’re working on. Just don’t go too fast, and don’t worry about her needing to “catch up.” Let her take her time, have fun, and show you what works for her.
If she’s less of a game person, you can explore working with blocks, or with crafts, to get things moving. There’s a book called Family Math that is chock full of ideas for activities and games to try. You can also check out our short list of activities and books to have fun with math at home.
In short, see if you can chuck out the worksheets for a little while, and focus on playing games that will be fun in their own right, and give her the arithmetic practice she needs. If she starts playing the games (or crafts) more expertly, you can (slowly) connect what she’s doing there to her school homework. But my hunch is that focusing on the games will be the best thing you can do for her.
This evening, we heard back from the same parent, and here’s the update, a few months later:
Thank you so much for your thoughtful response to my plea for help with my daughter’s inability to memorize math facts. I’ve taken a lot of your suggestions to heart and things have reached an equilibrium where I’m confident she’s moving ahead, but she’s not fighting back. She was so proud of herself the other day, saying “I never thought I’d solve a problem like … !” All is good in the math world these days!
…
I really tried to put your suggestion of separating “math” from “arithmetic” into practice. I explained to her that “arithmetic” is like learning how to read and that she hadn’t really even had a taste of the cool stuff math can do. A few times a week we find a video ([both my children] love Vi Hart and Numberphile) or a book (“Perfectly Perilous Math” is a favorite) to see the amazing things you can do with numbers. They loved it when they got to explain Fibonacci numbers to daddy! Also, we’ve replaced our evening math worksheets with 15 minutes of math “activities”. Her current go-to game is Colorku (a ‘color’ version of Sudoku) and I’m amazed by how fast she is getting.And thank you for your advice not to feel pressured to have her ‘catch up’. As a homeschooling parent who’s kinda winging it, it’s nice to have a reminder that ‘working at your own pace’ sometimes means going a little slower.Happy holidays!
It’s always great to hear that a child’s pathway in math has gotten a little less rocky and a little more in touch with what there is to love in the subject. Right as Thanksgiving week starts, there’s something to be thankful for.
This is a post about owning your mathematical experience, problem-solving, and flexibility. It also involves our goats—yes, we have goats—and a story of what happened when we took them hiking this summer.
What you need to know about our goats is that they hate to get wet (notice their bleats when they encounter the river in the video above). Still, if we, their herd, are on the other side of a river, they’ll figure out how to get to us.
Sid, the white goat, is a problem solver, and he takes full control of the problem at hand. When he crosses a river, he’ll stay still for a long time, considering his options, sometimes walking up and down the bank to scout for other ways across. Myshkin—the black goat—on the other hand, complains (via bleats) about the fact that he has to cross a river at all, then panics when he realizes he’s on one side and everyone else is on the other.
Here’s the thing, though: Myshkin is far more athletic than Sid; he can jump farther, higher, run faster, pull harder, etc. And yet, despite his athletic advantage, this river crossing goes much, much better for Sid.
Allow me to make an analogy here. We’ve got two approaches to problem solving here, one of which involves an owned, thoughtful, flexible approach, and one that involves avoidance and panic. This is one reason owning your mathematical experience is crucial: it makes math, and hence life, easier. Sometimes, the difference between easy and impossible is as simple as whether you feel you’re allowed to be a little flexible with a problem, and as a rule, we’re much more likely to mess with things if we feel like they’re ours.
Here’s a contrived example, which I normally wouldn’t mention, except kids see stuff like this all the time: without a calculator, determine the product 499 x 501.
Kid 1, who doesn’t own their mathematical experience, multiplies out this product using one of the long multiplication algorithms, and likely makes a mistake. When Conrad Wolfram talks about how we’re training kids to be third rate calculators, this is what he’s talking about (and you’d never trust the kid’s answer to this problem over the calculator’s).
Kid 2, who feels like they’re in charge, says, “Too bad they didn’t ask me 500 x 500, because that would be much easier.” Just in noticing that 499 and 501 are close to 500, this kid is already opening the door to something momentous: connecting hard new problems to easier ones you’ve already solved is at the heart of mathematics. This kid, who knows that 5 x 5 = 25 and therefore 500 x 500 = 250,000, now figures that the answer to the other problem must be close to 250,000. Maybe it’s even the same!
If Kid 2 is really on it, maybe they’ll check this out by doing some simple examples to see what happens when you make the numbers in a product one larger and smaller. Like:
1 x 3 vs. 2 x 2
2 x 4 vs. 3 x 3
3 x 5 vs. 4 x 4
and so on. And what this kid notices is that, amazingly, all the numbers on the left column are 1 less than the numbers on the right column.
1 x 3 = 3, which is one less than 2 x 2 = 4
2 x 4 = 8, which is one less than 3 x 3 = 9
3 x 5 = 15, which is one less than 4 x 4 = 16
Now Kid 2 is really excited. They’re raising their hand wildly, and they can’t wait to tell you that 499 x 501 must be one less than 500 x 500 = 250,000, that is, 499 x 501 = 249,999. Not only that, they want another one. They can tell you, they claim, what 499,999 x 500,001 is. They can tell you what 29 x 31 is. They’re all over it.
(They haven’t actually proven anything, just gone on the basis of a pattern, but let’s acknowledge that they’ve discovered something pretty cool here. Not only that, there are plenty of places to go from here.)
Here’s the thing about these two kids, though: Kid 1 has to contend with a much harder problem. Kid 2, you’ll notice, never does the tough long multiplication. There’s nothing in this whole though process that requires more than one-digit multiplication, plus adding on some zeroes at the end. What this means is that kids who don’t make connections to easier problems, who don’t feel enough in control of what they’re doing to take a step back and look at other ways it could have been or it’s connected to, those kids are actually doing math that is technically harder, as well as less illuminating and rewarding. This arithmetic isn’t just more interesting to Kid 2; the arithmetic is actually far easier.
In our analogy, Myshkin (the black goat) is Kid 1, better at straight arithmetic, but not thinking about how to set up the problem to his advantage; Sid is Kid 2, weaker when it comes to raw power, but in control and thoughtful about the situation. Kid 2, like Sid, finds the stepping stones that turns an impossible crossing into an exhilarating one. And Sid is the one who makes it dry across the river.
(PS You thought this blog post would be about this problem, didn’t you?)
The drug war is one issue that tends to be too hot for presidential politics. You won’t hear any questions at the debates about it, and you can be sure the candidates won’t be talking about it. But there’s a proposal in front of voters here in Washington State this November that has some pretty big implications for the drug war.
But let’s start at the beginning. Why am I writing about this issue on a math blog? The answer is that math is a subject that doesn’t merely exist in isolation. Learning math has political implications, and it’s an issue like this that helps us see it. And math helps us see the issue.
Consider, for example, the July article in the New York Times entitled Numbers Tell of Failure in the Drug War, which leads with this number: the price per gram of cocaine has gone down 74% since the beginning of the drug war; this is particularly sobering when you recall that the defining strategy of the drug war was to raise the price of drugs by cutting off supply.
Okay, so the price of drugs may not have gone up like they were supposed to (the story for cocaine is similar for almost all other drugs), but have we reduced the number of addicts? By this measure, too, the drug war has been a failure. Even as more money has been pumped in to “fight” drugs, the number of addicts has stayed roughly the same.
(Notice the discrepant labels on this graph? There’s some discussion of it here.)
Finally, the war on drugs is and always has been racist, and hideously wasteful. A recent report on marijuana use Washington State gave us some other telling numbers:
Although young African Americans and Latinos use marijuana at lower rates
than young whites, in the last ten years police in Washington arrested African
Americans at 2.9 times the rate of whites, and they arrested Latinos and Native
Americans at 1.6 times the rate of whites.
This racist application of justice has cost my state more than 200 million dollars.
The compelling thing about this issue, for me, is how the numbers tell the story so clearly. I remember hearing a story on This American Life where analysts at RAND came to the conclusion that we were making the wrong choices if lowering drug use was something we cared about: the drug war, as it has been prosecuted, has failed and is destined to continue failing, and creating waste and havoc as it does. (I couldn’t find the relevant radio program, but I did find this summary at wikipedia: “During the early-to-mid-1990s, the Clinton administration ordered and funded a major cocaine policy study, again by RAND. The Rand Drug Policy Research Center study concluded that $3 billion should be switched from federal and local law enforcement to treatment. The report said that treatment is the cheapest way to cut drug use, stating that drug treatment is twenty-three times more effective than the supply-side ‘war on drugs’.” Clinton tried to act, but got hamstrung by the politics of the moment in 1994. But that’s another story.)
Right now, Washington State has a proposition on the ballot to decriminalize possession of small amounts of marijuana. Personally, I hate pot. I’m not interested in smoking or ingesting it, and I tend to feel that people who use it are wasting their time. I find it a dopey and childish habit, and I find pro-pot culture idiotic. Nevertheless, it makes absolutely no sense for marijuana to be illegal. Even if this proposal doesn’t get everything right, I think it’s a step forward, a re-envisioning of how to discourage drug use in a public health and personal context, rather than a criminal one. The math tells us that the drug war has failed. It’s time for a new way forward.
So, I just voted yes on I-502. You may disagree with what I’ve said here. But if you argue against me, I hope you address the numbers.
I also hope to see The House I Live In. Jon Stewart’s interview with director Eugene Jarecki. This is a stunningly good interview.
We’ve got an exciting prospect in the works for elementary school teachers. This year, we’ll be offering professional development in mathematics in the form of Teacher Circles. Starting in October, these weekly gatherings will be places for 2nd-5th grade teachers to explore deep ideas in mathematics and math teaching. The full description is here. and elementary teachers can sign up now. The best part is that thanks to a grant from Washington STEM, we can offer these Teacher Circles at no cost; they’re free to participating teachers.
We’re pretty thrilled about it, and right now, we need to get the word out. That’s where we call on you. Do you know elementary teachers in the Seattle school district who would like this opportunity to work with us on taking their math classes to the next level? Do you know principals who support this kind of work for their teachers? Let them know! They can get information and sign up right away.
Last week was a very big week for mathematics.
First of all, Bill Clinton made arithmetic the centerpiece of his speech at the DNC. While it may not be new to let arithmetic affect policy, it has been absent from politics for some time. John Stewart hailed its return saying
I never thought I’d say this but I have missed you so much, math.
That’s the kind of thing we love to hear. The entire video is below. Warning: this plays 11pm on comedy central, and the language reflects that.
If that was all that happened this week, it would still be a big week. But, perhaps even more astonishingly, there’s been a credible, ambitious stab at a huge, unsolved mathematical problem wonderfully known as the abc Conjecture. The conjecture gives certain restrictions on the primes factorizations of numbers involved in the generic addition problem A + B = C. (Hence the conjecture’s name.) What’s so cool about it is that it seems to express a deep and almost distressing relationship between addition and multiplication (via prime factoring) that, for some reason, prevents certain kind of behavior. For example, the incredibly famous Fermat’s Last Theorem follows from the abc conjecture as a corollary. I saw a talk on this in graduate school, which was not too long ago, and the speaker essentially said that we have no idea how to even begin attacking this problem. Now, it may have been solved. The champion is Shinichi Mochizuki, and the details are here. If you want some deeper technicalities, you can find them here.
If that weren’t enough, Paul Lockhart’s new book, which I have been waiting for him to publish for 8 years, back when he gave me a draft that opened up some of the most beautiful mathematical approaches to classic problems I’ve ever seen, has finally been published. You can get a copy here. I’ll blog about this in more detail later, but if you’re interested in math or math ed, do yourself a favor and get yourself a copy now.
When we’re talking about a cultural interest in mathematics and mathematicians, this seems like a sign of a sea change: the reclusive Russian mathematician Grigory Perelman profiled in none other than Playboy magazine. Not a lot on the math, and it’s held out as something only a few hundred people in the world can understand, and Perelman is weird, of course. Still, he’s held up as a person who rejects the world in the manner of a beat poet, a kind of true cool. And it’s Playboy… I don’t know if it counts as cool, but at least it’s something that people read for the articles.
I just finished Malcolm Gladwell’s The Tipping Point. (I know, I’m over a decade late to the party. I’m a slow reader.) It was mostly pleasure reading, but towards the end I realized that what the book is about is not just how fads or epidemics spread, or how to create a best-selling product; it’s about how culture changes, and Gladwell’s hypothesis is essentially that certain elements—who’s proselyting the change, how compelling the change is, and the context the change occurs in—are the critical factors. Hush puppies tipped. Smoking tipped. Even suicide tipped in Micronesia (really!). So what would it take to make math tip?
In a way, that’s been our goal all along, which is why, in our own conversations, we haven’t restricted ourselves to assuming that we’ll only be working in education. The real goal is to change the culture around mathematics. Here’s the good news: I think it’s been happening for while anyway. The first mainstream breakthrough I remember was Good Will Hunting, and since then we’ve had A Beautiful Mind, Proof, Numbers, The Big Bang Theory, and more. When Steve Strogatz wrote pieces for the New York Times, they received hundreds of comments. Scientists are cooler than they’ve been in my lifetime. Is it happening? Is math tipping?
Here’s the counterpoint: when they talk about anything remotely mathematical on TV or radio, they still warn viewers or listeners not to run away from the media source. It’s still socially acceptable to describe yourself as a mathematical ignoramus. Bottom line: many people don’t know math, and are terrified by it. I don’t know how many, but I’d guess it’s a solid majority of the country.
So that’s our question: what can we do to tip our culture toward the pro-math direction, where it’s expected that you will learn math, where it’s understood that you don’t have to be frightened of it it, and where most of us will possess the facility to use it when we need it (and to know when we need it)? Our evolution at Math for Love has been: working with individual students; working with small groups of students; working with classes of students; working with teachers. That’s been evolution in the right direction: you help a group of students, that’s fine, but getting teachers excited about math and feeling that they own their mathematical knowledge and curriculum… that’s something else. It’s like the difference between giving a man a fish and teaching a man to fish (or better: teaching a math to fish and teaching a man to teach everyone in his village to fish).
Even though working with teachers gives us a broader reach and a better chance of making math tip, since they affect so many students, and can lead to a transformation of school culture, my gut tells me that there is some greater innovation, some leverage point we haven’t yet put our hands on. The hard thing about math is that it isn’t enough to see or hear it: you have to do it to feel what it feels like. I’ve always been impressed by my dad’s book because it doesn’t just tell you about the kinds of teaching he’s talking about: it actually gives you the experience of them. The closest thing I’ve seen to this in math is Paul Lockhart’s Mathematician’s Lament, but I don’t see that hitting the mainstream. What needs to happen? Do the cool kids need to start trying to find a simple proof of the 4 Color Theorem? Do we need a bad boy/girl mathematician (or scientist… a rising tide raises all ships) to cool it up in the media like Feynmann did for a while? (It’s hardly conceivable, even though the Onion made the connection.)
I’d love to end with a solution to this problem, but all I’ve got for a moment in the question: what needs to happen for people in our culture to think that knowing math is standard and being good at it is cool?
We are happy to announce that we’ve received an investment from Washington STEM to expand our work with teachers in Seattle area schools this coming year! We’ll be running math circles for teachers from schools that serve underserved and underrepresented kids, working to spread the love far and wide. We are still in the early stages of arranging the project, but we will keep the blog updated as the project unfolds.
You can read more about the other projects Washington STEM has invested in here.
Take note: Randall Munroe, the genius behind xkcd.com, has launched a new public service: people ask questions, and he answers on Tuesday. You have to see it to understand how amazing this is: his answers are funny, illuminating, and totally unexpected, even as they are absolutely true.
Here’s an example, where he determines how many Yodas it would take to transform the world to Yoda-based energy: http://what-if.xkcd.com/3/. What’s wonderful is that he has to consider so many elements that aren’t given in the question: the mass of an X-wing, the gravity on Degobah… and it comes together beautifully.
Or consider this one, where he calculates the odds of guessing your way to a perfect SAT. Again, what drives it home are the comparisons: how unlikely is it? He answers the question with lightning strikes.
This means that the odds of acing the SAT by guessing are worse than the odds of every living ex-President and every member of the main cast of Firefly all being independently struck by lightning … on the same day.
He publishes new What-Ifs every Tuesday. I hope to see these used in classrooms everywhere soon. After all, who wouldn’t want to learn more about what’s going on in this picture:
