1

## QAMA – the calculator that won’t make you lazy

I got a call months ago from someone with what sounded like like a bizarre idea: he claimed he’d invented a special calculator to use as a pedagogical device in the classroom. I was about to dismiss the call as being a crank, when I realized that what he was describing, if it worked, was potentially a brilliant idea.

I bought myself a calculator, and began to check it out. It’s called QAMA, and it’s now available as an app, which is a much better deal, and you won’t accidentally kill the batteries if you leave the calculator on, as I did after the first time I used it. Nevertheless, I’m blown away by QAMA, since it elegantly solves one of the central problems of using calculators in the classroom: the problem of students handing all the thinking to the calculator.

Here’s how it works.

You plug in whatever equation you want to solve. For example, I entered “65.86 x 21.”

Then you hit the equals button. And the calculator doesn’t give you an answer. And therein lies its genius, its usefulness, and also, according to my conversation with the designer, the great technical difficulty in creating it in the first place.

To get an answer, you have to enter an estimate. If it’s just an arbitrary number, the calculator won’t accept it. Your estimate must prove that you were thinking. And the calculator expects that you can work at a pretty decent level. It requires perfect answers for single digit multiplication for example—it is no help with memorizing your multiplication tables. For this particular problem, I figured that 1300 would be a decent estimate. I entered it, and the calculator showed me the real answer: 1383.06.

It can be fun to play around with how good the estimates need to be. I tried 13/21. First estimate: 1.6, and this turned out to be an excellent guess. Second estimate: 1.5. QAMA wouldn’t accept it. Third estimate: 1.55. That was acceptable. The promise of this innovation is no doubt obvious to middle and high school teachers. QAMA has reinforced in its architecture the process of thoughtful calculator use, by making the tool that much more difficult to use mindlessly. Here’s what students should do when they use a calculator:

1. Decide if the problem actually requires a calculator.
2. If it does, get a rough sense of what a reasonable answer might be, then enter the problem on the calculator.
3. Pay attention to whether the answer the calculator gave you makes sense.

Here’s what students too often do once they have easy access to calculators:

1. Grab a calculator whenever they have an arithmetic problem to do.
2. Take whatever comes out as fact, and move on.

I’ve seen eighth graders reach for a calculator to solve 100 – 98. I’ve seen college students accept total gibberish from their calculator after mis-keying, without considering whether the answer makes sense on a gut level. (“The swimming pool costs… 53 billion dollars.”) Any teacher who gives their students access to calculators knows who pervasive these problems can become. “Does your answer make sense?” we ask, repeatedly, and often to no avail. But QAMA prevents students from having the option to be lazy. Their motto is: “The calculator that thinks only if you think too.” And that seems true.

Personally, I’ve come down against having students use calculators until middle school (except for occasional use in 4th-5th grade). But I think QAMA could take all the worst parts (mindlessness, laziness, etc.) of calculator use out of the middle and high school classroom. I think they’re on to something.

1

## Book Reviews

I read pop math books. Quite a few, in fact. Also, sometimes publishers send me advance copies and ask for reviews. I generally read these too. What makes me a bad reviewer is that I then wait for 6 – 20 months before I actually write anything down.

Time to remedy this situation! Here are a batch of quick reviews for books on or related to math that I’ve read in the past couple of years.

##### How Not to Be Wrong: The Power of Mathematical Thinking

Jordan Ellenberg has written one of the finest books on mathematics in decades. How Not to Be Wrong belongs in the pop-math canon, alongside Simon Singh’s best works (The Code Book; Fermat’s Enigma) and Robert Dantzig’s Number: The Language of Science.

There are two qualities that make How Not to Be Wrong exceptional. The first is how much news it contains. So many math books are rehashes of classic stories: Zeno, Archimedes, Pythagoras, Newton & Leibnitz, Euler, Gauss, and so on. Read a few accounts of the history of math and the stories, fun as they are, start to run into each other. Ellenberg begins with Abraham Wald studying bullet holes in airplane fuselages during WWII and goes off in all sorts of new directions from there. I was shocked at how much I had never seen before, and how seamlessly Ellenberg ties together statistics, mathematics, and common sense.

Fourth from the top is How Not to Be Wrong

The second quality that makes this book necessary reading is the sense of humor. The asides and footnotes are laugh-out-loud funny, and Ellenberg is a masterful and delightful writer to read. Bill Gates picked How Not to Be Wrong as one of his top five recommendations for reading this summer, and it’s at the top of my list too. Don’t miss it.

##### Love and Math: The Heart of Hidden Reality

How I wanted to like this book. It’s name is almost identical to this website’s, and Ed Frenkel had been on a tear, speaking on the Colbert Report and other TV and radio shows about the book, which promised to share what he loved about math. I started reading hopefully, and the first five or so chapters didn’t disappoint. Frenkel’s story of learning math despite anti-semitism in Soviet Russia is compelling and readable. Soon enough he’s invited to the West, and the story loses its dramatic tension: Frenkel’s career heads up, and the sailing is smooth. Frenkel tries to create dramatic tension around whether an important mathematician might show up at a conference or not, but the stakes just feel too low.

Wisely, with little narrative left to mine from his own story, Frenkel pivots in the second half of the book mainly to explaining the math and the story behind the Langlands Program, an ambitious and collaborative mathematical undertaking. While points of this project are interesting, the complexity of the mathematics exceeds Frenkel’s ability (and possibly anyone’s ability) to explain it to a lay audience. I’d estimate the necessary mathematical background for much of the second half of this book to be roughly graduate school level, and the layperson who tries to read it may find themselves scared off.

Meanwhile, there are two undermining details that become more glaring as the book goes on. First, Frenkel’s presentation of the mathematical world is deeply male. The female characters appear as charming wives who serve tea to their hardworking mathematician husbands, then disappear. Frenkel seems to have no problem with this, and his video project (looking for the equation for love) in the last chapter doesn’t do a lot to present a more positive space for women in mathematics. While professional math continues to be dominated by men, it feels more important than ever to celebrate female mathematicians and make clear that women belong in the field. Frenkel seems happy with his Mad-Men-esque vision of the field.

Second, Frenkel’s self regard unbalances the story. He’s a master of the humble-brag, and the longer you read, the more you have a sense that the story he’s really interested in telling is the one about how great Ed Frenkel is. Frenkel is taking on more and more of a place in the conversation around popular mathematics, and I think he has something important to share about the passion and beauty of mathematics. If he can make a little more room for underrepresented groups and a little less room for himself, I think his contributions will be that much more valuable.

##### The Math Olympian

This is a peculiar and kind of wonderful book. It reads like a soap opera, almost: a sort of Slum Dog Millionaire for a female Canadian Math Olympian, who, in the course of a 5-question test, flashes back through all her preparation and through important life moments.

There’s some solid math throughout this book, and, compared with Frenkel, a very clear place for women in math, along with a clear-eyed view of some of the specific difficulties they might face. What makes the book exceptional, though, is the diverse picture it paints of great math mentoring, and the emphasis on what really matters in mathematics—not the contests, it turns out, but the work of doing math itself. A book

##### The New York Times Book of Mathematics

The Times sent me a review copy of their Book of Mathematics last year, and I’ve been slowly reading it since then. There’s a lot here: over 100 years of reporting on mathematics. Overall, it’s a pretty impressive collection. More than anything else, it’s amazing to see what they got right: in so many articles, they’re interviewing the pivotal players, and capturing the most important breakthroughs just as they’re happening. Reading through the book gives you a sense of what the news was in 20th century mathematics: chaos theory, cryptography, computers, mathematicians and their major breakthroughs (Wiles, Perelman, Erdos, Conway, Godel and others make appearances throughout the book). There are some whimsical sections too, like an interview with the real Monty Hall, who takes the writer to school.

If you want to get a sense of what the news in mathematics actually was this past century, this is a great place to start.

##### The Mathematics Devotional

Clifford Pickover has written a number of big, beautiful, coffee-table-grade books on mathematics and physics, and I’ve been a fan. But when his publisher sent me this one, I was skeptical. A Devotional? As in, read an inspirational quote and ponder a picture? Indeed, that’s exactly what this book is: one quote and one image per day of the calendar year. And yet, I’ve had it for over a year now, and find myself opening it up all the time, and using it exactly how it’s meant to be used. It’s exactly what it set out to be, and I continue to be a fan.

Here’s today’s quote: “The thing I want you especially to understand is this feeling of divine revelation. I feel that this structure was ‘out there’ all along I just couldn’t see it. And now I can! This is really what keeps me in the math game—the chance that I might glimpse some kind of secret underlying truth, some sort of message from the gods.” —Paul Lockhard, A Mathematician’s Lament, 2009

So there’s some reading to check out this summer! I’ll return now to my stack of books and start reading. Next summer is coming fast.

## Tiny Polka Dot has funded!

We’re thrilled to announce that our Kickstarter campaign for Tiny Polka Dot has funded! This means we’ll be bringing this game into the world before the end of the year.

There are still 66 hours in the campaign, and you can still support the project, help us reach our stretch goals to make the game even better, and get your own copy locked down.

A friend of mine—a homeschooling mom and long-time K-2 teacher—said recently that:

Every time we sit and play I think how really with pattern blocks, some beans and a Tiny Polka Dot set…  you could cover math from k-2 and have lots of fun doing it.

While it is impossible to know how these things will go in the long run, I’m hopeful that Tiny Polka Dot has will be the kind of game that ushers mathematical play into classrooms and families that much more quickly.

Speaking of which, Emily Grosvenor, author of Tessalation, just interviewed me (and some fantastic colleagues) in a piece on mathematical play in Parent Map. Take a look!

## Pyramid Puzzle

This Pyramid Puzzle was recently featured in a post in Forbes about our Tiny Polka Dot campaign.

Take two suits–that’s 22 cards, with 0 – 10 each occurring twice. The puzzle is to make a pyramid using 10 cards of those 22 cards, so that each number in the pyramid is the sum of the two below it. Here’s a near-solution: every card is the sum of the two dots below it; the only problem is that there’s no third 1 to go in the last space on the bottom. (Excess cards are on the right.)

I just received this email from a friend who’s been play-testing Tiny Polka Dot with her kids.

“[my daughter] couldn’t make it work [with 10] with two sets then decided “OK let’s try 9.” Off to verify 10 really doesn’t work”

Here’s the photo she attached with the email (note: spoiler below!)

I actually convinced myself that 10 couldn’t work at the top of the pyramid… for a while. Turns out, I was wrong! More ends up being possible with this puzzle than meets the eye.

But I love this puzzle for precisely the reason that it worked so well for my friend’s daughter: 10 doesn’t seem to work, so she takes a leap of faith and tries 9 at the top of the pyramid; the puzzle rewards the courageous step of trying an even harder puzzle!

Is it possible to put an even smaller number at the top of the Pyramid?

[Sidenote: you can now get Prime Climb and Tiny Polka Dot together at a big discount if you support our campaign. Pledge here!]
1

## What are the aims and goals of math education, K-2?

As part of the Math for Future Elementary School Teachers class we’re teaching at the UW, we regularly ask our students to reflect on what they’re learning in the class. This particular student reflection felt so dense and comprehensive that I thought it would be worth sharing here (with her permission, and her name removed).

The prompt: What do you think are the aims and goals of math education for grades K-2? Include topics, values, and outcomes.

What do you think?

So you want your kid to know math? Of course you do. Math is an important tool, used widely in many disciplines, and helps us make sense of our world. It’s also beautiful, fun, and interesting, especially for young children.

Kids are just entering the world of patterns and numbers, and their love of math is ready to bloom. They are ready and excited to count, classify, name, and look for patterns. But too often, parents unintentionally do damage, either by neglecting their kids’ mathematical development, or by pushing them too hard, too quickly. It’s important to find the right balance, but many parents aren’t sure how.

This is an issue now, as more attention is being paid to the development of a human intellect. Pre-Kindergarten education, Head Start, Common Core, and the achievement gap are all looming large in public discourse right now because we care so much about educating our children. Research has shown the best education begins at home, with a thoughtful approach to nurturing your child’s natural math instincts.

The following guide is adapted from our new math game deck for three to eight year olds, Tiny Polka Dot. These ideas are, essentially, the keys to nurturing the natural mathematical instinct that is growing in every child.

##### Guide for Grown-ups

1. Play! Play is the engine of learning for young children. Provide your kids with a rich environment to play in, and let them take the lead. What makes for a rich math environment?

• Blocks – pattern blocks, legos, tangrams, and other blocks for building, sorting, and playing are the best.
• Games – Classics like Uno, concentration, war, dots and boxes, and Tiny Polka Dot are great as soon as kids are ready for them. Winning is irrelevant at first, and many games can be played collaboratively. Keep it light, and have fun yourself!
• A mathematical perspective – a walk in the woods, a pile of buttons, an old egg carton… these all hold rich mathematical structure if you look at them with the right eyes. Any pile of assorted object can become an opportunity to group by color, by number of holes, or by size. Walks become a chance for counting steps or physical challenges that involve gut estimations and intuitions.

EXAMPLE
You: How many steps do you think it will take to get to that tree?
You: Let’s find out!
[You walk and count. The answer, it turns out, was 15.] You: How many steps to that next tree?
You: Let’s find out!

Math teachers in elementary, middle and high school know how hard it is to get kids to intuitively understand whether their answers make sense. Connecting numbers up to the world with fun guess-and-check challenges pays off big time later.

As for egg cartons, check out the patterns emerging from kids playing with colored plastic eggs and egg cartons at Math-on-a-Stick at the Minnesota State fair. (These taken from Christopher Danielson’s aptly titled blog post, Let the Children Play.)
You might not recognize this as mathematical play, but sorting, symmetry, and grouping by equal numbers is precisely the foundation of the more abstract mathematics that will come later in school. And most importantly, it’s fun!

2. Learning takes time! The process of learning and mastering a new skill can be slow and complex. This is not a test. Do not to rush your child or expect them to know something after they have seen it once or twice. You may find a child makes the same mistake over and over again. As long as they are having fun, trust that your child is learning! It’s easy to over-help. Take a breath, and make sure you are letting your child take as active a role in the game as possible. It doesn’t matter if they get the answer right. Because it isn’t a test.

• WRONG QUESTION: “Is my child smart?”
• RIGHT QUESTION   “How is my child thinking?”

3. Think out loud. Your child is imitating everything you do, and the more you can narrate your thoughts, the better a model of thinking you can be. Slow down, find reasons to count slowly and clearly, and use mathematical language to describe what’s happening in your head. Never say, “Mommy (or Daddy) is terrible at math!” If you don’t have the ideal relationship to math, now’s your chance take a fresh look.

The happy truth about doing math with your kids is that it’s way more fun than you’re expecting it to be. It’s not about right answers, and it’s not about speed. It’s about playing, counting, building, sorting, and studying the wonderful, colorful world around us.

If you’re interested in more ways to explore math with your three- to eight-year-olds, our new math game, Tiny Polka Dot, is on Kickstarter now. It provides multitudes of ways to play with number and pattern in a colorful and fun family game.

Because really, we should all be having way more fun.

## Tiny Polka Dot – the colorful math game for young kids

We’re happy to announce that our newest math game, Tiny Polka Dot, is now on Kickstarter!

Tiny Polka Dot is a math game deck for 3- to 8-year-old children and their families. Over the past several years we’ve designed series of games through our work with pre-K and K-2 teachers. Tiny Polka Dot is the culmination of that work: a deck of colorful cards that pulls together the best of those games.

If you or anyone you know would like this game to exist, please support the campaign now. Learn more here.

##### Why Tiny Polka Dot?

The most common question we get asked about Prime Climb is: “can my 4 (or 5 or 6 or 7) year old play this game?” And although some people have reported having fun playing the game with young children, Prime Climb was designed with older kids in mind.

We had some great card games to recommend, but the most mathematically relevant ones usually involved removing the face cards or a bunch of elaborate rules. (Cribbage jumps to mind.) And the design of standard playing cards makes simple games like memory or war or their variations less mathematical than they ought to be.

Why is there no deck of cards that’s expressly designed to bring out the mathematical nature of these games? Three- and four-year-old kids often need practice counting different arrangements of objects, while Kindergarten-aged kids transition from counting to adding and subtracting; the right series of games and a thoughtfully-designed deck of cards could back up the entire sequence of early numeracy.

And, in the pivotal moment that often occurs in these stories, we suddenly realized that we should build it.

And so we did!

If you have a 3- to 8-year-old in your life, please consider backing the campaign, and spread the word to folks you know who would be interested! We’re really excited about this game, and with your help we can get it produced before the year is out.

## Math Festival, TEDx Talk, Fraction Talks

So much is coming up!

First, Seattle’s Julia Robinson Math Festival is this Sunday. It’s a festival celebrating collaborative, beautiful, non-competitive mathematics. If you’re interested in signing your child up (4th grade or older), there’s still time! You can learn more about the festival here.

Second, my TEDx Talk, Five Principles of Extraordinary Math Teaching, if finally out. I’d love to know what you think of it.

And third, I’ve been having a blast exploring the new website fractiontalks.com. This is a tremendous resource, a kind of one-stop-shopping experience to find fraction images to use with students of all ranges of experience. For example, what fraction of each shape below is green/yellow/blue/etc.?

More coming soon, including a new Kickstarter game for 3-8 year-old kids. Stay tuned!

4

## Thoughts on linear equations

I recently received this email from a teacher I work with:

“Dan, I have a question for you. I just introduced my [pre-algebra students] to slope and then to slope-intercept form of linear equations and wanted to explore with them some word problems which could be written in that form. (Ex: . For babysitting, Anna charges a flat fee of $10, plus$5 per hour. Write an equation for the cost, C, after h hours of babysitting. What do you think the slope and the y-intercept represent? How much money will she make if she baby-sits 5 hours?)

Do you have any ideas how to make this kind of lesson more fun, hands-on and exploratory for students?”

Here’s my response:

“This is a great question. Fortunately, there’s a large community online that’s working to solve it. I’ve got some ideas too 🙂

Strategies for making linear equations more relevant, more interesting, more exploratory:

1. Same problems, slicker delivery
An example might be the 100×100 cheeseburger problem. Same idea as the babysitting problem, but real life, involves a menu, and a compelling premise.
2. Slick delivery, unanticipated result
You can sometimes grab students attention with a problem that seems easy, but has a twist, like this stacking cups activity.
3. Pared down delivery
For example, visual patterns has the same info and the same question each time: here’s steps 1-4, and you can get step 43 as a hint. Write the equation. Purely visual, so students can begin immediately, and there’s actually more thinking work for them to do. It’s harder for them to just use a recipe approach.
4. Give two options to compare.
For example, a Would You Rather structure, as in, “Would you rather charge $5 base rate plus$7 per hour, or $15 base rate plus$5 per hour to babysit? Defend with algebra.” This is a harder question, and involves actually having to make an argument, which is a more compelling, more social reason to do something, and usually generates a deeper understanding.
5. Use Desmos.
I know a number of teachers who really like what they can do with this tool in the classroom. It basically allows kids to explore and solve problems with their computers or tablets, in some creative ways. There are some clever lessons in the teacher portal.
6. Raise the ceiling.
If you were to show the sequence of dominoes in a growth pattern below, there are natural questions that you (or the kids!) might ask:
-How many columns will there be in the nth stage?
-How many dominoes will there be in the nth stage (double n dominoes)?
-How many dominoes will be in the tallest column of this organization in the nth stage?
-how many dots total on all the dominoes in the nth stage?

Some of these questions go beyond linear equations, but provide a natural stretch questions, and can actually help kids understand how to model with equations even better.”

The teacher who wrote me closed by say “I have a few ideas, but would love to hear yours.” I feel exactly the same way. What else goes on this list?

## Phi is the new root 2

My knowledge about the foundation history of irrational numbers was challenged today, and I’m pretty happy about it.

I had recently tweeted a Vi Hart video that gave a fun, geometric proof of the classic first proof of irrationality: $\sqrt{2}$ is irrational. If it weren’t, that would mean you could build a square that had integer sides and an integer diagonal, and that would allow you to build a smaller square with the same process. To get a contradiction, repeat until you run out of integers.

After I tweeted the video, I got a response claiming that…

I was somewhat taken aback. In source after source, I’ve seen $\sqrt{2}$ named as the first number ever proved irrational. Variations on the same proof abound. And here was a claim that $\phi$, the golden ratio, actually holds the rightful place in history as humankind’s first brush with “the unnameable.” There seems to be a historical argument; but how complicated is the proof?

In fact, it’s so wonderfully simple that there’s a pedagogical argument to be made for teaching that $\phi$ is irrational before we even mention the Pythagorean Theorem or square roots. You need to know how to find angles in regular polygons and chase them around diagrams, and know how Isosceles triangles work. The thrust of the proof is the same as for $\sqrt{2}$, but it sidesteps the parity argument that can sometimes feel less tangible.

Let’s imagine, as the Pythagoreans might have, that every number is rational. An equivalent way to state this is there is always some scaling of any pair of lengths that allows them both to be positive integers. (To the Pythagoreans, the relationship between any two lengths was identical to the relationship between two whole numbers, axiomatically.) So suppose we have a regular pentagon with integer side length a and integer diagonal b.
The ratio of b to a, is precisely the golden ratio, by the way. But we don’t even need to know what it is. We’ll just try to show that a and b can’t both be integers.

First off, chase some angles around and you’ll see pretty much every angle is either 36, 72, or 108 degrees. This gives a bunch of Isosceles triangles. It follows quickly that

$x = 2a - b$ $y = b - a$ [I’ll leave that piece as an exercise. It’s pretty satisfying to chase angles around and have everything come out nicely.] This implies that x and y are positive integers. But they are the side and diagonal of a regular pentagon again, so the argument repeats! And this is the crux of the problem: positive integers can get smaller for only so long before they run less than 1. (Just like the infinite regress that was hinted at in that error-laden but inspiring work, Donald in Mathemagic Land.)

Conclusion? The original pentagon couldn’t have been drawn with integer sides to start with. And that means it couldn’t have been drawn with two rational sides, or else we would have scaled them up to whole numbers. And that means the relationship between the side and diagonal of the regular pentagon is irrational.

And there we have it. Irrational numbers without actually dealing with numbers at all. Or evenness and oddness of numerators and denominators.

A delightful discovery. We’ll likely never know for sure what length the Pythagoreans proved irrational first, but that’s a strong claim for $/phi$ over $/sqrt{2}$.

Especially because, as Donald found out, the Pythagoreans were all over pentagons and the golden ratio.