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## Breaking and Remaking Lesson Plans

I just had an interesting lab session with a school district we’re working with this year. The teachers (K-2) want to extend the scope and dynamics of their lesson plans, and make sure they challenge their students. But the teachers also don’t have much free time, are, in many cases, new to the curriculum, and also feel that they need to be in alignment with the curriculum, for various administrative reasons. (Although the expectation from admin is that the curriculum will be supplemented. But still.)

The implications for me were that rather than demo one of my favorite Math for Love games or lessons, I wanted to model how one might break and remake a lesson plan. I wanted a process that would be quick and dirty, so teachers could implement ideas quickly; after all, they might have only a short time to review a lesson plan from the curriculum before teaching it.

The lesson that was given to me to demo was flawed in some pretty intense ways. For one thing, it was a lesson for 1st graders that was meant to be teacher-led instruction for a full hour. (Let that sink in.) The goal of the lesson was to have students connect story problems and subtraction equations. It included, in the middle, almost as an afterthought, the instruction to let students come up with a few of their own ideas for story problems, which the teacher would transcribe on the board and solve altogether.

Here’s the process I wrote down, which I think does a pretty good job of imitating the sieve I pass lesson plans through in my own mind.

View Fullscreen

I thought this was a pretty good draft, and a good introduction to breaking and refashioning lesson plans to fit your classroom better. I’d appreciate feedback, especially if you have some favorite strategies my list is missing.

For this particular lesson, I launched with a related warmup (Target Number), and built an example story problem with the class. In fifteen minutes, we had accomplished what the lesson had said should take forty minutes, and then I released the kids for what we decided should be the main event: writing and solving their own story problems, and challenging each other with the ones they’d written. By launching quickly and letting the students generate their own work, we’d also succeeded in raising the ceiling of the task, and many of the students showed that they were comfortable with much larger numbers than the class had explored up to that point, adhering, as it had, to the curriculum.

And more than that, it was fun. The students were proud of their work, and stayed engaged, even though the material, without their input, wouldn’t necessarily have held their interest. Here are some of the student’s problems.

After 15-20 minutes of students dreaming up and solving their own problems, we had plenty of time to wrap up and pose even more difficult problems, involving one-digit numbers everyone was comfortable with, but more complicated stories, involving addition AND subtraction.

Many teachers don’t have the time, expertise, or permission to go too far outside their curriculum. But the good news is that most lesson plans in your curriculum are built around some good idea, even if it’s only a kernel at the center. But there’s a filter you can put these lessons through that accentuates the good while straining out the bad.

You’ve seen my draft above. Try it out, and let me know how it works for you, and if there are changes that should be made to it.

## Thoughts on story problems

Story problems! They are the great bugaboo of math class, the problems everyone remembers hating, On the other hand, when educators think of “real-world” math, useful math, or motivated math, story problems are where they want to go. And this instinct makes sense. Story problems should be a fantastic resource in the classroom—a chance for reading comprehension, making sense of problems, modeling, and more—but instead they’re dreaded and ineffective.

The seedy underbelly of story problems was stunningly revealed recently in this video by Robert Kaplinsky.

Let’s take a minute and consider what happened here. Three quarters of the students who saw this question attacked it with a kitchen-sink strategy: just do some arithmetic with the numbers and maybe you’ll get it right. Why don’t they do what the other quarter did and say, “this doesn’t make sense”?

It’s possible that they don’t want to disappoint the questioner, and that they figure doing something is better than doing nothing. It’s possible they didn’t feel comfortable asking questions or expressing confusion. But look at that last girl describing why she decided to divide; this strategy doesn’t come from nowhere. These kids are doing story problems as they’ve been taught to do them.

In school, many teachers teach kids how to solve story problems as a sort of code. There’s a protocol:

• Step 1. Underline the numbers
• Step 2. Circle the important words, such as plus, minus, sum, product, difference, quotient, together, and, more, less, etc.
• Step 3. Create an equation using the numbers and the operations corresponding to those words. (If the operation is subtraction or division, we’ll subtract the smaller number from the larger, or divide the smaller number into the larger)

This kind of rubric for solving story problems is self-defeating. We’re basically turning the intuitively sense-making project of reading a story into another kind of encoded math project, devoid of meaning. There’s a subtle line here, because teachers want to help, and underlining or noticing key words isn’t inherently a bad thing. But to begin by sweeping aside all pretense of meaning in favor of a mechanical process is bad mathematics.

Not surprisingly, it ends up being self-defeating as well. Story problems get trickier as students get older, and when they do, these kinds of mechanistic strategies backfire big time. The fact that most of the eighth graders in the video still seem to approach problems in this way spells trouble for their future in math.

So what’s to be done? Story problems have been around for millennia, and though they’ve often felt a little contrived, I don’t think they’re going anywhere. And truly, they are a relatively untapped resource. How do we use them to better effect?

Here are some ideas. I’d love to hear yours too.

Idea 1. Use story problems, but don’t teach a rubric to solve them.
Drop the story-problem “strategy” and focus on helping kids draw pictures or models of the situation instead.

Idea 2. Create story problems that resist basic strategies.
Interestingly, it’s not that hard to write story problems that can’t be cracked with the rubric-based story-problem-solving strategy. I’ve seen these pop up on high-stakes tests at the end of the year, leaving teachers to feel cheated that the test was gamed with questions designed to fool their students. But what if all the questions always resisted the basic hack of getting the answer without understanding the problem?

There are a number of ways to create hack-resistant story problems. The simplest is to create problems that don’t follow the same basic structure, but can be better solved by understanding the situation or drawing a picture. Here’s an example of this kind of problem from our Summer Staircase curriculum. This is a sheet for 2nd graders. Simply adding more steps to the problem makes these virtually impossible to solve without understanding what’s happening. On the other hand, if you draw a picture or build a model, they’re quite straightforward.

Get the full worksheet here.
Idea 3. Create story problems with real interest as well as complexity.

The name “story problem” suggests a story. Why not tell a real story? Folks like Marilyn Burns and Greg Tang have been writing math story books for some time now (see a long list at living math.net). What’s nice is that these aren’t too hard to write, they get huge buy-in from the students, and they combine the fun of story time with a much deeper thoughtfulness regarding the math. Here’s one I wrote for our Summer Staircase curriculum, suitable for 3rd/4th graders.

View Fullscreen

Notice that the engagement created by a good story allows for a much greater complexity in the mathematical modeling. What’s the relevant info for each question? The meaning can’t be lost, and kids are tuned into the meaning because it’s a real story. Some teachers gave kids a chance to draw their own version of the monster, creating an opportunity for an interdisciplinary lesson, involving reading, math, and art.

Even if we skip the pictures, we get reading comprehension combined with mathematical meaning. The downside is that creating these kinds of stories is more work. But I can imagine a collectively-produced library of them.

Here are a few more examples of these long-form story problems.
Story Problem – The Ant and the Grasshopper
Story Problem – The Kite

Idea 4. Switch to Video
The 3-act math lesson is another way to grab attention and focus on meaning-making. There’s a lot to be said for this format (and a lot has already been said by others). Check out a lovely example here: The Cookie Monster.

Using video or rich images as a launch can be great. Really, though, this is a different animal than story problems, so I won’t focus on it here.

Idea 5. Use story problems as launches to complex tasks
This is another idea that stretches the very idea of a story problem. Consider a problem like the indecisive director problem.

View Fullscreen
This is a great project, but it resists a straightforward solution, and requires less of the modeling and simple arithmetic of the story problems above, and is more about digging in to a much deeper problem. I’m a huge fan of complex tasks, and personally think they should be much more represented in math class, but this feels like a different animal to me too.

Idea 6. Have students write their own story problems
There’s nothing like standing on the other end of a process to understand its inner workings. A student who finished all the problems in The Kite asked what she should do next, and I suggested she write her own question. Here’s what she came up with—I liked it so much I converted it into a challenge problem for the other kids. (Check out the lesson to see.)

Having kids write story problems is usually a great idea. The dangers are that it becomes another unmotivated exercise, and that their story problems may not be appropriate for others to solve—it can be tough to write a story problem of the appropriate difficulty! That said, there are great opportunities when students are involved in the back end of producing problems as well as solving them.

What’s your take on story problems? It seems that story problems are an untapped resource, and with the right approach, they could be leveraged in all sorts of powerful ways. I’m still hopeful about producing a free library of good story problems in the style of The Monster. I like the prospect of combining the meaning-making involved in reading comprehension and mathematics.

Thoughts?

## In Praise of the Open Middle with Pyramid Puzzles

Sometimes you hear the perfect word to describe something you were already doing, but didn’t quite realize you were doing. My most favorite recent case of this came from openmiddle.com.

We’ve all heard of open-ended problems, the genuine, intriguing problems that can take us on journeys to unexpected answers. A lesson based around an open-ended problem can be a beautiful thing, and done right, there’s hardly a more exciting educational experience you can have.

The trouble is, it doesn’t always go so well. Guiding a class through a discussion and exploration of an open-ended problem requires tremendous insight and understanding on the part of the teacher, and even the most experienced teachers sometimes have them go awry. Will this student’s comments lead in a productive direction, or just muddle the issue, and leave us all frustrated? Is this problem so hard that we can’t hope for more than a partial understanding of it, or is there some key idea that we’re all missing? When you have to make assessments on the fly, things don’t always go well.

One trick to coax these explorations in productive directions is to game them just a little bit. The teacher can go in with a plan, or an arc of where they expect the lesson to go. So they can “discover” something like the fact that the midpoints of quadrilaterals seem to form parallelograms and ask, “that doesn’t always happen, does it?”, all the while knowing that it is, in fact, true, and with a couple of good ideas about how to prove it. This little bit of lesson planning can greatly improve the chances of an individual lesson feeling like a success, even if it gives up a little bit of the total free of open ended lessons.

Enter the Open Middle. The idea of Open Middle problems is that the beginning (the problem) and the end (the answer) are both clearly well-defined, but the middle of the problem—the part where you look for the solution—is wide open. The middle is where you cast around for ideas, structures, tools, or just blind guesses about what’s going on. Because Open Middle problems have such a well-defined structure, there’s much less of a chance of them going off the rails. This makes them a fantastic tool for teachers who might be taking their first steps into less regimented math teaching, or who don’t feel like taking a chance on an exploration yielding a broad discussion that doesn’t go anywhere. (Read more about Open Middle problems here.)

I produced a number of these Pyramid Puzzles recently, and I think they’re great examples of Open Middle problems. it’s relatively easy to adjust their difficulty as well.

#### Pyramid Puzzle 1.

Each number must be the sum of the two directly below it in the pyramid. Fill in the blanks.

This is a puzzle that seems easy, but the solution is just out of reach. We can put a 7 above the 2 and 5, but what then? There’s no obvious next step, and our best guess is to take a stab at it, and see what happens. Some people like to work from top to bottom, and try taking a guess of what two numbers might go below the 30 (say, 12 and 18?), while some prefer to work up from the bottom, putting a number in the blank and going from there.

Suppose I put a 10 in the bottom blank. Then I’d have this:

A 10 on the bottom leads to a 51 at the top, which is a problem. But it’s also a useful mistake, since now I know the number in the bottom must be smaller than 10. One wild guess gives me traction, and it’s only a matter of time before I solve this problem.

#### Pyramid Puzzle 2.

Each number must be the sum of the two directly below it in the pyramid AND no number can appear more than once. Fill in the blanks with positive integers so that the top of the Pyramid is 20.

Bonus: Could the 20 at the top of the pyramid be replaced with a smaller number and the pyramid still be solved? Show how if it’s possible, or show why it’s impossible.

This is a much subtler puzzle, and solving the bonus problem in particular requires you to delve into the workings of these puzzles. Is there some kind of theory that can tell us what the maximum number at the top of the pyramid can be, given the rules that we’re filling them only with positive integers, and never the same one twice? What about differently-sized pyramids?

This is clearly the smallest number that can be atop a pyramid 2 stories high.

And this seems to be the smallest number that can be on top of a pyramid three stories high.

So what’s the smallest for four stories high, or more?

We’re in the territory of open-ended problems now, which is where I always seem to end up. But that’s the exciting thing about mathematical exploration: there are always questions pointing in directions you’ve never gone before.

Find more of our Pyramid Puzzles here, at our lessons page.

Check out openmiddle.com for more examples of Open Middle problems.

## Pazuju – the great new puzzle you never heard of

I remember years ago reading about Sudoku. Already popular in Japan, the reviewer predicted that it would be featured in newspapers as regularly as crossword puzzles. Since then it indeed had, as predicted, a meteoric rise.

Math educators tend to be fans of Sudoku and similar puzzles (especially KenKen). Logic puzzles motivate the same sorts of thinking we use when we solve math problems: if I do this, then what’s the result? What can I learn if I assume the opposite of what I think is true?

Still, I’ve never been a huge fan of Sudoku. I find the puzzle a little dry, and don’t usually bother with it.

Cut to a few months back, when a publisher contacted me about a new puzzle called Pazuju. Pazuju is a kind of marriage of Sudoku and Tetris, including both logical and geometrical elements. It was hard for me to imagine.

The publisher sent me a sample of the book, and I tried one out. Then another. And suddenly, I was hooked. What I found was a puzzle that had the same logical appeal of Sudoku, but with more variety and interest. In my opinion, it’s quite simply a superior puzzle.

Will Pazuju be featured in newspapers everywhere, and eclipse Sudoku? I don’t know; it’s always hard to know what will rise to the top. But frankly, I think it deserves to be played everywhere Sudoku is. If you find yourself with a little time to spend on puzzles, I highly recommend this one. (I’m especially partial to the smaller 6 x 6 versions, which are generally quicker, though can still be quite subtle to solve.)

You can play Pazuju puzzles for free here. There’s also a book and an app available.

## Math Circles for Teachers

.

We’ll thrilled to be offering math circles for elementary and middle school teachers in partnership with the Washington Experimental Math Lab at the UW. With their generous support, these unique professional development meetings are absolutely free. Clock hours will be available.

Elementary Math Teacher Circle

Middle School Math Teacher Circle

More details:

Meeting place will be at UW main campus – specific location TBA
Meeting time 5:00 – 7:00, or possibly a little longer, to accommodate dinner
Participation is free!

Our dates are as follows, most of which are the third Tuesday of each school month:

Sep 20
Oct 25
Nov 29
Jan 24
Feb 28
March 28
April 25
May 23

Questions? Contact Kristine Hampton at kristine.l.hampton@gmail.com

## Summer Staircase Retrospective Part 1

We recently wrapped up our most ambitious project ever, and as data on it starts to roll in, I thought I’d take a moment to share.

Summer Staircase stats:
—19 schools
—57 teachers
—2500 students (plus or minus)

This summer, we produced a math curriculum for Seattle Public Schools Summer Staircase, a six-week program in Seattle to prevent summer slide and, ambitiously, help kids like school.

Our job was to write a 6-week math curriculum for three grade bands: Kindergarten, 1st/2nd grade, and 3rd/4th grade (these are the grades the students completed in 2015-2016). We also trained the teachers in the spring, and provided support during the summer.

The curriculum was built on the idea that play is the engine of learning. We featured games, explorations, and story problems that were actually stories. Our goals were high engagement, differentiation, critical thinking, and productive disposition. This last one was maybe the most important goal. We wanted students to leave the program believing in the project of education, especially mathematical education.

I’ll share more about the details of the program soon, especially as we begin to parse the data. Browsing through the teacher feedback, I feel like we’re onto something incredibly exciting. The overwhelming response from teachers and site leaders—the summer “principals”— was that something great was happening in these classes.

For example, of the 30 out of 57 teachers who have so far filled out a survey:
100% would recommend teaching math in summer staircase to a colleague
97% would recommend our curriculum to other schools or districts

At first glance, the students seemed to improve on every measure—pre/post assessments, teacher observations of their math understanding, perseverance, sense-making, and argument skills, and even their enjoyment and engagement in math. I’m deeply excited about how using a play-based curriculum can engage an enormous range of students in an experience of mathematics that is both more fun and more rigorous. I’ll be sharing more about the details of the program, the types of lessons we wrote for the curriculum, and the outcomes as we crunch the numbers.

But for now, I wanted to share some of the less tangible growth that teachers reported seeing this summer:

“In their pre-tests students were only writing the answer to each question. I didn’t see any work or thought on their papers. In the post-tests, every students showed thinking and pictures or used counters. They were excited to show their strategies. That’s really exciting to see as a teacher!”
“I think they all (or most of them) learned that they could love school and learning could be fun. This was awesome.”
“For many of them, the enjoyment of math was the single best growth they could have had. They definitely deepened their math understanding of certain concepts. Learning how to win and lose games graciously was huge for them.”
“They all shared a valuable experience in learning with each other (and the teaching staff- we, too learned a great deal), and being a part of the collective and individual student growth!”
“They grew in confidence, grew in their ability to talk about math and share ideas, and growth was evident as they played games and treated one another with respect.”
“[This program] renewed my faith and love in learning and teaching.”

And this from a site leader:

“The kids were so happy and had fun, and LOVED telling me how they figured something out. So even if the scores didn’t move that much, I know that the kids will return to school with a greater acceptance of math, and without that, “I hate math,” attitude.”

Will the scores move? We’ll find out. But I’m very, very hopeful. I think we’re on the cusp of something big.

## An Interview with Emily Grosvenor, author of Tessalation!

Tessalation! is a new children’s book about a little girl who discovers tessellations in the outdoor world. I backed the project when it was on Kickstarter earlier this year, and my book just arrived. Both the drawings and the writing is beautiful, and it is, to my knowledge, the only book for kids about tessellations.

Emily Grosvenor, the author of Tessalation!, and I have been interviewing each other about tessellations, the mathematical projects that stick with us from elementary school, and big words for little kids. A lightly edited version of this conversation is below.

The website for the book is here.
Tessalation! is also on Kindle here.

____________________________________________

Dan: Why did you choose to write a book about tessellations?

Emily: I’d say tessellations chose me. I’m a magazine writer and memoirist, but one day I got it in my head that I was going to write a children’s book and sat down to do it. I think most parents have this compulsion — when you’re reading a dozen every night you start to see the world in picture book ideas. From there, I thought about what I loved as a child and what I love now. I had discovered tessellations in a 4th grade gifted class and remembered having a blast making them. But I also love tessellated pattern as an adult. All I had to do was look around my home and its patterned curtains and pillows and floors to see I had an obsession. All that remained was the challenge of creating a book where tessellations were organic to the story. Patterns are soothing to look at.

D: Your description of the love of patterns cuts close to my own feelings. I used to imagine the imaginary ball that would ricochet around the room just right to hit some chosen spot, like a trick pool shot. Tessellations are similar—the shapes that align perfectly so foreground and background flip back and forth. There’s a visual beauty there that’s easy to fall in love with. For me, mathematical love is like that: the recognition that ideas fit together with the same sort of elegance and precision as shapes, so that everything meshes perfectly. There’s a deep satisfaction and also a sense of awe that can arise when all the pieces fit. Elation, you might say.

I’m curious about your original encounter with tessellations in fourth grade. Do you actually remember the lessons you did with tessellations? What about that experience grabbed you?

What do you think determines whether or not lessons “stick?”

D: That’s a great question, and one I’m often preoccupied with. I remember certain lessons and experiences from elementary school too, and I do think there are some hallmarks that distinguish the memorable ones from those we forget.

I think having control over the experience is critical to making an experience memorable. This is why art can be so compelling: we get to be the actor, the doer. We choose where the line goes, what the colors are, and even if we can’t always make it look like what we imagined, we’re still in charge. Mathematical experiences can involve the same kind of creative ownership, and when they do, they stand out.

Looping this back to tessellations, I taught a class for 2nd and 3rd graders on tessellations a little while ago. It eventually centered on the question of whether you can create a polygon with any number of sides that still tessellates. Can you make a 17-gon that tessellates? A 31-gon? A 99-gon? The problem is really fascinating. It starts with the more artistic project of just finding examples that do tessellate, and there’s a fair amount of coloring and decorating polygons to make cool looking designs. Eventually, we start discovering certain “moves” that change the number of sides of our polygon in a predictable way, and still produce something that tessellates: joining two polygons together, for example, or adding a bump to an existing tessellating shape. Suddenly we have families of tessellating polygons with 4, 8, 12, 16, 20, etc. sides, for example. And then there’s a very clear goal: how do we get those missing ones?

I don’t know for sure, but I suspect this problem, and problems like it, are memorable. There’s a clear answer, but constructing the argument (and all the tessellations) calls for a tremendous amount of choice and autonomy from the students. The fact that the products are so pretty is nice. And the arguments are just as beautiful, even though they’re harder to draw pictures of.

But back to the book! Do you have a vision of how it will be used? A good book for bedtime reading, or for classroom use, or to launch a rainy Sunday craft project at home, or for a walk in the woods? Or is it for all of the above?

E: Tessalation! definitely works best in the classroom or as an introduction to an afternoon activity at home. I am already getting emails from teachers who are using the book as a jumping off point to a discussion of patterns, tessellations and pattern-making. One of my backers hasn’t got the book yet in her hands but printed out her digital PDF for just that.

I specifically designed the book to have entry points for several age groups. For younger kids, such as preschoolers, just finding Tessa within the tessellations is excitement enough. My three-year-old, Griffin, points out tessellations wherever we go.
Older kids will want to make their own and may respond to some of the other, more complicated ideas in the text. For example, on one page I write about bees dancing where to imbibe. What does imbibe mean? Why are they dancing?
Children 3-8 have a natural obsession with animal life and the outdoors and love learning about how nature works. Conventional wisdom in children’s books holds that you use the simplest words possible, but if I want my children to know that the back of a turtle is called a carapace, and it works with the rhyme, I’m going for it!
As for getting kids outside, I have a strong fondness for the international nonprofit Hike It Baby and have been developing a Tessa Hike to do at Hike It Baby meetups. Basically, you read the book, gather objects on your hike, and then make patterns with them in the parking lot. Fun!

D: One more question for you. Tessalation! include lots of long words. Why use advanced words in a children’s book? And are there implications for math learning here?

E: I’ve long bristled at the thought that young people can’t handle big words. The publishing market is designed within evermore defined reading age groups, and understandably so. A teacher or parent who goes looking for a story or book kids can connect with will inevitably search out one that has the themes, language and subject matter appropriate for that age.

I will always be a fan of the simple story well told in simple language — the Llama Llama books are still big in our house — but I also know that children will take ownership of words that they hear. So when my 3-year-old tells me he just found a tessellation on the bottom of his new shoe, it is because he has been exposed to the word and the visual of a tessellation.
Here’s another 4th grade story for you. I remember doing a book report on Mischievous Meg by Astrid Lindgren and using the word “fabricate” to describe the main character’s penchant for storytelling. The teacher marked it with a little note that said: “Your word?”
I was highly insulted — was she asking if my mom had helped or suggesting that that couldn’t be my word?
One of my all-time favorite children’s books is Bubble Trouble by Margaret Mahy. Here’s an example, from the climax of the book’s action:
“But Abel, though a treble, was a rascal and a rebel,
fond of getting into trouble when he didn’t have to sing.
Pushing quickly through the people, Abel clambered up the steeple,
With nefarious intentions and a pebble in his sling!”
When my older son was young, he simply basked in the rhythms of the language, but as he got older he started asking what the words meant. With language, exposure is important. How would you rather learn a word — spoken by your parents, in your favorite book, or on a worksheet in Middle School?
As for Tessalation!, the reason it sat in a drawer for a year and a half was because I got some early feedback that it was a book for highly literate children, and I might think about making it more accessible to all readers. I sat on that for a while. It’s valid criticism. But in the end, I plowed ahead with my original goal, which was to make the cleverest book I could. Kickstarter was a good way to connect with people who respond to that.
____________________________________________
Thanks, Emily Grosvenor, for a wonderful conversation, and best of luck with the book!
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 wonderful book to read, especially for math teachers and mentors interested in improving their math-educational craft.

##### 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 Lockhart, 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!