1

This Week’s NPR Puzzle: Extra Challenge

This week’s Sunday puzzle on NPR is a classic from Sam Loyd. Here’s Will Shortz:

This is one of the “lost” puzzles of Sam Loyd, the great American puzzlemaker from the 19th and early 20th centuries. It’s from an old magazine with a Sam Loyd puzzle column. The object is to arrange three 9s to make 20. There is no trick involved. Simply arrange three 9s, using any standard arithmetic signs and symbols, to total 20. How can it be done?

I played around with this puzzle this morning, and indeed, there is a way to solve it using only standard arithmetic (the four operations, parentheses, decimal points). However, as I was playing around, I found a second answer as well, using square roots and factorials.

Can you find both answers?

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Goodbye hexagon, hello 6-gon!

A colleague of mine once remarked how strange it is that while the Greeks talked about 6-cornered shapes and 4-sided shapes, we talk about hexagons and quadrilaterals. Why is it, aside from the historical accident that it is, that we persist in making people learn Greek to talk about shapes they see everyday?

And quadrilaterals and hexagons are the easy ones. What’s a 7-sided polygon called? It’s either a heptagon or a septagon—I never remember (do you?). What about a 12-sided polygon? That’s a dodecagon. Thirteen sided? No idea—no one ever taught me that one.

PolygonsOur vernacular around polygons is tied to an ancient system of numeration that not even experts know. We’ve created a system where we can speak properly about only a select subset of polygons: triangle, quadrilateral, pentagon, hexagon, octagon. Those prefixes denote the numbers 3, 4, 5, 6, and 8. And while there are certainly some of you who know more, I don’t think we ever bother teaching more than this. It’s like teaching inches and feet, and not bothering with miles.

We couldn’t convert to metric in the US, but we can do something even easier when it comes to polygons: name them by number. Forget the name of the icosikaitetragon? Just call it a 24-gon. Heptagons and decagons are 7-gons and 10-gons. We could even call hexagons 6-gons.

The advantages are immediate and enormous. First, every polygon now has an easy, instantly recognizable name. We’ve removed the barrier of Greek between ourselves and shapes. Second, we’re reminded of the defining trait of the thing when we name it. It’s why we call it a red-breasted robin instead of a Turdus migratorius.

Do you agree? Before you answer, let me add one more point: mathematicians use this nomenclature already. We even say n-gon instead of polygon, just so we can decide what n is later, or use the variable in equations.

We could have gone further. The prefix -gon is just Greek for -angle, as in triangle (3-gon). And while 5-angle and 9-angle have a certain poetry, I like the staccato of 5-gon and 9-gon. And it’s not so bad to keep a little Greek in there.

Sure, it’s nice for students to know the word triangle. And you can argue that learning vocabulary for the polygons is fun. But do we really need to add barriers around mathematical objects when we could just call them what they’re called? Do kids need to know quadrilateral or heptagon to relate to 4-gons and 7-gons?? If you’re a math teacher, you can start calling pentagons 5-gons and decagons 10-gons tomorrow.

And the beauty of it is, your students will know exactly what you mean.

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The Four Questions

Math Play Venn Diagram

We’ve argued for a long time that the real experience of mathematics is inextricably tied to play. But if you’re a parent or teacher, you’ve seen kids play in mathematically irrelevant ways. How do we hit that sweet spot of mathematical play?

One way is to recognize mathematical questions and ideas when they arise from the play itself. Another is to subtly introduce them, without sacrificing the play when you do. So it isn’t, “Stop playing so we can do these flashcards.” Instead, it’s “I wonder how many Jenga blocks you could stack on a single block without it falling.”

We’ve found that there are four central questions that can help uncover or motivate mathematical play.

1. How many?
2. How much?
3. What kind?
4. What if?

Each of these questions highlights a topic from mathematics, and arguably these same questions are the ones that continue to motivate modern mathematical research. Roughly, the correlation goes:

1. How many = questions of number
2. How much = questions of measurement
3. What kind = questions of classification & geometry
4. What if = questions of logic & imagination

For younger kids, the first two questions can often be simplified into a “which is bigger” or “which is more” kind of question, which can avoid counting if they’re not ready to do it.

Once you have these four questions at your disposal, opportunities for mathematical play start popping up everywhere. A walk around the block can be a chance to play the game of estimating steps and then checking whether you were right. (“How many steps do you think it will take to reach that tree?”)

Or an argument over who has more juice can become an experiment to see which cup actually holds more liquid. (How much juice can it hold? Which cup holds more? How can we figure that out?”)

Little kids will automatically classify and sort. What’s fascinating to me is how central this question is throughout mathematics. It seems like every field begins with a question of what kinds of objects are possible. What kinds of pentagons tessellate? What kinds of of polygons can you make by putting squares next to each other?

As for What If questions, cartoonist Randall Munroe has practically built a career out of them. The freedom to assume that the rules or the setup is different is one of the keys to owning your mathematical experience.

If you’ve got a favorite question you ask to find that sweet spot on the Venn diagram, let us know!

A last thought: perhaps the Venn diagram above isn’t to scale. Maybe it should be more like this:
Math Play 2

The Problem We All Live With

The problem we all live with

Back in 2006, I had the chance to see Jonathan Kozol when he visited Seattle touring his new book, The Shame of the Nation. The country, he said, had more educational racial segregation in 2006 than it did in 1968, when his first book, Death at an Early Age came out.

While the US rejected the doctrine of “separate but equal” in the landmark 1954 Brown vs. Board of Education Supreme Court decision, racial segregation in schools continues to be a fact of life. It almost never comes up. That’s the reason that the This American Life episodes, linked below, are so relevant. Integrating schools has been shown to be one of the most effective way to close the educational outcome gap between students of different races; it is also one of the only options for school reform not on the table.

As the beginning of school rises up on the horizon, these radio shows couldn’t be more timely. They are required listening for anyone who cares about race, class, education, and the future of the American democratic experiment.
The Problem We All Live With – Episode 1
The Problem We All Live With – Episode 2

After that somber note, here’s news on Math for Love in September. Our Sunday class registration is open for K-8th grade; Prime Climb has won another award (it’s fourth!); we’ve wrapped up our work on the Seattle Summer Staircase—a fantastic project once again—and got a brief vacation before this year kicks off. We’ll be running professional development with more schools than ever this year, as well as expanding our partnerships with libraries, pre-Ks, and possibly embarking on some new, exciting partnerships (more on these later).

We’ll have lots more news to share as the year progresses. Here’s a little inspiration for the beginning of the school year that made me smile.

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5 principles of extraordinary math teaching

We’re just finishing up a massive project of creating a supplementary curriculum for Seattle’s Summer School program. We realized that the spirit of the lessons was even more important than the content. To this end, we designed the activities to encourage students to own their mathematical experiences, to give kids an opportunity—and a reason—to fall in love with math.

So we introduce our lessons with this list of the 5 principles that you can use in your math teaching to make the classroom hum. We wanted to share the list here, even though school is almost out. Let us know if these principles speak to you, and if there’s anything you think we’re missing.

1. Give students time to struggle

Students learn by grappling with mental obstacles and overcoming them. Your students MUST spend time stuck on problems. The more a teacher steps in to solve a student’s problems, the less the student learns. This is not to say you shouldn’t be involved in their process at all. Learn how to identify when your students are productively stuck—i.e. unable to answer the question but still making progress by making various attempts at understanding the problem—and when they are unproductively stuck—i.e. giving in to despair and hopelessness about the problem.

Productively stuck students need little more than a bit of encouragement, reflection, or the occasional prompt from a teacher (best offered in the form of a question, such as “What have you done so far?” or “Have you tried ____?”). Unproductively stuck students need help scaffolding the problem, by rephrasing the question, identifying learning gaps, and possibly backing up to a more concrete or simpler problem. For both, time is critical: prioritize giving students the time required to let their perseverance flower.

2. Say yes to your students’ ideas

Doing math is creative work. It requires making connections between distinct concepts, translating knowledge into new contexts, and making intellectual leaps into unexplored territory. These are the hallmarks of creative thinking, and this is exactly the kind of capacity we want our students to develop. Creative work is hard, though, and becomes especially hard when the process of creative work is received with skepticism and negativity. When a student is working on a hard math problem, they are in a delicate place full of uncertainty, and a lot of the time the ideas they will have are wrong, or at least not exactly right. Many teachers want to point this out immediately to a student who is tentatively putting forth what to them is a novel idea on how to make sense of a math problem. However, to have an idea shut down means the student misses out getting to see why their idea might or might not work, and more importantly, they miss out on the exciting process of following wrong ideas into deeper understanding. We want our students to practice coming up with ideas and following them, even down rabbit holes, to see what they can discover.

As a teacher, one of the best ways to support the creative growth of students is to say yes to their ideas. That doesn’t mean confirming the correctness of an idea, but it does mean refraining from pointing out the wrongness. Instead, encourage students to test out their ideas for themselves. Say yes to the creative act and respond “I don’t know—let’s find out!”

3. Don’t be the answer key

Most students will avoid hard work if they suspect there is an easier way. (Most people do this too. It’s an efficient strategy for handling a complex world with an abundance of information.) Unfortunately, there is no substitute for hard work when developing the mind. Students need to struggle with concepts themselves if they are going to understand or master them. They will not struggle if they believe that instead, they can ply the teacher for the answer. The teacher needs to avoid being seen as the source of all knowledge in the classroom.

Rather, the teacher is the orchestrator of the classroom, setting up learning opportunities in which the students come to possess their own knowledge through grit, patience, and hopefully joy. Instead of using your knowledge to confirm to students when they have answered a problem right or wrong, encourage students to reference their own understanding of the problem and the mathematics behind it. If they don’t have the conceptual models at hand to check their understanding, help them build what they need.

4. Questions, questions, questions

Practice asking questions. Practice launching your lessons with questions and interacting with your working students by posing questions. Give your students opportunities to ask questions, and find ways to show them you value their questions. You can do this by using their questions to guide a lesson, having a special “Questions” board in the classroom, or making time for students to think of and write questions in a math journal.

Not all questions will be answered, and that’s okay. (You are not the answer key, remember?) More important than answering all the questions is learning the practice of asking them in the first place. Students benefit from having the classroom be a place of questions. Questions keep the math classroom active, engaging, and full of surprises. For many students, developing the habit of asking questions about math, and seeing the teacher ask questions about math, marks the point in their elementary math lives when math truly comes alive.

5. Play!

Seriously. The more a teacher models a positive and excited disposition toward learning and especially mathematics, the more students will begin sharing in the fun. Find the parts of math that you love, and share your joy with the students. Look for opportunities to keep play at the center of the classroom: for example, introduce games to students by playing them (rather than just explaining them); give students an opportunity to play freely with math manipulatives; and be willing to play along when students try changing the rules of a game to invent their own variation.

Avoid false enthusiasm: students know the difference. Find out how to get excited about math, and give yourself permission to play. Maybe for you this means being attentive to patterns, or finding really juicy questions to start a lesson with, or spending time making your own mathematical discoveries (remember how good an aha! moment feels?). Develop your own relationship with math and your students will benefit.

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Math that makes worlds

From the May 18 New Yorker article World Without End, by Raffi Khatchadourian:

The design allows for extraordinary economy in computer processing: the terrain for eighteen quintillion unique planets flows out of only fourteen hundred lines of code. Because all the necessary visual information in the game is described by formulas, nothing needs to be rendered graphically until a player encounters it. Murray compared the process to a sine curve: one simple equation can define a limitless contour of hills and valleys—with every point on that contour generated independently of every other. “This is a lovely thing,” he said. “It means I don’t need to calculate anything before or after that point.” In the same way, the game continuously identifies a player’s location, and then renders only what is visible. Turn away from a mountain, an antelope, a star system, and it will vanish just as quickly as it appeared. “You can get philosophical about it,” Murray once said. “Does that planet exist before you visit it? Sort of not—until the maths create it.”

This article gets to something fundamental about the mathematical experience for me: even when you’re making the rules, the rules talk back, and give you worlds to explore that you couldn’t even have conceived of. The description of the programmers being drawn into exploring their own, unfathomable creation resonates; that’s the story of mathematics from the beginning. They found a way to make it visual and more broadly experiential.

Hopefully the game, No Man’s Sky, will be fun for everyone who plays it!

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Quick Physical Games for the Math Classroom

We hold these truths to be self-evident, that kids need to move around, and creating opportunities to move during math class can pay off in spades.

Therefore, we have a collection of some of our favorite math/movement quick activities to share. These are especially good for K-4, though they’re adaptable to older and younger grades too. They provide a dose of movement, fun, and mathematical practice in an abbreviated time frame–perfect for station breaks and transitions.

If we’re missing any of your favorites, let us know! A PDF with Common Core tagging is available at our Lessons page.

Teacher-led Games

  • Groups (2-5 minutes)
    The teacher calls out a number (3), and the students have 10 seconds to get themselves into groups of that size. It might be impossible for everyone to get in a group every time, but each new number gives everyone another chance.In the basic game, just call out single numbers. Once students get the gist, you can call out addition or subtraction problems (i.e., “get into groups of 7-4.”)

    Don’t forget to call out a group of 1 and a group of however many students are in the entire class at some point in the game.

  • Stand Up/Sit Down (2-5 minutes)
    The rules are simple: if the teacher gives the number 10, students stand up. Any other number, they sit down. The trick is, the teacher will say things like “7+3” and “14 -5” (pick appropriate sums and differences for your students to solve mentally). This is a great game to try to “trick” the students by standing up or sitting down on when they should be doing the opposite.There are endless variations. For example:
    -stand when the number is larger than 5; sit if it is 5 or below
    -stand when the number is even; sit when it is odd
    -stand if the digit 1 appears on the number; sit otherwise.
  • Bigger/Smaller/Equal (2-5 minutes)
    If the teacher says a number greater than 10, students expand their bodies to take up as much space as they can (while keeping their feet firmly planted on the ground—no running around). If the teacher says a number less than 10, students shrink their bodies to take up the least space they can. If the teacher gives the number 10 exactly, students hold their body neutrally and make an equals sign with their arms.As before, the teacher moves to sums and differences once students get the rules.
  • Rhythmic Clapping/Counting (2-5 minutes)
    The teacher claps/counts out a rhythm. Students imitate the rhythm of the clap and the count.
  • Skip Counting with Movement (2-5 minutes)
    Make up a movement that comes in 2, 3, or more parts. Whisper the first parts, and call out the final move loudly.
    Example: Windmills. Whisper “1” and touch your right hand to your left foot. Whisper “2” and touch your left hand to your right foot. Call out “3” and do a jumping jack! Continue counting like this up to 30, calling out the multiples of 3 and whispering the numbers in between.Example: http://mathandmovement.com/pdfs/skipcountingguide.pdf
  • Circle Count (2-5 minutes)
    Stand in a circle and try to count off as quickly as possible all the way around the circle. Start with 1, then the student on your right says “2,” and the student on their right says “3,” and so on until the count comes back to you. Challenge the kids to go as quickly and seamlessly as possible.When everyone can do this proficiently, count by twos, fives, tens, or threes. You can also start at numbers greater than 1, or try counting backward.

Student-pair Games

  • Finger Speed-Sums (1-5 minutes)
    Students meet in pairs with one hand behind their back. On the count of three, they each put forward some number of fingers. Whoever says the sum first wins. Then the pair breaks up and each person finds a new person to play with. Advanced players can use two hands instead of just one.
  • Finger Speed-Differences (1-5 minutes)
    Same as speed-sums, except whoever find the difference between the two numbers first wins.
  • Five High Fives (1 – 2 minutes, or longer with the exploration)
    Students try to give a high-five to five different classmates. When they’ve gotten their five high-fives done, they sit down. This game is part mystery: sometimes it will be possible for everyone to get a high-five; sometimes not. The difference (which the teacher knows but the students don’t) is that it is only possible if there are an even number of people giving high-fives. Try this game at different times and let students guess whether they think everyone will get a high-five or not. Why does it only work sometimes, not always?If you make it four or six high-fives instead of five, then everyone will be able to get their high-fives every time.

Why we love these games

Getting kids moving is a win-win. Movement refreshes your students while giving them another take on math concepts. These games are super quick and super fun for everyone.

Tips for the classroom

  1. Make sure kids never feel ashamed if they don’t already know the right answer. You can also tweak competitive games to make them collaborative.
  2. You enthusiasm is critical in these games. Figure out your favorites, and expand on them, or get the students to come up with their own variations. If you’re into them and having a good time, the kids will have a good time too.
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Counting Collections and Dots and Boxes, fractional version

I just rewrote our write up of Counting Collections, and reclassified it on our Lessons page as a Foundational Activity. The reclassification was motivated in part by a conversation with a Kindergarten teacher, who mentioned that she had been having the kids in her room count collections every Friday. “I help kids who need it,” she said, “but honestly, they’re so focused and engaged, I could probably disappear and the class would still run all right without me.” When lessons work this fluidly, it is easy to feel that you aren’t doing enough as a teacher. In fact, it’s the opposite: the necessity of heroics, theatrics, and great feats of charismatic teaching can be a sign that the activity you’re pushing at the kids might not be a good fit. The easy path can sometimes be the best one, especially when the teacher is underworked because the students are doing the heavy thinking.

Speaking of student inspiration, I was playing Dots and Boxes with a student last week, and he made a play I hadn’t seen before: instead of connecting two dots with one edge, he made two half edges. Suddenly, it was a new game. I made four quarter edges; he played three third-edges. Then we started connecting using multiple fractions: 1/2, 1/3, and 1/6. This fractional version of Dots and Boxes goes like this. On your turn, you have to add a total length of 1 unit to the board. If you complete a box, you get a point, and an extra unit length to add before your turn is over. Whoever completes the most boxes wins.

So after 8 turns from Blue and 7 from Red, a game might look like this.

Dots and Boxes 1

 

It’s Red’s turn. What move can Red make? Any full edge will give Blue a box to complete. But what about adding half-edges?

Dots and Boxes 2

Now where can Red go? Adding halves will give Blue a box, but Red can add three thirds!

Dots and Boxes 3

Does Blue have a move that won’t cost a box? I’ll let you figure it out. This game is totally new to me, but it’s clear that it will end (every turn adds a unit length of line segment to the board, after all), and this particular game won’t end in a tie either. I think it’s more likely to end in a blowout for the winning player.

Try out the game and let me know how it goes!

Cheryl’s Birthday Party, Meta-logic, and the known unknown

I almost missed the Cheryl’s Birthday Party internet phenomenon this week. An awkwardly written logic problem went viral, and the internet was abuzz with attempts to solve it. Check out the NYTimes treatment of the origial pnroblem, and the afterparty.

The problem is an example of a metalogic puzzle. Logic puzzles usually consist of organizing what you know and don’t know to suss conclusions out of ambiguous seeming clues. Metalogic puzzles are a different sort of beast: they require you to imagine what characters know and don’t know to solve problems. In a metalogic puzzle, a character not knowing something can be critical information.

Here’s some examples of my favorites.

The Three Hats.

Three princesses are taken prisoner by an ogre. For sport, the ogre puts them in a row from eldest to youngest, so that each sister can only see her younger sisters. Then he blindfolds them and says, “I have three white hats and two black hats. I will put one hat on each of your head. If any one of you can tell me the color of your own hat when I take off your blindfold, I’ll let all three of you go.”

The ogre takes the blindfold off the eldest princess. She looks ahead at the hats on her two younger sisters, thinks, and finally says, “I don’t know what color my hat is.”

The ogre takes the blindfold off the middle princess. She looks ahead at the hat on her younger sister, thinks, and says, “I don’t know what color my hat is.”

Then the ogre takes the blindfold off the youngest princess. She looks ahead into the woods–she can’t see anyone else’s hat. Then she thinks, and says, “I know what color my hat is.”

What color is her hat? How did she know?

I love this puzzle, because it doesn’t seem like there’s enough there to answer it. The secret lies in imagining yourself in the head of each princess as they imagine themselves in the head of their sisters. It requires a fantastic leap of imagination. I’m not going to solve it for you, but consider:
What does the eldest sister know about her sisters’ hats?
What does the middle sister know from the fact that the eldest sister doesn’t know her own hat? What does it tell us that even with this information she doesn’t know the color of her own hat?

I’ve recently come up with my own series of metalogic puzzles. I don’t think these are too hard if you can get past the fact that they seem so devoid of usable information to start.

The Known Unknown

Abby and Bill play a game, where they each think of a whole number and try to guess each other’s number by asking each other questions which must be answered truthfully.

Game 1

Abby and Bill each pick a number in the 1 to 30 range.
Abby: Is your number twice my number?
Bill: I don’t know. Is your number twice my number?
Abby: I don’t know. Is your number half my number?
Bill: I don’t know. Is your number half my number?
Abby: I don’t know.
Bill: I know your number.
What is Abby’s number?

Game 2

Abby and Bill each pick a number in the 1 to 40 range.
Abby: Is your number twice my number?
Bill: I don’t know. Is your number twice my number?
Abby: I don’t know. Is your number twice my number?
Bill: I don’t know. Is your number twice my number?
Abby: I don’t know. Is your number twice my number?
Bill: I don’t know. Did you think we chose the same number?
Abby: We didn’t.
What are their numbers?

Finally, here’s a classic that I’d say is a level more difficult, and seems even more impossible to solve than the previous ones. It’s definitely doable though–you just need to remember who knows what, and write it all down.

The Surveyor’s Dilemma

A surveyor stops at a mathematician’s house to take a census.

Surveyor: “How many children do you have, and how old are they?”
Mathematician: “I have 3 children. The product of their ages is 36.
S: “That’s not enough information.”
M: “The sum of their ages is the same as my house number.”
S: “That’s still not enough information.”
M: “My eldest child is learning the violin.”
S: “Now I have enough information.”

What are the ages of the mathematician’s children?

Play is where love begins

I recently wrote a piece for the New York Times Numberplay blog on what we do to help people fall in love with math. I thought I’d include it here.

__________________________
No matter whom we work with, our initial goal is for them to have an authentic, mathematical experience; that is the first step to helping anyone — teachers, students, parents, kids, you name it — develop a love for math. The question then becomes, what does authentic mathematics look like, and how can you get people to experience it, especially if they’ve been turned off from the subject?

The answer is play. Play is the engine of learning, and our minds are wired to play with mathematics. When people are able to play with mathematics, they learn more profoundly and expertly than they would otherwise. The enemy of play is fear, and insistence that others do things your way, the “right” way. If you share mathematics as something you love, a gift you and everyone else gets to play with, and have the discipline to reserve judgment, what you end up seeing is that people of all ages will quickly take up the invitation to play with you, and soon they are doing math with a level of creativity, interest and rigor that you might have thought would be impossible.

All we need to do to open people up to mathematics then, is give them permission, and the right circumstances, to take ownership of their mathematical experience and begin playing. There are critical elements to provide:

A worthy mathematical question or problem, with a low barrier to entry and a high ceiling,
A safe atmosphere, free of judgment,
Time, and whatever encouragement and support is necessary to coax people into playing.

I think the Broken Calculator problem qualifies as the right kind of mathematical question, potentially, though working with teachers or students I usually change the question to: What numbers are possible to get on the calculator and what numbers are impossible? That way, people can mess around and get positive feedback. You made 12 on the calculator? Great! That tells us something new — what else is possible? That’s what I mean by a low barrier to entry. But the problem is designed to resist easy answers, too (unless you’ve worked on this kind of question before), so it is unlikely that anyone without training would be able to say, “Here’s the answer. Now what?”

I recently demoed a pared-down variation of the Broken Calculator problem in a third grade classroom (without square roots) as part of a teacher professional development partnership. I was worried that I’d have to sell the problem to the students, but the opposite was true: The moment the kids saw that they were in control of how they approached the question, they couldn’t get enough. Kids know how to play.

In the end, play really is the magic ingredient. Einstein called play the highest form of research; happily, play is also the best way to win anyone over to math. Play is where love begins.