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.

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.

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!”

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.

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.

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.

]]>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!

]]>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.

**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.

**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.

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.

- 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.
- 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.

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.

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?

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

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!

]]>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.

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.

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.

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?

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.

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?

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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.

]]>Recently, in an art gallery in Ballard, I saw the amazing painting above. (Check out the artist’s website here.) I love this kind of mathematical art–the tessellation in the background a kind of blanket that subsumes the floor and clothes of the people in the picture. It’s a meeting of mathematical structure and organic complexity.

We’re going to be exploring all kinds of beautiful geometric structure this spring and summer! Registration is open for Math for Love‘s upcoming:

These can fill up fast, so sign up now if you would like your student to join us!

Read on to learn more (or skip to the bottom for the Problem of the Moment).

Math for Love classes are a chance to learn beautiful, powerful mathematical ideas from mathematicians wholove to teach.

Sign up your K-8th grader now for a spot in our classes at the Phinney Neighborhood Association, starting Sunday, April 19.

Our theme this spring is *Making and Breaking Conjectures*.

Find out more here.

The mission of the Julia Robinson Mathematics Festival is to inspire students to explore the richness and beauty ofmathematics through activities that encourage collaborative, creative problem-solving.

Join us Saturday, April 4th for this noncompetitive celebration of great ideas and problems in mathematics. Held at the HUB on UW’s campus, and open to all students grades 4 – 10.

To learn more and sign up, click here.

To volunteer, click here.

Price: $10 – 15. *Free and reduced registration is available. Use the discount code “scholarship” to get an additional 50% discount.*

Made possible with financial support from the UW Math Department and the Puget Sound Council of Math Teachers.

We are expanding our popular Summer Math Camp from last year, with four 1-week sessions in Seattle and Bellevue.Registration is open now. Sign up your 8-10 or 11-15 year old for one or more sessions, and see what will unfold in these playful mathematical explorations of shape and structure.

Learn more here.

A Fault-Free Rectangle is a rectangle made of dominoes that contains no horizontal or vertical “faults,” that is, lines that would allow you to pull the rectangle apart into two rectangles.

A student recently constructed the fault free rectangle on the right. Is this the smallest fault-free rectangle possible (not counting the single domino)? If so, what is the next largest fault-free rectangle you can build?

- Prove that

It feels, as Prof. S says in the video, like a beautiful conjecture.

I highly recommend trying to come up with a proof. There are many (54!), and I’ve come up with about seven since I saw the video. More intriguing, though, is the question of whether this type of thing happens in any other cases. Let’s look at a picture that summarizes the most surprising part of this problem.

It is not particularly surprising that . The shock is that as well. Are there other right triangles we can draw on a grid that have this property? The answer, it turns out, is yes.

*Prove that .*

Let’s go into even greater generality. Suppose we have two rectangles with integer side lengths. (Everything that comes later will refer to the picture below.)

**Big Question 1**: For what collections A, B, C, and D will the ?**Big Question 2**: Given any A and B, does there always exist a C and D so that ?

Here are some of the answers I’ve discovered so far:

A = 2, B = 3, C = 5, D = 1.

A = 3, B = 4, C = 7, D = 1.

A = 4, B = 5, C = 9, D = 1.

*Prove the examples above all satisfy .**Define a pattern in the numbers above. Will always work?*

Here’s another observation about the list of numbers above. ! What could that have to do with things?

**Wild Conjecture**: If , then .

*Prove or disprove the wild conjecture.*

Here’s another sequence of solutions where :

A = 3, B = 5, C = 4, D = 1

A = 5, B = 7, C = 6, D = 1

*Prove that these sets of A,B,C,D satisfy .*

What happens if we sum the squares? Once again, is double . This is evidence in favor of the Wild Conjecture. Not nearly a proof.

Notice what’s happened here. We began with an isolated question about a cool relationship between specific angles. By asking the next natural questions, we have very surprising variations of that original relationship, and an entirely different pattern emerging in the squares of the sides of the rectangles. True understanding comes not from solving one problem, but in solving families of problems in multiple ways, and following the natural questions as far as we can.

I’ll leave you with a suggestive diagram for a proof of the original three square problem.Here we have the original 1 by 2 right triangle (AEB) next to a scaled up 1 by 3 right triangle (CDB). Can you see why their two small angles (the ones at B) sum to 45 degrees? Do you see how this picture could generalize?

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Part 1 ends with a mystery, including a $100 bounty for anyone who can find a counterexample to the trick. If you know a student (or are a student) who thinks you can find a way to break this trick, let me know if you do.

]]>This Saturday session runs for six sessions, from October 18 – November 22. This session’s topic: Games, Logic and Arithmetic.

**Kindergarten & 1st grade
Section 1 **12:05pm – 12:55pm Sign up now!

**2nd & 3rd grade**

12:05pm – 12:55 Sign up now!

**4th & 5th grade**

1:05pm – 1:55pm Sign up now!

**6th, 7th & 8th grade**

2:05 – 2:55pm Sign up now!

This new Sunday session runs for eight sessions, from October 5 – November 23. Games, Logic and Arithmetic will be the topic for these new Bellevue classes too.

**2nd & 3rd grade**

10:00 – 10:55 Sign up now!

**4th & 5th grade**

11:00 – 11:55 Sign up now!

**6th, 7th & 8th grade**

12:00 – 12:55 Sign up now!

Math Circles meet before or after school, and highlight our favorite games, puzzles, and mathematical ideas. Dates and times are variable. Click the signup link to learn more, or check with your school.

APP at Lincoln — Sign up now!

Catharine Blaine K-8 – Sign up now!

McGilvra Elementary School – Sign up through the school

Queen Anne Elementary – Sign up now!

Thurgood Marshall – Sign up through the school.

View Ridge Elementary 2nd & 3rd grades – Sign up now!

View Ridge Elementary 4th & 5th grades- Sign up now!

West Woodland Elementary – sign up through the school