Damult Dice Division

Quite some time ago (9 years!?) I invented a quick little classroom game called Damult Dice. It’s a dice game played with three dice. You roll, choose two to add, and multiply the sum by the third. In general, you’re trying to get as many points as you can per turn, though readers suggested many variations and improvements!

This week I received an email from a teacher named Christine who had spun off a variation she called Damult Dice Division. I thought it was a clever version, and wanted to share it here.

How to Play (the basic version)

On your turn, roll 3 dice. Choose 2 to make a 2-digit number, and divide that number by the number on the remaining die. Your score is the quotient, rounded down to the nearest whole number. You get a +10 point bonus if the quotient is a whole number (i.e., if there’s no remainder when you perform the division).

Example. You roll 2, 5, 6. You have the following options for moves:

  • 25 ÷ 6: 4 points
  • 26 ÷ 5: 5 points
  • 52 ÷ 6: 8 points
  • 56÷2 = 28. Plus the 10 point bonus is 38 points!
  • 62÷5: 12 points.
  • 65÷2: 32 points.

I like that 10 point bonus, because it prevents an algorithm (take the 2 biggest numbers and divide by the smallest) from determining your choice every turn, which is the weakness of the original Damult Dice for extended play.

Here’s Christine’s writeup and score sheet. She plays to 400 points, but I think playing to 150 or 200 points is probably sufficient.

My Variation of Damult Dice Division

In my original writeup of Damult Dice, Jason Buell noted the downside that while one person takes a turn the others in the game have nothing to do. So here’s a competitive variation of Damult Dice Division that gives everyone something to do.

How to Play Damult Dice Division All-Play

Take turns as the roller. Roll three dice. Everyone writes down a division problem formed by taking a 2-digit number and dividing by the third, along with the answer (either in decimal, fraction, or remainder form). The roller gets to score their move as per the original game, by taking points given by their answer, rounded down to the nearest whole number, and taking a 10 point bonus if there was no remainder.

After the roller has scored, all other players reveal the equation they wrote down. If a player was the only one to write down a certain equation, they get to score it. If two or more players wrote down the same equation (including the roller), they don’t score any points. The roller’s equation cancels out others as well.

Example turn

In a six-player game, it is player 1’s turn to be the roller. Player 1 rolls 2, 5,6. Everyone writes down an equation using those numbers.

The roller reveals their equation: 56 ÷ 2 = 28. This comes out to a whole for a 10 point bonus, so the roller gets 38 points.

Next the other players reveal their equations.

  • Player 2: 65 ÷ 2 = 32.5 (0 points)
  • Player 3: 65 ÷ 2 = 32.5 (0 points)
  • Player 4: 56 ÷ 2 = 28 (0 points)
  • Player 5: 26 ÷ 5 = 5.2 (5 points)
  • Player 6: 25 ÷ 6 = 4.5 (0 points)

Players 2 and 3 both wrote the same equation, so they don’t score any points. Player 4 wrote the same equation as the roller, so doesn’t score any points. Player 6 wrote an incorrect equation, so doesn’t score any points. But player 5 was the only player to write that equation, and it’s correct. So player 5 scores 5 points.

This version should get every student writing equations, and also gives everyone something to do on every turn. And of course, you can play with 1-sided dice, or with four dice (forming a 3-digit number to divide by a 1-digit number) for a more advanced game.

Does this look like something you would play with your students? Let me know how it goes!

Informed citizens need numeracy


I recently wrote about the three-fold nature of math education. The goal, I wrote should be:

  • To give everyone a baseline understanding of numeracy.
  • To give everyone at least a few glimpses of genuine mathematical beauty and power.
  • To allow those who might want to go on in mathematics-intensive fields like science, computer science, engineering, mathematics, etc. the preparation they need.

Yesterday I was listening to the radio, and happened to hear something that drove home just how important that baseline understanding of numeracy is, and why we need it for an informed electorate.

Exactly what numeracy looks like is a matter of some debate. Some folks argue that numerical literacy is basically achieved by the end of fifth grade—basic fractions, percents, operations, etc. (assuming students have learned what was being taught). Some say you need a deeper understanding of ratio and proportion, or algebra, or even geometry or polynomials, or probability and statistics. The details are important to iron out eventually, but the main point is the numeracy is analogous to literacy. We want students to know how to read, be able to easily read what’s around them, and, ideally, want to read. They may not read James Joyce or Dostoevsky, but inability to read shouldn’t prevent people from interacting with the world and having access to what they’re interested in understanding.

The same is true for math. People need to be able to read what numbers, equations, and statistics are telling them without fear or impediment. They need this for their own lives and future. But they also need it to be citizens.

Why informed citizens need numeracy: Nadine Woodward’s statistical lie

I was listening to the radio recently, and heard an interview with Spokane mayoral candidate Nadine Woodward. I’d heard that some people were skeptical of her proposals to handle homelessness punitively, by forcing homeless people to choose between addiction programs and jail time. But having not heard her speak before, I was curious to listen to her explain what was actually involved with her proposals on homelessness.

Woodward’s experience as a TV anchor was clearly serving her in the interview. She sounded like a paragon of thoughtful policy. She talked about how she came to the plan on addiction programs vs. jail time by exploring what had worked in other cities. But when the interviewer asked her about how her policy would deal with homeless people who weren’t addicted to anything, Woodward delivered a devious lie, a classic of what Charles Seife calls proofiness.

Her answer was that the vast majority of homeless people are addicts, and would hence benefit from being strong-armed into addiction programs. And then she rattled off the following statistics.

  • Percentage of homeless people with alcohol addiction: 38%
  • Percentage of homeless people with drug addictions: 26%
  • Percentage of homeless people with mental illness: 20-25%

Total percentage with any of the above: 84 – 89%. Right? Right?

Wrong.

At this point, I started yelling at the radio.

Can you spot the lie?

The Lie Unpacked

There’s the logical problem, which is that people with mental illness may not belong in the addiction program, but that’s more of a detail from the numeracy perspective. Let’s give her the benefit of the doubt and assume she’s got a program that’s great for addicts of any stripe, as well as people with mental illness.

The fundamental deception lies in the assumption that you can add those percentages together. What Woodward is suggesting that there are three distinct groups: mentally Ill, drug addicts, alcohol addicts.

Woodward’s deceptive suggestion of these groups

If every homeless person belongs in at most one of these groups, then you could add the percentages together, since there would be no overlap.

But the picture could be like this.

If every group was a subset of every other

The assumption here is that every mentally ill person is also a drug addict, and every drug addict also has a problem with alcohol. In this case, only 36% of homeless people would be candidates for Woodward’s proposal. Clearly, this is super unlikely. But it shows the nature of the lie.

Probably the real picture looks something like a classic Venn diagram, with various overlaps. Going to the extremes of no overlap vs. all overlap, we get a sense of the range of possible percentages of homeless people with one or more of these problems. That range, if the original statistics were right, goes from 38% – 89%.

The real picture probably looks like this, more or less

More overlap means a smaller overall percentage of people in any of the categories. So if mentally ill people are more likely to be addicts (a reasonable assumption), we should expect the total of people in all three categories to be smaller in total, since the overlap is larger. Knowing nothing else, I’d estimate the percentage of homeless people in one or more of the three categories is roughly 50%.

And here is the heart of the deception: Woodward commits statistical sleight of hand by adding the percentages up, and claims that about 90% of people are in the category of her program. In actuality, it’s more like half. That’s a huge difference, large enough to call into question her entire program.

More than that, it calls her honesty and integrity into question. This kind of dishonesty is, in a way, the worst sort. It’s based on real facts, but manipulates them to produce counterfeit results. They ring true despite being false. And using statistics like this means that you look at the data, don’t like what it tells you, and use a false version of it to make your case. And just refuting it makes me focus on homeless people who are in the categories she’s talking about, which leaves you with a mental image she’s trying to plant in spite of myself.

The interviewer should have caught this and pushed Woodward on it instantly. Without journalists who can catch these lies in the moment, we rely on a numerate electorate. The only problem is, we’re not there yet. But I hope this example makes the need clear. A numerically literate electorate is vital in order for democracy to function properly.

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What we mean when we say “Anyone can do math”

We need to unpack the phrase, and attendant phrases, that are so popular today, and that are in some ways so radical and unintuitive that we both believe and disbelieve them at the same time.

  • Anyone can do math
  • Everyone is a mathematician
  • You’re good at math (and don’t know it)
  • There’s no such thing as a math person. Everyone can do math!

And so on. These are correctives, and important ones, to another, earlier set of problematic (and faulty) axioms, that assumed the world is divided up into “math people” and “I’m-not-a-math-person” people. There are multitudes who believe they can’t do math when they suffer only from corrosive classroom experiences. But too unthinking an embrace of these taglines is problematic too.

The current excitement around growth mindset in classrooms around the world is meant, partly, to prevent math class from being a place where you get identified as a person who either has or doesn’t have the “math gene” (another discredited concept), and sorted accordingly into the appropriate track. Then students who are fast and know their facts are fast-tracked into more challenging and interesting mathematics, while folks who are slower or don’t have the facts down are placed in lower, slower tracks, and get the message that they don’t belong in the subject.

And yet we have a way of overcorrecting. Growth mindset is effectively a positive and useful outlook, but right now there’s a risk it gets overapplied (and under-understood) and becomes another educational fad that backfires in implementation.

When we say anyone can do math, what do we actually mean?

If we’re saying that everyone is equally talented mathematically, then we’re lying. And kids know this. You know it too. There are people who have unusual insights or abilities in mathematics. Some (e.g., Ramanujan, Nash, Turing, Johnson) get their own movies. And speaking of movies, that anyone can do math line has a counterpart in the movies, in Pixar’s Ratatouille. There, the line is anyone can cook.

“Not everyone can become a great artist- but a great artist can come from anywhere”

Ego, from Ratatouille

Ego’s parsing of the phrase anyone can cook is not obvious, and it’s not really the primary meaning of the phrase. The truth is, there are really three meanings all wrapped up there: anyone can learn to have the joy and pleasure of cooking in their life, even if they don’t become a master chef. Some people will get serious about it. And the visionaries who change the way we think about the art can come from anywhere – lock them out of the field and we all suffer.

This is what we have to mean when we insist that anyone can do math. For it to be more than an empty platitude, or a blatant falsehood, we have to be precise.

What does anyone can do math really mean?

  1. Everyone is capable of mathematical literacy. In other words, everyone has the capacity to learn the foundational mathematics that allow them to understand and participate in our (increasingly data-heavy) world. Everyone is capable of doing arithmetic, understanding fractions, percents, basic algebra and graphing, basic probability and statistics, and should be able to read a graph in a newspaper or hear a statistic on the radio without getting flustered. They should know that they have the ability to understand the vast majority of the math that surrounds them in the world if they decide to put in the work. This means they should have the numeracy to participate as citizens in our society, and also to pursue the career path of their choice. (It is shocking how many people literally give up on their dreams because it requires them to take too many math courses.)
  2. Everyone deserves to see some beautiful ideas of mathematics. Just like we send students on field trips to museums and have them read great poems and novels, part of their human inheritance is exposure to breathtaking mathematical ideas. (I’ve written about this extensively before, but if you feel like spending twenty minutes on a specific example, check out a 3blue1brown video, like this one on Hilbert Curves.) The fact that people respond with panic rather than wonder is a sign that we’re doing something wrong.
  3. A great mathematician can come from anywhere. We all have biases about what mathematicians are supposed to look like, and also what students who are “good at math” are supposed to look and act like. We need to teach like anyone and everyone in our classroom could have a gift for math that’s about to manifest… because they just might, and we may never know unless they’re given the opportunity.

This is what I mean when I say that anyone can do math. Not that everyone is equally talented (which is a lie), or equally interested in the subject (another lie). I used to say that Math for Love was dedicated to giving people a chance and a reason to fall in love with mathematics, but I know full well that not everyone will, which is fine.

What we should all be shooting for is a world where everyone is mathematically literate, and where fear or anxiety around mathematics doesn’t prevent people from doing the things they dream of doing. Everyone should see some beautiful mathematical ideas and know what it feels like. And if we can do that, we’ll also see great mathematical arising from all corners or our society and classrooms. Because there are kids who have a gift for or love of mathematics who we’re not reaching yet.

Not everyone is equally gifted in mathematics. But there are reasons to teach like everyone could be.

Rich Learning in Math Class

When I lead professional development, I focus on easy-to-implement changes first. Using openers and games are usually my first takeaways for teachers. When I’ve spent longer with them, I move to rich tasks.

I think of rich math tasks as the heartbeat of mathematical thinking, and essential to any classroom. They’re the best way, in my opinion, to offer real math—and the opportunity to thinking like mathematicians—to students.

They’re also tough to implement. They’re simple in a sense, but not easy, and they take practice. I’ve also learned that teachers often find them daunting at first. The good news is, with the right support, they can get comfortable using them in the classroom. Here’s a pre/post survey on comfort with rich tasks from a Math Teacher Circle series I just wrapped up.

To me, this is exciting. Our best tool to offer students rich learning experiences is teachable and learnable.

A new PD video support for rich tasks

This last November I flew to Australia as part of a grant to produce a video series on using rich tasks. The work was in partnership with an innovative math curriculum developer I’ve been collaborating with called Maths Pathway. The goal was to create resources that teachers anywhere would be able to use to support a move into using rich tasks in their classroom.

This series is now available. You can find the entire series, including specific lesson write-ups and video launch ideas, at Maths Pathway, or at Math for Love. Here’s the introduction.

Update: Mathematical Games in the Classroom

Registration is now open for my upcoming webinars on Mathematical games in the classroom. Register today for early-bird pricing!

Mathematical Games preK – 2nd Grade
Mathematical Games 3rd – 5th Grade

I’m thrilled to be offering my first-ever webinar, on how—and why—to use mathematical games in the classroom to best effect. Hosted by Christina Tondevold’s Build Math Minds website.

Christina and I made this video that shares my three traits to look for in great classroom math games. I also share an incredibly easy and fun game that you can use with no resources whatsoever.

https://www.youtube.com/watch?v=O4O_rvY6Xto


There are a whole slew of great webinars to check out as part of this series. I’m excited to be a part of it. I hope you can join me!

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Mathematical Games in the Classroom

I’m thrilled to be offering my first-ever webinar, on how—and why—to use mathematical games in the classroom to best effect. Hosted by Christina Tondevold’s Build Math Minds website.

Christina and I made this video that shares my three traits to look for in great classroom math games. I also share an incredibly easy and fun game that you can use with no resources whatsoever.

Early bird registration opens January 16, but you can get much more info and preregister now at the links below!

Preregister for pre-K – 2nd  grade webinar, Feb 19
Preregister for 3rd – 5th grade webinar, Feb 26

There are a whole slew of great webinars to check out as part of this series. I’m excited to be a part of it. I hope you can join me!

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How Do You Make Math Fun?

I was recently asked to be on a panel discussion online, along with a few others with an interest in recreational mathematics. The topic was how do you make math fun?

Because of time zone differences, I ended up writing a fairly detailed first post on the panel. I thought it would be of interest to readers of this blog as well. You can see the entire panel discussion here.

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Part of me wants to say you don’t have to make mathematics fun, because it already is. Or rather, it can be fun. It can also be frustrating, illuminating, elegant, baffling, challenging, and addictive. The question probably needs to be “how do you make SCHOOL math(s) fun?” Or possibly, “how do you make school math(s) meaningful and motivated?” And a typical answer to that is you make it more like real mathematics.

But I’m not sure that’s sufficient as an answer. It’s feeling like there’s something new that’s happening in mathematics education, and it has to do with crafting experiences that are more likely to be engaging, more likely to be playful, and more likely to be social. Even if these existed occasionally, making them more ubiquitous actually changes how people experience the subject.

When people are young (say, 2 – 8), mathematics tends to be a source of joy. Kids seem to be drawn to ideas about number, shape, pattern, and structure in a similar way they are drawn to language. They learn through experimentation, play, and repetition, and the exposure to mathematical ideas is fundamentally empowering. I think we need to create frameworks that imitate how young kids are drawn into mathematical thinking. Mine looks like this:

  1. Spark their curiosity. Get them engaged in an irresistible mystery. This means letting questions hang in the air without answers.
  2. Support their productive struggle. People learn by trying to make sense of things that aren’t obvious. This can be frustrating, but we need to let the struggle belong to the student. If we take it from them, we take the satisfaction and joy as well.
  3. Let students own the experience. A chance to reflect or share can let students see what they’ve done, and how far they’ve come. If we’re just concerned about them having the right answer, we communicate that their understanding and ownership isn’t what’s important. So we really have to give them space to take ownership of the process and the ideas that come from it.

One very important thing to note is that play supports all of this. For mathematics, play is the engine of learning. When you’re in a playful state, you’re more likely to be open to curiosity, more likely to struggle, and more likely to feel a sense of ownership.

So for parents as well as teachers, and especially for primary grades, I’d say the most vital advice is to play with mathematics. Playing games is great. Playing with blocks is crucial, especially for young children, since there’s a physical intuition that gets built that ends up providing fundamental analogies for mathematics. Just living with questions and providing a space for questions to live is very powerful.

The second thing I’d suggest is to change your fundamental question from “do you know the answer?” to “how are you thinking about this?” Worry less if your kid has reached whatever bar you think they need to reach. Instead, let yourself be curious about what’s actually happening in their mind. Mathematics has been called supercharged common sense. If we teach people to ignore their intuition and follow nonsensical steps to arrive at answers, we’re doing a deep disservice to them, and damaging their foundation for mathematical thinking long term. Don’t be answer-driven. Be sense-driven.

Will all this make mathematics fun? Sometimes it will. But hopefully the real shift is in letting mathematics be playful, challenging, empowering, meaningful, and motivated.