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.

Wizard Standoff Game & Lesson Plan

I just received an email from a teacher named Dustin Stoddart, who turned the Wizard Standoff Riddle I created with TED-Ed into an interactive classroom game. This is an appealing way to explore the intuition behind the probability and game theory of the original riddle. I’m sharing the original riddle and Dustin’s lesson below.

Thanks, Dustin!

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Fibonacci-like number sum puzzle

I just add a fascinating conversation on twitter, and I made a video to pose it to you. In particular, if you’ve got upper elementary or middle school students (or high school, or college), and want to explore whether this pattern keeps working, I’d love to hear how it goes.

Here’s the original tweet, and my video synopsis below.

 

Letter to a student – Fibonacci numbers and Lucas Numbers

Last spring I received a letter from a student who wanted to know more about me, and more, especially, about Fibonacci numbers. I wrote him back, and shared a bit more about Fibonacci numbers and their twin sibling, the Lucas numbers. Fibonacci numbers get a lot more attention, but, like real and imaginary numbers, there are many things about them that remain invisible until you put both together.

I’m including the letter below, and I hope you enjoy reading it, or sharing it with students in your life who might be interested in the Fibonacci numbers. Eddie Woo recently included some beautiful Fibonacci-related images in his TEDx Talk. These images, the “magical” connections are the bait we mathematician-educators use to draw people toward our topic. (Mystics do this too.) But you really see the magic and the meaning when you dig into them and do the math.

 

Here’s the letter.

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Dear M-,

Thanks for writing. I’m happy to hear about your passion for Fibonacci numbers! They deserve your excitement.

I grew up in Olympia, Washington, and have loved math for as long as I can remember. I’m a big fan of graph theory and combinatorics, which is the mathematics of how things connect, and counting. I’m quite familiar with Fibonacci numbers, though, and I’ve explored and even taught about them.

Here’s something really important about the Fibonacci numbers: they’re only half of the picture. There’s another sequence called the Lucas numbers that go like this:

1, 3, 4, 7, 11, 18, 29, …

(see what’s happening there?)

The Lucas numbers are like the twin sibling of the Fibonacci numbers. They really belong together, and you can see things when you put them together that are invisible with just one.

For example, have you ever added every other Fibonacci number together?

The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, …
The sum of every other Fibonacci number is: 1 + 2 = 3, 1+ 3 = 4, 2 + 5 = 7, 3 + 8 = 11.
It’s the Lucas sequence! It was in there all along.

What if we add every other Lucas number together?
1 + 4 = 5, 3 + 7 = 10, 4 + 11 = 15, 7 + 18 = 25, 11 + 29 = 40, …

Notice anything about the sequence 5, 10, 15, 25, 40, … ?

For starters, all those numbers are multiples of 5. If we divide them by 5, we would get… (drumroll) the Fibonacci sequence again.

Once you have these two sequences, you can play away in almost any way you want to, and there’s some connection to find. Take the sums of every third, fourth, or fifth Fibonacci number or Lucas number. Or try the differences! Or square the numbers and add them together. Or cube them and take the difference. You won’t always be able to find a connection, but you’ll be surprised how often it works out.

Here’s one more to notice, just so you see how weird this all is. Write both sequences, one above the other, and then take the product of each number on top by each number on the bottom.

1,  1,  2,  3,  5,  8,  13
1,  3,  4,  7,  11,  18,  29
1,  3,  8,  21,  55,  144,  377

Notice anything about that bottom row? It’s every other Fibonacci number.

This is all fun, but I’m not explaining why this happens. I don’t have a book to provide that does, but if you feel like you can hack it, there’s a great video/essay series from James Tanton that you can get at. He talks fast, and some of the math is a little higher level, but it’ll give you a lot of cool ideas and connections to explore.

As for me, I’m working on all kinds of puzzles and games, most of which are about math but not about the Fibonacci numbers. There are a lot more neat connections to discover, though, and if you’re willing to put in the time to follow your passion, and don’t mind being stuck for a while when the going gets tough, I think you’ll be happy with what you get out of the exploration. Some of the connections related to Fibonacci numbers seem almost like magic. For example, if you want to know, say, the 50th Fibonacci number, just take the golden ration (approximate 1.618), raise it to the 50th power, divide by the square root of 5 (approximately 2.236), and round off your answer to the nearest whole number. It seems like magic, but it all makes sense once you develop the right tools, and put the time in. Once you do, you get to be the magician.

All the Best,
Dan