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Jason Dyer writes a very important post highlighting Tiny Games, a listing of games you can play quickly, almost anywhere, with only limited materials. He then pivots to ask about tiny math games.

Could one make an all-mathematics variant — mathematical scrimmages, so to speak?

His post, and Tiny Games, are important because they reject an article of faith of the blended learning and flipped classroom movements, that students must learn and practice the basic skills of mathematics before they can do anything interesting with them.

For example, here's John Sipe, senior vice president of sales at Houghton Mifflin Harcourt, talking about Fuse, their iPad textbook:

So teachers don’t have to “waste their time” on some of these things that they’ve always had to do. They can spend much more time on individualized learning, identifying specific student needs. Let students cover the basics, if you will, on their own, and let teachers delve into enrichment and individualized learning. That’s what the good teachers are telling me.

This is a bad idea. People don't mind practicing a sport because playing the sport is fun. It's easy to tell a tennis player to practice 100 serves from the ad side of the court, for instance, because the tennis player has mentally connected the acts of practicing tennis and playing tennis. The blended learning movement, at its worst, disconnects practice and play.

Take multiplication of one- and two-digit numbers for instance.

If you need to learn multiplication facts, one option is to watch a video and then drill away. Or we can queue up all that practice in a tiny math game that'll have students playing as they practice:

Pick a number. Say 25. Now break it up into as many pieces as you want. 10, 10, and 5, maybe. Or 2 and 23. Twenty-five ones would work. Now multiply all those pieces together. What's the biggest product you can make? Pick another. What's your strategy? Will it always work? [Malcolm Swan]

Easy money says the student who's practicing math while playing it will practice more multiplication and enjoy that practice more than the student who is assigned to drill practice alone.

Jason Dyer helpfully highlights two examples of tiny math games, Nim and Fizz-Buzz, but he and I are both struggling to define a "tiny math game." The success of the Tiny Game Kickstarter project indicates serious interest in these tiny games. I'd like to see a similar collection of tiny math games. Here's how you can help with that.

1. Offer Examples of Tiny Math Games

This may be tricky. We all have games we play in math class. What distinguishes those games from "tiny math games?"

2. Help Us Define "Tiny Math Games"

This may be a better starting point. I'll add your suggestions to this list. Here are some seeds:

• The point of the game should be concise and intuitive. You can summarize the point of these games in a few seconds or a couple of sentences. It may be complicated to continue playing the game or to win it, but it isn't hard to start.
• They require few materials. That's part and parcel of being "tiny." These games don't require a laptop or iPhone.
• They're social, or at least they're better when people play together.
• They offer quick, useful feedback. With the multiplication game, you know you don't have the highest product because someone else hollers out one that's higher than yours. With Fizz-Buzz, your fellow players give you feedback when you blow it.
• They benefit from repetition. You may access some kind of mathematical insight on individual turns but you access even greater insight over the course of the game. With Fizz-Buzz, for instance, players might count five turns and then say "Buzz," but over time they may realize that you'll always say "Buzz" on numbers that end in 5 or 0. That extra understanding (what we could call the "strategy" of these tiny math games) is important.
• The math should only be incidental to the larger, more fun purpose of the game. I think this may be setting the bar higher than we need to, but Jason Dyer points out that people play Fizz-Buzz as a drinking game. [Jason Dyer]

What can you add to our understanding of tiny math games?

2013 Apr 17. Nobody wanted to tackle the qualities of tiny math games, which is fine since you all threw down a number of interesting games. I'll be compiling those on a separate domain at some point soon.

Featured Tweets

Jason Dyer elaborates on his contribution above.

The line between "math that is game" and "game that is math" is pretty thin.

— Jason Dyer (@jdyer) April 16, 2013

However, students can still smell the former a mile away.

— Jason Dyer (@jdyer) April 16, 2013

Other teachers go in on the lie that students need basic skills before they can do anything interesting in their disciplines:

@ddmeyer The truth of this is shown in language. Kids don't study grammatical rules (neither did I when I learnt Japanese).

— Rustin Selvey, CFA (@rustinselvey) April 16, 2013

.@michellek107 @ddmeyer Similar 'lie' in Rdg.Kids must lrn all letters & sounds before guided rdg can begin.

— Michele Corbat (@MicheleCorbat) April 16, 2013

@ddmeyer similar "lie" in Music Education. Too many think that students must learn notation b4 composing music. #mused

— Michelle Baldwin (@michellek107) April 16, 2013

@ddmeyer similar "lie" in social studies/history. Many think students must learn facts before analyzing concepts, trends, & effects."

— Emily Jolley (@emilybjolley) April 17, 2013

2013 Apr 24. Jason Dyer elaborates in another post.

once upon a time I would kill for a great lesson idea, then I started reading the internet and have too many

— Jonathan (@ultrarawr) April 12, 2013

If the deluge of interesting problem-based material on the Internet overwhelms you, as it does Jonathan Claydon, Geoff Krall's curriculum maps are a great place to start. He's taken the Common Core's scope and sequence documents and combed the Internet for items that fit. He's included a few of my own items, some items from the Shell Centre, along with a lot of great lesson ideas I'd completely forgotten. Bookmark it. Throw him some love in the comments.

I was walking with my wife along the River Corrib in Galway last weekend when we got into an argument that lasted the rest of the walk. I'll present our two arguments and some illustrative video. Then I'd like you or your students to help sort us out.

Argument A: It would be much harder to swim to the other side of the river in the fast-moving water as in still water.

Argument B: It would be just as easy to swim to the other side of the river in the fast-moving water as in still water.

I hope this gets as out of hand for you and your students as it did for me and my wife.

Featured Comment

Scott Farrar:

This excellent question exhibits a quality that is not found often in math curricula: it has the "specificity sweet spot": it is specific enough for a student to answer, but non-specific enough for every kid to agree on the answer. Students making different assumptions will have different responses, thus creating a real mathematical argument.

1. Bill McCallum runs a "standard of the week" contest through his Illustrative Math Project, the goal of which is to illustrate what the different Common Core standards look like in student tasks. I submitted a task for 8.4.F, linear modeling, which was accepted. It's called "Graduation" [pdf].
2. Key Curriculum Press posted the Ignite talks from CMC-South. I did five minutes on the question, "When will I ever use this in the real world?"

2012 Feb 18. Patti Smith used this task in class. Fun feedback:

My students finally understood the meaning of y-intercept as something more than "when it all began". They also understood slope to mean rate – or how fast they read the names – rather than rise over run!

Two things I'd do if I were still doing the job instead of just talking about it:

Set Up The Expected Value Spinner

I don't think people who understand expected value understand how hard it is for other people to understand expected value.

Let's say I roll a die. I ask if you want to bet on an even number coming up or a five. You're bright. You pick the even number. It has a 3/6 shot versus a 1/6 shot for the five. But what if I said I'd pay you $150 if the even number comes up and $600 for the five. What if I said I'd keep on giving you that same bet every day for the rest of your life? This is where expected value steps in and puts a number on the value of each bet, not its probability. The expected value of the even number bet is (3/6) * $150 or $75. The expected value of the five bet is (1/6) * $600 or $100. The five bet will score you more money over time.

This is tricky to fathom in gambling where superstition rules the day. ("Tails never fails," betting your anniversary on the pick six, blowing on the dice, etc.) So one month before our formal discussion of expected value, I'd print out this image, tack a spinner to it, and ask every student to fix a bet on one region for the entire month. I'd seal my own bet in an envelope.

I'd ask a new student to spin it every day for a month. We'd tally up the cash at the end of the month as the introduction to our discussion of expected value.

So let them have their superstition. Let them take a wild bet on $12,000. How on Earth did the math teacher know the best bet in advance?

BTW: You could make an argument that a computer simulation of the spinner would be better since you could run it millions of times and all on the same day. My guess is that your simulation would be less convincing and less fun for your students than the daily spin, but you could definitely make that argument.

Host A Steepest / Shallowest Stairs Competition

Tonight's homework: Find some stairs. Calculate their slope. Describe how you did it. Take a picture.

Your students should then determine whose stairs were the steepest and the shallowest and you'll post those photos at the front of the classroom. You'll make a big fuss over them. Then you'll post a bounty for stairs that will knock them off their perch.

One interesting thing about slope is that it doesn't have a unit, so you don't need a measuring tape or a ruler to calculate it. Anything your students have on hand will work, including their hands.

Be prepared for a contentious discussion about the difference between the tallest steps and the steepest steps. It's possible to design steps that are extremely shallow but too tall for anyone to climb up. Wrap your students' heads around that one.

Be prepared also for students who can't shake the sense that math is here every time they climb up a new set of stairs.

What a cool job the rest of y'all have.

[photo credits: moyogo, vulcho]

2012 Jan 17: Useful description and modifications from James Cleveland.