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AWonderfulGame 08 Jul 2005 - 21:42 CarolynJohnston

AnneDwyer has a wonderful math game for kids that she wrote about on her wiki page.

The kids pick the number of digits (we usually start with 5). They put 5 dashes on their paper. I turn over 5 cards in a deck one by one. They have to decide where to put the numbers. Then each kid reads their number to me while I put it on the white board. The kids with the highest number wins.

For some reason, they love this game. On the next round, we go up one digit. Today, we went all the way up to 100 million.

It's a great game.

  • They gain familiarity with large numbers. They get a lot of practice with reading large numbers out loud and hearing large numbers read out loud while it is being written on the board.

  • They have to use strategy. In some games, we have a lot of high numbers at first which every kid puts in the same place. Then, they winner is the determined by the numbers in the ones and tens place. Conversely, sometimes we have a lot of low numbers in the beginning. Then the winner is determined by the highest digits. Much more interesting is when we have medium and low cards. Then, they have to do a lot more thinking about where the cards go.

  • There are very concrete results from this game that allow us to explore numbers even further. In one game, 5 out of 8 kids had the same highest number. So we talk about why and when does this happen? In one game, we had one winner that was a lot higher than anyone else. When does this happen?

We have a gang of kids that run semi-wild in our neighborhood in the summer. They are very mixed in age (ranging from 7 through 11). I have thought about corralling the whole lot of them and bringing them in to teach them all some math together; it would do them all some good to work on it over the summer, and Ben would enjoy his math sessions more if he shared them. I'm a little stumped, though, about how to teach a wide range of ages and interest levels simultaneously.

I'd love to collect some more math games that are as simple and elegant as this one is, especially games that might appeal to a broad range of ages, and (like this one) start a math session off on the right foot.

FunWithVertices 14 Jul 2005 - 00:10 CatherineJohnson

This is fun!

I think it may have come from Carolyn's friend, Charles Martin.....

CognitiveHoles 19 Jul 2005 - 16:27 CarolynJohnston

Bernie and I were talking tonight, and he told me a story that worried me a bit.

Ben came to visit us at work the other day, and wanted to get a snack from the vending machine. So he went into his dad's office and asked for some money. Bernie gave him a few coins, and Ben went into the snack room, picked out what he wanted, and put his money into the machine; but he didn't have enough. So he came in and asked for more; but he couldn't tell Bernie how much more he needed. He didn't seem to have much sense of how much more he needed, either.

Well, it wouldn't be the first time we came across this sort of gap in his understanding. We have a sort of a family byword for these things, very much like Catherine and Ed's no-common-sense-y; we call Ben's gaps his Cognitive Holes. They are located in unexpected places -- they're generally about something, like handling coins, that you think is very easy by comparison with other things he can do, like long division. And they tend to be very big gaping holes in his knowledge, and at first they were very frightening. But we come across them less often now than we used to, and we've found that once we know they are there, we can remediate them pretty quickly.

So I thought this was another run-of-the-mill Cognitive Hole.

Well, you tackle these by filling them in. Ben and I were ready for a change from what we've been doing lately, anyway (introductory equations, solved by adding and subtracting). We've been doing them all week, and struggling, and we finally got a 'click' a couple of nights ago (those babies are practically audible, aren't they?), and last night when he took his section test he got a 100. So tonight, when it was math time, instead of doing algebra, I got out some coins.

I had 3 quarters, and a dime. "OK, you're at our work, and you want a snack, and these are the coins I have", I told him. "The snack you want costs 60 cents. Which coins do you take?"

He went for two of the three quarters, and the dime. Good. "How much do I get back from the machine?" I asked. Nothing: good.

"OK, your snack costs 40 cents". He goes for the two quarters: he tells me the machine returns a dime.

"The snack costs 80 cents." He takes all the coins, and tells me the machine returns 5 cents.

In short, he passed my common sense test with flying colors, and Math Time was fun and a breeze for once. So what the heck was happening the other day? In short, what part of this Cognitive Hole we think we've uncovered am I not mapping correctly?

Tomorrow, we try it a little differently; we'll simulate the precise problem we had the other day with the snack machine at work. I'll give him too little money, tell him the snack costs a certain amount, and get him to tell me how much more he needs.

There may in fact be no Cognitive Hole, this time, just some situational rigidity. This is the deal with smart people on the autism spectrum; sometimes they know what they need to know, they just stiffen up when it comes time to apply it in the real world.


MathTalkInTheCar 01 Aug 2005 - 16:50 CarolynJohnston

We took the kids to a bar tonight, as it happened. Colin (17) is into playing the bass these days; he has a band that he plays with during the school year. I have a friend at work who is a hot guitar player and who just joined a classic rock band, and he was playing his first gig tonight, and they were letting kids stay through the first set, so we went to see him. It was a long drive for us -- all the way out to Greeley. The place was an authentic roadhouse with motorcycles parked out front, and the food was good -- it was Cajun food, and very authentic given that we were not in Cajun country but in Greeley, Colorado, home of the Feedlot You Can Smell All The Way To Denver.

On the way home, Colin asked us about the difference between the median, the mean, and the mode of a data set, and what each of them is good for. This is, of course, the sort of thing we love to pontificate about. He then told us that he felt he had never really quite gotten the idea of a function, and asked us to explain it.

It's a smart kid who understands what he doesn't understand. Most adults can't do that very well.

Actually, most kids coming into calculus classes are confused by functions. A function is just a black box; you put in an input, and get out an output. What makes it a function is that, when you put in the same inputs, you always get the same outputs. You can't put the same number in the black box and get 2 one time, and 5 the next.

Most texts teach functions using formulas to define the functions; all the functions kids see look like f(x)=3x-5, or g(x)=x/6. But functions don't have to have formulas to go with them; they can defy description by a formula. The only rule is that if you put in the same input multiple times, you get the same output, every time.

The reason kids confuse formulas with functions is that it's hard to define functions that don't use formulas, even though in real life we encounter them all the time. When a function totally defies description with a formula, we often resort to trying to describe it with only a couple of numbers, such as the mean, median, and standard deviation (this is how the whole field of statistics arises).

We played a 'figure-out-the-function' game on the way home from Greeley. Bernie and I would think of a function, and Colin and Ben would give us numbers for inputs, and we would then tell them the output. They'd then try to guess the formula we were using to define the function.

They are both aces at extracting patterns. If anything, Ben would try to generalize from too little data; once he guessed, after one try, that the function was 'add 2'; he'd given me a 2, and I'd come back with 4 (the function I'd thought of was squaring; he got it on the next try). Bernie was giving Colin some functions that are so simple they trip up students with their obviousness, like the function that returns the same number you give it, and the one that returns '3' no matter what you give it. He gave Colin one function that was so bizarre you can't describe it with a pattern.

Ben knew more about functions than I thought, even piping up with "that's the constant function 3" at the appropriate moment. Did they do functions one day for 5 minutes in Everyday Math? Well, he was definitely on the ball that day.

DanKOnMakingMathInteresting 30 Jul 2005 - 16:50 CatherineJohnson

Great comment from Dan K!

  • Go bowling. Ignore the automated system, and keep score manually. Then, work through the calculation for some counter-factual cases (“What would my score have been if I hadn’t missed that @#$! spare in the fourth frame?”). Try to figure which one roll would have boosted your score the most if it would have knocked down all the pins.

  • Check the standings. Develop the formula for computing “magic numbers” for clinching the division in baseball. Just please don’t tell me how small the Cardinals’ magic number is to eliminate the Cubs.

  • Follow the market. Each person picks five stocks to watch. Invest your pretend portfolio in them. Track their performance throughout the month of August. Figure out how to plot their daily performance on a graph, comparing their performance to the Dow, the NASDAQ, and the S&P 500. Trade into other stocks along the way.

  • Try Mathmania. Look for interesting problems in the Mathmania booklets put out by Highlights publishing. These periodicals are probably aimed at 4th or 5th graders, but you can upscale some of the problems by trying to describe them using algebra.

  • Look at MATHCOUNTS. The MATHCOUNTS web site ( They’ve got a “problem of the week” archive (with solutions!) that you can browse through. These problems are often topically related to current events. They’re designed to interest kids, so maybe some of them will succeed with your kid. MATHCOUNTS is for math-oriented middle schoolers, so it will challenge most high school students, too.

  • Graph the logical flow. Develop a flow chart—or pseudo-code, if you’re already into programming—describing scoring in tennis. Nest a loop for point scoring within a loop for set scoring. Sometimes deuce is an infinite loop.

  • Play Jeopardy. Write up your own problems and arrange them in categories. This could be a lot of work, depending on how hard you make the problems. Don’t be too strict about answering in the form of a question.

  • WARNING: High risk of failure. Plan a math rally around the yard or neighborhood. Students must solve clues in the form of math problems to find out, say, which envelope to open to get the next clue. Then they must determine which direction to walk to find the next clue. If you open the wrong envelope (or box, or whatever), you lose points, but it then tells you what would have been correct, so you can get back on the right track. If this turns out to be fun, that’s great. If, however, the kid thinks it’s bogus, then you’ve invested a lot of time to end up looking pretty foolish.

I think I'm going to start a user page for this subject....I've forgotten how to set it up so anyone can edit it--Carolyn, do you want to do that?

I'm going to start it from the User Page Index.

FreeGeographyGamesFromDoug 10 Oct 2005 - 23:47 CatherineJohnson

Christopher is going to think he's died and gone to heaven when he sees this site.