Tuesday, August 25, 2009


Good thing to be thinking about as school starts. I'm a bit skeptical about Dan Pink, but several people I know and respect have a high opinion of him. It feels sometimes like he's selling creativity.

But on his recent TED talk, he reports (link to his talk) some social psychology experiments that help me understand a big divide in teaching: incentives in classroom management. A simplification of his point would be that incentives work best on simple, direct tasks. On more challenging problems rewards don't work and, in fact, impede. What does work? Choice! Autonomy! (Conditions of Learning chalk up another point.)

Plus he's pretty entertaining.

Picture click leads to a Florida reading teacher site.

Friday, August 21, 2009

Riddles and Reasoning and Math Teachers at Play 14

When is a carnival full of problems? Besides like every circus movie ever?

The new carnival is up, hosted this week by Susan Van Hattum at Math Mama Writes.

My favorites include the Math Recreation post on origami and a clever lesson using statistics to catch cheaters which also uses bad jokes.

The bad jokes thing reminded me of a lesson I use with riddles about reasoning.

Reasoning and Riddles
The framework David Coffey and I use for reasoning, based on the NCTM process standards of course, is:
Mathematicians are engaged in reasoning when they:
-Make sense of something (sorting, understanding a problem, interpreting a representation)
-Make a conjecture about something (initial answer, plan of attack, possible relationship)
-Make an argument for something (justification, verification, proof)

I then give the students a list of riddles and ask them to figure out the answers. As we look at their answers, and more importantly, how they got their answers, they generate lots of examples of making sense, making conjectures, and arguing for why their answer fits.
(General riddles and Halloween riddles are posted at my faculty page. Click the links for the pdfs.)

We then explore a more math-centric riddle (it's usually a geometry class):

Four Sided Riddle

1) Taking the clues for a mystery shape in order, put a checkmark next to the last clue you need to know exactly the type of shape that the mystery shape is. Then explain your answer.
1. It is a closed figure with four straight sides.
2. It has two long sides and two short sides.
3. The two long sides are the same length.
4. The two short sides are the same length.
5. One of the angles is larger than one of the other angles.
6. Two of the angles are the same size.
7. The other two angles are the same size.
8. The two long sides are parallel.
9. The two short sides are parallel.

2) Using one less clue than your answer to number (1), draw a shape that satisfies all those clues BUT is different than the mystery shape, or explain why this cannot be done.

There is also a nice Van Hiele connection here as students at different levels approach this task very differently.

Dinosaur Comics are perfectly qwantzian. Click the cartoon to see it full size, click the link to get to the web comic's home. (T-Rex does not always subscribe to human norms of taste and good form, obviously, so, at your own risk.)

Eventually I'll work all my favorite webcomics in here.

Tuesday, August 18, 2009

Running Out of Options

This game was a favorite for Paul Rettinger, a brilliant grad school room-mate I had at Penn State. He went from a Master's of Fisheries and Wildlife to a law degree. Amazing guy. I was lucky to ever win.

Last Letter Loses

Players take turns adding a letter to a word. The first player to be forced to spell a word (at least three letters) loses. A player can challenge the previous turn if they think there is no such word. If there is such a word, the challenger is out. If there is not, the challenged player is out.

1- A
2- A-M (doesn't lose because less than 3 letters)
1- A-M-E (P would lose, even if you're thinking of 'amplify', since amp is already a word.)
2- A-M-E-R (thinking of america)
1- (thinking they're in trouble) A-M-E-R-I
2- (happily) A-M-E-R-I-C
1- (inspiration strikes) A-M-E-R-I-C-I
2-(thinking what? that's not a word!) Challenge
1-Americium! I still would have lost but thought you might not remember that word. (It's an element.)

In that example Paul would have been player 1, snatching victory from the jaws of defeat.

(From a repository of free educational clipart, click the image.)

That inspired the following game, which I use to introduce prime number decomposition.

Last One Loses

Players take turns breaking down a number by multiplication. The first player starts with a whole number. The next player makes a string of two numbers that multiply to give the first. The next player then can break down one of the two numbers, making a three number string. The last player who can break it down is the loser of the game. Numbers chosen must be able to be broken down more than twice. A player may challenge if they disagree with a breakdown, or if they say it's the end but it's not. Players may not reuse starting numbers. 1 may not be used in the breakdown. You can't use a number that's been used this session.

Example 1:
A- 24
B- 3x8
A- 3x4x2
B- 3x2x2x2 - augh! (loses)

Example 2:
A- 112
B- 2x56
A- 2x2x28
B- 2x2x4x7
A- 2x2x2x2x7 - curses! (loses)

Instructional uses:
Have students keep track of which numbers make first player lose, and which numbers made second player lose. When the data is collected, students will see that the same number almost always has the same effect. (Although there's usually a number that was mis-factored.)

Pose the questions: what if you break the number down in a different way? Is it always the same number of steps? Is the end result always the same?

Several important ideas about the prime decomposition will come out immediately, including the idea of prime numbers! Also why 1 is not a prime. (And 1 is NOT a prime.)

I allow calculators for numbers bigger than 144. Follow up investigations can be things like: find a number that takes 6 steps and is between 500 and 1000, or trying the Sieve of Eratosthenes.

Sunday, August 16, 2009

More Math Correlates to Higher Income

The blog where I saw the study referenced: Free Exchange at the Economist.

The study the blog is citing: Joshua Goodman

The cartoon that teaches correlation:

(XKCD is occasionally profane, almost always funny, and frequently geeky. Clicking the cartoon leads to the site.)

Thursday, August 13, 2009

Ken Robinson Answers

I've written about Ken Robinson a few times (One and Two). The idea of creativity in mathematics was a sub-theme for my summer calculus class. Students at the end felt that some of the open-ended assignments (projects of their choice), non-standard problems (like the mobiles) and emphasis on problem solving helped open them up to creativity in math. But they suggested more specific demonstrations of how to be creative. (Boy, is that insightful.)

TED occasionally has question and answer sessions with their speakers who really ignited something with their presentation, and Sir Ken recently did this. (Here's the article.) He addresses math specifically:

"If you want to promote creativity, you need, firstly, to stimulate kids minds with puzzles and questions which will intrigue them. Often that's best done by giving them problems, rather than just solutions. What often happens in classrooms is, kids sit there trying to learn in a drone-like way things of not much interest that have already been figured out.

The best math teachers I know, like the best English teachers, are always giving kids puzzles. They're given things to work on where math skills are required but may not be the focus of the activity. There giving them problems to solve. Or they're made to engage with age-old mathematical problems. For example, I'm thinking about the problem of latitude. How do you go about measuring the planet? I mean, somebody had to do that. How do you do it? Professional mathematicians have such a cornucopia of fascinating puzzles, questions, proposals and conundrums. A great math teacher really has endless opportunities to stimulate kids minds and get them engaged with things they'd probably never thought about before. Rather than just giving them techniques." -Ken Robinson

It's not hugely original, but it's nice to get confirmation of things we believe from outside sources. He touches on engagement several times in the Q&A, and I do believe that's the central issue in teaching, and I love to ponder what is the key for math. Going to more and more of these reading conferences, I am insanely jealous of the teachers who talk about the book that turned a student on to reading. Problems don't seem to have the same effect.

Friday, August 7, 2009

Math Teachers at Play 13

The new carnival is up at Blog, She Wrote. The Money Games are there, as well as the generally nice idea of runnign a store for money practice at the Pumpkin Patch. I like some of the puzzles at Critical Thinking Puzzles. The card trick that's linked really makes you think about how much information there is in arrangement.

Tuesday, August 4, 2009

Money Games

Here are my two favorite money games. Change for the Better is based on a James Ernest design. He's the genius behind Cheapass Games (don't be put off by the name), and the game this is based on, Fight, he used to have on his business card. It really has some non-trivial strategy and thinking to it. Later I made the connection - or one of my preservice teachers did - with Smart, the excellent poem by Shel Silverstein. The other game I think I invented, Make It Take It. The idea was from a group of teachers who wanted students to be forced to find non-standard combinations of coins, instead of always taking 27 pennies, for example. That suggested a dwindling resource game to me. It's poissible to combine both with some visual representations of money, which is a nice support for struggling students. The handout is here, if you like worksheets or some of the representation support.

by Shel Silverstein

My dad gave me one dollar bill
'Cause I'm his smartest son,
And I swapped it for two shiny quarters
'Cause two is more than one!

And then I took the quarters
And traded them to Lou
For three dimes -- I guess he don't know
That three is more than two!

Just then, along came old blind Bates
And just 'cause he can't see
He gave me four nickels for my three dimes,
And four is more than three!

And I took the nickels to Hiram Coombs
Down at the seed-feed store,
And the fool gave me five pennies for them,
And five is more than four!

And then I went and showed my dad,
And he got red in the cheeks
And closed his eyes and shook his head--
Too proud of me to speak!

Change for the Better

Materials: Each player needs 1 quarter, 2 dimes, 3 nickels, and 4 pennies.

Rules: Play in groups of 2 to 6. Each player takes a turn. On their turn they put in one coin. They can take out a combination of coins that is less than the value of what they put in. For example, if you put in a dime (10¢) you can take back up to 9¢ – if it is there. Play continues until only one person has money left.

Instruction: Beginning players should just concentrate on the moves of the game. After students have gained some experience with the game, they can try recording their games to translate to symbolic representation. The data collected can then be examined for patterns.

Make It, Take It
a money game for 2 players or teams

Materials: Play coins or coin pictures or cards, amount cards. Record sheet if desired.

Play: Put the coins in the center. Shuffle the amount cards and make a stack. Players each turn over an amount card, and the player with the smaller amount goes first. On subsequent turns, players turn over an amount card, and see if they can make that amount with the coins. If they can, they take the coins. If they can not, it’s the other player’s turn. Play until all coins are gone, or both players in a row can’t make their amounts. The winner is the player with the biggest total value of coins they collected.

Recommended starting amounts – 4 quarters, 6 dimes, 8 nickels, 10 pennies. Other amounts can be used. Teachers can add amount cards for more complicated amounts.
Players can roll two dice to determine the amount. (Note the dice variation requires more pennies.) Advanced play allows people to make change with the coins they’ve collected. For example, trading a dime from the center with two nickels they have taken before.
Players can use dollar value charts to keep a running total.

Bill and Keenya have been playing for a few turns.
Bill turns over 12 cents and takes two nickels and two pennies.
Keenya turns over 25 cents, but there are no quarters left. She takes five nickels.
Bill turns over 50 cents and can not make it.
Keenya turns over 6 cents and takes a nickel and a penny.
Bill turns over …

As with most games, it is recommended to play a game with teacher vs. the whole class to launch the game. Emphasize the variation in ways to make an amount by soliciting other possibilities from the students. Ask questions like “what card would be good to turn over next?” or “what card would leave me with no possibilities?” If someone is stuck, encourage good sportsmanship in helping them figure out a way to make the total. If that doesn’t seem to be working, or you are worried about their ability to make the amounts, students can play in a team of two vs. another team of two.

Many students will try a place value approach first, taking dimes and pennies. This will rapidly run them out of one or the other, forcing them to find other amounts. The amount cards concentrate on values that can be made with one, two or three coins, though several can be made with many more coins.

In summary, the teacher may wish to have students share their strategy for figuring their total at the end of the game. It is important to summarize by having students describe how they knew if they could make an amount or not. Another interesting discussion to start is if there is a strategy for better ways to play the game – is there an advantage to using fewer or more coins to make your moves?