Sets, Planets, and Comets

From the article “Sets, Planets, and Comets” that recently appeared in College Mathematics Journal.

Set is an enjoyable—even addictive—card game that challenges players to identify

certain visual patterns. A mathematically rich game, it provides ample opportunity for

students and teachers to ponder combinatorial, algebraic, and geometric questions. Part

of Set’s appeal is that once the fundamentals of the game are understood, it is nearly

impossible to resist investigating its structure, whatever one’s background. We con-

centrate on the geometry, introducing interesting objects we call planets and comets,

which lead to an elegant variation on the game.

The full article can be found here: http://www.maa.org/sites/default/files/pdf/pubs/SetsPlanetsAndComets.pdf

Symphonic Equations: Waves and Tubes

This excellent and engaging video describes how sine and cosine functions can be applied to music.

A Review of WuzzitTrouble: an app for math education

Most apps and computer games that claim to assist with the development of mathematical knowledge only focus on rote memorization. There’s certainly a place for rote memorization, but I’ve been very disappointed with the paucity of games that encourage mathematical creativity beyond, say, immediate recall of the times tables.

Enter WuzzitTrouble, a new app that was developed by Keith Devlin, a professor of mathematics at Stanford and one of the great popularizers of mathematics today. An introduction to WuzzitTrouble can be seen in this promotional video:

One minor complaint about WuzzitTrouble is that the first few levels are so easy that it’s easy for children to low-ball the game… in much the same way that the first few levels of Angry Birds are utterly easy. (My other complaints is that the game only assume one user, so that a parent can’t play the game without affecting a child’s settings.) However, the level of difficulty does eventually increase. Here’s another promotional video showing how to solve Level 1-25:

Here’s a sampling of some of the higher levels. Remember that the wheel has 65 steps along the circumference, as shown in the above picture and videos.

Level 2-5: Using cog wheels of size 5 and 9, pick up keys at 23 and 36 and prizes at 27, 45, and 55.
Level 2-15: Using cog wheels of size 5, 7, and 9, pick up keys at 11, 16, and 21 and prizes at 32 and 42.
Level 2-25: Using cog wheels of size 5, 9, and 16, pick up keys at 24, 48, and 59; prizes at 11 and 37; and avoid a penalty at 64.
Level 3-3: Using cog wheels of size 3, 4, and 5, pick up keys at 7, 17, and 27 and prizes at 12 and 22.

In the words of their promotional materials:

At InnerTube Games, we set out to design and build mobile casual video games and puzzles that can attract and engage a large number of players, yet are built on fundamental mathematical concepts and embed sound mathematics learning principles.

We start with one simple, yet powerful observation. A musical instrument won’t teach you about music. But when you pick up an instrument and start playing – badly at first – you cannot fail to learn about music. And the more you play, the more you learn. In fact, using that one instrument, you can go all the way from stumbling beginner to virtuoso concert performances. It’s the music that changes, not the instrument. In modern parlance, the instrument is a platform. And (well designed) platforms are good for learning because they make the learning meaningful and put the learner in charge.

InnerTube Games does not build video games to “teach mathematics.” Rather, we build instruments which you can play, and we design them so that when you play them, you cannot fail to learn about mathematics. Moreover, each single game can be used to deliver mathematical challenges of increasing sophistication.

Our vision for learning design is to build the game around core mathematical concepts and practice so it looks and plays like the familiar casual games on the market. As a result, you won’t be able to see the difference by playing the first few levels, or by watching someone else play. It’s the educational power under the hood that makes our games different.

We’re not making a secret of the fact that our games are math-based. It’s not “stealth learning;” it’s a form of learning through action that the brain finds natural, having much in common with what educational researchers call embodied learning.

Wuzzit Trouble is our first puzzle to reach the market. It is built around the important mathematical concepts of integer partitions–the expression of a whole number as a sum of other whole numbers–and Diophantine equations. At the easiest levels of the puzzle, these provide engaging practice in basic arithmetic, leading to arithmetical fluency.

But that’s just the start. Integer partitions and Diophantine equations are major areas of mathematics, still being worked on today by leading mathematicians.

Freeing the Wuzzits won’t take you into those dizzy realms—at least in the initial release, which comes loaded with puzzles aimed at the Elementary and Middle School levels. But as you progress, you will face challenges that increasingly require higher-order arithmetical thinking, algebraic thinking, strategy design and modification, optimization, and algorithm design, all crucial abilities in today’s world. Getting three stars can require considerable ingenuity.

As you attempt to free each Wuzzit and maximize your score, you will be developing and applying valuable conceptual, analytic thinking skills that sharpen your mind—all without lifting pencil to paper.

As educators and former educators, all of us at InnerTube are very aware of the importance of learners meeting agreed standards. In its initial release version Wuzzit Trouble provides natural learning in the following areas of the US Common Core Curriculum:

*Grade 2, Operations & Algebraic Thinking #2

*Grade 2, Number & Operations in Base Ten #2, #8

*Grade 3, Operations & Algebraic Thinking #1, #4

*Grade 4, Operations & Algebraic Thinking #5

*Grade 6, Number System #5, #6

But we don’t want anyone to play our game purely to hit those Common Core markers. We want you to play it because it’s fun and challenging. Improvement in those CC areas comes automatically. Just like learning music by playing a musical instrument!

The analogy that I prefer is playing basketball. When young children are first learning to play basketball, there’s a place for learning how to dribble, how to pass, how to shoot free throws, etc. (These are analogous to learning how to add, subtract, multiply, and divide.) But children don’t just learn skills: they also go out and play. That’s where the WuzzitTrouble app fits in: it offers children a chance to just play with mathematics and enjoy it.

More references:

http://profkeithdevlin.org/2013/09/03/the-wuzzits-free-at-last/

Review: Wuzzit Trouble

From high school math teacher to quarterback for the Dallas Cowboys

I’ve never been a fan of the Dallas Cowboys, but Jon Kitna remains one of the good guys of the NFL. After retirement, he went to work as a math teacher and football coach at his high school alma mater, getting students with learning disabilities to understand algebra (and thus be prepared for higher-level math classes in later years).

After the unfortunate injury to starting quarterback Tony Romo, the Dallas Cowboys called upon Kitna for emergency service. He plans to donate his one-game salary back to the high school.

Reference: http://bleacherreport.com/articles/1901693-jon-kitnas-salary-decision-proves-his-return-is-noble

The Monty Hall Problem

In 1990 and 1991, columnist Marilyn vos Savant (who once held the Guinness World Record for “Highest IQ”) set off a small firestorm when a reader posed the famous Monty Hall Problem to her:

Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, “Do you want to pick door #2?” Is it to your advantage to switch your choice of doors?

She gave the correct answer: it’s in your advantage to switch. This launched an avalanche of mail (this was the early 90’s, when e-mail wasn’t as popular) complaining that she gave the incorrect answer. Perhaps not surprisingly, none of the complainers actually tried the experiment for themselves. She explained her reasoning — in two different columns — and then offered a challenge:

And as this problem is of such intense interest, I’m willing to put my thinking to the test with a nationwide experiment. This is a call to math classes all across the country. Set up a probability trial exactly as outlined below and send me a chart of all the games along with a cover letter repeating just how you did it so we can make sure the methods are consistent.

One student plays the contestant, and another, the host. Label three paper cups #1, #2, and #3. While the contestant looks away, the host randomly hides a penny under a cup by throwing a die until a 1, 2, or 3 comes up. Next, the contestant randomly points to a cup by throwing a die the same way. Then the host purposely lifts up a losing cup from the two unchosen. Lastly, the contestant “stays” and lifts up his original cup to see if it covers the penny. Play “not switching” two hundred times and keep track of how often the contestant wins.

Then test the other strategy. Play the game the same way until the last instruction, at which point the contestant instead “switches” and lifts up the cup not chosen by anyone to see if it covers the penny. Play “switching” two hundred times, also.

You can read the whole exchange here: http://marilynvossavant.com/game-show-problem/

For much more information — and plenty of ways (some good, some not-so-good) of explaining this very counterintuitive result, just search “Monty Hall Problem” on either Google or YouTube.

Source: http://www.xkcd.com/1282/

Mathematics and The Price Is Right

I just read a very entertaining article on the use of game theory for improving contestants’ odds of winning the various games on the long-running television game show “The Price Is Right.” Quoting from the article:

On a crisp November day eight years ago, I took the only sick day of my four years of high school. I was laid up with an awful fever, and annoyed that I was missing geometry class, which at the time was the highlight of my day. I flipped on the television in the hope of finding some distraction from my woes, but what I found only made me more upset: A contestant named Margie who was in the process of completely bungling her six chances of making it out of Contestants’ Row on The Price is Right.

Many contestants fail to win anything on The Price is Right, of course. But as I watched the venerable game show that morning, it quickly became clear to me that most contestants haven’t thought through the structure of the game they’re so excited to be playing. It didn’t bother me that Margie didn’t know how much a stainless steel oven range costs; that’s a relatively obscure fact. It bothered me, as a budding mathematician, that she failed to use basic game theory to help her advance. If she’d applied a few principles of game theory—the science of decision-making used by economists and generals—she could have planted a big kiss on Bob Barker’s cheek, and maybe have gone home with … a new car! Instead, she went home empty-handed…

To help future contestants avoid Margie’s fate, I decided to make a handy cheat sheet explaining how to win The Price Is Right—not just the Contestants’ Row segment, but all of its many pricing games. This guide, which conveniently fits on the front and back of an 8.5-by-11-inch piece of paper, does not rely on the prices of items.

The full article can be found at http://www.slate.com/articles/arts/culturebox/2013/11/winning_the_price_is_right_strategies_for_contestants_row_plinko_and_the.html.

Mathematics and College Football

For years, various algorithms (derisively called “the computers” by sports commentators) have been used to rank college football teams. The source of derision is usually quite simple to explain: most of these algorithms are too hard to explain in layman’s terms, and therefore they are mocked.

For both its simplicity and its ability to provide reasonable rankings, my favorite algorithm is “Random Walker Rankings,” published at http://rwrankings.blogspot.com. Here is a concise description of this ranking system (quoted from http://rwrankings.blogspot.com/2003_12_01_archive.html):

We’ve all experienced befuddlement upon perusing the NCAA Division I-A college football
Bowl Championship Series (BCS) standings, because of the seemingly divine inspiration that must have been incorporated into their determination. The relatively small numbers of games between a large number of teams makes any ranking immediately suspect because of the dearth of head-to-head information. Perhaps you’ve even wondered if a bunch of monkeys could have ranked the football teams as well as the expert coaches and sportswriters polls and the complicated statistical ranking algorithms.

We had these thoughts, so we set out to test this hypothesis, although with simulated monkeys (random walkers) rather than real ones.

Each of our simulated “monkeys” gets a single vote to cast for the “best” team in the nation, making their decisions based on only one simple guideline: They periodically look up the win-loss outcome of a single game played by their favorite team, and flip a weighted coin to determine whether to change their allegiance to the other team. In order to make this process even modestly reasonable, this random decision is made so that there is higher probability that the monkey’s allegiance and vote will go with the team that won the head-to-head contest. For instance, the weighting of the coin might be chosen so that 75% (say) of the time the monkey changes his vote to go with the winner of the game, meaning only a 25% chance of voting for the loser.

The monkey starts by voting for a randomly chosen team. Each monkey then meanders around a network which describes the collection of teams, randomly changing allegiance from one team to another along connections representing games played between the two teams that year. This network is graphically depicted in the figure here, with the monkeys—okay, technically one is a gorilla—not so happily lent to us by Ben Mucha (inset). It’s a simple process: if the outcome of the weighted coin flip indicates that he should be casting his vote for the opposing team, the monkey stops cheerleading for the old team and moves to the site in the network representing his new favorite team. While we let the monkeys change their minds over and over again—indeed, a single monkey voter will forever be changing his vote in this scheme—the percentage of votes cast for each football team quickly stabilizes. We thereby obtain rankings each week of the season and at the end of the season, based on the games played to that point of the season, by looking at the fraction of monkeys that vote for each team…

The virtue of this ranking system lies in its relative ease of explanation. Its performance is arguably on par with the expert polls and (typically more complicated) computer algorithms employed by the BCS. Can a bunch of monkeys rank football teams as well as the systems in use now? Perhaps they can.

Using this algorithm, here’s the current ranking of college football teams as of today. (With great pride, I note that Stanford is ranked #4.) These rankings certainly don’t exactly match the latest AP poll or BCS rankings, but they’re also still reasonable and defensible.

Hollywood Hates Math

Dan Meyer spliced together scenes from various movies where knowledge of mathematics is denigrated. Since a big part of my job is instilling confidence in my students that they can indeed succeed in my classes, it’s a little depressing to see that I have a big opponent in popular culture.

This video has the occasional PG language and innuendo, while I prefer to keep my classes rated G to every extent possible. Some time ago, Dan was kind enough to post the original movie sources for this clip, and someday I might edit down this clip to something that I would be comfortable showing in class.