Rudbekia Hirta has worries about problems that turn her students off math:
Learning Curves: Contest Problems: I've also been given a book produced at the University of Delaware in 1986 called "Resource Problems to Enhance the Teaching of Mathematics." Here's a randomly selected problem:
In a 5 by 12 rectangle, one of the diagonals is drawn and circles are inscribed in both right triangles thus formed. Find the distance between the centers of the two circles.
I can't use problems like this. Most of the problems in the book are what I would categorize as contest problems for general-population high school students. These might be fine for high school students who already like math, but those are not my students. I'll definitely have a bunch of pre-engineers who need to learn an ambitious amount of calculus in a short period of time (dictated by the departmental syllabus), and the other bunch is an unknown. The regular gen-ed class would mutiny over this sort of problem (these are the students so averse to doing unfamiliar problems that they will leave questions blank on the exam and leave early). The honors version of the gen-ed class does not have a math placement prerequisite -- just an overall standardized test score -- so I have no information about how good they are at math. I'm guessing, though, that if I were to give them puzzle problems that they'd be more appreciative of the LSAT puzzle problems or problems from the quantitative section of the GMAT.
This book also features word problems about completely fanciful nonsense situations:
Sally is having a party. The first time the doorbell rings, one guest enters. If on each successive ring a group enters that has two more persons than the group that entered on the previous ring, how many guests will have arrived after the 20th ring?
It's problems like this that make my weaker students hate word problems....
So what are some problems that have clear and immediate payoffs, or that will be of interest to somebody with a normal level of curiosity? This question is of interest to me as well, for two reasons:
- I want the sophomores and juniors I teach to understand that math is a useful tool--which means assigning them problems that they can do and understand the substantive payoff.
- I want to persuade the kids that the payoff to math is high.
I used to have a wiki that I hoped would become a place where people could contribute and edit interesting math problems. But, alas, it got caught in the great wiki-crash of 2004, as the spam-robots became more and more sophisticated. I had to shut it down.
This is as far as I had gotten:
One Hundred Interesting Math Calculations: How do you convince adolescents that there is a big long-run payoff from math? Teaching them (mine at least) that there is a huge short-run payoff from reading and a huge medium-run payoff to writing is easy. But math is harder.
- [World War II Bomber Pilot Survival Odds]
- [How Many Extraterrestrial Civilizations Are There?]
- [Gravity and "Weighing the Earth"]
- [Economic Growth Since 1500]
- [Exponential Growth and Human Populations]
- [How Much Blood Is There in the World?]
- [Julius Caesar's Last Breath]
- [The Birthday Fact]
- [The False-Positive Problem]
- [The Grass-Is-Greener Paradox]
- [The All-Knowing Alien Paradox]
- [Repeating Decimals]
- [Introduction to Compound Growth]
- [Elementary Ballistics: The Kinematics of Falling Bodies]
- [Elementary Ballistics: What Goes Up Must Come Down]
- [How Rich Is Fitzwilliam Darcy?]
- [The Clock Hands Problem]
- [Sunscreen, or the Freak Mutant Near-Albino Problem]
- [The Distributive Law, or the Get-Out-of-the-Way Problem]
- [The Federal Reserve Problem]
- [The Ancestor Problem]
- [Strategy Secrets of ENRON]
- [The Muddy Parent Problem]
- [The Kissing Problem]
- [Understanding "Risk Arbitrage"]
- [Orbiting the Earth]
And here are some suggestions from others for problems it would be interesting to write up:
Suggestions For Entries:
- How long can Moore's Law go on? Starting from the average distance between atoms in a silicon crystal, find the time when chip features will be (supposedly) one atom wide.
- Intro to counting and combinatorics. Suppose there were 14 (or 12) cards in one suit. Suppose there were 5 (or 3) suits in a standard deck of playing cards. How would the relative ranking of poker hands change? They don't all scale the same way. Do most of the work by cancellation, so you don't have to perform a lot of the (tedious, error-prone) multiplication.
- Simple bits of probability, especially conditional probability from games -- card games (poker), dice games (craps), whatever. For example, understanding why it is harder to make your point the hard way (with a pair) when it is 8 rather than 4. As a grad student, I spent a lot of time teaching basic concepts to undergrads (at MIT!) that I mastered in middle school because i thought about the games i spent my time playing.
- Consider the mathematics of triage versus parity policies as described by Garrett Hardin in Chapter 4 of "Promethean Ethics," University of Washington Press, 1980.
- If you try something unlikely a few times, you might fail every time -- but it's commonly said "Even if the odds are a thousand to one against you, you try it a thousand times, you're sure to get it." Right? Wrong. If the odds are 50-50, and you can have two tries, you've got a 75% chance of a win. But it's downhill from there. One in a thou chance with up to a thousand tries? Only a 63.23% chance of a win. One in a million over a million tries? You're down to 63.21%, and it keeps dropping from there. How low can it go?
- Xeno's paradox came about because the Ancient Greeks did not know how to sum an infinite series. I've always used it to illustrate the concept of limits approaching infinity, because it puts the complex math on the side of common sense.
- For "Exponential Growth and Human Populations" set the end point as filling up the Americas by a colonizing group of 100 people, it's more interesting.
- Richard Dawkins has an interesting calculation on human ancestors in the book "River out of Eden" (1995). Figuring 20 years per generation, calculate the number of ancestors you have 2000 generations ago if none of your ancestors appear more than once in your family tree (no inbreeding).
- If the kids are into science fiction, have them work out dimensions of their favorite space ship based on extrapolation from the sizes of particular features (e.g. if the bridge of the Enterprise is so many feet across, how long is the whole ship?). Have them make upper and lower bound estimates to teach them error margins. It's not so great for web page presentation, though...
- Small business economics. Next time you and the kids are at the ice cream shop or other restaurant, have them work out the typical number of customers per hour (from typical customers-in-store and customer-visit-time). From this and the amount spent by a typical customer you get typical revenue. Guess at employee wages and commercial space leasing costs. Ask them why the place closes at 9 instead of staying open all night.
- Bridges of Konigsberg. Requires an illustration. The fundamental problem of graph theory.
- Predator/Prey? balancing over time.
- The different coin problem. N coins or objects of the same weight, one object of a different weight (in the simpler form lighter or heavier is known, in the slightly more difficult, just that it is "different"), a scale, and a limited number of weighings. Teaches binary group comparions. (Similarly, the switchback problem - You are at a fork in the road. You know your destination lies an unknown distance from the fork down one fork. What is the fastest way to surely find your destination?)
- Gabriel's Horn. A mathematical object with finite volume, but infinite surface area. Thus you can conclude that if you wish to paint Gabriel's horn, it's much wiser (and less costly) to fill the horn with paint than to try to coat the outside. Full appreciation will require calculus experience. [Link] [Link]
Gabriel's Horn, alas! is too sophisticated for my purposes--but it is wonderful:
Gabriel's Horn - Wikipedia, the free encyclopedia: Gabriel's Horn (also called Torricelli's trumpet) is a figure invented by Evangelista Torricelli which has infinite surface area, but finite volume. Gabriel's horn is formed by taking the graph of y = 1/x, with the range x ≥ 1 (thus avoiding the asymptote at x = 0), and rotating it in three dimensions about the x axis...