Chad Orzel writes about interesting mathematical calculations...
"What's the goal?" he asks. I guess the goal is to come up with a set of problems, puzzles, and calculations that are (a) fun to do, (b) teach interesting facts and relationships, and (c) convince pre-teenagers and early teenagers that math is worth learning because it is a useful tool.
Posted by DeLong at December 17, 2003 09:16 PM | TrackBackUncertain Principles: Brad DeLong asks the all-important question "Can there possibly be as many as one hundred interesting math problems?"
Well, OK, strictly speaking, he's assuming that there are at least one hundred interesting math problems, and is compiling a list as an aid to getting his kids to see the utility of math. This makes it more of a proof-by-contradiction problem: since he's asking for help from his readers, he may be close to the end...
Part of the problem with trying to compile such a list is defining just what counts as an interesting math problem. A bunch of the list to date are more puzzles than calculations, a few others are demonstration exercises, a few are Fermi Problems, and four are freshman physics problems (numbers 3, 14, 15, and 23). Very few of them bear any resemblance to math as practiced by mathematicians, and the physics ones come close to failing the "interesting" criterion.
Of course, there's a tight constraint placed on the set of possible problems by the fact that they're meant to inspire kids.... The other main problem also has to do with goals. It's not entirely clear what Brad's after-- if the goal is to demonstrate that mathematics makes a useful tool for other disciplines, these are good examples, but I wouldn't expect them to awaken a deep love of the subject for its own sake. [Nah. Too late for that. Not going to happen.] Inspiring an interest in pure math is a trickier business.... Nevertheless, here are a couple of suggestions (see also Andrew Northrup's list):
More Freshman Physics. The current list includes a couple of kinematics problems, and a couple of force problems. If you've already deemed objects sliding on inclined plans to be an Interesting Math Problem, there are reams of problems involving the same sort of math-- tension forces in ropes suspending signs, connected object problems, etc. Two specific suggestions:
1) The Kansas City Walkway Collapse. 114 people were killed in the collapse of a hotel walkway because the engineers in charge of the project didn't realize that changing the way they suspended the walkway would double the stress on one of the components. It's a freshman physics level calculation, and not doing it cost 114 lives.
2) The car-in-the-mud problem. A much more frivolous, but kind of interesting problem: If your car is stuck in the mud, you can dramatically magnify the force you can exert on it to get it out by tying one end of a rope to the car, the other end to an immobile object (a tree, say), and pushing the middle of the rope sideways. It's just a vector addition problem (combined with the fact that you can't push with a rope), but it's kind of cool.
Stochastic Processes. There are a bunch of probability problems on the list already, so why not do something about random walks? The "drunkard's walk" variation with a hard wall on one side is easy enough to set up, and reasonably interesting. You could also talk about Stephen Jay Gould's use of it. Or maybe not, if you don't want hate mail from biologists.
It might be possible to do something with Brownian Motion, too, maybe by way of the Brownian Motor idea. (Check out the nifty applet...)
Tiling the Plane. There's a whole field of math devoted to covering infinite surfaces with odd shapes, including a wealth of material on non-periodic tiles (including nifty puzzles). I'm not entirely sure what you'd do with that (unless you're planning to remodel your bathroom), but it would let you get in the amusing anecdote of Roger Penrose vs. Kleenex.
Least Time Problems. You can actually develop pretty much all of optics starting from the idea that a beam of light will always follow the path between two points that requires the minimum travel time (together with the idea that the index of refraction changes the speed of light in a medium). Yeah, OK, calculus of variations is a bit much for kids, but simple variants like the lifeguard problem (scroll down to Feb 3) could work for the "Hundred Interesting Problems" list.
Mathematical Induction. Math as it's actually practiced relies heavily on the idea of proof, so it might be worthwhile to work in some sort of proof technique, and induction springs fairly naturally from common sense. It's a little hard to see how to show that it's useful in everyday life. Perhaps you could explain how all horses are the same color...
The Car Talk Problem. Here's one that requires calculus, taken from an episode of Car Talk (as related by a colleague):
A truck driver called in to the show, and explained that he had a broken gas gauge on his truck, and was reduced to measuring fuel levels with a dipstick. The gas tank was in the shape of a horizontal cylinder with an opening at the top in the middle of the cylinder. It's trivially easy to recognize totally full and totally empty, and half full is not much harder (fuel to half the distance between the top and bottom of the tank), but if you want to mark the quarter-full and three-quarter-full points, where do you put the marks (as fractions of the diameter d of the tank)?
And that ought to be enough suggested math problems to prove a point of some sort.
A suggestion for the list of interesting math problems (hope you haven't already used this one or discarded it):
Let's make a deal. Monty Hall tells you that behind one of 3 closed doors lies a wonderful prize, and worthless prizes lie behind the other two. He lets you pick the prize behind one of the doors. He then opens one of the other two doors, revealing one of the worthless prizes. Monty offers you the chance to change your choice to the remaining closed door. Show that switching your choice after the elimination doubles your odds of winning from 1/3 to 2/3.
This one's not so tough on the actual math, but it does involve a proof that can be done by brute force, and it shows the value of guessing logically.
-paul
Why would you get hate mail from biologists? GOuld's use of the random walk as described in that review seems perfectly unobjectionable to me. A random walk will cross any finite barrier in finite time.
Posted by: dsquared on December 18, 2003 03:04 AMIt is never too late to turn kids on to mathematics. I was well into my teens before I developed an interest in the subject.
Posted by: Abiola Lapite on December 18, 2003 03:28 AMI hvae very little sympathy for the point that "very few of them (the 100 problems) bear any resemblance to math as practiced by mathematicians." Why should they? For mathematicians (professional and amature), pure math may be the cat's pj, but math is vastly useful. To the rest of us, being able to figure out how long a trip might take (in a boat, through a wide variety of currents, tacking through one leg of the trip), how much food and water to bring, how many books to carry, and what the risk is of running our of the birth control device of choice along the way, makes math a heck of a thing to know how to do. Math is far more than merely what is "practiced by mathemeticians."
Posted by: K Harris on December 18, 2003 04:36 AMGetting kids interested in applied math, vs getting them hooked on pure math is like sports, IMHO. It's good for the kids to be interested in [participating in] sports, and will make their lives better. And it's very hard pick this up later in life.
However, your children will not be professional athletes. If they have a lot of natural talent, practice a couple of hours every day through jr. high school, and get professional coaching, they'll probably be in the middle of the pack when trying out for the varsity high school team.
Similarly, if your children are very smart, and very interested in mathematics, they'll be in the middle of the pack when applying for a math graduate program. Considering the world-wide competition they'll face at a good school, they might not even be in the middle of the pack.
However, keeping active in math during their undergraduate years will give them a leg up.
Posted by: Barry on December 18, 2003 05:21 AMWhy would you get hate mail from biologists? Gould's use of the random walk as described in that review seems perfectly unobjectionable to me. A random walk will cross any finite barrier in finite time.
When I Googled for a passage about that book, I came up with an amazing number of pages that used it as a springboard to talk about how Gould was an imbecile, and didn't understand evolution at all. I think this is an outgrowth of his feud with Dawkins et al.. (A surprising number of the comments had a sort of "Ewwww" tone to them that gave them a sort of Scopes Monkey Trial flavor.) I'm not sure how many of those were actual biologists, but I have gotten the impression that Gould's ideas are moderately controversial.
As for math-as-mathematicians-know-it, that's part of the question about goals. If the goal is to get them to see math as a useful tool, then these are great problems, but you run the risk of convincing them that math is nothing but manipulation of numbers for other purposes.
I've seen this (indirectly, as I'm not in a math department) a few times already at the college level-- some students come in saying that they like math, meaning that they're good at manipulating numbers. This does not mean that they'll be happy as math majors, though-- you only have to get a couple of courses into the math sequence before the numbers disappear completely.
Why would you get hate mail from biologists? Gould's use of the random walk as described in that review seems perfectly unobjectionable to me. A random walk will cross any finite barrier in finite time.
When I Googled for a passage about that book, I came up with an amazing number of pages that used it as a springboard to talk about how Gould was an imbecile, and didn't understand evolution at all. I think this is an outgrowth of his feud with Dawkins et al.. (A surprising number of the comments had a sort of "Ewwww" tone to them that gave them a sort of Scopes Monkey Trial flavor.) I'm not sure how many of those were actual biologists, but I have gotten the impression that Gould's ideas are moderately controversial.
As for math-as-mathematicians-know-it, that's part of the question about goals. If the goal is to get them to see math as a useful tool, then these are great problems, but you run the risk of convincing them that math is nothing but manipulation of numbers for other purposes.
I've seen this (indirectly, as I'm not in a math department) a few times already at the college level-- some students come in saying that they like math, meaning that they're good at manipulating numbers. This does not mean that they'll be happy as math majors, though-- you only have to get a couple of courses into the math sequence before the numbers disappear completely.
Apologies for the double post, and the first paragraph in that should be italicized, but the comment system ate the tags.
Do you information (that you can reveal without embarassment to family members other than yourself) as to whether any of your problems so far have interested your children?
Posted by: Martin on December 18, 2003 08:20 AMThere are more good suggestions at Chris Genovese's Signal + Noise:
http://signalplusnoise.com/archives/000341.html
Can't think of any suggestions right now, but there are sometimes interesting problems at http://www.nrich.maths.org, which is a kids' maths education project run by the University of Cambridge.
Posted by: Mark Adams on December 18, 2003 09:40 AMI would suggest the book "In Code: A Mathematical Journey" by Sarah Flannery (and her father), the young girl who nearly set the encryption world on end if not for a minor flaw in her methodology. The book is written from her point of view as a teenage girl in the man's world of mathematics, and it's a very fun read. There are lots of interesting problems, though many are kind of brain-teasers rather than math problems--though there are sections on modulo math which is vital to encryption.
Posted by: Stoffel on December 18, 2003 09:48 AMHere's another, on "Curvature and Polyhedra":
http://www.theculture.org/rich/sharpblue/archives/000108.html
Posted by: Rich on December 18, 2003 12:43 PMI'd use Stock Market Maths.
Start with the absolute basics of Simple and Compound Interest.
Then add currency fluctuations ... ie "You can borrow $1m for 3 years at 3.3% in the US, or you can borrow $1m for 3 years at 4.6% in Euros. What chance of 10% appreciation per annum in the US dollar would there need to be to make borrowing in Europe worthwhile ?"
Next, go to Growth Stocks and PE Ratios ...
You have $100 000 to invest, and 2 choices (well, 3 including fast company, slow horses and tequila, but I dont think your Father would approve. Even though he does work at Berkely).
Joe-Bob's Pipeline company is boring, and ships 100 Mcf of natural gas from Louisiana to Texas each day. It earns $1m a month, and it gives $10m a year to it's shareholders each and every year, keeping $2m for admin and pipeline maintainence. It trades at a PE of 8.
WhizzCorp is exciting, and currently has $2m of earnings a year, but which is growing by 25% a year. Once they get up to $100m in sales, they'll pay $10m a year to shareholders. Right now, it has a PE ratio of 50.
In exactly 7 years, your Uncle Bob, a psychotic old cracker out for revenge against you, will destroy all references to you on the NYSE (etc), meaning you will no longer own, and thus get no value from the stocks at that point. If you ever try to sell the stocks, Uncle Bob will find you, so it's a dividend-only renumeration policy for you.
Given that in 7 years you will want money to buy your first Ferrari with, which company will pay you more dividends over that time for the $100 000 you have to invest (which, incidentally, as an unimportant side note, we can assume you stole off Uncle Bob) ?'
Posted by: Ian Whitchurch on December 18, 2003 01:26 PMwrt Ian's allegedly boring Joe-Bob's Pipeline Co, this item dredged up from http://www.consumerwatchdog.org/ftcr/st/st002462.php3
...
Joe Bob Perkins, Reliant Resources Inc. Perkins resigned from his position as COO and president of the company's wholesale division last week. Perkins, who became a lightning rod during the California energy crisis for his company's profiteering in the state, resigned, along with Shahid J. Malik, president of Reliant Energy Services. The departures followed the revelation that Reliant engaged in massive volumes of sham electricity trades with CMS Energy Corp., which then re-sold the power to Reliant. This practice has the effects of inflating a company's apparent status as a market player and of inflating prices by making electricity appear to be more in demand than it actually is. At one point during the California energy crisis, Joe Bob Perkins chastised Californians: "Reliant hopes the Governor and others will stop these baseless accusations and focus on true solutions to California's energy shortage. We are now being singled out because we believe in an open market." Secretly manipulating energy sales volumes to deceive consumers and investors is, of course, a hallmark of an open market.
I rather suspect that Reliant owned a bit of pipeline here and there...
Creative economists are scarey.
Creative accountants are scarier.
}d-Q
Posted by: Ecomonist on December 18, 2003 04:37 PMOn the applied side, I draw your attention to Zalewski and Allen's *Shaping Structures: Statics*, where real structural engineering problems are addressed through graphical methods. It's a bit specialized to architecture students, but it might be just the thing for people who want to see the mathematics applied to something concrete. And it is very cool that the book's methods cover the actual very cool structures they use as illustrations.
See
http://shapingstructures.com/home.html
http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471169684.html