I have never tried to count the times I have read a newspaper article explaining that students are bored with math that has no visible practical application, and follows with an example of a teacher, or club, that rectifies the situation in some novel and engaging way.

In the present case a class has built a sculpture that resembles a graph of a modulated wave motion. Of all the practical, real-world
applications of mathematics! It is as practical as a snowman.

Why doesn’t anyone ask for real-world applications of table tennis? What a bore any game must be, that has no real-world application! Why do kids stand for it? Ping-pong again? Ugh.

But I can think of something: Let’s all make a model of a ping-pong ball in the school yard, seventy feet high, blocking all the entrances and thus preventing all us students from entering the (ugh) school. Then we can take our fishing poles and torn straw hats out from under our beds and, with the hats on our heads and fishing poles over our shoulders, all traipse together down the dusty road to Norman Rockwell’s house.

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An anonymous math department chairman reports on his own strategy for cutting down on the workload. He believes that one of the most important determinants of a successful career is luck. So each year, he randomly rejects half the applicants without even reading their folders. That way, he eliminates the unlucky ones.

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A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory…Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music”. It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music…are considered very advanced topics and generally put off till college, and more often graduate school.

As for the primary and secondary schools, their mission is to train students to use this language—to jiggle symbols around according to a fixed set of rules: “Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key…One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way.”

…

Sadly, our present system of mathematics education is precisely this sort of nightmare.

So begins Paul Lockhart’s scathing critique of how mathematics is taught in this country, A Mathematician’s Lament. The book is an expansion of Lockhart’s essay of the same title. I encourage you to read the essay, buy the book, and share your thoughts in comments.

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Today I read it. Wow, was I wrong. This is Caitlin Flanagan at her blistering best. I’ll offer you a few choice quotes, but my real recommendation is to leave now and go read the entire piece.

With the Edible Schoolyard..the idea of a school as a venue in which to advance a social agenda has reached rock bottom. This kind of misuse of instructional time…has been employed to cheat kids out of thousands of crucial learning hours over the years, so that they might be indoctrinated in whatever the fashionable idea of the moment or the school district might be. One year it’s hygiene and the another it’s anti-Communism; in one city it’s safe-sex “outercourse” and in another it’s abstinence-only education.

Does the immigrant farm worker dream that his child will learn to enjoy manual labor, or that his child will be freed from it?…If this patronizing agenda were promulgated in the Jim Crow South by a white man who was espousing a sharecropping curriculum for African American students, we would see it for what it is: A way of bestowing field work and low expectations on a giant population of students who might become troublesome if they actually got an education.

Until our kids have a decent chance at mastering the essential skills and knowledge that they will need to graduate from high school, we should devote every resource and every moment of their academic day to helping them realize that life-changing goal. Otherwise we become complicit—through our best intentions—in an act of theft that will not only contribute to the creation of a permanent, undereducated underclass, but will rob that group of the very force necessary to change its state.

There’s much more where that came from. Why are you still here?

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Question 6. When Eve works, she produces exactly one apple per hour. Adam is completely unproductive and can produce nothing at all. Eve’s income is taxed at a flat percentage rate, with the proceeds delivered to Adam. What determines the optimal tax rate? What does “optimal” mean here, and what philosophical justification would many economists give for adopting this tax rate?

To make the problem concrete, you can assume that both Adam and Eve, if it were both possible and necessary, would be willing to work up to 1 hour for 1 apple, up to 2 hours for 4 apples, up to 3 hours for 9 apples, and up to x hours for x² apples. Now what is the optimal tax rate? (Your answer should be a number.)

Question 7. Jack and Jill play a game. First, each flips a coin. After seeing their own coins (but not each others’), each player (separately) says either “Red” or “Black”. If they name opposite colors, then the Black-sayer gets $4 and the Red-sayer gets nothing. If both say Black, then they both get either $5 (if both flipped heads) or $10 (otherwise). If they both say Red, then they both get either nothing (if both flipped heads) or $20 (otherwise). Assume both players play optimally. If Jack flips heads, what is the probability that he says “Black”? What if Jack flips tails?

Edited to add (in response to a comment from Ron): Assume that neither Jack nor Jill says either Red or Black with probability zero.

Question 8. The five Dukes of Earl are scheduled to arrive at the royal palace on each of the first five days of May. Duke One is scheduled to arrive on the first day of May, Duke Two on the second, etc. Each Duke, upon arrival, can either kill the king or support the king. If he kills the king, he takes the king’s place, becomes the new king, and awaits the next Duke’s arrival. If he supports the king, all subsequent Dukes cancel their visits. A Duke’s first priority is to remain alive, and his second priority is to become king. Who is king on May 6?

Question 9. Suppose the government mails every taxpayer a check for $300. Under a variety of assumptions, discuss the short run and long run effects on a variety of economic variables such as output, employment, the interest rate and the trade balance.

Question 10. Suppose you want to study the effect of education on wages. You have wage data for 100 pairs of siblings, where one member of each pair attended college and one didn’t. Based on these data, you make some estimates. Now you learn that all 100 pairs of siblings are in fact twins. Does this increase or decrease your confidence in your results? Make some arguments in both directions.

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I thought my readers here might be interested in seeing the written exam. It’s by no means comprehensive; entire areas of economics are omitted. Instead, it’s supposed to test core material and ways of thinking that I believe should mostly be second nature to any top economics graduate.

Where necessary, I’ve translated some of these questions from the original economese to something approximating English. Occasionally, a little has been lost in the translation, but not, I think, too much.

There were ten questions on the exam. I’ll post five today and the remaining five next week.

Here, then, is Part I:

Question 1. When the price of peanuts rises, Frieda reduces her root beer consumption. If Frieda’s income rises, will her root beer consumption go up or down?

Question 2. Bananas cost $6 apiece, except for members of the banana club, who pay $2 apiece.

• Given full knowledge of Thomas’s prefereces, explain how you’d compute his willingness to pay for a membership in the banana club.
• Given knowledge only of Thomas’s demand curve for bananas, explain how you’d estimate his willingness to pay for a membership in the banana club.
• Under what circumstances is your estimate an overestimate? Under what circumstances is it an underestimate?

Question 3. Snidely Whiplash owns all the grocery stores and all the houses in the Yukon Territory. He charges a competitive price for groceries, and rents the houses at the highest price residents (who are all identical) are willing to pay. (If he charged any more, they’d all leave town). True or False: If Snidely raises the price of groceries, he’ll have to lower the price of housing, so he’ll be no better off than before.

Question 4. Discuss the consequences for economic efficiency of giving your father a Barnes and Noble gift card, under various assumptions about how he uses (or doesn’t use) the card.

Question 5. Rank these taxes in order of how much you’d dislike paying them:

• A tax on consumption
• A tax on wages
• A tax on income (including wages, interest and dividends)

Assume that the tax rates are adjusted so that your total tax bill is the same in each case.

(Edited to add): At the request of a reader, I’ve also posted the questions in the orginal economese.

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Stanford professor Carolyn Hoxby reports that in the college admissions market, the big change over the past 40 years is students’ increased willingness to travel far from home—not surprising since the costs of long distance travel and communication have fallen dramatically over that time. The main effects are these:

• The top colleges (meaning the top 10%) have gotten far more selective, because they’re now drawing from a far broader base of applicants.
• Most other colleges (well over half) have gotten far less selective, because the pool of local applicants is shopping elsewhere.
• This change in students’ willingness to travel provides a complete explanation for the increased selectivity of top colleges; in fact, without it, they’d have become slightly less selective.
• As a result of these trends, the student bodies at the best colleges have gotten much stronger and the student bodies at the weaker colleges have gotten much weaker.
• Strong student bodies demand and receive a lot more resources—better faculty, better courses, better facilities, better living arrangements. This is partly because stronger students are more willing to pay for such amenities in the first place and partly because stronger students graduate, earn higher incomes, and make bigger donations.
• Therefore students at the best schools, in addition to being much stronger than they used to be, are also getting a much better deal. Tuition has risen, but not by nearly as much as the value of the college experience. The difference is huge: At the very top schools, student-oriented expenditures exceed tuition by almost $80,000 per student per year.
• All that student-oriented expenditure pays off for students, and it pays off most for the very best students, who learn a lot more and ultimately earn a lot more as a result.

Here’s a (rather startling) graph (taken from Hoxby’s paper) to show what a great deal students at the top schools are getting, and how much it’s changed since 1970. The bottom line is that getting into the very best schools is both a lot harder and a lot more valuable than it used to be.

The full paper is here. Hat tip to Tyler Cowen, who found this before I did. Arnold Kling has a slightly different take.

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