Thoughts on Teaching: 2

Curving Grades: Gordon Tullock’s Solution

Ideally a grade in a course one professor teaches represents the same level of accomplishment as the same grade in a course someone else teaches. Often it doesn’t, since different professors have different standards. The usual solution to that problem is to impose a grade curve on the professors, a fixed distribution of grades, sometimes with some flexibility in the details. That makes it hard to get an A in a class that only the best students take, easy in one populated mostly by the worst students.

My friend and ex-colleague Gordon Tullock, one of the most ingenious people I have had the pleasure of knowing, had a solution to this problem. You start with some measure of student ability; in a law school you could use grades in the first-year classes that everyone has to take. At the beginning of the second yea r youbase the curve for each class on the distribution of first year grades of the students taking it. In each semester thereafter recalculate student ability, adding in the information provided by the previous semester’s classes, and repeat.

Most undergraduate colleges do not have a first year where everyone takes the same classes so need some other way of starting the process, perhaps SAT score or high school GPA. Alternatively they could start everyone even, giving all the first quarter classes the same curve, and use the grades students get to gradually improve the measure of student ability. Mechanically speaking it should be easy enough to do in a world of computers and spreadsheet programs but I do not know of any school that has done it.

Adam Smith on Laptops in the Classroom

Professors have a mixed view of student laptops in the classroom. They are useful tools for taking notes and, connected to the internet, can be used to quickly research things relevant to classroom discussion. They can also be used to exchange email or instant messages, play games, read books, do any of a wide variety of things unrelated to and distracting from what is supposed to be going on in the classroom.

Some years ago I had the pleasure of sitting in on a class taught by a colleague who did a brilliant job of keeping his students' interest and attention — and mine — while covering material usually considered less than entrancing. I was at the back of the classroom and so could see quite a lot of laptop screens. With one or two brief exceptions there was no color on them, which I took as evidence that they were being used to take notes, not to browse the web or play games.

Whether to permit students to use laptops connected to the web during class is a new variant on the old question of whether class attendance should be compulsory, since the net lets one be physically in one place, virtually in another, attend class while corresponding with your friends or reading the newspaper. There are other ways of doing that — some of us remember reading concealed books during boring high school classes or retreating into thoughts unrelated to what we were supposed to be learning — but the new technology provides a more convenient tool for the purpose.

On the subject of compulsory attendance, I cannot improve on the words of Adam Smith:

No discipline is ever requisite to force attendance upon lectures which are really worth the attending, as is well known wherever any such lectures are given. Force and restraint may, no doubt, be in some degree requisite in order to oblige children, or very young boys, to attend to those parts of education which it is thought necessary for them to acquire during that early period of life; but after twelve or thirteen years of age, provided the master does his duty, force or restraint can scarce ever be necessary to carry on any part of education.

(Wealth of Nations Book V Chapter 1 Part 3 Article II)

As demonstrated by my colleague.

How not to Teach Math — or Economics

A conversation I once had with my younger son, frustrated over his undergraduate math course, reminded me of my longstanding objection to how math, and for that matter economics, are often taught. Theorems are proved with a rigor that is more than the students really need — especially in economics, where rigorous proofs can be applied to the real world only by combining them with non-rigorous models. The rigor is not only more than the student needs, it is more than any save the ablest students can understand. It is one thing to follow a proof step by step. It is a different and much more difficult thing to hold the proof in your head and understand why it is true.

My usual example of the problem is the failure to teach students of calculus why the fundamental theorem, that integrating and taking a derivative are inverse operations, is true. It is possible to give a non-rigorous but intuitively persuasive proof of the theorem in about five minutes, one that any student who understands what the two operations are can follow and has a good chance of remembering. One of the commenters on my blog did it in 38 words¹ but it is easier with a drawing. 

As best I can tell, very few of the students who take calculus, even at a good school, are ever shown the proof;  I would be surprised if more than one in fifty, a year after taking the course, could reproduce the more rigorous proof that they were, presumably, taught. To check the former  impression I asked my wife for her experience. Her response was that she was taught calculus twice, the first time at a good suburban high school by an incompetent teacher, the second time at a top liberal arts college. To the best of her memory, she was never shown the proof. I got more recent evidence interviewing high school seniors who had applied to Harvard, something I have done several times as an alumni volunteer. They are all bright students, many have taken AP Calculus and gotten a top grade. I ask them if they can show me why integrating and taking a derivative are inverse operations, why the derivative of the integral of a function is the function. I do not think any yet has been able to.

It is common at good schools to complain about cookbook mathematics, memorizing the sequence of steps to solve a problem without ever understanding why it works. It is, I think, an almost equally serious mistake to present a branch of mathematics in the form in which professional mathematicians structure it after all of the original work in that particular field is done. Not only is it a form in which almost no student not qualified to become a professional mathematician can understand it, it is a form that gives a highly misleading picture of how mathematics, or other forms of theory, are actually done.

I am not a mathematician but I am an economist and know by direct observation how the original parts of my work were done. The process did not start with a step by step proof but with an intuition of how some set of ideas fit together, what characteristics the solution to a problem ought to have. Only after I had groped my way to what was (hopefully) the right answer did I, or someone else, go back and make the argument rigorous. About forty years after my first book was published, I wrote a third edition. Part of what it consisted of was filling in the blanks, working out in more depth and more detail ideas whose essence I understood then and still believe, in most cases, were correct.

Alfred Marshall, arguably the figure most responsible for the creation of neo-classical economics, commented in a letter on the relation between mathematical and verbal arguments in his field:

But I know I had a growing feeling in the later years of my work at the subject that a good mathematical theorem dealing with economic hypotheses was very unlikely to be good economics: and I went more and more on the rules---(1) Use mathematics as a short-hand language, rather than as an engine of inquiry. (2) Keep to them till you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life. (5) Burn the mathematics. (6) If you can't succeed in 4, burn 3. This last I did often.²

There is much to be said for that policy. Mathematics is a more precise language than English but also a language farther from the intuition of almost all of us. If you have the math and cannot translate it or think of anything in the real world it corresponds to, it is quite likely that you do not understand it.

I sometimes referee journal articles. Occasionally I get one where, if you translate the math into words, it makes no sense, is arguably insane. The author or authors probably had doctorates in the field they were writing in. But they were manipulating symbols, not ideas.

When my daughter transferred to Chicago after two years at Oberlin, she was seriously considering majoring in economics. After taking an economics course, she decided on her alternative major. The reason was not that she does not like economics; she audited several of my courses while a home schooled student of high school age, one of my articles contains an idea that I credit in a footnote to her, and since graduating college she has edited several of my books. The reason was that the course was mostly about mathematics not economics.

I discussed her experience with a professor at that university of whom I have a high opinion, someone on the short list of people who, when they disagree with me, cause me to seriously consider that I may be making a mistake. He also had a daughter taking economics at the same school. He agreed with  my daughter's judgement, that the courses were teaching mathematical rigor instead of economic intuition.

What matters is not remembering but understanding. If you have memorized a proof but cannot explain why the result is true, you have been wasting your time.

“The fish trap exists because of the fish. Once you've gotten the fish you can forget the trap. The rabbit snare exists because of the rabbit. Once you've gotten the rabbit, you can forget the snare. Words exist because of meaning. Once you've gotten the meaning, you can forget the words. Where can I find a man who has forgotten words so I can talk with him?” (Zhuangzi)

Cargo Cult Teaching

The son of friends of ours is required by his teacher to spend twenty minutes a day reading and report on doing so. The theory, presumably, is that since regular reading correlates with desirable outcomes, the way to get those outcomes is to compel children to read. Since someone who obeys the rules will be cutting the book into twenty minute chunks instead of reading right through it, the likely effect is to teach him that reading is a chore to be done only under compulsion.

On second thought, it’s worse than a cargo cult. Building fake airplanes doesn’t make it more likely for Pacific islanders to get western goods but at least it doesn’t make it less likely.

Years earlier, we observed an analogous mistake in a different context. Our home schooled daughter, considering a career as a librarian, volunteered to work without pay at a large local library. After a week they thanked her and told her that her term of volunteering was over. Pretty clearly, their assumption was that she was volunteering either because her high school required her to or else to get something to claim to have done on her college application; it was now someone else’s turn.

Wanting to volunteer in order to do useful things is evidence of desirable personality traits. Wanting to volunteer because someone will reward you for it is not. She found a smaller library that had a use for her services and worked there for a couple of years.

For a third example, consider one of the problems I have observed in my experience of historical recreation in the Society for Creative Anachronism. Too often, people in the SCA see being historically accurate is something you do because other people are pressuring you to do it or in the hope of getting rewards and status. The result is “documentation” whose purpose is not to figure out how something was done in period but to find some excuse to claim that what you want to do, or at least something vaguely similar, was done in period.

In each case the mistake is the same, the attempt to create the effect without its proper cause. To fake it.

One commenter on my blog responded:

Trying to get the cause by pursuing the effect is one of the characteristic behaviors of the villains in Atlas Shrugged, in domains ranging from economics to sex.

Doing it Right: Scav at the University of Chicago

Every year a sizable fraction of the student body at Chicago spends most of several days on a scavenger hunt claimed to be the largest such event in the world. It starts with a list of some two or three hundred things to be found, done, solved. Competition is by teams, often although not always associated with a dorm, typically a hundred to two hundred people — students, alumni, friends and allies — dividing up the problems of the hunt.

The lists of requirements for past hunts can be found online. For example, from 1999:

Item 240. A breeder reactor built in a shed, and the boy scout badge to prove credit was given where boy scout credit was due. [500 points]

And a pair of physics students did it.

Most of the items are less difficult than that one.

Examples:

22. Our plants have been listening to the Plantasia album for a few months now, but we’re worried they’re getting bored. Record 5 minutes of My Bulblet, My Bulblet and Me, an advice podcast by plants, for plants. Submit to mbmbam@uchicago.edu before Judgment. [6 points]

36. Get a basketball player to perform a baptism. Get dunked on, baby. [8 points for a collegiate basketball player, 21 points for an NBA/WBNA basketball player]

37. We were told that all RAs are given the option to live in Vue53 over the summer for the low price of $500. We were also told that this means that there’s just a bunch of RAs living in a dorm alone together for the whole summer. We were disappointed to hear that no one has capitalized on this by making a Big Brother-esque reality show about these RAs living together. [8 points]

Some things must be done outside of Chicago, so a team will need a road trip group. For example:

43. At the Lew Wallace Study and Museum [In Crawforsville, IN], the Traveling Circus picks up two souvenirs for home: Flat Lew and a seed packet from the library outside. [8 points]

Some are puzzles that require research to solve:

30. According to a 1987 rent advertisement at the Givins Castle, the castle could be rented by a school or what other institution that required large grounds? [4 points]

No grades, no curriculum, no classes. Just people doing things for fun — many of which will turn out to be educational.

1. “One can start with Rieman's definition of the derivative, draw little rectangles under the curve, and show that derivating the cumulative integral is equivalent in the limit to looking at the height of the curve on that point.”

2. Which left me wondering how much of the economics of the next century went into Marshall’s fireplace.