I came across a really interesting paper on the MAA webpage the other day,* a mathematician’s lament* by Paul Lockhart, a school teacher in New York. I think many of the points are relevant to university maths and stats teaching, and even those that are not add perspective to how we deal with students on service courses in especial, since they may suffer from dyscalculia.

Here’s an exerpt from the first page:

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, let alone composing an original piece, are considered very advanced topics and are generally put off until college… Of course, not many students actually go on to concentrate in music, so only a few will ever get to hear the sounds that the black dots represent. Nevertheless, it is important that every member of society be able to recognize a modulation or a fugal passage, regardless of the fact that they will never hear one. “To tell you the truth, most students just aren’t very good at music. They are bored in class, their skills are terrible, and their homework is barely legible… they just want to take the minimum number of music courses and be done with it.”

Here is my take on the main points:

- Maths is about problem solving, which is intellectually fun.
- But maths teaching often involves memorising formulæ and applying them over and over again, in other words, a class of problems is outlined, a solution for those problems provided, and then students get to practice applying that solution. This is not fun.
- Once they leave school, most people don’t need to know all the mathematics they were taught, so we should focus on teaching them problem solving (genuine problem solving) rather than “pointless definitions”.

I wonder what messages we can take from this when it comes to teaching stats at university level? Should we refrain from reproving old theorems? Should exams focus exclusively on solving new problems (rather than redoing some questions from tutorials with minor changes)? Should we scrap lectures?

2010/04/09 at 10:52

I am all for solving word problems in exams; they should try to have no mathematical symbols, so the student must relate the problem to theory.

Some theorems are important and their proofs quite accessible, so should be taught. The student doesn’t need to prove the Cramer-Rao lower bound, but should know why the LS estimators are unbiased.

Mathematics educators have thought very hard about pedagogy for decades; we have much to learn from them, if only because probability is mathematical. But statistics also peculiarities that escape the mathematicians. Some examples:

(a) To apply Pythagoras’ Theorem, you measure the two perpendicular sides of a triangle, and say a = ??, b = ??. Therefore the hypotenuse c is ???. If y is an observed value of a random variable Y, we don’t put Y = y. This is difficult, but essential for a proper understanding of statistical inference.

(b) Some random variables are observable, some unobservable. For example, we measure an unknown constant c with error: Y = c + e. e is an unobservable random variable, since c is unknown. But Y is observable. The data are observed values of IID copies of Y. This is more difficult than (a), but also essential.

(c) Another model could have both c and e being random, both unobservable, while Y is still observable. This is more sophisticated than (b), but (perhaps?) essential to Bayesian inference.

I will be more than happy if our majors (with or without Honours) understand (a) and (b). (c) is a bonus.

2010/04/09 at 11:11

There’s always a strong temptation when writing an exam to ask the student, in essence, have you remembered this fact? (A simple fact, such as definition of AIC, or a complex one, such as proof of somesuch theorem.) By so doing we ask: Have you assimilated the factual information of the syllabus? It’s much harder to ask “why” questions, or questions that can only be answered by thinking “outside the box”. But wouldn’t it be great to do so, and to change how we teach to cover less information and more process?

2010/04/09 at 11:19

Yes, that’s a good suggestion. After all, we would like to think we are teaching some really good deep stuff. Then should it be a surprise that to check if the learning takes place, will take more work? For a start, mixing both types can ease the pain of transition for both setters and takers.