What should be in a maths class?

I have recently (well, in the past two years) read two very interesting essays on the teaching of mathematics. At first glance, they seem to be almost diametrically opposed, but I tend to find myself agreeing, overall, with the thrust of both. The essays are Paul Lockhart's A Mathematician's Lament and Conrad Wolfram's TED talk on mathematical education. A couple of quotes from each which provide a brief summary.

From Wolfram's talk (trancript available here):

I want to see a completely renewed, changed math curriculum built from the ground up, based on computers being being there, computers that are now ubiquitous almost. calculating machines are everywhere and will be completely everywhere in a small number of years. Now I'm not even sure if we should brand the subject as math, but what I am sure is it's the mainstream subject of the future.

From Lockhart's Lament:

The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.

So, Lockhart thinks we should be teaching mathematics as an art form, and Wolfram thinks we should be introducing more computers into mathematics lessons so that people can concentrate on doing the bits that are actually useful. Which of them is right? Well... both, but mostly Wolfram.

The question we have to ask ourselves before we can even begin to compare the two essays is why do we teach mathematics at all? So far as I can see, the only sensible justification for having mathematics as a subject that everyone in the world should be taught up to a relatively high level is because it's useful. You can't do physics, or engineering, or any sort of science, or do derivatives trading, or even decide which mortgage to get, without knowing quite a lot of mathematics. For this reason, everyone should be taught the basic mathematics that they need to know in order to do these things (or the basics they need to know in order to learn the specific maths they wnat to use).

Note that one corollary of this mode of thinking is that most of the mathematics you learn probably shouldn't be taught in a maths class. It's much easier to learn how to get from acceleration to speed than it is to learn how to differentiate a function. Yes, it is useful to then point out the possible generalisations (getting from acceleration to speed is the same as getting from jerk to acceleration) but I don't see any reason why these topics can't be introduced concretely. I personally have serious trouble doing any calculus that I can't do using physical intuition, and I think it's fair to say that I am an above-average student when it comes to learning maths. Calculus should be taught when you're doing engineering, statistics should be taught when you're analysing the results of experiments, graph theory should be taught when you're trying to solve scheduling problems.

Lockhart, on the other hand, seems to think that we should be learning mathematics because, essentially, mathematics is awesome. I happen to agree with him that mathematics is an exceptionally beautiful art form. I'm happy to sit back and bask in the glory of Cantor's diagonalisation argument, or the ingenuity of Karp's reductions between NP problems, but I'm not sure that I'm willing to contend that everyone should be forced to. Yes, if you want to be a mathematician you have to learn that mathematics is actually an art, but most people who study mathematics don't want to be mathematicians, and most people who study mathematics *shouldn't* want to be mathematicians. For these people, learning about the art of mathematics is little more than an intellectual curiosity, on a par with learning about Titian or Shakespeare.

In other words, Lockhart is right, inasmuch as we want people to study mathematics for it's own sake. Wolfram is right, inasmuch as we want people to study mathematics because it's useful.

Now, I happen to tend strongly towards the idea that the only things we should be teaching in schools are things which are potentially useful, but that obviously isn't the prevailing wisdom - everyone in this country is still forced to do an English Literature GCSE. Lockhart-style mathematics is a perfectly good substitute for art class, or critical theory. Wolfram's mathematics is a necessary prerequisite for doing just about anything else.

Philosophy of Education

I just came across this on Rosemary Bailey's website. While I think the ideals are nice, I think in practice I disagree with just about all of it:

My vision of a university was succintly described by David L. King, writing in The Times Higher on 9 April 2004. It is:

• “the belief in a community of scholars and not a confederacy of self-seekers;
• the idea of openness and not ownership;
• the professor as a pursuer of truth and not an entrepreneur;
• the student as an acolyte whose preferences are to be formed, not a consumer whose preferences are to be satisfied.”

I believe that university students should be able to be confident that they are being taught by people who are immersed in the subject in other ways than teaching. I collaborate with a range of scientists on the design of their experiments and the analysis of their data, so I teach Statistics. I still prove theorems in Combinatorics and Algebra, so I also teach those subjects.

I've no idea what a "confederacy of self-seekers" is, so I'm not going to address that point. I don't think anyone believes in a professor as an entrepreneur, one thing that I think people can sensibly expect professors to be, at least in the current system, is teachers. Incidentally, I happen to think that Rosemary is a very good teacher, and one who thinks a lot about doing the best for her students, so I hope she won't be offended by any of this.

The problem I have mostly is with the last point in the bulleted list: "the student as an acolyte whose preferences are to be formed, not a consumer whose preferences are to be satisfied.". Now, that might be a nice model for a university if the point of going to university was to learn about the subject you were studying. The problem is that for the vast majority of people who go to university currently this just isn't the case. They go because in order to get the better paid jobs you have to have a little piece of paper which said that you went to university. They go to university to get this piece of paper and, and I think this is the important part. This system is not the students' fault.

The students in our department are not there because they are particularly interested in mathematics and (and I think this is something that academics sometime struggle with, and I will almost certainly get round to making a separate post about this one day) the vast majority of them are not interested in pursuing a career in academia. The overwhelming majority of students are at university to get a degree because it will help them get a better job. And they are right. In the world we happen to live in people do need degrees to get good jobs, and universities are there to supply these degrees, and to make sure that the degrees are a useful signal of ability for employers.

It might be that the vision of the university described by David King is a good way to train people who want to obtain the sort of detailed knowledge of an academic subject one needs to do research, but it absolutely is not the case that 50% of the population want, or need to obtain that sort of detailed knowledge of any subject. So, while we have a system where something pushing 50% of young people go to university (and, remember, pay the fees that pay the salaries of the people who work at the university).

Most people are there to get a degree because getting a degree is necessary prerequisite to doing whatever it is they actually want to do. Like it or not, the job of the teaching staff in a university is to help them get that degree. If you don't like it, by all means do your best to change it, but while you're accepting money from the university as it exists now, I'm afraid I think you are morally obligated to teach in the way that is appropriate for the majority of students who are there now. More to the point, if you don't like the current system, you're going to have to figure out another way to pay for researchers - at the moment research is cross-subsidised by an awful lot of socially innefficient teaching.

I personally would love to live in a world where universities could all be like Cambridge and Oxford, and where the vast signalling game that is university education as it exists now didn't exist. I admit that this world would probably be a world in which less "blue skies" research got done, and I find it hard to believe that would be a particularly bad thing.

This post is getting long, and I'm not sure my thoughts on the last paragraph of Rosemary's statement are fully-formed, so I won't write much about that for now. I will however finish with one thought that I have on the issue: it would be a remarkable coincidence if the best teachers in the world also happened to be the people who were best at doing original research. I happen to think that I personally am quite a lot better at the former than the latter. I'm not going to give an example of someone whose abilities are skewed in the other direction, but I'm almost certain everyone who has studied or worked at a university already has one in mind.

Things they should teach in school part 2: Touch-typing

I mean, really, this is a gimme isn't it? You spend 6 hours a day for at least 10 years going to a building where they are supposed to be teaching you basic life skills, and they don't even teach you how to use the single piece of equipment that you're going to spend more of your time using than any other (disclaimer, I just made that statistic up, but I can't even think of another candidate, so I'm going to go ahead and assert it as fact).

It took me about 10 hours to learn to touch type, very possibly less, and it must save me several hours a week in time spent not looking at the keyboard and, more importantly, not having to correct mistakes that I didn't notice because I was looking at the keyboard instead of the screen.

I propose that teaching kids to touch type would be more useful than literally everything that they learn in school after the age of eleven. Anyone got any counter examples?

University is not about learning

With a generous hat tip to Robin Hanson for the title, and to just about every blog on the planet for the link, I'm going to quote the following paragraph from this article, from a man who writes other people's assignments for a living:

I've written toward a master's degree in cognitive psychology, a Ph.D. in sociology, and a handful of postgraduate credits in international diplomacy. I've worked on bachelor's degrees in hospitality, business administration, and accounting. I've written for courses in history, cinema, labor relations, pharmacology, theology, sports management, maritime security, airline services, sustainability, municipal budgeting, marketing, philosophy, ethics, Eastern religion, postmodern architecture, anthropology, literature, and public administration. I've attended three dozen online universities. I've completed 12 graduate theses of 50 pages or more. All for someone else.

Now, let's say university actually was about learning, let's say the reason that people got degrees was because they came out with useful skills at the end of them and let's say that unversity grades were  good measure of people's skill at the subjects in which they get their degrees. Is it really likely that any one person could be good at all of those things? Good enough to get a high-level - graduate level, in fact - degree in all of those things?

No, university, like all places where people get educational labels, is a signalling game (I think Paul Graham is probably more accurate on what school is about). I've been meaning to write about this for a while, and might finally get round to it soon, but I just thought this quote was too much to pass up.

Things I wish I'd read earlier: I Pencil

I'm going to start an occasional series in which I'll post links to several articles and books that I wish I'd read earlier (or if they're in copyright, links to somewhere you can buy them). Mostly because if you're reading this, and you haven't read them, you're probably going to enjoy them.

Also, I kind of plan to post something sometime about what I think they should teach in schools (and which parts of the current curriculum they could afford to miss out) and I think most of this belongs there.

I Pencil

I think by the time I finished school, I had just about heard of the Invisible Hand, but I'd never been taught about it, and didn't *really* understand what it meant. The genealogy of a pencil is an incredibly powerful illustration of what it means. I challenge anyone to read it and not be amazed by the power of the price system (why don't more people cite *that* as a proof of God's existence?). The fact that I can walk into a shop whenever I want a loaf of bread and be sure that I'll find one on a shelf really is a miracle - everyone should be reminded of that every now and then.