by Ian Stewart
This cover-up can go undetected for a long time, but it falls down in the face of sums like 2546 + 9773.
Multiplication affords another example. For a time, you can think of, say, 4 × 5 as 4 + 4 + 4 + 4 + 4 and fall back on your understanding of addition (even using counting). But when faced with 444 × 555, you need something more sophisticated.
The interesting thing is that the strategies that fail in the long term are precisely the ones that we use to teach these new concepts in the short term. We relate numbers to counting, often using real “counters.” We relate multiplication to repeated addition. There’s nothing wrong with this. Mathematics builds new ideas on old ones. It’s hard to see how else you could teach it. But eventually the training wheels have to come off the bike: students have to internalize the new idea.
David calls these process-cum-concepts “procepts.” A procept can sometimes be usefully viewed as a process, and other times as a concept, a thing. The art of mathematics involves switching effortlessly from one of those viewpoints to the other. When you’re doing your research, you don’t even notice the switch. But when you’re teaching, you have to be aware of it. If one of your students is having trouble with a new procept, the cause may be a past failure to “proceptualize” one of the processes involved. So your job as a teacher is to backtrack through the series of ideas that leads up to the new one. You’re not looking for the first place where your student fails to answer a question. You’re looking for the first place where they can answer it only by using some simpler idea as a crutch.
In British elementary schools, the educational establishment has managed to get this spectacularly wrong. We now have a highly prescriptive “national curriculum,” and teachers—quite literally—check hundreds of boxes to mark the student’s progress. Can they count to five? Check. Can they add five to three? Check. The assumption is that what matters is their ability to get the answer. But what really matters is how they get the answer. I’m old-fashioned enough to believe that either way they have to get the right answer; no easy grades for “method” from me. But I am absolutely certain that checking a series of boxes is not the way to teach anyone mathematics.
18
The Mathematical Community
Dear Meg,
Now that you are on the verge of becoming a fully fledged member of the mathematical community, it’s a good idea to understand what that entails. Not just the professional aspects, which we’ve already discussed, but the people you will be working alongside, and how you will fit in.
There’s a saying in science fiction circles: “It is a proud and lonely thing to be a fan.” The rest of the world cannot appreciate your enthusiasm for what seems to them a bizarre and pointless activity. The word “nerd” comes to mind. But we are all nerds about something, unless we are couch potatoes who have no interests except what’s on TV. Mathematicians are passionate about their subject, and proud to belong to a mathematical community whose tentacles stretch far and wide. You will find that community to be a constant source of encouragement and support—not to mention criticism and advice. Yes, there will be disagreements too, but generally speaking, mathematicians are friendly and relaxed, provided you avoid pushing the wrong buttons.
Pride is one thing, loneliness another. My experience is that today’s public is much more aware than it used to be that mathematicians do useful and interesting things. At parties, if you admit to being one, you are far more likely to be asked, “What do you think about chaos theory?” than be told, “I was never any good at math when I was at school.” In Jurassic Park, Michael Crichton says that today’s mathematicians no longer resemble accountants, and some are more like rock stars.
If so, this is very bad news for rock stars.
Even if people ask you about chaos theory at parties, it is still unwise to explain your latest theorem on semicontinuous pseudometrics on Kähler manifolds to the guy in a leather jacket. (Though nowadays he might turn out to be a mathematician. But don’t count on it.) So, despite the public’s newfound tolerance of math, there will be occasions when you want to be with people who understand where you’re coming from. Such as just after you’ve finally proved the semicontinuous case of the Roddick–Federer conjecture on the irregularity of Kähler manifold pseudometrics in dimensions greater than 34.
Science fiction fans go to conventions (“cons,” as they say) to talk to other science fiction fans. Whippet breeders go to whippet shows and compete with other people who breed whippets. Mathematicians go to conferences to hang out with other mathematicians. Or they give seminars, or colloquia, or just visit.
Our first vice chancellor, Jack Butterworth, once said that no university was worth anything unless a quarter of its faculty was in the air. He intended this literally: air travel, not intellectual high flight. The best way to advance the cause of mathematics is to meet other mathematicians.
If you are lucky, they will come to you. The University of Warwick, founded in the 1960s, became a world-class center for mathematics because from day one it held symposia, year-long special programs in some area of math. (I was once told that “symposium” means “drinking together,” a theory that cannot be rejected out of hand.) But it’s a good idea, and more fun, if you go to them. Mathematics, like all the sciences, has always been international. Isaac Newton used to write to his counterparts in France and Germany, but today he could hop on a budget flight and meet them.
Mathematicians get together, usually over coffee; Erd~s said that a mathematician is a machine for turning coffee into theorems. They share jokes, gossip, theorems, and news.
The jokes are mathematical, of course. There is a lengthy compendium of classic mathematical jokes in the January 2005 issue of the Notices of the American Mathematical Society, and its contents are a vital part of your mathematical culture, Meg. There is, for instance, a Noah’s ark joke. (Actually, my favorite Noah’s ark joke is biological: a cartoon. The rain is coming down in sheets, the ark is loaded with two of every kind of animal, and Noah is on hands and knees grubbing around in the mud. Mrs. Noah is shouting from the ark, “Noah! Forget the other amoeba!”) Anyway, the mathematical Noah’s ark joke goes like this:
The Flood has receded and the ark is safely aground atop Mount Ararat; Noah tells all the animals to go forth and multiply. Soon the land is teeming with every kind of living creature in abundance, except for snakes. Noah wonders why. One morning two miserable snakes knock on the door of the ark with a complaint. “You haven’t cut down any trees.” Noah is puzzled, but does as they wish. Within a month, you can’t walk a step without treading on baby snakes. With difficulty, he tracks down the two parents. “What was all that with the trees?” “Ah,” says one of the snakes, “you didn’t notice which species we are.” Noah still looks blank. “We’re adders, and we can only multiply using logs.”
This joke is a multiple pun: you can multiply numbers by adding their logarithms. Other jokes parody the logic of proofs: “Theorem: A cat has nine tails. Proof: No cat has eight tails. A cat has one more tail than no cat. QED.”
Mathematicians tell each other theorems. Quirky ones, like the “ham sandwich theorem”: if you have a slice of ham and two slices of bread, arranged in space in any relative positions whatsoever, then there exists a plane dividing each of the three pieces exactly in half. Or the recently proved “bellows conjecture,” which says that if a polyhedron flexes (as, remarkably, some can), then its volume doesn’t change. But there is often a sting in the tail: “Proved that? OK, now do it with n objects in n dimensions.” Sometimes they tell each other conjectures, theorems not yet proved and that for all they know might be false. My favorite is the “sausage conjecture.” For starters, suppose you want to wrap a number of tennis balls in plastic film. What arrangement has the least surface area? (Assume that the film forms a convex surface: no dents.) The answer is that if you have fifty-six balls or fewer, they should be placed in a line to make a “sausage.” If you have fifty-seven or more, then they should be clum
ped together more like potatoes in a string bag.
In a four-dimensional analogue, the breakpoint is somewhere between fifty thousand and one hundred thousand. With fifty thousand balls they form a sausage. With one hundred thousand they clump. The exact breakpoint here is not known.
Here is the full conjecture: Three and four dimensions are misleading. Prove that in five or more dimensions, sausages are always the answer, no matter how large the number of balls may be.
The sausage conjecture has been proved in forty-two dimensions or more.
This is bizarre. I love it.
There will be gossip. Nowadays it may be about the topologist who ran off with her secretary, or the messy divorce of two well-known group theorists, but that’s a recent development that I trace to the bad influence of television. Traditionally, gossip is about who is in line for the Chair of Abstract Nonsense at Boondoggle University, or do you know anyone who has a postdoc position going for a young functional analyst like my student Kylie, or do you think Winkle and Whelk’s purported proof of the mass gap hypothesis has any chance of being right?
There will be serious news. As I write, a major topic of conversation is the latest information on Grisha Perelman’s alleged proof of the Poincaré conjecture. Has anyone found a hole in it yet? What do the experts think today? This is really exciting because the Poincaré conjecture is one of the great open questions in mathematics, second only to the Riemann hypothesis. It all went back to a mistake that Henri Poincaré made in 1900. He assumed without proof that any three-dimensional topological space (with some technical conditions) in which every loop can be continuously shrunk to a single point must be equivalent to a three-sphere, the three-dimensional analogue of the two-dimensional surface of an ordinary sphere. Then he noticed the absence of any proof, tried to find one, and failed. He turned the failure into a question: is every such space a three-sphere? But everyone was so sure that the answer had to be yes that his question quietly turned into a conjecture. Its generalization to higher dimensions was then proved, for every dimension except 3, which was disappointing, to say the least. The Poincaré conjecture became so notorious that it is now one of the seven millennium problems selected by the Clay Institute as the most important open questions in mathematics. Each problem carries a million-dollar reward for its solution.
In 2002 and 2003 Perelman, a rather diffident young Russian with a physics background, published two papers on the arXiv (“archive”), a website for mathematical preprints, with the offhand remark that they not only proved the Poincaré conjecture, they also proved the even more powerful Thurston geometrization conjecture, which holds the key to all three-dimensional topological spaces!
Usually this kind of claim turns out to be nonsense, but Perelman’s idea is clever and comes with a good pedigree. His trick is to use the so-called Ricci flow to deform the candidate space in a manner closely analogous to how space-time deforms under Einstein’s equations of general relativity. And that’s the snag. To understand the proof properly, you need to know three-dimensional topology, relativity, cosmology, and a dozen other hitherto unconnected areas of pure math and mathematical physics. And it’s a long and difficult proof, with plenty of traps for the unwary. Moreover, Perelman followed the time-honored Russian tradition of not giving all the details. So the experts, who have been working through his ideas in seminars all over the world, are understandably wary of declaring the proof correct. But every time someone finds what might be a gap or a mistake, Perelman quietly explains that he’s already thought of that and why it isn’t a problem. And he’s right.
It’s gotten to the point where, even if the proof turns out to be wrong, the correct things achieved along the way are of major significance to mathematics. And as I write, the experts seem to be nudging ever closer to the view that the proof really does work. Keep your ears open over the coffee, Meg.
As your career develops, the worldwide mathematical community will be increasingly important to you. You will become part of it, and then you will have a home in every city on Earth.
Just arrived in Tokyo? Drop by the nearest university, find the math department, walk in. There will be at least one person you know, or who knows you by your work even if you’ve not met before. They will drop everything, call their baby-sitter, and take you out on the town for the evening. They may have a spare room, if you forgot to book a hotel. They will set up a seminar so that you can present your latest ideas to a sympathetic audience. They may even be able to drum up a small financial contribution to your airfare.
You don’t get to fly business class, though. Or to sleep in a hotel suite. (Not yours, at least.) Math operates on the cheap and cheerful principle. I sometimes wish we didn’t undervalue ourselves in this manner, but it’s ingrained habit and it is far too late to change it.
It is of course more civilized and more organized to e-mail the University of Tokyo math department in advance. The result will be similar.
If you get on well with your host, they will invite you back. As you and they climb the career ladder, both of you will start being invited to conferences. Then you will find yourself organizing conferences, which means that you can invite everyone you want to talk to. There is some kind of “phase transition,” so that over a period of about a year, you will go from being invited to no conferences to being invited to far too many. Be selective; learn to say no. Learn sometimes to say yes.
There are big conferences and medium-sized conferences and small conferences. There are special conferences and general ones. The big, general ones are great for meeting people and trawling for jobs. Every four years, the International Congress of Mathematicians is held somewhere in the world. I last went when it was in Kyoto, and there were four thousand participants. I saw a lot of Kyoto, met lots of old friends and made some new ones, and learned a little bit about what people outside my area were doing. The family came too, and they had a whale of a time exploring the city and its surroundings.
I much prefer smallish, specialized meetings with a specific research theme. You can learn a lot from those, because almost every talk is on something that interests you and is related to what you are working on at the moment. And once you’ve been in the business for a few years, you will know almost everyone else who is attending. Except for the youngest participants, who have only just joined the community.
Welcome, Meg.
19
Pigs and Pick up Trucks
Dear Meg,
Assistant Professor, indeed. I’m proud of you; we all are. At an excellent institution, too. You’re a professional mathematician now, with professional obligations. And it occurs to me that I’ve been so busy offering advice about what to do in various circumstances that I’ve left out the other side of the equation: what not to do. Now that you have a tenure-track position, you will be taking on more responsibility, so you will have more to lose if you foul up. There are plenty of ways for mathematicians to make complete idiots of themselves in public, and nearly all of us have managed it at some stage in our careers. People make mistakes; wise people learn from them. And the least painful way is to learn from mistakes made by others.
The longer you stay in the business of mathematics, the more blunders you will inevitably make; this is how experienced people gain their experience. I have witnessed, and committed, plenty of mistakes myself. They can range from writing the wrong equation on the board to mortally insulting the president of your university at some significant public event. Be warned. You will no doubt invent some new mistakes of your own; occasional embarrassment is the natural human condition.
Most of my advice will be obvious. An assistant professor who wishes to be tenured at her university must find out what the requirements and expectations are, and then meet them. If you are expected to have published two papers beyond the subject of your dissertation and instead publish one paper, coach the math club, direct the study-abroad program in Budapest, obtain a major research grant, and win the teacher-of-the-decade prize,
you may be denied tenure. Take care to be polite to your superiors, unless you have excellent reasons not to and want to change jobs. Be polite to everyone else, when they deserve it and even sometimes when they don’t. If you disagree with some decision or argument, make your point concisely, clearly, and without implying that the opposing view is insane, even when it is. Honor your commitments, whether they are tutorial sessions, office hours, examination grading, or plenary lectures at the International Congress of Mathematicians. If you agree to sit on a committee, turn up for its meetings. Listen to the discussion. Contribute, though not at length. Generally, remember that you are a professional, and behave like one.
Some mistakes, on the other hand, are obvious only after you’ve made them. There is a persistent story at Warwick University that I ended my first ever undergraduate lecture by walking into the broom cupboard. It is time to set the record straight. Yes, I admit that it was a broom cupboard, but it was also the emergency exit from the lecture hall. I had assumed, without finding out ahead of time, that when the students left the hall by the main doors, I would be able to leave by what looked like a side door. But when I tried it, I found myself surrounded by buckets and mops. Worse, I discovered that the only way to leave the building by that route was to push open an emergency exit, which would set off an alarm. I had noticed the EXIT sign over the door but had failed to spot the word “emergency” above it. So I was forced, rather sheepishly, to emerge from the so-called broom cupboard and join the students as they walked up the stairs to the back of the hall and out the main doors.