Letters to a Young Mathematician (12 page)

Dear Meg,

Excellent news! Congratulations on the postdoc position. I’m delighted, though not surprised: you deserve it. The research project on visual processing in fruit flies sounds very interesting, and it overlaps your interests pretty well, even if you haven’t been exposed to the biological aspects before.

You should count it as a bonus that the position includes some teaching duties. You’ll find that teaching math to others improves your own understanding. But it’s only natural to be a little nervous, and I’m not surprised that you think you are “not at all prepared” for your teaching responsibilities. Many people in your position feel that way. But the nerves will vanish as soon as you get started. You’ve been in classrooms all your life, you’ve observed several dozen teachers at length, and you have strong opinions about what makes a course good or bad. All this is preparation. It’s important that
you not let a lack of confidence delude you into taking this part of your job too lightly.

A good teacher, like my Mr. Radford, is worth his or her weight in gold. Good teachers inspire their students, well, some of them. Correspondingly, bad teachers can put students off a subject for life. Unfortunately, it is much easier to be a bad teacher than a good one, and you don’t have to be
really
bad to have the same negative effect as someone who genuinely is totally awful. It is far easier to destroy someone’s confidence than to help them regain it.

Teaching matters. It’s not just a boring necessity that pays for the excitement of research. It is your opportunity to pass on your understanding of math to the next generation. Many good mathematicians enjoy their teaching, and work just as hard at it as they do at their research projects. They feel a great pride of ownership in their courses.

It is not unusual for a research idea to occur to you while you are preparing a course, or teaching it, or setting a test on it. I think this happens because your mind moves out of its usual “research” grooves when you are teaching, and you start asking new questions.

Here I must add a confession. It is now several years since I taught any undergraduates, because my position was changed in 1997 to allow more time for “public understanding of science” activities. Instead of the usual fifty percent teaching, fifty percent research, it became fifty percent research and fifty percent public lectures,
radio, TV, magazines, newspapers, and popular science books. This has had one beneficial effect: I’ve been able to employ teaching techniques you seldom encounter in an undergraduate lecture. The most memorable of these was the day I brought a live tiger into the lecture theatre.

I was delivering the 1997 Christmas Lectures, on BBC television: five hours of popular science, filmed “as live” before an audience of about five hundred mostly young people. This lecture series was initiated by Michael Faraday in 1826, and I was the second mathematician to take part.

One of the five lectures was on symmetry and pattern formation, and I wanted to start with William Blake’s poem—a cliché, but a good one nonetheless—whose opening stanza begins “Tyger! Tyger! burning bright,” and ends with “Dare frame thy fearful symmetry.” So, television being what it is, we decided to bring on a real tiger. How we found one is a story in its own right, but suffice it to say, we did. Nikka was a six-month-old tigress, and she entered the lecture theatre on the end of a chain held by two burly young men.

I have never heard an audience go so quiet, so quickly.

Nikka’s role went beyond poetic metaphor. The symmetry I had in mind was her stripes, especially the regular rings along her elegant tail. She was a true star:
she behaved beautifully, and we got exactly the footage we wanted.

I’ve never really been able to match that as a way to start a lecture.

What I lost by way of contact with undergraduates, then, I gained in contact with wildlife. The Mathematics Department didn’t suffer, because it was allocated a replacement teaching position, but I missed my regular interactions with undergraduate students. I gained in other ways, of course; mostly, I could allocate my time as I wished, which was wonderful. I still supervise PhD students, so some of my teaching has continued, but you should bear in mind that some of what I say may be outdated. In my defense, I did teach regular undergraduate courses for twenty-eight years before that, including two years in the United States, so I have some knowledge of the American system and how it differs from the British. The similarities are more important than the differences, but I’ll try to translate my experience into your context.

To my mind, the most important feature of good teachers is that they put themselves in the student’s position. It’s not just a matter of giving clear and accurate lectures and grading tests; the main objective is to help the student understand the material. Whether you are delivering a lecture or talking with students during office hours, you have to remember that what seems perfectly obvious and transparent to you may be mysterious and opaque to someone who has not encountered the ideas before.

I always tried to remind myself of that. When grading tests, it is so easy to start thinking, “I’ve taught them this stuff for twenty years now, and they
still
don’t understand it.” But each year brings new students, who encounter much the same difficulties as their predecessors, make the same mistakes, misunderstand the same things. It’s not
their
fault that you’ve seen it all before.

It’s actually to your advantage, Meg, that you haven’t seen it all before. Consider yourself fortunate, and exploit that. The students will feel comfortable with you because you’re close to their age, you’ve just been through the same mill that is now grinding them, and you haven’t grown bored teaching the same course several times. I can still remember my first few lecture courses vividly; teaching was easier for me then than it became ten years later. After a while, you know too much, and there’s the danger that you try to pass
all
of that knowledge and insight on to the students. Big mistake. They don’t have the same perspective that you do.
So the KISS principle applies: Keep It Simple, Stupid.
Stick to the main points, and try not to digress if doing so requires the students to understand new ideas that are not in the syllabus, however fascinating and illuminating they may seem to you.

The American system is more straightforward than the British one in this regard. Typically, there is a set text and an agreed syllabus—right down to page numbers and specific paragraphs to be included or not—so the
content is established and everyone knows it, or should.
But there’s still room for input from the teacher, and there’s a delicate balance between helping the students by putting your own stamp on the material, and confusing them by introducing too many extraneous ideas.

So before telling them something that may be outside the text, you need to ask yourself, if I were a student, who knew the textbook up to this particular page but nothing beyond, what would help me understand the material better? And the key step in coming up with a good answer to that question is to make sure you understand the material yourself.

Let me give you an example. The details are less important than the approach, which applies to many similar situations.

At some point early in your teaching career, one of your precalculus students is going to ask you why “minus times minus gives plus.” For instance, (- 3) × (- 5) = +15, not -15. And even though this topic should have been beaten to death in high school, you are going to have to justify the standard mathematical convention.

The first thing is to admit that it
is
a convention. It may be the only choice that makes sense, but mathematicians could, if they wished, have insisted that (- 3) × (- 5) = - 15. The concept of multiplication would then have been different, and the usual laws of algebra would have been torn to shreds and thrown out the window, but hey! Old words often take on new meanings in new
contexts, and there is nothing sacred about the laws of algebra.

There are two reasons why the standard convention is a good one: an external reason, to do with how mathematics models reality, and an internal one, to do with elegance.

The external reason convinces a lot of students.
Think of numbers as representing money in the bank, with positive numbers being money you possess and negative numbers being debts to the bank. Thus 5 is a debt of $5, so 3 × (- 5) is three debts of $5, which clearly amount to a total debt of $15. So 3 × (- 5) = -15, and no one seems bothered much about that. But what of (- 3) × (- 5)? This is what you get when the bank
forgives
3 debts of $5. If it does that, you
gain
$15. So (- 3) × (- 5) = +15.

The only other choice any students ever advocate is- 15, but that would leave you in debt.

The “internal” explanation is to work out a sum like (- 3) × (5 - 5). On the one hand, this is clearly zero. On the other, we can use the laws of algebra to expand it, getting (- 3) × 5 + (- 3) × (- 5). Since we’ve already agreed that minus times plus is minus, we deduce that consistency of the laws of algebra requires - 15 + (- 3) × (- 5) = 0, which implies that (- 3) × (- 5) = 15 (add 15 to each side).

The most we can assert in the first case is that
if
we want the math to model bank accounts, then minus times minus has to work out as plus. The most we can assert in the second case is that
if
we want the usual laws of algebra to hold for negative numbers, then the same goes. There is nothing requiring either of these things to be true. But it will certainly be more convenient if they are, and that is why mathematicians chose that particular convention.

I’m sure you can think of other, similar arguments. The important thing is not to say to the student, “That’s how it is. Don’t question it, just learn it.” But to my mind it would be even worse to leave them with the impression that there was never any choice to be made, that it is somehow
ordained
that minus times minus makes plus. All of those concepts—plus, minus, times— are human inventions.

At this point one of your more thoughtful students may remark that it
is
ordained, in the sense that there exists only one system of mathematics that proceeds from the counting numbers in a completely consistent way. You can respond that actually there are several extensions of the number concept—negative numbers, fractions, “real” numbers (infinite decimals), “complex” numbers in which - 1 has a square root . . . even quater-nions (in which -1 has many square roots but some laws of algebra fail). Each of these extensions is provably unique, subject to possessing certain features, but it is up to human beings to choose which features they consider significant. It would, for example, be possible to invent a new number system in which all negative numbers are equal, and it would be entirely consistent
from a logical point of view. But it would not obey the usual rules of algebra.

You can concede, if pushed, that some extensions seem to be more natural than others.

Discussions of this kind do not always work; ingrained misconceptions can be hard to eradicate. Even if they do work, you need to show your students where their intuition is going wrong.

Typically, when a student gets stuck at some point in the syllabus, the real problem lies elsewhere, some pages or courses or years back. Perhaps they don’t understand the relation between multiplication and repeated addition. Perhaps they understand it only too well, and can’t see how you can add minus three lots of - 5 together. It’s amazing how often teaching reveals hidden assumptions or unquestioned features of your own mathematical background. Whenever an existing mathematical concept is extended into a new domain, you have to abandon some of its previous interpretations and accept new ones.

You can survive new material in mathematics for quite a while without taking it fully onboard. My colleague David Tall has a theory about this, which I rather like. His idea is that mathematics advances by (conceptually) turning processes into things. For instance, “number” starts out as the process of counting. The number 5 is where you get to when you count the fingers (by which we’ll include the thumb) of one hand: “One, two,
three, four, five.” But in order to make progress, at some stage you have to stop going through the process of counting, and think of 5 as a
thing
in its own right. This is already useful when you start doing sums like 5 + 3.
But students can cover up their inability to turn counting into a thing by lining up the fingers of one hand with three fingers of the other, and then counting the lot: “One, two, three, four, five,
six, seven, eight
.”

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.