Review: Ian Stewart: Letters To A Young Mathematician

Letters to a Young Mathematician (Art of Mentoring) by Ian Stewart is a short, quick read with many interesting ideas. Ian Stewart wrote a series of letters to his niece, Meg, as she grew from a high school student to a tenured professor of mathematics. The book is his "attempt to bring some parts of A Mathematician's Apology up to date, namely, those parts that might influence the decisions of a young person contemplating a degree in mathematics and a possible career in the subject."

The book kept my interest because Stewart scatters his ideas about teachers throughout it. Otherwise I may have stopped reading. There simply is not enough mathematics of substance in it for me. Except in chapter five, Surrounded by Math, where he discusses "bird crystals" and about which I posted earlier. And except in what I call the Doublets chapter (chapter eight: Fear of Proofs).

Lewis Carroll invented the game of doublets in which you take a word (such as WARM) and change it, one letter at a time, to another word (such as COLD). Each time you change a letter you must have a real word. Stewart proves that at some stage you must have a word that contains exactly two vowels. There was a time when I played doublets, so this proof interested me. You have to consider W and Y to always be vowels in order for this proof to be valid. What interests me, and this is something Stewart never addressed, is that in all of his examples, the vowels are double vowels (OO, AA, etc). I spent some time finding doublets games on the Internet and while all the words in each game have two vowels, they are not necessarily double vowels. For example, WARM — WORM — WORD — CORD — COLD: if you accept that W is a vowel (see page seventy-three for Stewart's explanation), then WARM and WORM do have two vowels.

I found other examples where the vowels are not positioned next to each other:

GIVE to TAKE: GIVE — GAVE — RAVE — RAKE — TAKE

Strictly speaking, these are examples of Stewart's proof. But he never offered one of these as an example. Perhaps he should have.

Stewart writes about teachers: "The best teachers will occasionally, perhaps more than occasionally, make you feel a bit stupid." I am not sure, despite reading his rationale, that this is true. Other statements that he makes concerning teaching ring very true with me: "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. . . But the nerves will vanish as soon as you get started." I experience that "stage-fright" at the beginning of every college semester and public school academic year.

One reason that I enjoy teaching developmental mathematics at the community college is that I may be able to help someone discover that she can succeed at something she used to think beyond her ability. Stewart points out that we need to put ourselves in the student's position and help her understand the material. "[W]hat seems perfectly obvious and transparent to you may be mysterious and opaque to someone who has not encountered the ideas before." I have learned, through experience, that mathematics instruction must be kept simple: " 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." That has been a difficult lesson for me to internalize over the years. There seem to be thousands of interesting side roads that we can take in every lesson. Keeping our course objectives in front of us at all times can prevent us from straying off course and bewildering our students.

Validation by Stewart of what I have learned about teaching was not reason enough for me to read this book. It is math-lite. I suggest that if you are interested in it, that you wait for your library to purchase it.

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Photo Hunters: Long: ". . . and birds perch on the phone lines."

Photo: Sparrows on a Fence. Taken in New Haven, CT in October 2007.

The photo will open, horizontally, in a new window when clicked.

Text: Letters to a Young Mathematician. Ian Stewart. Basic Books: 2006. New York. pp 46-48.

"From a distance they look like sheet music, fat little blobs on rows of horizontal lines. There seem to be special places they like to perch, and it’s not at all clear to me why, but one thing stands out. If a lot of birds are perching on a wire, they end up evenly spaced.

"That’s a mathematical pattern, and I think there’s a mathematical explanation. I don’t think the birds 'know' they ought to space themselves out evenly. But each bird has its own 'personal space,' and if another bird gets too close, it will sidle along the wire to leave a bit more room, unless there’s another bird crowding it from the other side.

"When there are just a few birds, they end up randomly spaced. But when there are a lot, they get pushed close together. As each one sidles along to make itself feel more comfortable, the 'population pressure' evens them out. Birds at the edge of denser regions get pushed into less densely populated regions. And since the birds are all of the same species (usually they’re pigeons), they all have much the same idea of what their personal space should be. So they space themselves evenly.

"Not exactly evenly, of course. That would be a Platonic ideal. As such, it helps us to comprehend a more messy reality.

"You could do the math on this problem if you wanted to. Write down some simple rules for how birds move when the neighbors get too close, plonk them down at random, run the rules, and watch the spacing evolve. But there’s an analogy with a common physical system, where that math has already been done, and the analogy tells you what to expect.

"It’s a bird crystal.

"The same process that makes birds space themselves regularly makes the atoms in a solid object line up to form a repetitive lattice. The atoms also have a 'personal space': they repel each other if they’re too close together. In a solid, the atoms are forced to pack fairly tightly, but as they adjust their personal spaces, they arrange themselves in an elegant crystal lattice.

"The bird lattice is one-dimensional, since they’re sitting on a wire. A one-dimensional lattice consists of equally spaced points. When there are just a few birds, arranged at random and not subject to population pressure, it’s not a crystal, it’s a gas.

"This isn’t just a vague analogy. The same mathematical process that creates a regular crystal of salt or calcite also creates my 'bird crystal.'"

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