Sunday, December 23, 2007

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:


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.



  1. I must be missing something about the doublets game or proof (which I haven't read), because I think I came up with a quick counter-example:


    It just seems odd to me that any proof could exist, because it would rely on whether or not certain combinations of 4 letters were "real" words or not, a fact which might change over time.

    So, what's the gist of the proof?

  2. Aha, having taking a peek via "search inside" on Amazon, it appears that his theorem does not apply to all doublets, but only to the SHIP -> DOCK doublet, a key feature of this doublet being the change in position of the vowel. I couldn't read the whole thing on Amazon, but I suspect his proof applies to all cases where the vowel moves, but obviously does not apply to cases like my counterexample, where the vowel changes but remains in the same position.

    Stewart also seems to claim that you can include W as a vowel or not and the theorem will still work if you are consistent, which also rules out the WARM -> COLD example.

  3. Hi mathmom: I'm glad I caught your comment before I go out. I have acquired all of the chapter for you. It can be seen at my web gallery at:

    Page 72a: "you are allowed to change (but not move) exactly one letter, and the result must be a valid word."

    Page 73: "Alternatively, we can require Y and W always to count as vowels, even when they are being used as consonants."

    also: "A "vowel" will mean one of the letters A, E, I, O, U, and a "word" will be required to contain at least one of those five letters."

    So I *think* my WARM-COLD example is valid. I hope so. And I also think that your CORN-BALD example is a counter-example still.

    I enjoy your comments so much. Thank you.


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