How to Write About Mathematics

Occasionally I rant about the general awfulness of mathematics textbooks. If I were to express my major objection in the most charitable possible way, it is that most textbooks are written like reference books. They are usually very good at recording the basic facts of a subject and proving them with admirable rigor. If you just need to look up some elementary theorem or formal definition, then by all means consult a textbook. The trouble, though, is that textbooks are seldom written from the perspective of a student encountering the material for the first time.

If I were to express things more melodramatically, I would say that the problem is that textbooks seldom tell a story. They have no plot, no characters. And they should! Good mathematical writing is marked by a progression from the initial statement of a problem, through the rising action of our initial, fumbling attempts to solve it, reaching its climax with the solution itself, and then proceeding to its denouement in the form of a proper, rigorous proof. The best part is that the story never really ends, since the solution to one problem leads inevitably to new problems.

Over time, however, I have come to the sad realization that many of my colleagues do not share my view. A piece of mathematical writing with two consecutive sentences of exposition will be derided by many as too wordy. Books that are nothing but a sequence of dense, unmotivated definitions followed by equally unmotivated theorems are praised for being concise. It has happened many times that I have heard people gush about textbooks I would be embarrassed to use to level off a table.

Occasionally, though, I grow uncertain. Maybe I’m the crazy one, and those deathly-dull, jargon-filled, notation-fests are actually on the right track. That is, after all, how most mathematicians write. Who am I to argue?

Every once in a while, though, I come across a book that really does it right! I have just finished reading one such book. It is called Measurement, by Paul Lockhart. It is quite simply the best math book I have read in quite some time. Anyone thinking about writing a math textbook should be required to read it.

It is possible that I am a little biased. Lockhart was a post-doc at Brown when I was an undergraduate there in the early nineties. I had two courses with him, one in linear algebra, the other in complex analysis. He had a big influence on me, especially during my periodical crises of confidence about my future in mathematics. I often tell my students that math is easy, it’s math classes that are hard. Sometimes I would feel beaten down by the drudgery of my courses and start wondering if maybe I should pursue something else. Professor Lockhart played a big role in getting me over that. I mostly remember him as the funniest, but also the most lucid, math teacher I have ever had.

I am happy to report that he writes the way he teaches. The book has two main sections. The first deals with certain topics from Euclidean geometry, while the second discusses calculus. I found myself learning many new things, and looking at some old things in a new way. More than that, though, what impressed me was the naturalness of the discussion. Take the section on calculus, for example. At no point does he say anything like, “We shall now define the notion of the limit of a function at a point…” Not at all. Instead he describes a certain problem: How should we think about the motion of a particle through space? Then, with some clear thinking and a sequence of natural questions, he develops all of the standard ideas presented in the calculus sequence. In around 200 pages, he starts from scratch and gets to some fairly sophisticated ideas in multi-variable calculus. It all seems so natural and inevitable and interesting, and not at all like the tedious, symbol-laden, thousand-pagers we inflict on freshman math majors.

Here’s an example of something I had not really thought about before. Lockhart is contrasting the approach toward calculating areas taken by the ancient Greeks with the more modern approach we employ in calculus. The Greeks tended to use the “method of exhaustion,” in which irregular areas are broken up into a large number of more regularly shaped regions. The modern approach is to perturb the area a bit, and then to solve the differential equation that results from measuring how it changes. Lockhart writes:

I love the contrast between the ancient and modern approaches to geometric measurement. The classical Greek idea is to hold your measurement down and chop it into pieces; the seventeenth-century method is to let it run free and watch how it changes. There is something slightly perverse (or at least ironic) about how much easier it is to deal with an infinite family of varying measurements than with a single static one.

I know quite a bit about Greek geometry, and also quite a bit about calculus, but I had never thought about it that way. Very cool.

Anyway, if you have any interest in mathematics at all you should procure a copy of this book immediately. It is brilliant. Be warned, though, that I don’t mean to say the book always makes for light reading. Math, even when explained with perfect lucidity, is still challenging.

Here’s a short video of Lockhart discussing his book. I hope he writes a dozen sequels. Enjoy!

Comments

  1. #1 Neil Rickert
    March 26, 2013

    That video was good.

    The trouble with math textbooks, is that there is no market for them. The professor chooses the textbook, and is often motivated by a desire to impress his colleagues with the rigor used. The student, who has to actually buy the book, has no choice in which book is used. The normal market feedbacks are not there.

  2. #2 lordaxil
    March 26, 2013

    It’s worth emphasizing (as pointed out at end of video) that this is the same Lockhart who wrote “Lockhart’s Lament”, an extremely insightful and poignant critique of math education.

    http://www.maa.org/devlin/LockhartsLament.pdf

  3. #3 MNb
    March 26, 2013

    “I would say that the problem is that textbooks seldom tell a story.”
    As an experienced teacher math and physics I disagree. The problem with math textbooks generally is that they are not written on the didactical level of my pupils (12-16 years old). It would be nice if authors knew their Piaget and Vygotsky.

    http://en.wikipedia.org/wiki/Jean_Piaget
    http://en.wikipedia.org/wiki/Vygotsky

    At the age of 12-16 pupils are capable of some abstract thinking, but still need references to concrete things.
    Another feature typical of Dutch mathematical textbooks is that they adopt exactly one didactic principle and consequently apply it in every book and every chapter. That get’s boring. Kids want variation. Sometimes they want to do repetitive exercises to master a skill, like the distributive property. The next time they want to solve a brain teaser.

    “will be derided by many as too wordy”
    Here I agree with you. Math is a language and thus it should be possible to translate it into Dutch and English. I do it now and then all kids immediately see the advantage of math.
    The dense books as you describe them are not based on any single didactical princinple at all, so they suck indeed.

  4. #4 JimR
    March 26, 2013

    Video cleverly poses a problem or two, but not the answers. I would up ordering the book.

    I saw a bit about reversing teaching in lower grades, where the students watch video lectures at home and work on problems in supporting groups in class with the teacher there to help. Great praise for the effectiveness of this approach. I wonder if it really is that good. With The Kahn Academy and other online lectures, the quality of online lectures has gotten quite good.

  5. #5 Richard Simons
    March 26, 2013

    Few textbook tell you why a procedure is done. For a few years I was teaching math to adults and also substitute taught in high school. Whenever I had a pre calculus group I would briefly describe what calculus did and some of its uses. They all seemed to appreciate knowing where they were heading, but the only people who had any previous idea were those who had already taken some calculus. The adults in particular also appreciated hearing anecdotes about the people who came up with the concepts.

  6. #6 Greg Esres
    March 26, 2013

    I would say that the problem is that textbooks seldom tell a story.

    Very much agree. It would be very neat if we were told how the various theorems or techniques were stumbled upon, because that would 1) naturally present the material in the order that makes the most intuitive sense to the student and 2) be less intimidating to the student, and 3) might make the student actually care.

    Regarding comment #3, I think most math teachers are pretty unimaginative and are probably the last people I would consult with to figure out how to teach math better, unless they have demonstrated an exceptional ability to reach the math-phobic.

  7. #7 Poincare
    March 26, 2013

    Maybe we shouldn’t be too unhappy with these “bad” text books; they allow us to tell the the story and let the students use the books for reference and as a source of problems.

    Then there is Ramanujan who learned a lot from reading a book that was just a list of 5000 equations.

  8. #8 konrad
    March 27, 2013

    I don’t understand why he gave it a title that, if I saw it on the shelf in the math section of a bookstore, would actively prevent me from taking it off the shelf for browsing. In the video he explains that math is the antithesis of measurement. Luckily now I will know to look out for the book despite its title.

  9. #9 Ned Rosen
    March 27, 2013

    I think Poincare @7 has the main point about present day math texts: they are not (written as) books to be “read”, but as companion references in courses with class time for unfolding the story and homework time to master the mechanics.

    I admit to choosing texts for calculus and other standard courses (where the topics haven’t changed in 100 years) based on the collection of homework problems– that is the main interaction my students will have with the book.

    With coming changes in pedagogy such as MOOC, better stand-alone text books may appear, as they will need to be more engaging when there isn’t a live teacher to “tell the story.”

  10. #10 Jason Rosenhouse
    March 27, 2013

    konrad –

    I agree with you about the title. I snapped up the book as soon as I became aware of it, because I knew the author, but the subject matter was not at all what I expected.

  11. #11 Bruce W. Fowler
    Arab, AL
    March 31, 2013

    The selection criteria used by an instructor are often different from those of the student. In many cases the two sets are orthogonal. I can only recall two maths courses in my college days that had good, effective textbooks: one was taught by a mechanical engineer (MacRobert’s book on Complex Variables) and the other a service course on Green Functions (Stakgold.) The good books, the ones that I could mesh with, had to be found. One of the hapiest days of my life was when I discovered Ruel Churchill.

  12. #12 Hisa
    http://www.newsday.co.zw/2013/01/22/constitutional-breakthrough-no-guarantee-for-free-polls/
    April 1, 2013

    It

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