Friday, June 17, 2011

Mathematician's Lament - Paul Lockhart

Following on consistently in my habit of being the last kid on the block to discover something I recently fell upon “Mathematician’s Lament” by Paul Lockhart.

The history of the book is interesting in itself – it actually started life as a 25 page type written document that was presented to math guru Keith Devlin at a conference. Devlin was intrigued and impressed by the brief piece – so much so that he tracked down the author and sought permission to feature the article in his regular math opinion piece at Mathematical Association of America online site (a great site which will reward frequent visiting or follow this link  to Devlin’s column with a link to the PDF of the original piece). Publication in that forum lead to the publication of the book form – complete with some intriguing examples of what Lockhart considers “real maths”. There are some stylistic features of note – not the least of which is Lockhart’s homage to Galileo’s 1632 tome “Dialogue Concerning the Two Chief World Systems” which features two characters involved in discussion. At seemingly random times around the book Lockhart uses the same conversational format and even two of the character names, Simplcio and Salviati, to advance his arguments. This actually simplifies the discussion - no prose or description, just the two characters “talking” about issues related to current mathematics instruction.

However, the section I would like to share is where Lockhart describes math education via a dreaming musician early in the book.

“A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory. “We are helping our students become more comptetitive in an increasing sound-filled world.” Educators, school systems, and the state are put in charge of this vital project. Studies are commissioned, committees are formed, and decisions are made – all without the advice or participation of a single working musician or composer.

Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music.” It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music, let alone composing an original piece, are considered very advanced topics and are generally put off until college, and more often graduate school.

As for the primary and secondary schools, their mission is to train students to use this language - to jiggle symbols around according to a fixed set of rules....

… In their wisdom, educators soon realize that even very young children can be given this kind of musical instruction. In fact it is considered quite shameful if one’s third-grader hasn’t completely memorized his circle of fifths. …

In the higher grades the pressure is really on. After all, the students must be prepared for the standardized tests and college admissions exams. Students must take courses in scales and modes, meter, harmony, and counterpoint. “It’s a lot for them to learn, but later in college when they finally get to hear this stuff, they’ll really appreciate all the work they did in high school.” Of course, not many students actually concentrate in music, so only a few will ever get to hear the sounds that the black dots represent. “

And so it goes ...

Now, I have to say that I have now read the book twice – and I don’t agree with everything that Lockhart says. I’m still debating whether I think Lockhart got some things wrong – as well as an awful lot right – or if I’m simply not brave enough to agree with him; for to agree with Lockhart means that maths instruction MUST not only change but change dramatically.

For me a key aspect of the musician’s dream is the notion of providing “skills” now that a student will need later - even if they have little or no relevance to the students now. Lets think about this - we are actually saying to our students - we know this is irrelevant to you now - but we are going to teach it to you anyway. How did this become accepted practice? What learning theory does this reflect? Why do the conventions of the educational system outweigh the needs of our students?

“Mathematician’s Lament” is not a flawless work - but it does provoke some important questions; it is well worth reading and thinking about - and I’d suggest not just for math’s teachers but any teacher who considers themselves a “subject teacher” rather than a teacher of students.

Addendum:  The link provided above was not active as of 8/12/2011.   The document can still be accessed however by following this link.   


  1. Good stuff. I will have to buy the book and give it a good read. I really would like to understand Lockhart's views on what real math is, and how one can enjoy it.

  2. Hi Kawai Kat,
    Thanks for the comment - this really is a book worth reading about - and then thinking about - and then talking about.

  3. The pdf sadly has disappeared from the interweb, it seems. I do agree with the example of the musician who dreams of a world where the life has been sucked out of music, and will proceed from that starting point.

    When I taught my children, I did not use formal lesson plans; nor did I spend 45 minutes on a lesson; indeed, my typical "lesson" lasted only 5 minutes, and was more like a conversation or a puzzle than a "lesson."

    I asked my grandson - who was six at the time - to think about adding the numbers from 1 to 100. Instead of going one-by-one, adding 1 plus 2 plus 3, and so forth, what if we take a number from the end, and add 1 plus 100? and 2+99, and 3+98, and so forth? We'd have 50 pairs times 101. He immediately answered "5050" - which is correct.

    I then asked, "what about the even numbers from 2 to 100?"

    He realized that this would be twice the sum of 1..50, which would be twice 25*51 -> and replied "2550" in next to no time. This is not half of the previous answer, by the way.

    Obviously, he already knew a great deal more about math at that age than most first graders; in fact, I haven't even mentioned his facility with negative numbers, exponents, fractions, decimals, and binary arithmetic - by the age of 6.

    His education was not a series of formal lessons, or drill-and-kill; it was a series of brief playful thought exercises. He learned at an early age to play with numbers for fun, not to memorize facts and procedures by rote. If you were to ask how he derived any particular answer, you'd be amazed; he might casually mention that 2 to the 10th power is the same as (2 to the 5th) squared, which is 32*32, or 1024.

  4. Hi Terry,
    Your comments illustrate the point that Paul Lockhart is making – that there is a joy exploring and discovering mathematics that is too often missing in “traditional” math classrooms. Children can do much more with mathematics – if their classroom setting permits.

    I’ve found another link to the Lockhart pdf which you may enjoy reading.
    Cheers and thanks for your comment