Tuesday, June 9, 2009

Changing the Way we Teach Math - an Interview with Paul Lockhart of "A Mathematician's Lament"

I came across an essay called “A Mathematician’s Lament” and was so moved by it that I had to share it. The essay starts with a musician having a bad dream that his craft is now compulsorily taught in schools in the way that we teach mathematics currently—devoid of beauty and fun. It goes on to show how math could be taught as art, full of meaning and pleasure and discovery. The essay became a book and I recently interviewed the author, Paul Lockhart on the subject of math and education.

How did you come to be a math teacher? Was it the result of a positive math education or in spite of bad math teaching?

I didn’t do particularly well in math. I think I got a D in eighth grade. Math instruction then was what it continues to be: a training course in the manipulation of symbols without meaning, procedures to learn and tasks to perform. It’s a lot like being at the DMV. It is a thing you have to do. You might see the validity and necessity of it, but you derive no pleasure from it and it’s certainly not something you would choose to do.

I stumbled on math accidentally, through puzzles with my grandfather and in finding my dad’s college textbooks and putting them together. It was then that I saw the wonderland of structure and beauty. It is so sad that so few people know about that. So for me, a big part of being a mathematician is being a teacher.

The world of research mathematicians is one of infinite mystery, with brave, intrepid mental explorers. Then there is the world of school and they have nothing to do with each other. Few mathematicians are interested in the lowly world of elementary school mathematics. Some have tried to help, but found it an infuriating instrument of torture, and left. In a small space of time I try to get across what it is to be a mathematician in a real, natural, and meaningful way.

I started teaching at 18 in an after-school program. I fell in love with telling kids about cool ideas. I went to graduate school and became a math professor, which was fun for a while, but I grew to feel that the world of the university was rather corrupt. I am an idealist and I was disappointed to discover that the university was not an ivory tower for the pursuit of knowledge for its own sake, but a machine for profit. I didn’t want to be part of that world. Also, by the time I got my students they were 20 and they were ruined already. They were tragically incapable of starting over or engaging in the real mathematics’ charm, mystery and beauty. So I felt that teaching high school would give me an opportunity to get to them earlier. But they are still mostly dysfunctional as mathematical thinkers and their math wounds run too deep. I have gone progressively down the chain, and am now teaching first and third graders as well.

How is it that schools fail to teach math in a way that shares with others its beauty and art?

School takes something real and chops it up into meaningless pieces with no organic whole to it. Learning doesn’t happen when it is so disconnected. What goes on in schools is not about learning. Instead you study something, take a test, and a few months later you have no memory or concern about that thing you supposedly “learned.” School is authority compliance testing and in the end you get a certificate of endurance. What we call “smart” is the ability to memorize meaningless data and retrieve it like a trained seal. The whole school thing is pretty nightmarish. Even in an enlightened, affluent, private setting it’s still there. I cringe when I hear a fellow teacher complaining about cheaters. How many more signals do you need to see that this is not interesting, that they are not learning?

I walk into a third grade classroom and on the board is the kids’ whole schedule. There is no natural flow through the day; it’s all cut up into pieces. An eight-year old should not have an end-time to her water coloring! It should be a natural flow through the day with time for daydreaming—that is what living an intellectual life is! If she feels she’s come to the end of her painting, and then wants to stop and play, that is a beautiful thing. Our school culture is quite insane, actually. A third grader recently told me at the start of the year that she only wanted to learn enough in my class to get a good score on the SAT so that she could get into Yale.

Think about what we ourselves do as learners and notice that you learn what you care about—you will pursue what you want and you will remember it. The emphasis in schools is on teaching, not learning. Real teaching is an uncovering or re-awakening. It is inspiration and revelation, not evaluation. The meaning of “student” in Latin is “having zeal.” Learning is not important, it is fun. There is no more noble purpose than enjoyment, fun, love and care for others. If kids are having fun then they are learning. I don’t believe in the utility of education as it is now understood. School should be about learning, thinking, creating, dreaming and exploring. Right now it is an engine of worker production to supply human capital for the economy.

We all went through this awful math experience. Then people grow up, have a kid or two, and now they suddenly believe in it and want their kids to have this same miserable math education? I don’t get it. “This was boring and awful in eighth grade, and it has not been useful to me as a successful adult, but I really want my child to do it.” I would understand it more if parents were storming the barricades in protest! People who admit no math ability feel that their opinion about math instruction is valid. That is mystifying. I can’t dance, but I know that, so I don’t give any dancing advice.

There is a myth that math will get you economic security, so parents fear that their child will not be able to handle adding some numbers together. Man has the ability to count. Left to their own devices they will figure it out. It is too early for 6, 7, 8 year olds to understand an abstract decimal code and 13th-century technologies. A 12-year-old would be ready to understand this and, without the fear and loathing for math, they would learn it so quickly and willingly.

After reading your book, I feel I was cheated in my math education. How can math instruction look and feel different?

The simple conclusion I have come to is that it should be treated as an apprenticeship, just like an 18th-century carpenter who has a dozen apprentices learning the trade. As a master carpenter, you teach your apprentices to pay attention to various details. You foster an understanding of what this art is all about. You teach them to measure and saw, but also to regard their work lovingly and to feel a connection to it.

I try to create a mathematical community at each stage of development. Students are engaged in the practice of mathematics, meaning the questions and ideas that are being raised come from them, not me. The students write mathematical papers in prose and diagrams (not worksheets) to present their explanation of the problem they are working on. It is the greatest thing ever for them to discover the solution themselves. I want people to see the breathtaking beauty and be charmed and intrigued.

Aside for practical applications like tipping and balancing checkbook why do we need to learn math?

We don’t need to learn math, any more than we need to learn poetry and music. As for the practical world, arithmetic procedures are mechanizable. All you really need is a good sense of estimation and the ability to use a calculator. I teach third graders to use a calculator and balance a checkbook. We should not be training children to be arithmetic machines. We build calculators so we don’t have to do these boring tasks. Running a lemonade stand and having to make change is real, and that kind of thing is the best way to learn basic arithmetic.

Math is usually thought of as a left-brained, linear, logical activity. Would you comment on that?

Math is like music and art. You need to use everything you’ve got: imagination, intuition, rationality—your whole self. You find connections and shatter preconceived ideas. If you are a composer, there are things happening on every creative level.

Do you use textbooks? If not, what tools do you use.

Textbooks are an abomination. Just as an English class has you read directly from “As I Lay Dying,” written by the actual author/artist, Faulkner, we should also have math books written by working mathematicians, not a committee of “educators” working for a textbook company. I do not give assignments, quizzes, tests or homework. My students are happy about that, but they discover that it is actually far more serious and difficult work: it is 24 hours a day of obsession with triangles. It is immersion in the most fascinating thing ever.

To read the original essay, “A Mathematician’s Lament,” go to: www.maa.org/devlin/LockhartsLament.pdf

The expanded book version of A Mathematician’s Lament is available from www.amazon.com


  1. Now this is good stuff my friend!

  2. If you get a chance, read Ivan Illich's book "Deschooling Society." Truly a fascinating adventure into these same ideas. Thanks for a great interview, it made my day!

  3. This is exactly how I feel about HS mathematics. In HS I hated mathematics with every fiber of my being. It wasn't till I got to college I learned to appreciate then love mathematics for all it was. As a avid musician and volunteer band director on the side, I draw from my creative side and the critical thinking I gained as a musician more than anything i learned in an HS math class.

  4. Wow! As an art and technology teacher, I love this article. I begun to despise math quickly as a teen and got frustrated therefore, gave up on it! I wish more math teachers read and consumed this interview!

  5. I am a 15 year old that goes to high school, taking one of the courses in the "mindless manipulation of symbols" (Lockhart 6), Algebra II. I developed an interest in mathematics pretty much the same way Paul Lockhart did. I feel like I am being failed by this state of mathematics education because I have intuition and imagination, as I look into modern mathematics (sometimes research level) everyday (if possible), and make discoveries that blow my mind. I also feel that our students are capable of so much more than what school is bringing to them; they are talented, but sometimes school's procedures have the capability to obscure this.

  6. I go to a magnet school called the Liberal Arts and Science Academy. While most classes are doomed by the curriculum to be dry, I feel that my Algebra II teacher was really good at helping us discover 'real math'. Every few weeks we would have a 'challenge problem' where the task might be to prove that the square root of two was irrational. These were graded on how good the explanation was. In other challenge problems, we looked at modular arithmetic and RSA.
    Even in the current system, a good teacher really can make the difference.