Excerpts from the essays of Paul Lockhart
Paul Lockhart teaches mathematics at Saint Ann’s School in Brooklyn, New York.
From letter to Keith Devlin, Mathematical Association of America
On teaching mathematics to young children:
“I want them to understand that there is a playground in their minds and that that is where mathematics happens.”
From Lockhart’s Lament, Mathematical Association of America
Mathematics is an Art
… if the world had to be divided into the “poetic dreamers” and the “rational thinkers” most people would place mathematicians in the latter category.
Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depend heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood.
What math is
That’s what math is— wondering, playing, amusing yourself with your imagination.
What math is not
Math is not about following directions, it’s about making new directions.
“Good at Math”
If everyone were exposed to mathematics in its natural state, with all the challenging fun and surprises that that entails, I think we would see a dramatic change both in the attitude of students toward mathematics, and in our conception of what it means to be “good at math.”
Almost painful beauty
Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion— not because it makes no sense to you, but because you gave it sense and you still don’t understand what your creation is up to; to have a breakthrough idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it. Remove this from mathematics and you can have all the conferences you like; it won’t matter. Operate all you want, doctors: your patient is already dead.
In any case, do you really think kids even want something that is relevant to their daily lives? You think something practical like compound interest is going to get them excited? People enjoy fantasy, and that is just what mathematics can provide— a relief from daily life, an anodyne to the practical workaday world.
TEXTBOOK PUBLISHERS : TEACHERS ::
A) pharmaceutical companies : doctors
B) record companies : disk jockeys
C) corporations : congressmen
D) all of the above
It’s perfectly simple. Students are not aliens. They respond to beauty and pattern, and are naturally curious like anyone else. Just talk to them! And more importantly, listen to them!
you will never be a real teacher if you are unwilling to be a real person.
From “Lockhart’s Lament: The Sequel,” Mathematical Association of America.
One real teacher
Look. A child will have only one real teacher in her life: herself! I see my role as not to train, but to inspire and to expose my students to a wide range of ideas and possibilities; to open up new windows. It is up to each of us to be students – to have zeal and interest, to practice, and to set and reach our own personal artistic and scientific goals. Children already know how to learn: you play around and have fun and struggle and figure it out for yourself. Grownups don’t need to hold infants up and move their legs for them to teach them to walk; kids walk when there is something interesting in the room that they want to get to. So a good teacher is someone who “puts interesting things in the room,” so to speak.
Deal with the Devil
Suppose the devil were to offer you this deal: your child will get a perfect score on the English section of the SAT, but will never again read a book for pleasure. I would like to believe that no parent would make that deal. But how many would gladly shake the devil’s other hand? Math is not something we want our children to enjoy, it is something we want them to get through.
I certainly do care about measuring educational results. But what is an “educational result?” The twinkling eyes of my students, together with their heartfelt and beautifully expressed mathematical arguments are all the results I need.