“The first thing to understand,” says Paul Lockhart in A Mathematician’s Lament, “is that mathematics is an art.” He continues on to say that “Mathematics is the purest of the arts, as well as the most misunderstood.”
The twenty-first-century mind has a very difficult time getting around this. Math as art? To us, math is dry, logical and left-brained. It follows discrete rules and it is nearly impossible to imagine creativity playing any kind of role in the rote manipulation of numbers. To help clarify he quotes G. H. Hardy:
A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.
This definitely helps. Patterns conjure up images of shapes and colors, maybe even geometrical paintings by Kandinsky, Delaunay (shown above) or Mondrian. He adds his own example: a drawing of a triangle inside of a rectangle.
This is a creative problem, he says. Give this to students (high school students) and see what they do with it. Perhaps after much doodling someone will notice that if you bisect the rectangle where the vertex of the triangle touches it (this is easy to see but difficult to describe), the corresponding rectangles will be exactly halved by the remaining lines of the original triangle. Again, a visual makes it clear:
Give them a puzzle, let them play around with it, possibly even solve it and then tell them that this is where A=1/2bh (the area of a triangle equals one-half the base multiplied by the height) comes from. For Lockhart, this makes the case that math, like all human endeavors, has a history. Mathematical patterns, formulas and concepts do not arise fully-formed and bite-sized out of thin air as they do from a school textbook (“full-color textbook abominations” Lockhart calls them). They arose from years, sometimes centuries of inquiry and were fed by the sweat, tears and enthusiasm of many men and women. Math is a creative process, and to disassociate it from its past sucks all of the life and dynamism from it, as well as the enthusiasm from its students. Math needs context.
By removing the creative process and leaving only the results of that process, you virtually guarantee that no one will have any real engagement with the subject. It is like saying that Michelangelo created a beautiful sculpture, without letting me see it. How am I supposed to be inspired by that? (And course it’s actually much worse than this-at least it’s understood that there is an art of sculpture that I am being prevented from appreciating.
By concentrating on what and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument.
This is what math is all about; it is about puzzles and problems and discoveries (in some kind of narrative context), not rote memorization of facts and algorithms.
Alas, I’m only describing a fraction (no pun intended) of what Lockhart writes in his essay. He continues to lament, critique and inspire for another 100 pages. But hopefully this inspires you to order the book and read it. It is well worth the time and money.
Before I finish, I will leave you with one last quote from A Mathematician’s Lament. In each chapter Lockhart weaves in a Socratic dialog with Salviati and Simplicio. Simplicio (the reader) asks all of the questions that I wanted to hear Paul’s answer to and Salviati answers them. Here is a snippet of one of their conversations:
Simplicio: Then what should we do with young children in math class?
Salviati: Play games! Teach them chess and Go, Hex and backgammon, Sprouts and Nim, whatever. Make up a game. Do puzzles. Expose them to situations where deductive reasoning is necessary. Don’t worry about notation and technique; help them to become active and creative mathematical thinkers.
And this is from a current public school math teacher from New York!!
Coming up I will explain where I disagree (respectfully) with Lockhart’s view of art, and review his newest book Measurement which explains in more detail his conception of mathematics education.
I will also, from time to time post number games, puzzles, logic games and other math-related ‘whatever’ that I believe fulfills Lockhart’s vision of what a mathematics education for young children should entail.
But for now,