This little book, had me yelling, “YEA! HELL, YEA!” at every page.
The book’s evolved from this really beautiful lament, which is available online and has a newer section titled Exultation.
The article has the main thrust of the book and is worth your time. (as is the book, specially if you have kids, or you teach kids, or if you want to shape someone’s thinking about Mathematics)
Highlights from the book follow …
On the other side of town, a painter has just awakened from a similar nightmare …
I was surprised to find myself in a regular school classroom— no easels, no tubes of paint. “Oh we don’t actually apply paint until high school,” I was told by the students. “In seventh grade we mostly study colors and applicators.” They showed me a worksheet. On one side were swatches of color with blank spaces next to them. They were told to write in the names. “I like painting,” one of them remarked, “they tell me what to do and I do it. It’s easy!”
After class I spoke with the teacher. “So your students don’t actually do any painting?” I asked. “Well, next year they take Pre-Paint-by-Numbers. That prepares them for the main Paint-by-Numbers sequence in high school. So they’ll get to use what they’ve learned here and apply it to real-life painting situations— dipping the brush into paint, wiping it off, stuff like that. Of course we track our students by ability. The really excellent painters— the ones who know their colors and brushes backwards and forwards— they get to the actual painting a little sooner, and some of them even take the Advanced Placement classes for college credit. But mostly we’re just trying to give these kids a good foundation in what painting is all about, so when they get out there in the real world and paint their kitchen they don’t make a total mess of it.”
“Um, these high school classes you mentioned …”
“You mean Paint-by-Numbers? We’re seeing much higher enrollments lately. I think it’s mostly coming from parents wanting to make sure their kid gets into a good college. Nothing looks better than Advanced Paint-by-Numbers on a high school transcript.”
“Why do colleges care if you can fill in numbered regions with the corresponding color?”
“Oh, well, you know, it shows clear-headed logical thinking. And of course if a student is planning to major in one of the visual sciences, like fashion or interior decorating, then it’s really a good idea to get your painting requirements out of the way in high school.”
“I see. And when do students get to paint freely, on a blank canvas?”
“You sound like one of my professors! They were always going on about expressing yourself and your feelings and things like that—really way-out-there abstract stuff. I’ve got a degree in Painting myself, but I’ve never really worked much with blank canvasses. I just use the Paint-by-Numbers kits supplied by the school board.”
Sadly, our present system of mathematics education is precisely this kind of nightmare. In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done—I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.
SIMPLICIO: I don’t think that’s very fair. Surely teaching methods have improved since then.
SALVIATI: You mean training methods. Teaching is a messy human relationship; it does not require a method. Or rather I should say, if you need a method you’re probably not a very good teacher. If you don’t have enough of a feeling for your subject to be able to talk about it in your own voice, in a natural and spontaneous way, how well could you understand it? And speaking of being stuck in the nineteenth century, isn’t it shocking how the curriculum itself is stuck in the seventeenth? To think of all the amazing discoveries and profound revolutions in mathematical thought that have occurred in the last three centuries! There is no more mention of these than if they had never happened.
SIMPLICIO: But surely we want all of our students to learn a basic set of facts and skills. That’s what a curriculum is for, and that’s why it is so uniform—there are certain timeless, cold, hard facts we need our students to know: one plus one is two, and the angles of a triangle add up to 180 degrees. These are not opinions, or mushy artistic feelings.
SALVIATI: On the contrary. Mathematical structures, useful or not, are invented and developed within a problem context and derive their meaning from that context. Sometimes we want one plus one to equal zero (as in so-called ‘mod 2’ arithmetic) and on the surface of a sphere the angles of a triangle add up to more than 180 degrees. There are no facts per se; everything is relative and relational. It is the story that matters, not just the ending.
What is happening is the systematic undermining of the student’s intuition.
A proof, that is, a mathematical argument, is a work of fiction, a poem. Its goal is to satisfy. A beautiful proof should explain, and it should explain clearly, deeply, and elegantly. A well-written, well-crafted argument should feel like a splash of cool water, and be a beacon of light—it should refresh the spirit and illuminate the mind. And it should be charming.
There is nothing charming about what passes for proof in geometry class. Students are presented a rigid and dogmatic format in which their so-called “proofs” are to be conducted—a format as unnecessary and inappropriate as insisting that children who wish to plant a garden refer to their flowers by genus and species.
Calling into question the obvious, by insisting that it be “rigorously proved” (as if the above even constitutes a legitimate formal proof), is to say to a student, “Your feelings and ideas are suspect. You need to think and speak our way.”
Now there is a place for formal proof in mathematics, no question. But that place is not a student’s first introduction to mathematical argument.
if multiplication is something you are interested in (that is, making repeated copies of piles of rocks), you might also notice an unpleasant lack of symmetry. What number triples to make six? Why, two of course. But what triples to make seven? There isn’t any pile of rocks like that. How annoying!
Of course we’re not really talking about piles of rocks (or anti-rocks). We’re talking about an abstract imaginary structure inspired by rocks. So if we want there to be a number which when tripled makes seven, then we can simply build one. We don’t even have to go out to the garage and get tools—we just “bring it into being” linguistically. We can even give it a name like ‘7/3’ (a modified Egyptian shorthand for “that which when multiplied by three makes seven.”) And so on. All of the usual “rules” of arithmetic are simply the consequences of these aesthetic choices.
The point is that there is no reality to any of this, so there are no rules or restrictions other than the ones we care to impose. And the aesthetic here is very clear, both historically and philosophically: if a pattern is interesting and attractive, then it’s good. (And if it means having to work hard to bend your mind around a new idea, so much the better.)
Make up anything you want, so long as it isn’t boring. Of course this is a matter of taste, and tastes change and evolve. Welcome to art history! Being a mathematician is not so much about being clever (although lord knows that helps); it’s about being aesthetically sensitive and having refined and exquisite taste.
the mathematical landscape is filled with these interesting and delightful structures that we have built (or accidentally discovered) for our own amusement. We observe them, notice interesting patterns, and try to craft elegant and compelling narratives to explain their behavior.
It seems to keep happening! And it’s utterly beyond our control. Either this is a true (and surprising and beautiful) feature of odd numbers or it isn’t, and we simply have no say in the matter. We may have brought these creatures into existence (and that is a serious philosophical question in itself) but now they are running amok and doing things we never intended. This is the Frankenstein aspect of mathematics—we have the authority to define our creations, to instill in them whatever features or properties we choose, but we have no say in what behaviors may then ensue as a consequence of our choices.
These aren’t hairy, smelly hamsters with bloodstreams and intestines; they’re happy, free, lighter-than-air constructs of my imagination. And they are absolutely terrifying. Do you get what I mean here? So simple they’re scary? These aren’t science-fiction aliens, these are aliens. And they’re up to something, apparently. They seem to always add up to squares. But why? At this point what we have is a conjecture about odd numbers. We have discovered a pattern, and we think it continues. We could even verify that it works for the first trillion cases if we wanted. We could then say that it’s true for all practical purposes, and be done with it.
But that’s not what mathematics is about. Math is not about a collection of “truths” (however useful or interesting they may be).
Math is about reason and understanding.
We want to know why.
And not for any practical purpose.
Here’s where the art has to happen. Observation and discovery are one thing, but explanation is quite another. What we need is a proof, a narrative of some kind that helps us to understand why this pattern is occurring. And the standards for proof in mathematics are pretty damn high. A mathematical proof should be an absolutely clear logical deduction, which, as I said before, needs not only to satisfy, but to satisfy beautifully. That is the goal of the mathematician: to explain in the simplest, most elegant and logically satisfying way possible. To make the mystery melt away and to reveal a simple, crystalline truth.
Finally, I want to stress again that it’s not the fact that consecutive odd numbers add up to squares that really matters here; it’s the discovery, the explanation, the analysis. Mathematical truths are merely the incidental by-products of these activities.
Painting is not about what hangs in the museum, it’s about what you do—the experience you have with brushes and paint.
So where do math problems like these come from? Well, I’ll tell you: they come from playing. Just playing around in Mathematical Reality, often with no particular goal in mind. It’s not hard to find good problems—just go to the jungle yourself. You can’t take three steps without tripping over something interesting.
This is how math problems arise—just from sincere and serendipitous exploration. And isn’t that how every great thing in life works? Children understand this. They know that learning and playing are the same thing. How sad that the grownups have forgotten. They think of learning as a chore, so they make it into one. Their problem is intentionality. So let me leave you with the only practical advice I have to offer: just play! You don’t need a license to do math. You don’t need to take a class or read a book. Mathematical Reality is yours to enjoy for the rest of your life. It exists in your imagination and you can do whatever you want with it. Including nothing, of course.