Wolfram, computer-based math, “and making maths beautiful”March 6, 2012
I’ve had Conrad Wolfram on my mind for some time. He gave a TED talk on computer-based math a while back, and just gave an updated talk at Learning Without Frontiers. Here’s the link:
His premise is challengingly obvious: teaching should be motivated by good problems, and they should use every tool at their disposal to solve them. In fact, he goes–happily–the extra step of saying that even posing the problem should be part of the student’s responsibility. Here’s the breakdown:
In our education system, we spend most of our energy teaching step 3 (sensibly, from a pre-computer perspective: that used to be the limiting step). But, Wolfram says, now that we have computers, computation is a breeze. So, why waste our time learning ancient and defunct computer methods (think: long division on pencil and paper, quadratic equations, etc.)? Why not pose and solve real problems?
Wolfram’s real problems tend to be on the applied side. But he makes room for aesthetically motivated pure math questions too, like exploring the geometry behind the golden ratio (“What’s a beautiful shape”). The other ideas for motivating problems below run the gamut from social network theory to compression to cryptography.
I am intrigued by Wolfram’s message, and I find myself agreeing with a lot of what he says. I believe, for example, in motivated mathematics, and one way to motivate a math is with a great problem. He also is careful to say that certain fundamentals still need to be taught; estimation, for example, is critical in a computer-based approach, and rightly, since without it you won’t notice if you mis-enter your data your computer starts spurting out gibberish.
At the same time, I can already imagine how his message will be garbled as it gets translated into schools and curriculum.
More fundamentally, though, I think there’s a critical age cutoff in all this. I’ve gone back and forth on the question of calculators and computers, but for young kids, I see them do more harm than good. Kids shouldn’t need to reach for a calculator to do simple arithmetic, and the fact that they do if they’re available encourages lazy and disengaged thinking, rather than free them up to do more with their minds.
I’m pretty sure I could get on board Wolfram’s project if it kicked into gear in 7th or 8th grade. I’ve helped 11th graders to “learn” how to do polynomial long division of some nonsense with functions, and I’m left with the distinct sense that they are being asked to waste a whole lot of their time for no reason at all. If I, as a mathematician, don’t care, why should they? There are certain things that I appreciate students knowing (and long division and the quadratic formula are in this category), but are they so important?
I’m reminded of a moment when a student of mine was trying to find the diagonal of a regular 7-gon. This involved deriving a cubic polynomial. To solve it, all he had to do was plug it into the cubic formula:
Now tell me, what kind of human being would ever choose to work that out by hand when we could just enter it into Wolfram Alpha and get the answer instantly? And more, by getting the answer instantly, you keep your eyes on what you actually want to know–what’s motivating you–rather than waste your time with a pointless calculation. What would mathematicians in the old days do? We know the answer: hire a human computer. (The deeper observation is that the cubic formula, horrible as it may be, was actually a breakthrough tool to allow cubic equations to be solved without much more agonizingly difficult work. It was cutting edge tech in the 16th century.)
What I’d like to see, I guess, is a K-6 curriculum with no computers or calculators whatsoever, followed by a middle and high school (and college) program that looks a lot like what Wolfram is proposing.
Can we get the implementation right, is the question. As soon as you introduce computers and video into the class, there are so many ways for teachers and students to get lazy. You need to be motivated, and disciplined.
A parting thought. When I first taught differential equations, it was to an audience of engineers and scientists who needed the tools, but didn’t necessarily need to know how it worked. I wrote down on the board almost exactly what Wolfram noted in his talk:
The goal of differential equations is to predict the future.
Here’s how it works:
Step 1: Translate your situation into a differential equation
Step 2: Solve the differential equation
Step 3: Translate the solution back into the real world.
I was chained to a curriculum that demanded that we spend the majority of our time on Step 2 (and I didn’t even mention posing your own questions in the context, though it was already very important to me). What I told them, again and again, was that understanding steps 1 and 3 is what would make them successful scientists, engineers, and problem solvers. Insofar as I demanded from them that they do that, I think I taught them something they could use. The rest was academic.