Here’s another argument that I hear: that if you use computers it dumbs math down. This one is really frustrating. Somehow the idea has come about that, intrinsic to the use of computers, everything turns into mindless button pushing, intellectually all dumbed down. But if you do stuff by hand, it’s all very intellectual and brain training. Do we really believe most students studying math right now think it’s anything other than fairly mindless? Most of the time what they’re actually doing is running through a bunch of calculating processes they don’t understand for reasons they don’t get. Mindless or not, at least those processes had real practical use 50-100 years ago--they were the only way of calculating. But now they don’t, almost nobody actually uses them anymore outside education.
So let’s be clear about one thing: the mindlessness base we’re starting from with hand-calculating math is pretty low right now. And, far from thinking computers will dumb math down, I actually think, they don’t understand for reasons they don’t get. Mindless or not, at least those processes had real practical use 50-100 years ago--they were the only way of calculating. But now they don’t, almost nobody actually uses them anymore outside education. So let’s be clear about one thing: the mindlessness base we’re starting from with hand-calculating math is pretty low right now. And, far from thinking computers will dumb math down, I actually think, correctly applied, they can do quite the opposite. I think computers are the greatest tool for conceptually understanding math. As I’ve said, they liberate you from calculating to think at a higher level. But like all tools, they can be used completely mindlessly--for example making endless multimedia presentations. There was one I saw which aimed to use a computer to show people how to solve an equation by hand--all the steps you would take by hand. Now, maybe that’s good if you’re exciting about learning that, but it seems to me that it’s completely backwards for mainstream math. Why are we getting humans to learn with a computer how to solve an equation by hand that the computer should be solving anyway for them? They should be setting up the problem that the computer then solves; and working with the result. So what I’m arguing for is open-ended use of computers.
Use a computer as an open-ended tool as much as possible where the students are trying much harder problems. And if you really want to check that computers haven’t dumbed down math, look in the outside world. Look in the real world. Do we honestly believe that science and engineering and all other things that
depend on math have somehow got conceptually simpler since computers were introduced? Absolutely not. Computers have allowed them to drive far further forward, have allowed them to become much more conceptual because people can get rid of the calculating step and get the computers to do it. So let’s be quite clear where problem is now. What’s dumbed down is the complexity of a math problems; it’s not that computers will dumb math down.
So look, I believe computer-based math is a critical reform and it’s not an optional extra. It’s also a reform that has much more importance and resonance with many more people than it did in the past; the technology is much better too and close to ubiquitous; but with all that it’s certainly extremely difficult to pull off. However difficult, it is vital. Let’s recognize that it’s critical part of moving economies forward. I think it can take us from a knowledge economy to what I call a computational knowledge economy where high-level mathematical thinking is widespread---for many not just the few--and those abilities rather than just base knowledge that one would term knowledge economies as having are driving the economy forward. The country to do this first will leapfrog others. And there’s opportunity for developing as well as developed countries: their less developed math education infrastructure is easier to reform, faster.
Let me be clear that I do not think this is an incremental kind of reform. Here’s what happens if you walk rather slowly over a chasm is typically this. You go right into it. So what you need to do is start with a very high initial velocity (of course solving the differential equations correctly before you do) and jump over and hopefully get to the other side. Let’s go for it!
Finally, I’m not even sure if the subject that one’s talking about here is called math. Is the naming wrong? But whatever it’s called, let’s be quite clear: it’s the mainstream subject of the future.
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