 # Communications of the ACM

BLOG@CACM

## Teaching Kids to Feel the Math With Computers

The world has changed.  Computers are everywhere.  Many children start using computers at a young age.  Yet, how we teach our kids has barely changed.

Let's say we were going to start over.  Let's redesign how we teach our kids.  If we assume that everyone has a computer, how would math education change?

Conrad Wolfram gave an interesting TED talk recently, "Teaching Kids Real Math with Computers", that explores how math education could change now that we have calculating devices at our fingertips.

Some excerpts:

What is math? What do we mean when we say we're doing math, or educating people to do math?

I think it's about four steps, roughly speaking, starting with posing the right question. What is it that we want to ask? What is it we're trying to find out here?

The next thing is take that problem and turn it from a real-world problem into a math problem.

Once you've done that, then there's the computation step.

And then finally, turn it back to the real world. Did it answer the question?

The crazy thing right now, in math education, we're spending about perhaps 80 percent of the time teaching people to do step three [the computation step] by hand. Yet, that's the one step computers can do better than any human after years of practice. Instead, we ought to be using computers to do step three and using the students to spend much more effort on learning how to do steps one, two and four--conceptualizing problems [and] applying them.

Math is not equal to calculating ... [Years ago] calculating [by hand] was typically the limiting step, and not often it isn't ...  Calculating ... [is a] chore ... It's a means to an end, not an end in itself ... It's the thing you'd like to avoid if you can, like to get a machine to do ... Computers allow us to do that.

What I really think we gain from this is students getting intuition and experience in far greater quantities than they've ever got before. And experience of harder problems--being able to play with the math, interact with it, feel it. We want people who can feel the math instinctively. That's what computers allow us to do.

We can reorder [the curriculum] by how difficult it is to understand the concepts, however hard the calculating ... Calculus has traditionally been taught very late ... [because] it's ... hard doing the calculations ... [but] many of the concepts are amenable to a much younger age group ... [such as] talking about what happens when you increase the number of sides of a polygon to a very large number .... I want to see a completely renewed, changed math curriculum built from the ground up, based on computers being being there.

Conrad also suggested that we should teach programming much earlier, saying, "If you really want to check how you understand math, then write a program to do it." Programming not only gets students to understand how to convert a real-world problem into a math problem and lets them zip through the calculation chore, but also it allows students to explore variations on the problem, yielding a more complete understanding of whole classes of problems and how minor changes in a problem might result in drastically different outcomes.

On having students explore math problems more deeply using computers, Jon McLoone adds a compelling example to Conrad's talk in his blog post, "Do Computers Dumb Down Math Education". Jon shows how much more deeply he can explore the problem of calculating where an artillery shell will land if computers are doing the chore of grinding through the calculations.  He not only runs through the standard example that neglects air resistance, but also shows how much the answer varies as he adds a few increasingly realistic models of air resistance.

One problem Conrad touches on in his talk is that the exams and evaluations would also have to change.  Teaching children to feel math and use computers as a tool to solve problems may best prepare them for today's world, but, if the standardized tests at all levels of U.S. education are checking for calculating speed over problem solving, the curriculum change will look like a failure.  For more on that issue, please see my earlier BLOG@CACM post, "Computers Are a Tool, Even in Education".

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