Comment by WheelsAtLarge | Hacker News Reader

Comment by WheelsAtLarge | Hacker News Reader

Yes, it can. The biggest failure in math education is that we spend an inordinate amount of time solving equations without understanding how to apply them to solve real-world problems. Thanks to the calculator we can now spend less time figuring out how to solve x+1=7 and put that time on how to understand when the equation is needed in the real world or how to create an equation that will help solve a real-world problem.

The same can be said about calculus. Given the abilities that computers give us, it's much better to teach applied calculus than to teach students how to solve equations by hand. Calculus is a tool, and it should be available to as many people as possible.

For most students, Calculus should be seen in the same light as drills. We understand when we need a drill so we get one and use it but we never have to go out and learn how to build one just because we have a need to use it. Calculus should be similar. We need to learn when we need it and then use a computer to get an answer that helps solve a problem that advances our needs.

Students don’t like “real world” problems. They say they want more of that but when you actually do those types of problems they complain or do poorly on them. Word problems are even more confusing to students than non-word problems.

The vast majority of so called real world problems aren’t things people who use math in their jobs actually do.

I guess you are referring to the "Steve has 17.5 credits, how many pizzas can he buy?" type problems?

For me, when learning calculus it was that it seemed pointless. They were teaching me to do this mechanical task, but why? Why not another task like increasing every other number in the equation by 6? It wasn't until I learnt about velocity and acceleration that it all started to make sense. The task of differentiating/integrating seemed far less important than the understanding that functions have derivatives and anti-derivatives and what that means.

I'm sorry, but if you don't know how to solve x+1=7, you don't actually understand maths.

I agree, but how far do you take this? Do you say if you don't know the Peano axioms you don't understand maths?

No.

Why?

Because knowing how you derive arithmetic from axioms isn't necessary for mathematical modeling. Mostly only logicians, theoretical computer scientists etc . need to know about PA. Euler and Gauss didn't know about PA (it hadn't been invented yet), clearly they were still great mathematicians.

I made it an easy equation to make a point. Pick your favorite multivariable set of equations. Depending on the situation a computer can solve it in a fraction of the time and with better accuracy than a human with pencil and paper. And it can create thousands of variations in a matter of minutes.

> I made it an easy equation to make a point

It was a bad example then. Obviously there are equations that are difficult to solve by hand and you don't need to remember all tricks in order to do successful modeling, but if you're not able to solve x+1=7, it means you don't even understand what an equation is, so you won't be able to use it for modeling anything.

Maybe that's fine? I mean I'm sure a lot of people who fix stuff up in their homes know very little of the physics behind their hammers, but they can still use them just fine.

Maybe a good tool does not need to be understood on a deeper level to be used.

If we gave students more exposure to maths as a tool to be used, rather than arcane formulae and symbolic manipulations, they could build an intuition and appreciation for it that allow them to use it even if they cannot perform the mechanical transformations of equations on their own.

I'm not saying x+1=7 is a good example of that -- my son, who is not even literate, could answer that in the blink of an eye. But recently I needed to get the implied marginal probability out of a partition of conditional probabilities, i.e. solve for p in something like e = ap + b(1-p). Could I have done it manually? Sure. Did I plug it into a symbolic solver? Absolutely.

For me the insight lay in (a) knowing that was the shape of the equation I knew, and (b) that I needed to rearrange it to get p as a dependent variable. Actually going through the motions is not that important, in my mind.

> I'm not saying x+1=7 is a good example of that.

Exactly, it's not. I'm not arguing with what you're saying, I'm arguing with what GP wrote.

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