stoutfellow

Today in calculus I introduced the definition of the derivative of a function, and worked several examples. One of my students then asked whether, if they knew how to just read off the derivative without using the definition, they could do so on the test.

I responded at length.

I talked about the difference between learning to push symbols around and learning to think in a particular way.

I talked about the difference between being able to drive a car and being able to fix one.

I talked about the difference between trained seals and human beings.

I don't know how many, if any, of them understood my drift. There were a few grins, but I think they may have had more to do with my use of the word "hardass".

Now, I don't really agree with Robert Heinlein's famous list of Things Every Human Ought To Be Able To Do. I don't regard an inability to differentiate functions as a disqualifier from the human race. However, if you take Calculus I from me, you will learn about derivatives from the ground up, and you will by God learn to compute them from the definition.

That is all.