In case you come here only on Wednesdays when I write about homeschooling, you might also want to check out Tech Tuesdays (a series that goes back several years). In yesterday’s Tech Tuesday I wrote about computational complexity and gave an example of a program that would take hundreds of thousands of years to run. It turns out to have been an example of exponential growth which happens to, well, grow a lot! That reminded me that I have been meaning to write about rates of change more generally.
Calculus is one of those math topics that really divides people. They tend to either love it or remember it as the topic where they gave up on math. It occurs to me that a lot of that has to do with how calculus is taught. It is generally presented as something quite abstract and the homework consists of taking lots of derivatives of complicated expressions (and later integrals). I can safely say that I have never since school taken another derivative and if I had to do it today, I would rely on something like Wolfram Alpha for it. But it is also true that the concept of rates of change is incredibly useful and in fact pops up in my work all the time.
So I think it is essential for everyone to learn the following three core ideas in a pragmatic fashion:
1. Appreciate the vast difference between linear growth and exponential growth.
2. Be able to relate linear change to addition/subtraction and exponential change to multiplication/division (and percentages).
3. Grok that there is a rate of change of the rate of change and relate that to acceleration and deceleration (e.g. of a car).
I fear that for most people the teaching of calculus winds up obscuring rather than elucidating these ideas. I will probably pull together a presentation for our kids on the topic and share it here. Once the basic ideas are understood there are many interesting refinements possible, such as other rates of change (polynomial, factorial) and the notion of an asymptote.
