I recently found the webpage for Harvard Calculus, and one thing I think is very cool is this “rule of nine” that shows nine different ways we might approach a math problem:
algebraically, analytically, geometrically, historically, graphically, numerically, conceptually, psychologically, as well as experimentally.
What the heck does that mean? This may require some calc knowledge, but for something like finding the derivative of sin(x) at x=pi/3, you could think in these ways:
|algebraically||we know sin’=cos and cos(pi/3)=1/2|
|analytically||we know sin(x) = x-x^3/3! + x^5/5! – … and so sin'(x) = 1-x^2/2! + x^4/4!-… = cos(x) so that sin'(pi/3)=1/2.|
|geometrically||sin(x) is the height of a right angle triangle with hypotenuse of length 1. The rate of change of this length in dependence of the angle can be seen geometrically.|
|historically||the derivative can be derived from Euler’s formula exp(i x) = cos(x) + i sin(x) which has the derivative i exp(i x) = i cos(x) – sin(x). Comparing real and complex parts shows cos’=-sin, sin’=cos.|
|graphically||draw the graph of sin(x) and determine the slope at x=pi/3|
|numerically||take a small number 1/1000 and compute 1000 sin(pi/3+1/1000)-sin(pi/3)) which gives 0.499567.|
|conceptually||since sin(x) increases for increasing for acute angles, the result is positive.|
|psychologically||my teacher does not like to assign problems with irrational numbers as answers. The result should be a simple rational number. Because 0 and 1 are out of question, the next reasonable result is 1/2 ……|
|experimentally||here is an esoteric experiment: why not use Fourier series and differentiate that series. To make it interesting, take f(x) = |sin(x)| since sin(x)=|sin(x)| around pi/3 …|
One thing they note, though, is that if you try to approach a problem from too many ways at once, you’re going to overwhelm yourself before you even understand anything. So we should pick three, maybe four approaches, to make sure we really get something without going overboard.
Jo Boaler (yes, I know, I keep going on about her) has adapted this for algebra, and suggests that for algebra problem, we should have students try to see if they can express it in four of the five following ways:
in words, in pictures and diagrams, in tables, in graphs, and in symbols.
How many ways can you think through this problem?