Recently a friend asked me to help her learn enough math to take the GRE’s. My response was to give her the first problem that I thought she should be able to solve before we discussed anything else. It was a very simple problem from the perspective of a mathematician, but one that is not simple enough to solve only using the approaches to problem-solving that are usually taught in American high schools. Specifically, I asked my friend to find a and b such that,
This equation poses a problem for many students who’ve only taken American high school math classes, because you have to accept that the “solution” to this equation is not a single pair of numbers, but an infinite set of numbers. If you can handle that idea, you should be able to see pretty easily that you can pull the a over to the lefthand side of the equation and then factor it out to get
As long as b is not 1 — which can never occur in an solution to this equation –, you can divide out by (b – 1) to get
which is an explicit formula for all of the possible solutions to this equation. You can pick any b and the formula will give you the corresponding value of a; because this works for all b not equal to 1, you have an infinite set of solutions.
The math itself is bland: what is interesting about this example is the question, “why are we not teaching our children to think mathematically?” We are teaching them to memorize explicit formulas for solving certain algebraic equations, but they are not learning a general approach to solving equations. Why has the substance of mathematics been lost and replaced with the shell of mathematics?