Great new NYTimes Column
Last week, the New York Times (or at least it's online website, I haven't read the hard version in a while) began running what looks to be a very promising column focused on mathematics. Steven Strogatz' weekly column can be read here.
The theme of the column is something that I've thought about a lot. The author's intent is to teach mathematics in a way that is basic and obvious to people who may not remember much math or may be afraid of it.
In my opinion, math is taught somewhat poorly the vast majority of the time. For most, math is a fixed set of algorithms and rules that one preforms to add two numbers together, or to divide longhand, or to find what x equals. Many get by having learned these algorithms, whether they like it or not. I won't pretend to be any better. For the majority of my time studying math, that's what I learned. Ironically, it wasn't until I was learning much more advanced math that I was able to relearn the simplest math and really understand the logic behind it.
In other words, it wasn't until I was learning about complicated structures, groups, rings, fields, that I was able to think about the simplest forms of math (addition, subtraction, multiplication) in abstract ways. I'll give an example.
A fun exercise is to try to understand the logic behind using base-10, or rather to do the opposite and show how arbitrary it is. For the average person, the idea of a base-10 number system is so ingrained in their brain that they can't even conceive of anything else. Base 10 has to be right because it makes the most sense, they think. But when you think about it and write out exactly what using a particular base means, you get a much better understanding of things that you've taken for granted.
(Base-10 just means that we have decided to use 9 symbols and 0 to describe all numbers. Instead of labeling them 1-9, let's call them a-i, so as to distinguish between when I mean a number and when I mean an symbol invented by man to represent a number. An arbitrary number N can be represented by:
$latex N = a * 10^0 + b * 10^1 + c * 10^2 + d * 10^3 $
etc. We would call N abcd. And that makes a lot of sense to most people. But N is just an abstract concept that represents some amount of things, and it exists outside of any base. It can be represented in an infinite number of ways using any arbitrary base and any number of symbols to represent the numbers of that base....)
Anyway, before I ramble off too much, my overall point is that math should be taught, even from an early age, as something that is natural and intuitive. It should be built from the ground up in ways that make sense, so that people can see WHY we do certain things, and not just how.