Why has there not been any progress in the way we teach?

The latest buzz word in education is grit. Grittier students learn better. Well maybe, but shouldn’t there be a discussion of better ways of presenting the material? We have been teaching and learning since before we were human and, near as I can tell, pedagogy has remained the same.

We can do better. My particular interest in math. Here is a way of teaching the formula for arithmetic series that I have on my Web page.

Consider the numbers 1, 2, 3, 4 and 5. Without doing any calculation, take a guess what the average is for these numbers. Most people, would guess 3, because it is in the middle. Assuming that 3 is the average, how could we use that to find the sum. Easy, just multiply by 5. Add the numbers to verify.

How do we find the sum from 1 to n? What is the average? Some students may initially say n/2, but the example shows that is not right. We take the average of the first and last numbers, (n+1)/2. It then follows that the sum is n(n+1)/2.

This of course is not a proof. In an advanced class, the next step would be to try to find a proof. Inspired by what was done, someone may notice that if we write the numbers 1 to n and then write the numbers underneath from n to 1, we get n+1 in each column. The important point is that we were able to guess at a formula using intuition. Who’d have thought, mathematics and intuition? At the highest levels of math, it happens all the time, both in determining what to prove and in finding a method of proof.

I am certainly not the first person to see the formula for arithmetic series in this way, so why is the formula simply taught without explanation? Not only is the process more interesting, but the formula becomes easier to remember. Sometimes it seems that the teaching method is chosen so as to make the material seem as dull as possible.