Home > A Level, Commentary, Maths, Teaching > The apprentice has become the master…

## The apprentice has become the master…

Over the last three weeks this sentiment that is present in so many films has been showing itself in my classroom. Specifically during my year 13 further pure lessons. It has been a real joy to see the superb mathematicians my students have become, especially given the fact that their teaching prior to A level had left a bit to be desired. We have come a long way since they started year twelve with major gaps in their knowledge which we have had to fill in on the way.

Recently we have been covering further complex numbers. When we came to de Moivre’s theorem I was planning the intro lesson and I decided that I would prove the theorem for them so they could see why it worked. However, when it came to teaching the lesson I changed my mind. I gave them a brief biog of de Moivre (not required course material I know, but I find it interesting and so do they), then I showed them the theorem he came up with. I then asked them to prove it by induction (they have already covered proof by induction). They managed rather quickly to apply their knowledge of induction and prove de Moivre’s theorem for all positive integers. I was astonished by the speed they managed this so instead of showing them how to prove it for all negative integers I asked them if they could work it out. Again, between them they blitzed it. It was a joy to watch, and it was brilliant to see the enthusiasm from all of them as they bounced ideas off each other and came to their answer. As an extra, I told them Euler came up with a much simpler proof using a different formula we had already proved (Obviously Euler’s formula, but I didn’t tell them that) and asked them to see if they could come up with that. Again, they blew me away with the speed they managed this.

Again, while looking at loci in the complex plain I put up some examples on the board to go through, and they managed to guide me through them with minimal (if any) input from me. At one point one of the class asked “Why when you get rid of the modulus is it just, ‘x2 + y2 ’, where is the 2ixy, and why isn’t the y2 negative?” and before I could say anything one of the others gave an explanation that was phenomenally detailed but still crystal clear.

The third instance of this though, and probably the strangest, was when we were going through the C3 and C4 papers they had sat (none of them got what we were expecting so we wanted to know what went wrong). One of the lads was spotting his own mistakes and correcting them far, far quicker than I could manage. He had corrected them all to perfect answers (dropping only a couple of marks) within an hour. This was both quite brilliant, and quite infuriating, as he clearly knew his stuff and should have had at the very least 90%!