Dr Ron Knott in the Department of Mathematics at Surrey University is not a name I recognised, but reading his resume, I now realise that I have heard him talk a few times about Mathematics on Radio 4, both on Simon Singh’s 5 Numbers series, and in Melvyn Bragg’s In Our Time podcast.

I was looking for some information about exact values of trigonometric ratios, and came across his most informative site. I was extremely pleasantly surprised to discover that for some values the trigonometric functions give exact solutions in terms of phi, the golden ratio, among other information.

For example, did you know that the cosine of 27 degrees is exactly a half of the square root of (two plus the square root of (two subtract phi)). (One day when I finish writing my own equation display movies, I’ll write that out in a prettier way, Dr Knott’s website tries a little harder than I do). I love that the number 27, which clearly wants to be prime so much it tricks generations of children into thinking it is, the square root of two and the golden ratio are connected inextricably through the circle-based cosine function. Fantastic!

The whole page, indeed the whole of his site in general, is steeped in extremely interesting, and relatively accessible mathematics with Fibonacci numbers, Egyptian Fractions and so on and so forth. It’s mostly a site for KS4 and beyond (14 years old +), with most material for the older students. Some of it is not for the faint-hearted. However, it is a valuable resource for mathematicians of all hues, and well worth a look.

John Mason, Professor of Mathematics Education at the Open University, was the closing speaker at this year’s MA-ATM combined mathematics education conference.

The central theme of the talk was the importance of reflecting upon experiences in order to unify them. We don’t often seem to learn from experience alone Prof. Mason exhorted; we learn incrementally, unifying experiences with one another, gradually gaining awareness of a greater whole.

Unifying experiences does not happen by chance; or rather, it should not. As teachers we must act as faciliators of this reflective process. One way to achieve this is to offer students familiar patterns of problems which are then developed and extended, and the richness drawn out of them through reflection.

The associated prensentation to the talk, with examples of such activities can be downloaded from Prof. Mason’s website, or by clicking on the link below:

Powerpoint: Closing Plenary at the MA-ATM Conference

Of course, this presentation does not do justice to an impassioned talk from such an experienced educator!

There was a great deal of truth in Prof. Mason’s talk. One of the main things that I will take away from the talk was my reflection during the talk that as a teacher I found myself too often concentrating on the activity in quantity, searching to find new activities and new ways of approaching a topic. I would find them on the internet, or in books, or by asking others. By contrast, the activities considered in Prof. Mason’s talk could be found only by taking the mathematics seriously and doing the mathematics for yourself. He searched for activities in quality, searching for ways of reusing the familiar but provoking new thought within that structure. From this, students are in a position to draw upon previous experiences, become familiar with areas of mathematics, and begin to unify their experiences. That is a lesson worth (re)learning!