I’ve written before about the beauty of mathematics and how it is sometimes difficult to see unless you have reached a certain level of understanding.
With this in mind it has been interesting “watching” some of the edu-twitterati as they journey through higher levels of mathematics than they are familiar with. Aside from observing the sheer joy exhibited as an erstwhile incomprehensible melange of algebraic notation swirled into focus and understanding, I see this as an opportunity for a number of intelligent educators to comment on processes that have enabled them to get to grips with topics that may previously have confounded them.
Personally I have always been very lucky. From a very early age Mathematics has never seemed that hard to me. I have no known family background in the subject (ok, my grandad was a bookies runner, so that perhaps explains the ease with probability) and managed to soak up all the maths that was thrown at me. Some of the pedestrian stuff I’ve never really mastered. For example, I still don’t know my times table by heart. I know most of them, but tend to derive, for example 9 x 8 from knowing what 9 x 9 is. The 12’s are still a bit of a mystery. At the other end of the scale one of my earliest forays into programming was to write a routine for the ZX-81 that inverted n x n matrices to help me with solving problems in regression analysis. O, happy days.
Now, there is never going to be just a single thing that enables an individual to understand maths. But when I think back to primary school there is always one thing that stands out for me. We had a balance beam in the classroom. The beam was not dissimilar to this one on the NRich site, except ours was a small grey plastic thing, about 50cm high and the beam was about 75cm long. Along the beam were the numbers 1 to 10, and each number had a hook by it. On the hook we could hang oblong shaped piece of flat steel. So we could hang 2 pieces on the 10 on one side and 4 pieces on the 5 on the other and see that the beam balanced. We could get even more complex than that – 1 on the 10 and 3 on the five on one side, 3 on the eight and 1 on the the one on the other. Balance! This helped me understand the links between numbers better than hours of examples in work books.
I think it went further than that. It helped me to visualise the nature of equality, which is a key underpinning of many areas of mathematics. Later, I found problems in mechanics, particularly those involving moments, a lot easier to grasp compared to my fellow students (they soon caught me up, mind).
So, where am I going with this? Its simple. Maths is too often seen as a cerebral activity to be carried out entirely in the abstract. In my experience, both as a learner and a teacher, for most mortals that abstract has to be grounded somewhere in the physical world. One of the roles of the teacher is to help each student find out where to ground themselves.
There are some, a few, for whom mathematics can exist entirely in the mind. For the rest of us we need some reality to link it to. For me, it was the balance beam. And I do believe that once each person finds their personal balance beam, they will also more easily find the beauty.