Have you ever noticed how equations look far more complicated and hard to understand than the concept they represent?
I sometimes get myself stuck having to read other people’s work (it’s the ‘peer review process’) and when I first read it, I am often utterly confused, like a person stumbling around a dark room they’ve never been in before. However, because I am expected to make intelligible commentary, I soldier on until I understand what is being said.
Once you understand something, it is hard to remember what you felt like before you understood it. How did that equation look the first time you saw it? I have been thinking about this…
Let’s consider ‘equations’ – a common part of many technical documents. I have found that I always overestimate how clever or useful the equations really are when I first see them. So what does this mean?
It means that using equations to help teach people we risk turning them off by giving them the impression that the work is harder than it is.
Let me give an example:
Maxwell’s wave equations. These are considered (rightly) to be an cornerstone of physics, as they model the behaviour of waves in the inter-related electric and magnetic fields. When I first read them, they were ‘greek’ to me, literally. Here’s a small one:
Obviously, you need to know more to understand what they are about. You need to know what each symbol represents – and you need to know what the operators (the × in this case) actually do. For anyone who has not specifically studied maths at university would then need to backtrack quite far, because in this case the ‘×’ is not the ‘×’ most folks know and love, its the ‘cross product’ which applies to vectors. That even leaves most science graduates cold, draining the joy of discovery for a few hours or days while you go away to learn (or remember) what the heck that means.
But is it all worth it? Is the complexity of partial differential equations and matrix multiplication really required in order to understand what the equation is describing?
Of course not!
So why are equations always wheeled out to ‘explain’ phenomena? This is a failure of teaching. Of science communication. Surely concepts can be explained much better by the use of anecdotes, metaphors & illustrations?
Scientists working at the bleeding edge of science have to be very precise in their logic, and when communicating with one another, equations are undoubtedly very efficient ways to describe hypotheses. And so, while they are good ways for experts to relate, they make it harder for newbies to “break in”, and are dreadful teaching tools.
The Maxwell equations really just describe how waves propagate in a medium – and really its just the full 3-d version of waves in a slinky, or ripples in a pond. The equations, while drawing on complex (and difficult) maths, are describing something the human brain already has an intuitive grip on, because we’ve seen it!
I’m not suggesting we could do away with equations – they are valuable in the predictions they make for those who already understand what they represent – I am just suggesting that equations should be de-emphasised, and only dragged out when the student starts to feel the need to describe the phenomenon mathematically.
So my message to all university lecturers and text-book writers is: describe a phenomenon with the use of analogy, please!