The other morning, I decided to run through an area in my neighbourhood that I’d never explored before.
I had decided to take a different path to the route I normally take, and I saw an interesting direction so I decided to follow it.
I got a little bit lost, but when I found myself, I realised that this would make an excellent metaphor for my relationship with maths.
When you’ve got a new, weird question that you’ve never seen before, it’s like coming across an interesting path so you follow it with faith. Faith in your navigation abilities and faith in the road (that it won’t lead you somewhere dodgy).
It’s a new, interesting direction, the road is pretty straightforward so even though you’ve never seen this part of town before, you know what to do. You still maintain a general sense of direction according to where you came from.
You’re taking in the new buildings, the new feeling of this part of town. It leads you to places you’ve never seen before, and it’s all new and beautiful and you’re seeing it for the first time. You decide “It’s not as bad as I thought!”
A few twists and turns later, however, your sense of direction starts to falter and the less sure you are of yourself.
Just as you start to doubt that you’re going in the right direction, you notice an interesting footpath that seems green and leafy, in stark contrast to the brick and tarmac of the street. You decide to follow it, all sense of direction gone.
You round the corner and there! The park that you used to visit as a child unfurls itself before you. You stand there, feeling the wind on your face, happy and relieved.
You see that you’ve entered through a gate in a corner that you never noticed before.
Running down along the edge of the park, you notice another gate leading in the direction of home that you had no idea existed so you exit that way, hoping for a shortcut.
The park has re-orientated you and gave you a sense of confidence that you could return home safely and soon.
Keeping to your sense of direction, you keep going through places you’ve never seen before, but this time you are certain you know the way home. Rounding a few more bends and taking a small path through a hedge, you suddenly emerge through a side footpath on the straight main road that leads you home.
If you’ve got no idea what I’m talking about, I’ll explain it using a question I helped my friend with.
No joke, I spent about 10 minutes staring.
I had no idea what the question was going on about. However, it looked like a really cool question and I was in the mood to try something new, so I decided to take this path that I had never seen before.
That is the first step to discovery: deciding to explore.
I messed around with the expression using my general sense of direction until I got this:
I was still feeling pretty confident.
However, I started to get a little worried at this point as I had no idea what to do. I looked around at the other questions and figured that this topic was under Binomial Expansion.
I suddenly realised that one section – the brackets raised to a power, could be expanded binomially, and that was my moment of entering the park I visited as a kid: I learnt binomial expansion in GCSE, but I thought it only worked with positive whole numbers!
Thus I entered this park through a gate that I never knew existed.
However, I applied the formula I did know from GCSE and expanded out the brackets as far as x^2 as required, thus exiting the park of binomial expansion through the strange, undiscovered gate of negative fractional binomial expansion.
I then knew what direction I needed to take this, so I multiplied the brackets together and that was when I emerged onto the home stretch: all I needed to do then was find a, b and c, bringing me home to the answer.
This is how I approach new questions that I have no idea how to do: there’s also no shame in pulling out Google Maps to look up where you are if you’re really lost, just as there’s no shame in looking it up on the internet.
However, keeping your eyes locked on Google Maps on the way home and not looking up is the same as copying down the answer without understanding – you’ll miss all the beautiful and unexpected bits of the journey, and you’ll just forget the way back next time.
Do you understand what I’m going on about? Has this made you seen maths in a new light? Or do you think I’m just talking nonsense? Do let me know in the comments!