Learning maths is hard for everyone at some point, and virtually everyone who’s ever been through a modern education has had the experience of not understanding a mathematical idea.

It’s frustrating, and it can seem like the gap is insurmountable. One thing I realised in the later years of my undergraduate degree is how essential it is to look at things from different perspectives. Reading another take on a topic, often just one small comment makes you realise what you’ve been missing and then everything clicks into place. Suddenly the impossible seems possible!

I think this is essential to remember when tutoring, as looking at things from a different perspective is often the only way to ensure a deep understanding of a topic.

For instance we can think of differentiation from a symbolic perspective (x^2 differentiated is 2x), in terms of the slope of a tangent (the slope of the tangent to x^2 at x=3 is 6), or in terms of a rate of change, and each one informs the other in a myriad of different ways.

At university there are, literally, tens of ways the concept of differentiation is enriched, and each time you gain a new insight into the fundamental idea. For problem solving, it is essential to be able to switch from the conceptual level (differentiation yields the rate of change of a quantity) to the practical (x^2 differentiated is 2x) and making this connection is not always easy.

For a student learning about differentiation, I think it is important to emphasize both of these levels, so that the duality between the conceptual and the practical is firmly established early on.

With the new A-level syllabus, the government has sought to focus less on rote learning so being fluent with ideas as well as calculations is more vital now than ever. In my own studies, I often find I only understand difficult concepts by going over and over them in different ways, and finding new ways to represent old ideas is a key part of this.

As much as this is important for ideas, it is also important for solving problems. This is illustrated by the following (rather macabre) problem which I will quote:

Two trains 200 miles apart are moving toward each other; each one is going at a speed of 50 miles per hour. A fly starting on the front of one of the trains flies back and forth between them at a rate of 75 miles per hour. It does this until the trains collide and crush the fly. What is the total distance the fly has flown?

Those with a background in mathematics may sigh with irritation upon hearing this read to them. To work it out it seems we will be forced to solve many awkward equations which relate the speed of the train to the speed of the fly, and then the answer will come from a long drawn out calculation.

However, there is a trick. Actually all we need are two things: the fly is moving at 75mph, and for 1 hour. So it must fly a total distance of 75 miles. The whole business with the trains is actually just a distraction, and the answer was staring right in the face from the get go.

Whilst A-level questions are not designed to trick you (as this one was), the lesson learnt is clear: often there is a way of looking at a problem that is vastly easier than the most obvious one, and the goal is to find this perspective.