Ushtrime te Matematikes

The implication here is that we slaved over triangles at school and it was completely useless, because we never need triangles in ‘real life’, whereas we do need to understand taxes and so it would have been much more useful to study how to do taxes in school, rather than all that pointless stuff about triangles. (This is admittedly more widely relevant in the US where everyone has to do a tax return, unlike in other countries like the UK where it’s done automatically for people in regular employment.)

This meme makes me sad in many different ways at once. First of all, because it has an element of truth in it: many things we do in maths at school are not things that will ever be useful in daily life. Or rather, they won’t be directly useful, and I suppose that is the real point. The point is that ‘useful’ can mean a rather wide variety of things, and we spend too long focusing on maths as ‘directly useful’ while simultaneously teaching maths that is not directly useful.

There are two ways we could remedy that. One approach would be to teach maths that is directly useful instead. I suppose this would mean things like taxes, mortgages, inflation, debt repayment, budgeting. I, personally, think that sounds awfully boring. It’s also very limiting. Because if you teach ‘How to Do Taxes’, then it’s not really applicable to anything except doing your taxes. Likewise there aren’t many things that work quite like a mortgage, so understanding mortgages is not extremely helpful to anything except understanding mortgages.

One of my favourite questions I’ve ever been asked at the end of a public maths talk came from a six-year-old girl in Panama. She asked: ‘If maths is everywhere, why do we have to go to school to learn it?’ This question encapsulated what I find wonderful about innocent questions, at both a mathematical level and also at a meta-level of language: my Spanish is exceedingly rusty, but I was able to understand her question as posed in Spanish; however, I had no chance whatsoever of being able to provide an answer in Spanish, so had to rely on the interpreter for my response.

Innocent questions in maths are like that: they can be very easy to pose and very easy to understand, but extremely difficult to answer.

To me, the point of formal education, as opposed to life education, is to accumulate knowledge from generations and generations of humans without having to go through the entire process ourselves to learn it ‘from experience’. Yes, some things can only really be learnt from experience, like, perhaps, how to deal with grief. But even in that I have been helped immeasurably by an expert psychologist and all the formal knowledge she brought from the field; however, one part that you can only learn by experience is how you as an individual are going to respond to the pain and also to the interventions.

So the question then isn’t ‘Will I ever use this exact thing in my life?’ but rather ‘In doing this, am I developing myself in some way that will be beneficial later?’ I find that this latter definition of ‘useful’ is more … useful. It’s also more relevant to why we do maths. Thus, if we’re doing algebra or thinking about triangles or prime numbers, the point isn’t that we will need algebra or triangles in our future daily life; the point is that we’re developing our thinking in a way that will enable us to think more clearly about daily life in the future.

We’re going to look at various different motivations for the maths we do. This isn’t just about why we do maths, it’s about why we do it in the way that we do it. There are some deep guiding principles at work, stemming from our view of maths as ‘the logical study of how logical things work’. An important part of studying things logically is to take it very slowly and understand what the basic building blocks of the situation are, and how they interact with each other. We’ll also see that understanding the principles at work doesn’t just help us get ‘the right answer’ (although it might also do that); it helps us understand more situations at once, and helps us move onto understanding much more complicated situations analogously, by a process of mathematical generalisation.