junior school

high school

Periodically during tax season a meme goes round saying something like this:

I sure am glad we studied triangles

every time triangle season comes round

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.

Really this all comes back to why we do maths education, which comes down to why we do maths, as well as why we do education; and this comes down to why we do anything in life.

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.

We can get a glimpse of so much more during formal education than we could if we just waited for the experiences to happen to us. This does provoke the question of why (or whether) that’s a good thing, and I’ll come back to that.

So, personally, I believe it’s most powerful for formal education to address things that are not too closely related to real life, but that are very broadly transferable instead. General foundational skills, if you like, rather than very specific ones.

That’s a very brief account of why we do education; what about why we do maths? Why do we do anything?

People do things because they’re useful, or because they’re fun, or possibly because there will be some dire consequences if we don’t do it. (I realise that this does not include darker motivations like revenge, anger, hatred.)

Perhaps fun is useful? This comes back to my earlier point about different meanings of the word ‘useful’. There’s the rather utilitarian version of direct usefulness, but then there’s the other version that is more transferable. So rather than ‘I am doing this thing that I can then do to great effect in my life’, it’s more like ‘I am doing this thing which is exercising my brain in a particular way so that I can then use my brain to great effect in my life.’

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.

Now, there are instances where the exact thing we’re studying will be useful in our future lives. There was another meme that went round during the COVID-19 pandemic, depicting a maths teacher teaching a class about exponentials, and some bored students saying ‘When are we ever going to need this in life?’ Unfortunately, when the pandemic began it would have been really helpful if more people had understood exponentials in advance. Instead, when scientists tried to point out that it looked like things were going to get really bad, because of how exponentials work, far too many people thought they were scaremongering or making things up when you ‘can’t predict the future’.

So I’m not trying to say that school maths is never or should never be directly useful. We’ll see that some things that mathematicians did mostly for fun actually turned out to be very directly useful later – it turns out that humans are not very good at predicting what will be useful 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.

To address the question of why 1 isn’t prime, we need to think harder about the principles of prime numbers rather than just their definition. And the principle is that they are basic building blocks for numbers.