Maths until 18

I learned more from a three-minute record

At the start of the year, Prime Minister Rishi Sunak proposed that everyone in the UK should study maths until the age of 18. This provoked a fair amount of derision on Twitter and elsewhere, from the kinds of people who generally have a reflex “Tories bad” response - despite the fact that the UK is something of an international outlier on this front. It was particularly funny to watch people having meltdowns about “data-entering robots” when the same policy had been part of Ed Miliband’s Labour Manifesto in 2015.

But of course, it’s one thing to declare an aspiration of maths until 18, it’s another thing to turn it into implementable actions, and there are lots of ways it could be done. I actually had an online chat in April with someone working on this, and I agreed to help offer ideas. But as I haven’t heard back since, the following is what I would have suggested.

General principles

I think there are two key things here: audience and delivery.

AI image generated in Substack, prompt “18 year olds learning fun maths”

1. There are essentially three categories of people: those who are taking A Level Maths already, those who are re-sitting GCSE Maths having missed their grades, and those who are studying non-maths subjects (whether other A Levels or a range of other qualifications including apprenticeships and vocational training). This policy is really aimed at the third group, because the other two are doing some form of maths already. In other words, the topics need to appeal across the board to a wide range of people, only united by the fact that they reached a certain level at maths but know they don’t want to do more. Tricky!

2. There is already a national shortage of maths teachers, and so whatever is proposed really needs to be deliverable by some combination of non-specialists and online resources. But I don’t think anything I’m suggesting should be outside the realm of people with qualifications in economics, science, geography and other subjects.

I think it’s important to say that this isn’t about A Level Maths for everyone, it’s probably a couple of hours a week to gain a wider appreciation of the landscape. I don’t even think it’s necessarily about doing calculations yourself. The analogy I would use is GCSE Latin: there you don’t need to be able to write in Latin, only to read and understand it. In the same spirit, I don’t think you necessarily need to know how to calculate a confidence interval yourself, but you should know what a confidence interval is and how to interpret it.

The final thing is that I don’t see this as a culture war Arts v Sciences thing. I think it’s vital for people not to be siloed, for “arts people” to have some understanding of maths and science and for “science people” to be able to write in coherent sentences and have some appreciation of arts and literature. In fact the final set of topics I suggest below are designed to reduce these barriers, and could be usefully studied by A Level maths students as well. This is perhaps somewhat in the spirit of a traditional “liberal arts” education, but with the boring Euclidean geometry taken out and replaced by something fun and interesting.

Topics

When considering what people ought to know about maths, I’m obviously clinging reasonably close to the fact that I just wrote a 75,000 word book on that subject! As a result many of these things below are in Numbercrunch (in a lot more detail!), but they are of course discussed elsewhere too.

I would tentatively split the subject up into three roughly equal-sized topics:

Topic 1: Numbers and growth

The COVID pandemic perfectly illustrated the need for a greater degree of streetwise thinking around numbers. For example, it became clear that many people would benefit from a greater familiarity with the principles of data visualisation, both as a consumer and as a creator. Being able to understand what stories a graph is trying to tell (and what tricks might be used by nefarious actors to use graphs for disinformation purposes) is a key part of 21st Century digital literacy.

Indeed, I would recommend that students practice a variety of the kind of “Numbercrunching” techniques that I advocate in the book, to try to build sense-checking radar into their consumption of news sources. Key skills here might include the ability to spot bogus or misleading statistics, the ability to contextualize them using the tricks that I described here, and practice in using tools such as Fermi estimation.

Finally (of course!) I’d argue for some coverage of different kinds of growth and evolution of processes. Many of the arguments around the pandemic would have been much easier if people were more familiar with the idea of exponential growth. However, this topic goes well beyond that, with the idea of multiplicative change being important more generally. It underpins the idea of inflation - and it seems clear that people aren’t as familiar as they should be with the idea that inflation falling and prices falling aren’t the same thing. Indeed, exponential growth and logarithmic scales arise in a variety of contexts, ranging from Richter scales for earthquakes to the pH scale of acidity, and being familiar with change across a range of orders of magnitude is extremely valuable.

Topic 2: uncertainty and randomness

A key skill in making sense of the world is understanding the roles of randomness and uncertainty. As I describe in Numbercrunch, many of the systems that we deal with in everyday life are complicated enough that it makes sense to think of them as somewhat random, and we need to be able to capture the role of chance in general.

For this reason, I think students should understand these ideas not just at the level of expected value and spread of outcomes (though these are important), but also through some appreciation of the role of extreme values in contexts such as climate change or natural disasters.

But in general, issues such as vaccine misinformation have shown us the importance of understanding probability. This is not just a matter of understanding simple coin flips (though this helps!), but it is key to understand how events relate to each other - that the probability of a heart attack given that you are vaccinated is not the same as the probability of being vaccinated given that you had a heart attack. So, even if they’ve seen it already, I think more intuition around Bayes theorem and associated issues is always a good thing.

One more topic where randomness matters is in terms of surveys and samples, and the uncertainty contained within them. Ideally students would already know that taking a larger sample will give a more accurate answer - but it’s important to understand the square root rule, whereby quadrupling the number of people interviewed only doubles the accuracy. And more than this, its important to understand that this depends on the sample being truly representative - having a larger but unrepresentative sample will only make you home in on the wrong answer.

Further, in this context, to make sense of clinical trials and opinion polls, it is important to be familiar with the idea that data is random, and so every reported outcome is likely to be somewhat imprecise. It is valuable to be able to understand the language of confidence intervals and p-values, and to distinguish between significant and insignificant results (while understanding that a large rise in relative risk may correspond to a small rise in absolute risk).

Topic 3: Maths and other subjects

This module would discuss synergies between mathematics and various arts and humanities subjects including history and literature. The former can be discussed not just through the famous data visualisations of Florence Nightingale and Napoleon’s troops, but also through topics such as Enigma and the German Tank Problem. The latter has been superbly discussed in Professor Sarah Hart’s recent book Once Upon a Prime, which could provide a reading list on its own. However (wearing my information theorist’s hat) I have to mention the links between Thomas Pynchon’s writing and the mathematical concept of entropy (it’s probably not a coincidence that the name of the character Tyrone Slothrop in Gravity’s Rainbow is an anagram of “Entropy or Sloth?” - though I’m not sure that book is necessarily suitable for a school audience!)

Another arts-based topic with deep and beautiful links to maths is of course music. The idea of the Pythagorean Comma, and the resulting issues around tuning, is something that everyone should know about. But there are many other mathematical links that would reward study: whether aleatory music (composed with a random element to it), mathematical patterns in the works of composers such as La Monte Young, Conlon Nancarrow and Steve Reich, or the use of unconventional time signatures by bands from Radiohead to Pink Floyd.

Finally, there are many mathematical links to visual art itself, for example through classical topics such as perspective and the use of the Golden Ratio to express perfect proportion. The work of M.C. Escher is of course full of mathematical tricks, such as exploring the effect of tiling in hyperbolic and spherical geometries. Tilings in general are a fascinating topic, with the underlying symmetries creating beautiful patterns in Islamic art, complemented by the development of non-periodic tilings by Roger Penrose and the recent discovery of the einstein tile. And of course I’m sure that students would like to learn some of the mathematical principles underlying the generation of AI art, like the one higher up in this post!

Of course, other people will undoubtedly disagree about the choice of topics. In fact, the more voices weighing in, the better (for example what might employers or university lecturers in other academic subjects like students to know?). But hopefully this gives a sense of some things that everyone (adults or 18-year-olds!) ought to know, and which could form the basis of a new syllabus. If you found this interesting, please do share it, and if you don’t subscribe to Logging the World already then please do so!