Book review: A Compendium of Mathematical Methods

Mathematics can be riddled with snobbery and elitism. But this book is a lovely tiny step towards the democratisation of the subject, says Lucy Rycroft-Smith

Lucy Rycroft-Smith

A Compendium of Mathematical Methods

A Compendium of Mathematical Methods

Author: Jo Morgan
Publisher: John Catt
Details: 344pp; £18.00
ISBN: 978-1912906604

The past three conferences and roundtables I have been to – high-level, future-of-maths-education-type shenanigans with smoked salmon and good coffee – have all been peppered with the word “non-mathematician”. 

Whether it’s part of self-deprecating confessions, a way of snidely categorising parents, or just dismissively referring to many pupils, this word, defining as it does by exclusion and deficit, bothers me.

What does it mean to be a mathematician? Mathematics and mathematics education is riddled with snobbery. Time and again, I have witnessed insecurity play out as contempt, anxiety as egotism, privilege as smuggery. 

I don’t think most of the people who display this behaviour are terrible humans, just that they are desperate, and all too often desperate contexts call for disparaging measures.

I suspect that we say we are not mathematicians in a panicked attempt to prevent someone else from thinking it first.

Reserved for the elite

I once had dinner with a well-known mathematician and asked him the same question. “We need to reserve the term ‘mathematician’ for the elite,” he said. 

“Because otherwise…anyone could be one, right?”


Anyone can be a mathematician. That doesn’t mean that everyone is. But, if we can’t start with the premise that a mathematical identity should be open to anyone, then we are not holding to the principle of equal access. 

But this is not the central issue: it is a second principle, that of low threshold, that I can see is of difficulty to many. 

“But I worked extremely hard to get here,” they might complain. “Why should I throw down the ladder now? Isn’t this about standards?”

Part of their identity

Three reasons. One: we need maths teachers, desperately

Two: if you ask people what started them on the path to their future career, they will almost always tell you stories of that teacher who told them they had talent in something, who helped them believe that it could be part of their identity and reconciled successfully with their other characteristics. 

We know that girls and minority groups are being kept out of, and discriminated against in, STEM careers. If we don’t start welcoming them early, giving them the same kind of encouragement and support those with advantages have had all their lives, we’ll never know what might be possible. 

Three, and most importantly: we need to be consistent with our definitions. If part of being a mathematician is that you are purposively doing maths then it follows that many more people should rightly be called mathematicians than any of us may have thought. 

But the verb matters the most here. What could it mean to do maths, and how do mathematicians do it in a way that may be distinct from others?

As simple as paying attention

To be an artist is to explore, create, solve problems and critique. To be a scientist is, perhaps surprisingly, the same. It is about grain size. It comes down to paying attention: it is that simple. 

We can all be an evanescent artist or scientist by simply by choosing to pay attention. Further, we can all practise these things, choose to do them more and more, and make them habits, becoming a habitual artist or scientist. And these identities intersect in beautiful ways: paying attention makes us pay more attention.

To be a mathematician, then, is also to explore, create, solve problems and critique. Paying attention to maths – and not just doing it, but doing it with intent, thought, purpose, maybe sometimes love – makes you a mathematician. 

Importantly, it may mean stepping back and letting others be mathematicians when it feels irresistible to laugh or crow or gloat instead. 

To be part of a mathematical community should be to remember how it feels to find something for the first time and allow everyone that pleasure. It should involve being humble and kind and acknowledging that even the most simple of mathematical concepts – in fact especially those – need the most care and attention to understand.

A lovely, tiny step in the democratisation of maths

In the back of A Compendium of Mathematical Methods, Jo Morgan says: “Imposter syndrome is a very real thing for women in mathematics, and I’m fighting it with all my strength.” 

I had to swallow hard at that. According to my definition, Morgan is very much a mathematician, and her book about the beautiful detail of mathematical methods strongly encourages you to be the same – “read it…with a pen and paper to hand so you can have a go”, she says in the introduction. 

This is not a small contribution to the world – it is another lovely, tiny step in the democratisation of mathematics, and the fact that it is written by a practising maths teacher who also happens to be female is a source of great joy to me. 

She has been a wonderful ambassador for mathematics for some time, with her quality resources, and this book adds to her efforts on improving mathematics teaching by actually doing something useful to help. 

Because this delightful book is all about paying attention to mathematics: the kind of mathematics we might have taught a thousand times and stopped paying attention to. The kind of mathematics that feels like ritual or rote. The kind of mathematics that some elitists might deride as “easy”, but in fact is the hideous obstacle causing hundreds of thousands of pupils to fail their exams each year


Many of the “different” methods shown here are separated only by the particular way they are written, or tiny mutations in the way they notate the deep structures underneath, and Morgan says this clearly: “It is simply the surface structures and process that differ.” 

As we begin to dig into the methods, with Morgan as an expert guide, those fine details start to feel more and more interesting, and the historical notes add to this interest. 

Demystifying the building blocks of mathematical computation by examining them in detail is really, really important work, and I take my hat off to Morgan for doing it.

This book can and should act as an accessible text to teachers, parents and, indeed, pupils. 

It’s not exhaustive, but it covers some important elements of working on and with the four operations, equations, surds, factors, percentages and fractions, taking as its rough basis the English secondary mathematics curriculum.

I have a few small criticisms. Firstly and most importantly, and because this is a book looking at historical methods, some of the language used could have been looked at with a more critical lens. I would have found an index very useful here too. Also, it is, like almost all education books, not beautiful. 

Finally, the work of connecting the concepts in the book is left as an exercise to the reader. 

I am biased – this is the bulk of my professional work at Cambridge – but I would have liked more mapping, more commentary and more insight into how the whole fits together.

Reading this book alone won’t make you a mathematician, but reading it in the way Morgan intends will certainly help. 

Lucy Rycroft-Smith works in communications and research for Cambridge Mathematics and is the presenter of the Tes Maths podcast. She tweets @honeypisquared

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Lucy Rycroft-Smith

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