*How I Wish I'd Taught Maths*is a clear exposition of pedagogical methods, informed by the latest cognitive science, but it is also sadly blinkered in its vision of mathematics.

Barton begins with some prolegomena, looking at the differences between expert and novice learners ("expert and novice learners don't just know different things, they think in a fundamentally different way from each other"), the limitations of working memory, the importance of keeping the focus on what is to be remembered, the non-existence of "natural mathematicians", and maths anxiety (which "anyone can suffer from").

Turning to motivation, he argues against real life contexts as motivation ("rarely the best way to go"), and suggests that, while intrinsic motivation is good, a combination of external rewards and sanctions is often more practical. More fundamentally, he argues that motivation derives from achievement and success, rather than the other way around.

He argues for explicit instruction rather than any kind of guided discovery or exploratory learning, that inducing cognitive conflict is only useful when students have the background necessary to learn from it, and that conceptual understanding should come alongside or after procedural fluency ("how before why").

There follows some general background on focusing thinking and improving cognitive performance: reducing distractions and removing redundant information, simplifying presentations and effectively combining talking, text and images, using goal free problems, and so forth. It is possible for students to be thinking hard, but about the wrong things.

Self-explanation is not just a diagnostic tool for teachers but a powerful learning tool. Barton explains how to create opportunities for students to self-explain and how to prompt them to do that effectively, with minimal interruption to a lesson.

Barton highlights the importance of worked examples, explaining not just how to use them effectively but how to design them, with careful choice of examples (and non-examples), suitable variation, and progression. A similar attention to the details needs to be paid in constructing practice questions.

A chapter on deliberate practice argues that processes can often be broken down into sub-processes which can be taught separately. This can be effective for novices even if it involves drills that are apparently detached from any actual exam question. And Barton argues for giving students access to answers, to make sure they are learning the right thing and to encourage self-explanation.

Barton argues that problem-solving is not a skill, that doing it needs domain-specific knowledge, and that the best way to teach both problem-solving and independent learning is to teach everything else solidly first ("develop inflexible knowledge"). Similar problems can be batched together. Students can struggle with a problem without learning anything.

"Purposeful practice" involves students approaching material they are already familiar with from different directions, practising the key procedures involved while at the same time making broader connections.

Formative assessment — and in particular regularly asking and responding to diagnostic questions — is central to effective teaching: it offers the only reliable way to track student understanding. And he suggests that carefully constructed multiple-choice questions offer the most efficient way to do this.

The final chapter looks at long-term memory and the role of testing. "Desirable difficulties" can help by making retrieval hard enough to reinforce long-term retention. Interleaving, or mixing up different topics rather than doing them in blocks, can also help. Well designed tests are a powerful tool for learning, not just assessment, and frequent low-stakes quizzes are particularly effective; a "pretest effect" means that it may even be useful to test students on material they don't yet know.

Each of these chapters, or in some cases sections within them, is pitched as a kind of story, with Barton moving from "What I used to think" to "What I do now". They are also clearly structured, and in most cases they can be read (or re-read) by themselves. And classroom implementation is kept central: generalisations are always connected to concrete practice.

Each section also comes with a "Sources of Inspiration", listing further reading and references. I've only looked at a few of these, but as far as I can tell Barton is accurately representing the cognitive psychology he draws on. (Though of course the bigger concern with this kind of psychology is that it might be possible to assemble equally compelling evidence supporting alternative conclusions.)

Though *How I Wish I'd Taught Maths* is aimed at secondary school
mathematics teachers, it's actually much wider in application.
The examples are from mathematics, but the broader ideas are relevant
elsewhere — it might be easier to transfer them to physics than to
history, but some of them apply even there. And Barton's examples
are from secondary school, but I'd say most of the ideas he presents,
other than the classroom specifics, are relevant to university
students and many of them to older primary school children.
Some of them now inform my informal teaching with my six year old
(though I think application to younger children is more limited,
with other cognitive constraints at these ages). And, completely
outside any classroom context, I found some of *How I Wish I'd
Taught Maths* relevant even to my self-study of mathematics —
I came away with a better understanding of why some textbooks work
better for me than others, for example.

*How I Wish I'd Taught Maths* also seems to me extraordinarily
blinkered. Barton is like a carpenter who has developed a fantastic
toolkit for shaping wood, but is so taken up with that that they have
forgotten that carpentry might also involve style or aesthetics —
and so focused on implementing the plans they are given that they
don't stop to consider whether those could be improved on.

Most fundamentally, there is no discussion of what is being taught or of what the purpose of teaching it is. Barton seems to have internalised "teaching to the test" to the point where he doesn't even feel the need to mention it, but his only measure of success appears to be how well his students do in their exams. And the procedural skills necessary for that seem to have largely replaced any actual mathematics.

The English Key Stage 3 National Curriculum states that:

"A high-quality mathematics education therefore provides a foundation for understanding the world, the ability to reason mathematically, an appreciation of the beauty and power of mathematics, and a sense of enjoyment and curiosity about the subject."

But most of that seems to have disappeared by the wayside, or been pushed into the future. Barton makes no suggestion that teachers should attempt to convey anything about the structure of mathematics or its history: "showing students that maths is an integrated and connected subject" is mentioned only in passing, as an incidental benefit of interleaving. And attempts to connect mathematics to its applications are dismissed.

Students' questions about the value of what they are doing are valid and important, even if answering them won't help them do better in their exams. They should learn what mathematics is and how it works and how it came to be, and about the role mathematics plays in the sciences and more broadly in human efforts to understand the world, not because it will motivate them to work harder but because those are important things for them to know. (And students are not the only ones who should be asking questions. The curriculum and testing regime are accidents of educational history, not God-given, and teachers should be criticising them so they can be improved.)

The limitations of Barton's approach seem clearest when it comes to problem-solving and independent learning. Yes, having the relevant domain-specific knowledge is necessary to understand problems, let alone solve them, but that's not everything; Barton seems to be falling back here on where his toolkit works. It's not that problem-solving and independent learning can't be learned or taught, but that this can't be done by the methods he favours: this is where students need conceptual understanding more than procedural competence, where they need to move away from direct instruction and manage with less guidance, and where intrinsic motivation really matters. More generally, Barton offers no advice on turning novices into experts. (The closest he comes is with purposeful practice, but there he insists that "the focus is always on the practice", making this a poor way of teaching broader connections or new ways of thinking.)

By situating his criticism of other ideas as self-criticism, Barton allows himself to make arguments from anecdote that would seem entirely unfair in any other context. That he once carried out amusingly ineffective performance stunts in class is hardly a generalisable argument against trying to make mathematics exciting or trying to enthuse students. Similarly, there are indeed artificial and confusing and ineffective ways to link mathematics to real world applications, but that's no more an argument against doing that than the existence of confusing and ineffective ways of teaching Pythagoras' Theorem is an argument against teaching that.

Motivation from success is great, but limited by the fact that much perceived success in a class is relative. Also, given any task — even sorting grains of sand by size or shape — it's possible to find some pleasure in doing it well. But some things are actually more exciting than others, and Barton doesn't seem to grasp that mathematics is one of those, ranking with art and literature as one of humanity's great creative domains.

Barton seems uninterested in whether his students enjoy what he's
teaching them or how many if any continue to study mathematics
afterwards. Someone taught entirely following the approach of *How
I Wish I'd Taught Maths* could come away capable of passing their
exams but with no idea what mathematics is or why anyone would want
to study it. In the unlikely event they are motivated to get that
far without other drivers (parents or extra-curricular inspiration),
I'm picturing them in an Oxford admissions interview, being asked
questions like "Do you have a favourite mathematician?" and "What's
a bit of mathematics that really excites you?", and not being able
to come with anything more than "I'm really good at integration
by parts". That may seem like an elite problem, but it's even
worse for less privileged children with no chance of experiencing
mathematics in contexts outside school: they are the ones most
likely to go away with a lifetime aversion to "mathematics" —
after forgetting, within months, everything they knew at sixteen
about solving simultaneous equations or doing angle calculations,
no matter how well-drilled into them that was.

So if you're a mathematics teacher, or otherwise involved in
teaching mathematics, you should read Craig Barton's *How I Wish
I'd Taught Maths*: there's a wealth of useful information in it.
But you should also read Paul Lockhart's *A Mathematician's Lament*
(an abridged version of which is freely available online), think
about what you are trying to achieve, and maybe find yourself a
balance between expertly hammering a bunch of procedural skills
into your students and passionately conveying to them the excitement
and wonder of mathematics.

December 2019

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