How I wish I’d taught maths

Craig Barton
Jan 24, 2018 · 11 min read

The following is an excerpt from the Introduction to my book “How I wish I’d taught maths: lessons learned from research, conversations with experts, and 12 years of mistakes”. The book is available from Amazon, and all good (and evil) book retailers, but my publishers, John Catt Educational, are currently offering it to all Diagnostic Questions and Medium readers at the discounted price of £15. Simply visit the John Catt website here and enter the code DQ15 where prompted on the SHOPPING CART page for the discount to apply. Thank you.

I thought I knew it all

As a maths teacher, I have been quite successful.

I know that is not exactly the most humble way to kick things off, but I feel it is important to establish the context in which this book is written. So please forgive me the indulgence of giving my own trumpet a heartily blow for just one paragraph.

At the time of writing, I am in my 13th year of teaching mathematics. I was one of the youngest Advanced Skills Teachers (ASTs) ever appointed in the UK, giving me the opportunity to work closely with teachers and students across dozens of schools. For the last 8 years I have been the TES Maths Adviser, with a responsibility for sharing the best practice and resources with the largest professional network of maths teachers in the world. I am a member of the AQA Expert Panel, responsible for giving advice on and assessment and support materials at a national level. I am the creator of two popular maths websites:, which has received over 20 million visits from teachers, students and parents from across the globe; and, which contains the world’s largest collection of multiple-choice diagnostic questions, and is used in over 80% of UK secondary schools and in more than 50 countries. My teaching has been judged as Outstanding in four successive Ofsted inspections in two different state schools, and each one of my classes has met or exceeded their challenging targets in national exams in every year I have taught. I have written two maths textbooks, had the honour of delivering workshops on teaching across the UK and all over the world, including Bangkok, Nanjing and Kuala Lumpur, and I have worked directly on a national project with the Ministry of Education and the British Embassy in Cambodia. I have even been asked to pose for the odd selfie.

I do not list my achievements to make myself feel good (well, not entirely, anyway). I do so to illustrate just how far it is possible to come without really having a clue what you are doing.

If someone had asked me to write a book about maths teaching two years ago, it would have been simply unrecognisable from the one you are (hopefully) about to read. It would probably have been entitled “The Beauty of Rich Tasks”, “The Power of Mathematical Discovery”, or “The Ultimate Guide to Teaching Problem Solving”, and no doubt featured a picture of me looking very happy with myself on the inside cover. It would have consisted of the open-ended tasks and ideas that I have developed over many years and used with thousands of students. It would have been full of me exclaiming how much my students enjoyed these activities, the insights they made, the problem solving skills they developed, the independent learners they became, and the results they achieved. I would have extolled the benefits of discovery learning, inquiries, projects, puzzles, student-centred learning, and of me as a teacher taking a back-seat (I probably would have used the phrase “the teacher should be the facilitator of learning” more than once). The one noticeable absence from this hypothetical book, of course, would have been any research to backup my claims. And if someone had pointed out this tiny omission, I would have replied with a patronising smile and explained “I don’t need research, I know it works”.

If really pushed for some external justifications for the practices I had always employed, I could very quickly point to Ofsted. When I was observed as part of our school’s inspection during the last week of term in December 2015 (Happy Christmas!), the inspector walked into my room to find a group of my Year 9 students stood at the front of the class, interactive whiteboard pen in hand, explaining the intricacies of rearranging algebraic equations to the remaining twenty-five students, whilst I sat at the back of the room playing on my phone. My “teaching” was judged to be Outstanding.

Teacher-led lessons were dull, uninspiring and bad for learning. Everyone knew that — well, apart from those teachers who refused to turn on an interactive whiteboard, plan adequately for their kinesthetic learners, or engage with Kagan group structures. But they were a dying breed. I always found it a bit weird that those teachers seemed to consistently get good results and students seemed to really enjoy their lessons, but surely that was despite their archaic, ineffective teaching methods, not because of them. To use the old Ofsted language, such teacher-led lessons were “satisfactory” at best. My lessons were better than that — flipping heck, I was better than that — and I saw it as part of my wider responsibilities to ensure I helped other teachers develop along similar lines.

My Mid-Career Crisis

Towards the end of 2015, I began recording the Mr Barton Maths Podcast. This show gives me an opportunity to interview people from the world of education who interest and inspire me, and I have been fortunate enough to speak to the likes of Dylan Wiliam, Robert and Elizabeth Bjork, Daisy Christodoulou and Doug Lemov.

If my guest is a practicing teacher, one of the first questions I ask them is to describe a lesson they have taught recently. I want to know every detail, from how they decide what to teach, how they plan, what resources they use, what happens as the students first enter the room, how do they assess, model and differentiate, how much independent and group work is involved, right through to the very end of the lesson. Above all, I want to know why they make these choices.

The types of lessons my guests describe vary widely. In 2017 Andrew Blair described a lesson built around an inquiry with the students determining to direction the lesson took, whereas that same year Dani Quinn talked in detail about a very teacher-led lesson encompassing drills and rigorous practice. The thought my guests put into the planning and delivery of their lessons is mind-blowing. Indeed, when I asked Kris Boulton to describe how he plans a lesson, it took him two hours to answer. And whilst there are key differences between the lessons themselves, they all have one thing in common — my guests can justify each and every decision they make.

These interviews made me realise that I could not. My only justification for the things I did was that it was obvious. It was obvious that teachers should talk less and place the responsibility of learning on the shoulders of the students. It was obvious that we should teach students to problem solve, and to allow them to discover things for themselves. It was obvious that we should to let students struggle. It was obvious because I had always done it that way. If I was a guest on my own show, I would have been reduced to a bumbling wreck.

I knew I needed to do something about this.

My guests often mentioned books and various bits of research they had read that informed their practises and methodologies. I had never really read any educational research, but I made a note of each reference and promised myself I would get around to them one day — maybe after I had planned another amazing lesson on problem solving.

The Day my Life Changed

What I am about to say next may sound ridiculous, but I promise if anything it is an understatement: the day I started reading educational research was the day my life changed.

It all began with Daniel Willingham’s Why Don’t Students Like School, which led me onto his ridiculously good series of Ask the Cognitive Scientist articles for the American Federation of Teachers and the associated research papers he quoted. I devoured them all with eyes wide open and jaw firmly on the floor. Next came Cognitive Load Theory, which I was introduced to via the incredible blog of Greg Ashman and our subsequent conversation on my podcast. I was familiar with the work of Dylan Wiliam through the development of my Diagnostic Questions website, but I read and re-read everything I could get my hands on from the great man. I didn’t think life could get any better — and then I came across the work of Robert and Elizabeth Bjork. Good God.

One thing led to another, and before I knew it I had read well over 200 hundreds books and research papers. I would wake up in the middle of the night, my head buzzing with ideas — my wife is incredibly understanding, but even she refused to indulge my enthusiasm for the Pretest Effect at 4am on a Thursday morning. I summarised all my reading and my practical takeaways on my website, and began mentioning them whenever I gave talks to teachers.

I started trialling out ideas in my classroom, and the effects were immediate. My reading of the Redundancy Effect (Section 4.7) and my subsequent conscious effort to shut-up more at key points in the lesson caused one concerned Year 11 to ask if I was feeling okay today. Even after a few weeks I sensed a change. We were getting through more work in class. My students seemed more confident with the concepts we tackled. I felt like I was actually teaching.

But the more I read and the more I experimented, the more questions I had. So, I read more, spoke to more people, tried out more stuff with my students and colleagues, and wrote more takeaways. By the end I had over 100 hours of interviews, 1000 PowerPoint slides and more than 100,000 words of notes written.

This book is the result.

The Intended Audience

I’ll be honest — this book has been created for maths teachers.

I don’t know if maths teachers are a bit of a funny breed (in fact, I definitely do know that we are), but every time I sit through some generic cross-curricular training session, I always end up thinking to myself “yeah sure, but how would that work in maths?”. I know this reflects incredibly badly on me, but as soon as I hear my colleague from Geography tell me about a problem-solving strategy that I could use when teaching solving quadratic equations, or a my colleague from P.E offer up wisdom on using group work that I could employ when investigating the finer points of conditional probability, I tend to switch off.

So, it would be incredibly hypocritical of me to suggest that this book has any worth whatsoever outside of the boundaries of maths teaching. English teachers may find strategies in here that they can use when studying Shakespeare, and there may well be nuggets that History teachers can use to when contrasting the validity of sources of the First World War, but I genuinely do not know if that is true, and I am not going to claim it is so. I believe maths is different (special, you might say), and this book has been written by a maths teacher, for maths teachers.

Whilst I am successfully losing readers, I will also confess that the majority of my thirteen years teaching experience has been spent in secondary schools working with students between the ages of 11 and 18. Whilst I have spent a significant amount of time in primary schools, working both with students and teachers, I am by no means an expert in that field. I hope the strategies I write about can be transferred into a primary setting — and the feedback I have had when teachers have trialled these is that they can be — but I want to be open and honest and say that they have mainly been written from a secondary perspective.

That said, I hope that no matter your age, experience, mathematical background, type of school, role, responsibility, or favourite number, there will be plenty in this book that you find useful.

The Structure of the Book

I have tried to make this book as user-friendly as possible, appealing to the time-poor teacher who has five minutes spare on a weeknight, through to the rare occasion where we might have a chunk of time free to do some reading on a Sunday morning or a holiday.

The book covers twelve key themes, with each theme further broken down into ideas. Each idea consists of four sections:

What I used to do
Here I explain my previous beliefs, where they came from, and how they impacted what I did in the classroom. At times this makes for painful reading, and I can assure you it was equally painful to write. Please don’t think too badly of me!

Sources of inspiration
Here I will give my sources of inspiration, which will be research papers, books, blog posts or interviews. The original versions of almost all quoted research papers are linked to directly from the following page on my website:

My takeaway
This is my summary of the key points from the sources of inspiration that are pertinent to the issue under discussion. You may well disagree with my interpretation, or simply want to dive in for greater detail, so I encourage you to study the original source and reach your own conclusions.

What I do now
This explains the ways I have changed my approach based on my takeaway. It is how I wished I had taught maths throughout my career, and how I plan on teaching it going forward. I want this book to be full of practical strategies that you can use right away, and this is the section where you will find these.

It is possible to dip in and dip out of this book as you please, jumping to the sections and issues that interest you most. However, I must warn you that I have thought carefully about the order I present these ideas, and it is often the case that later sections of the book will refer to ideas introduced and developed earlier on.

Ten things I used to Believe

In order to whet your appetite for what follows, here are ten things I used to believe were true for at least the first 10 years of my career, and which I no longer do. Each of these will be covered at some point in the book.

  1. The best lessons have little teacher-talk and lots of student-talk
  2. Where possible students should “discover” things for themselves
  3. We can teach problem solving
  4. Effective differentiation means giving students different work to do
  5. The maths we teach should be relevant to our students’ lives
  6. Students should always know why they are doing something before they learn how to do it
  7. The more feedback we give students the better
  8. Tests are predominantly tools of assessment
  9. Doing lots of past papers is the best way to prepare for an exam
  10. If students are struggling, then they are learning

Hopefully I am not alone in having believed these things.

Anyway, I have built this up far too much. It is time to get cracking. I really hope you enjoy this book. At times it may prove uncomfortable reading, and not just because of my poor grammar, but because it may challenge techniques and ideas you have held for years. You will certainly not agree with all of it — flipping heck, I am not sure I agree with all of it myself — but I hope it will make you think.

This is how I wish I’d taught maths.

That was an excerpt from the Introduction to my book “How I wish I’d taught maths: lessons learned from research, conversations with experts, and 12 years of mistakes”. The book is available from Amazon, and all good (and evil) book retailers, but my publishers, John Catt Educational, are currently offering it to all Diagnostic Questions and Medium readers at the discounted price of £15. Simply visit the John Catt website here and enter the code DQ15 where prompted on the SHOPPING CART page for the discount to apply. Thank you.

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