How I Wish Id Taught Maths Lessons learned from research, conversations with experts, and 12 years of mistakes

How much do I want to read more? 8/10

What a powerful book. It question everything about teaching (maths).
This guy thought he already had teaching all right, until he read a lot of books and articles. This changed his knowledge, his beliefs, his psychology of teaching.
The nice thing is, for each of his old belief, he list the references he read about, his takeaway, and how it changed his teaching.


Craig is the creator of two of the country’s most popular maths websites:, and


"I know of no other book that presents as much usable research evidence on the dos and don’ts of mathematics teaching in such a clear and practical way."
-- Dylan Wiliam

"It makes us wish that young people the world over might have the good fortune to find themselves in classes that incorporate Craig’s insights."
-- Robert A. Bjork and Elizabeth L. Bjork

FOREWORD - Kristopher Boulton

Ray Dalio is the of the world’s most successful billionaires, and he makes the point that ‘smart people are the ones who ask the most thoughtful questions, as opposed to thinking they have all the answers’.


I thought I knew it all

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.
I would have extolled the benefits of discovery learning, inquiries, projects, puzzles, student-centred learning.

My mid-career crisis

I have been fortunate enough to speak to the likes of Dylan Wiliam, Robert and Elizabeth Bjork, Daisy Christodoulou and Doug Lemov.
one of the first things I ask of 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…

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 let students struggle. It was obvious because I had always done it that way.

The day my life changed

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.
I was familiar with the work of Dylan Wiliam 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.
before I knew it I had read well over 200 books and research papers.

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 that day.
My students seemed more confident with the concepts we tackled. I felt like I was actually teaching.

The intended audience

this book has been created for maths teachers.

The structure of the book

The book covers 12 key themes, broken down into ideas. Each idea consists of four sections:

Ten things I used to believe

here are ten things I used to believe were true for at least the first ten years of my career, and which I no longer do.

  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.

A word of warning

Despite my best efforts to avoid them, the selection of sources of inspiration and my interpretation of them have undoubtedly been influenced by the following:

The Confirmation Bias

The tendency to search for, interpret, favor, and recall information in a way that confirms one’s preexisting beliefs or hypotheses.

The Halo Effect

When an observer’s overall impression of a person, company, brand, or product influences the observer’s feelings and thoughts about that entity’s character or properties.

I simply love the writings of Daniel Willingham, John Sweller, Dylan Wiliam and the Bjorks. I think those people are geniuses. They are my heroes.
my first instinct when someone criticises their work is to go into denial.

The Dunning-Kruger Effect

Where persons of low ability suffer from illusory superiority, mistakenly assessing their ability as greater than it is.

For about ten years I was happily teaching in ignorant bliss. I firmly believed I was an Outstanding teacher, I knew what was best for my students, and I had no real interest in anyone telling me otherwise.

The more I have read, the more people I have talked to, and the more things I have tried out, the more I realise just how clueless I really was.
I am a firm believer that the only way to improve is to confront your weaknesses head-on, acknowledge them for what they are, and strive to improve. This book is my attempt to do just that.


1.1. A simple model of thinking and learning

What I used to think

in 12 years of teaching, I never really considered how my students think and learn.
kind of like a doctor not being overly concerned about the workings of the body.

My takeaway

How students think

For Willigham (2009), thought occurs when you combine information in new ways, and successful thinking relies on four factors: information from the environment, facts in long-term memory, procedures in long-term memory, and space in working memory. If any one of these factors is deficient, thinking will likely fail.

Long-term memory

For Mccrea (2017), long-term memory may be thought of as our mental model of the world: a map we construct ourselves.
Long-term memory represents what we know and who we are and informs how we act.
All of the information stored in long-term memory resides outside of awareness, lying there patiently until it is needed, before entering working memory and becoming conscious.
=> knowledge.

When I say I want my students to have a good knowledge of fractions, I mean I want them to know relevant facts, such as what a numerator is.
I also want as much of this knowledge as possible to be automated, so students know instantly that a half of 24 is 12 and that to add fractions you need a common denominator, without imposing any strain on their working memories.

This knowledge is stored and organised in long-term memory in schemas.
There is no limit to how complex schemas can become, or how many can be stored in long-term memory.
long-term memory is not just a vast databank of knowledge, but an integral component of all cognitive activity.

Working memory

Working memory is best viewed as the place where thought occurs. It is all about the here and now. Unlike long-term memory, working memory has a finite capacity.
for instruction to be effective, care must be taken to not overload the mind’s capacity for processing information.

Thinking takes place in working memory, focused on the interplay between the environment and what we retrieve from long-term memory. The more knowledge we have stored and organised in long-term memory, and the more of this knowledge that is automated, the easier thinking is and the more we can think about. Knowledge helps students take in more information, think about new information, and remember new information.

How students learn

learning is ‘a change in long-term memory’.
if nothing has changed in long-term memory, then nothing has been learned. Working memory is the vehicle that instigates this change.

knowledge ‘is constantly evolving and decaying as a result of our thinking and interaction with the environment. Our long-term memory is more like a forest than a library’.

learning happens when people have to think hard , and indeed making changes to long-term memory is likely to be effortful.

From working memory to long-term memory

Students learn new ideas by reference to ideas they already know.
A learner holds information in working memory, and then makes connections between that information and knowledge already stored in long-term memory.
‘understanding is remembering in disguise’
it is taking correct old ideas from long-term memory, getting them into working memory, and rearranging them in a new order to make new connections.

From long-term memory to working memory

When information is successfully retrieved from long-term memory into working memory, its representation in long-term memory is changed such that it becomes more accessible in the future.
Using our memories changes our memories. ‘retrieval is a powerful memory modifier’
Thus, the act of retrieval can result in learning.

What I do now

Thinking how can I design my teaching to ensure:

  1. …my students are thinking about the right things?
  2. …what they think about is transferred to long-term memory?
  3. …once it is in long-term memory, it can be accessed?
  4. …it can be applied successfully to different situations?

1.2. Experts and Novices

What I used to think

I believed I could help novice learners become experts simply by treating them as experts.
I would model how experts think, give them the same work as experts, and encourage them to struggle.
‘I am going to treat you like adults, so you had better behave like adults’.