How to Teach or Things I’ve got Wrong

This is a draft of a presentation, an edited version of which I intend to share with my colleagues in September.

I am going to talk about how to teach, well, how to improve as a teacher by reflecting on one’s practice. In particular I am going to discuss how I’ve been wrong, and how my reaction to being wrong has helped me to become a better teacher.

There is a social and professional stigma to standing here and saying I have got things wrong. I am encouraged by our Headteacher’s advocacy of Matthew Syed’s book ‘Black Box Thinking’. I stand here in the spirit of this quote from the book:

“Learn from the mistakes of others. You can’t live long enough to make them all yourself.”

I mean this in the sense of the particular example I am going to use of how I have got something wrong, but mainly in the broader sense of us all as teachers being open about making mistakes, discussing them, and using them to inform and improve our teaching.

My expertise as a teacher is concentrated on how students learn maths at school. I will talk about how I’ve been wrong in one of my strategies, why it was reasonable to think I was right and how I came to realise I was wrong.

I’m a mathematician, and I share this attribute with Andrew Wiles. Andrew Wiles is the British mathematician who solved a problem that had perplexed the best mathematical minds for three centuries: Fermat’s Last Theorem. Lets watch him talk, I think in a pleasing way, about being a mathematician:

Andrew Wiles talks about Maths (first 1 min 45 seconds only)

For Andrew Wiles, for me, for some of my friends who studied maths at university, the metaphor of a dark room is perfect to describe how we best learn maths. I have heard myself say “the best way to learn maths is in the dark”. I’ll give you an example, I follow a number of mathematicians on Twitter, and sometimes they post puzzles. Here’s a good one:


The square on the right has an area of 1 unit. What is the area of the square made in the middle?

I like the puzzle, and its simplicity illustrates nicely the idea of ‘the dark’ in doing maths. There is very little information to point you in the direction of how to solve it. As Andrew Wiles says, one might bump into something before the illumination of how to solve it comes. The solution is only reached with a lot of difficult mathematical thought, and for me, it is this kind of difficultly that propels me along in extending my understanding of maths. I will come back to its solution in a bit.

Now, my thinking behind the strategy for my students learning maths that I am now saying is wrong, is based on this kind of mathematical thinking. If Andrew Wiles describes doing maths in this way, and I learn maths well this way, and even one of the objectives for the national curriculum is for students to learn to solve problems, then it seemed sensible for me to use problems in the classroom. I can describe the strategy that I now think is wrong, as a ‘problem solving’ one.

For example, if I want a Y7 class to learn about area, then this problem seemed to me to be a good one in comparable terms to the problems that help me to learn:

Fences 1Fences 2

I judged this to be a good level of difficulty for a Y7 class and assumed that students in Y7 learned in a similar way to me and Andrew Wiles. This is not a huge assumption to make, there is probably a smaller gap in ability in maths between a Y7 student and me, than me and Andrew Wiles.

However, now I think I’m wrong. The reason is because of the difference in thinking between ‘novices’ and ‘experts’. In this reading, me and Andrew Wiles are similar in our expertise, and both very different from a Y7 student.

Lets talk about novice and experts. One could think of the journey of learning anything: maths, the cello, history, driving, … as being one from where you start from a position of knowing very little if anything: a novice, to one where you understand everything: an expert. Although I can’t think of a situation where someone actually knows everything about a subject, even Andrew Wiles has maths that he is still learning. I can set these as poles in a learning spectrum:

novice expert 1

For the purposes of this discussion, I will crudely break this up into recognisable chunks based on our schooling system:

novice expert 2

David Didau states that “experts and novices think in qualitatively  different ways”, I’ll talk about who David Didau is a bit later. Now, a Y7 student learns maths in much the fashion of a novice, whereas myself and Andrew Wiles and any maths graduate learn maths more in the fashion of an expert, this is despite there being a closer gap in subject knowledge between me and a Y7 student than me and Andrew Wiles. It is the fact that I have enough subject knowledge to have strong existing mental schemas, or representations of mathematical knowledge to build upon, and Y7s, to a huge extent, do not. I can represent some differences between a novice and an expert:

novice vs expert

At some point on one’s journey towards expertise, the best way to learn becomes like that of the expert, but until that point, it is wrong to consider that novices learn in a similar way to experts. From the point of view of the novice, this is called ‘the expert reversal effect’: one learns best in a certain way until one gains a level of expertise perhaps typified by strong, comprehensive mental schemas, and then that way of learning in effect reverses.

Back to the puzzle about area. I posted it on facebook and some solutions were offered: Charles Melvin suggested it was one 25th of a unit, Martha, an English teacher who used to work here, thought it was one sixteenth. An old friend from school, Joel, who also studied maths at university came up with this answer:

puzzle answer.jpg

He was right although I’d got something slightly different: ‘7 – 4root3’. Now, you have to know quite a lot about maths to understand that Joel and I had got the same answer. He’s a nice guy and he said that my answer was ‘nicer’, even if his was presented in a ‘nicer’ way, and he’s right, but that is an expert maths conversation.

Albeit that this was a conversation on facebook, and some of the participants might not have given the problem more than a passing thought, I think it is helpful to highlight the similarities and the differences of the four of us. All four of us are thoughtful, educated adults, of the two who got the correct solution, and the two who didn’t, one from each pair (not me) studied at Oxbridge. So, if I want to exemplify my point about novices and experts, it is not general intelligence that is the issue, it is subject specific knowledge. Joel and I studied maths at university and are both in professions that require the use of maths, and the other two didn’t, and are not.

If I want my students to be good at ‘problem solving’ then I need to make sure that they have very good, secure subject knowledge. I think this is best achieved not by ‘problem solving’ lessons but by good teaching in the style of David Didau’s model for students becoming increasingly independent:

DD model.png


I want to finish this talk by clarifying that I don’t think this means that I’m doing a bad job. In fact I think a strength of mine as a teacher is my ability to reflect critically on what I’m doing. And the alternative to planning well thought out lessons where students solve problems is not an easier way of teaching. It’s not ‘teach by telling’, but perhaps it is ‘teach by explaining’.

To sum up:

1a) I am not a bad teacher because I make mistakes; I could be a better teacher by making less mistakes;

1b) Part of what makes me a good teacher is that I am reflective, and can acknowledge I make mistakes, which makes it possible for me to learn from them;

1c) Humility is important. If you’re thinking that I’ve made mistakes you would never make, well perhaps there are thing you are doing wrong that you haven’t considered.

2. I think I’m right, but I could be wrong in the things I have talked about today, if you disagree with me, I welcome constructive argument.

3. Back to David Didau. He has written brilliantly on the difference between novices and experts. One of the reasons I realised I had been getting things wrong was that I noticed on Twitter that he was saying things that didn’t make sense to me. I argued my case with him and it rapidly became clear that he knew far more than me. He even wrote a blog about one idea that I put to him.

I recommend reading his books. One of them has the rather apt title ‘What if everything you knew about education was wrong’.

Don’t take my word for it, this is what Dylan Wiliam has to say on it:

“In short, this is my new favourite book on education. If I was still running a PGCE programme it would be required reading for my students, and I can think of no better choice for a book-study for experienced teachers. Anyone seriously interested in education should read this book.”


Some notes

  • I also heartily recommend reading ‘7 Myths about Education’ by Daisy Christodoulou.
  • With respect to High Achieving Pupils (HAPs) I have argued that if one thinks a good way of teaching is good for HAPs, then it will be good for everyone. I think that I would now say that instead of HAPs, we should be thinking of novices and experts.
  • Related to the ideas expressed in this post is the idea of a knowledge based curriculum as opposed to a skills based one, Joe Kirby has written about this extremely well.
  • Jemma Sherwood has written well on the problem of ‘problem solving lessons‘.
  • I am not advocating that we explicitly teach students to learn from their mistakes, just that teachers do it in their own practice.
  • One of the images used in this post is from






5 thoughts on “How to Teach or Things I’ve got Wrong

  1. Good stuff Rufus!

    One point. You say, “At some point on one’s journey towards expertise, the best way to learn becomes like that of the expert…” I’m not sure if you know it but this is a very well trodden path in education research. It even has a name: the reversal effect. For novices, cognitive load is handled best when given worked examples, for experts the reverse is true.

    Because experts already possess effective mental representations, additional instructions are extraneous and add to cognitive load. Precisely as you’ve said, there’s a point at which cognitive load is handled better by reducing explanations. Exactly where that point might lie is probably different for every individual so trial and error is needed to proceed. My instinct says that identifying and effectively teaching threshold concepts could be the most efficient way to ensure novices learn sufficient subject knowledge to possess the characteristics of novices.

    Does that help? You might find the new book Nick Rose and I have written sheds some light on all this.

    Liked by 1 person

  2. […] I have a visceral aversion to the idea of committing something to memory before one knows why it is true. However, through my reading over the last year, in particular What if everything you knew about education was wrong? by David Didau, I am aware that it is always helpful to interrogate what feels right in education. I have been doing this recently, and many times my  strong feelings have been wrong, what has seemed to be self-evident has been exposed as erroneous when put under the microscope. […]


  3. […] and I had no time for inflexible knowledge, I’ve written about how I changed my mind on this here; and I consider it to be ideological on a systemic level: I did my PGCE at King’s College […]


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