The Understanding Paradox

No one wants students who don’t understand the meaning of their subject; we don’t want our students to merely regurgitate facts devoid of context, or for them to know how to answer questions in an exam yet have no idea what these things mean outside of an exam hall. And yet, on the path to understanding it is unavoidable that our students will often have to learn things that perhaps they feel they don’t fully understand, and will have to memorise things devoid of context. This is what I call The Understanding Paradox.

My view is that teachers attempting to bypass the memorisation and rote learning part of teaching in order to ‘teach for understanding’ can have disastrous consequences for students.

I want to illustrate my point by discussing trigonometry. You might remember it from school, and if you do, you probably remember SOHCAHTOA, the mnemonic device used by pretty much everyone in the UK to learn it. If you didn’t take Maths beyond GCSE or O-level then you might wonder what on Earth it was all about and I know many people feel that they have zero understanding of it.

I have heard it argued, including on my PGCE, that the idea that people who learned SOHCAHTOA reporting this lack of understanding of trigonometry makes the convincing argument that we shouldn’t teach it. Indeed, in a recent conversation on Twitter, a maths teacher said to me that anyone teaching SOHCAHTOA ‘should retrain to be a PE teacher’. There was a certain amount of opprobrium aimed at me for suggesting that it was a good method, and I empathise with that view as I used to feel similarly, mainly under the influence of my PGCE.

This is a faulty line of thinking though, using erroneous logic. Just because people who learned the method feel they don’t understand the topic, it doesn’t follow that we not teaching the method and teaching it in a different way will lead to understanding. The mnemonic has been used historically to teach the topic because it is a good method for students to learn trigonometry. This is backed up by research (although this wasn’t relayed to me on my PGCE). Learning it doesn’t hinder people from having a good understanding of trigonometry, what stops people is that they learn this and nothing else about the topic. It’s clear to me that in Y9 every teacher in the UK should embed the knowledge of SOHCAHTOA in the minds of their students, so that in Y10 they can expand on this knowledge and teach them more about trigonometry. This would avoid the situation that I often hear described by maths teachers

I try to teach them trigonometry properly through similarity and the unit circle and then those students who don’t get it, well I teach them SOHCAHTOA in the Y11 Feb half-term.

Here, by denying students the chance to learn about trigonometry in a straightforward, simple way, teachers risk embedding misconceptions about the topic. When it is realised in Y11 they don’t actually know much about how to answer questions about the topic, they then have to rectify this by masking the misconceptions with the straightforward method they could have taught everyone in the first place.

What all this comes down to is the word understanding

My contention is that there is no objective measurement of whether someone understands something. People use the word as a subjective, post hoc judgement that arises once a critical mass of knowledge is accumulated about a topic. This critical mass will be different for different people. Hence, teachers can’t aim at getting students to understand a topic. What they can do is ensure students know as much as possible about each topic. This is a curriculum and sequencing issue. For example, in trigonometry, teachers should first ensure students are proficient with SOHCAHTOA and then teach them more about the topic, such as how it relates to similarity, and the wave functions.

This might not feel right, but it is the right way to teach. Lets not deny our students the best ways to learn just because it doesn’t seem to fit with our ideals of how we would like them to learn.


Why study maths?

If a child has a love of any arcane yet google-able subject they should, of course, pursue it – but for, say, the unmathematical, the sheer misery of compulsory maths GCSE is, in a world of calculators on smartphones, as crazy as compulsory sword fighting or barrelmaking.

– Caitlin Moran

Caitlin Moran is a brilliant writer; I have read a lot of her work and her words are thoughtful, wise and very funny. In this quote, she expresses a still widely held belief in the myth that knowledge isn’t needed in the smartphone world – a myth that has been comprehensively demolished by Daisy Cristodoulou. She also chooses to deride the compulsory study of maths, and that is want I want to explore in this blog: the social and cultural norm of asking ‘why study maths?’

There is a subjective-objective paradox to maths. Objectively, maths underpins much of the modern world:

  • mathematical models used in economics, finance and epidemiology
  • the maths of computer science driving most of the modern work place
  • the maths of engineering used in our transport and construction

Subjectively, unless you need it for your career, there is no necessity to know maths to be part of the modern world, we can make use of all the expertise of it without needing to understand the maths behind it, just as we all use electricity without having to think about the science behind it. I think this is what Moran is getting at. Take the utility of maths out of the argument and are you left studying an ‘arcane’ subject such as Ancient Greek? Moran would have it so, and I can see her point. Both Maths and Ancient Greek strike me as being wonderfully rich and stimulating subjects to study; I can see the equivalence. But perhaps where they diverge is in their scope.

Let me offer up some of the greatest thinkers of all time: Plato, the father of philosophy, who had inscribed above the entrance of his Academy ‘let no one ignorant of geometry enter here’, and for whom the Platonic solids are named; Descartes, the father of modern western philosophy, who developed analytic geometry and for whom Cartesian Geometry is named; Bertrand Russell, one of the founders of analytic philosophy, and author of The Principles of Mathematics.

Mathematics has progressed by deeds of the most dazzling intellects in our history. It is a jewel in humankind’s cultural heritage. My case for learning maths rests on this argument for the intellect, undoubtedly it can help one’s career prospects, but lets appeal to its grander status.

So why the arguments against it? Perhaps we are not giving enough attention to the wonderful history of maths. Perhaps at KS3 we need to find time and space in the curriculum to teach its beauty and scope, to teach students the quality of their intellectual inheritance.





Learning by heart

This is an overlay of my previous post called Explicit Memorisation

A while ago I delivered a morning briefing to my colleagues, I spoke without notes and without a PowerPoint or similar. I had learned what I had to say by heart.

Today I want to talk about our students learning by heart, or memorising, or what is sometimes pejoratively called rote learning.

I’m going to start by reciting a poem I have learned by heart.

Now, there are two reasons I believe that learning by heart is important for our students:

  • the more you know, the easier it is to know more; and
  • increasing your knowledge base in your long-term memory can increase your IQ.

It was Hirsch who first made the argument that the more one knows, the easier it is to know more. He called this The Knowledge Deficit and the argument relevant to our secondary school goes something like this: students arrive at our school with different levels of knowledge, often a higher knowledge base is positively correlated with higher SES, and what happens is that the students who know more find it easier to learn even more, and conversely for the students who know less. Therefore, the gap between the knowledge of our students only grows whilst at our school. The gap that students arrive to our school with, with us giving all our students the same excellent education, inevitably grows larger by the time they leave.

I want to be clear here, I don’t want to close this gap by reducing the knowledge of our highest achievers, and I believe there’s a real danger of that happening with any strategy to reduce the gap. The important point is that we need to close any gaps by bringing the knowledge of those at the lower end up to those at the top, and we need to do it as quickly as possible on their arrival here.

I’ll bring in here the idea of IQ. IQ is positively correlated with a huge amount of success factors in life, and what is not widely known, is that it measures two types of intelligence: crystallised intelligence and fluid intelligence. Fluid intelligence which is perhaps related to our working memory can be thought of as the ability to solve novel problems or ‘think on one’s feet’. It’s been said that Fluid intelligence is almost impossible to increase. Whereas Crystallised intelligence, which perhaps is linked to our long-term memory, can be increased. By increasing the crystallised intelligence of our students we can increase their IQ. So what is crystallised intelligence? In short it is the knowledge base of our long term memory. The more it is stocked, the better our crystallised intelligence; it will include our vocabulary, and for example, mine includes the poem I recited.

So we have the idea that a well stocked mind makes it easier to learn more, and can improve one’s IQ. James Colver has spoken here before about memory and cognitive science, from where we get a nice definition of what it means to learn something: learning is a change in the long-term memory and the implications for what we do in the classroom: memory is the residue of thought. I would argue that everything we do with our students is with the goal of changing their long-term memories i.e. learning. In class, we get our students to think about what we want them to learn, and it this thought which becomes memory.

What I’m talking about here is slightly different though, I’m talking about closing the knowledge gap by pinpointing knowledge that we want every single student to learn, and then getting all students to explicitly memorise it. This involves the whole school embarking on a four-step process:

  1. Identify and organise the knowledge we want every student to learn
  2. Give this to students along with clear, precise ways in which they can memorise it
  3. Make time for the students to memorise it
  4. Give students frequent low-stakes quizzes on the knowledge at strategically thought out intervals.

My argument is that by influencing every single one of our students to commit the most important knowledge of our subjects to heart, we can close the attainment gap and make all our students more intelligent. It is the right thing to do.

Explicit Memorisation

Recently I’ve learned a poem by heart for the first time ever. I found it a hugely enjoyable thing to do, and also quite difficult. This now becomes part of my crystallised knowledge.

It is only recently that I’ve become aware that intelligence is thought of in two ways: fluid and crystallised. Nick Rose’s article Are these the 7 Pillars of Classroom Practice? put me on to Schooling Makes you Smarter by Richard E. Nisbett which discusses how IQ actually measures two types of intelligence:

  1. Fluid intelligence: perhaps the ability to solve novel problems using one’s working memory; and
  2. Crystallised intelligence: one’s store of knowledge about the world in the long term memory.

David Didau has (obviously) written about this and his blog is as usual essential reading. The second part of that blog puts forward the idea that although it is very difficult to increase one’s fluid intelligence, crystallised intelligence can be improved. It can be improved by knowing more things. Hence, school can and does increase the IQ of its students by getting them to know more.

This ties in with blogs by Joe Kirby which I have been reading over the last couple of years, advocating students memorising knowledge by using knowledge organisers, in fact his school have students memorising/ learning by heart knowledge for all their subjects every day. It is a knowledge based curriculum.

Back to my memorisation of the poem. Memorising it was hard. Learning things by heart is hard. To ‘rote learn’ as some may have it is not an easy option. Lets look at the detail of how I did it and how it can be done at school. Again, this is basically all from Joe Kirby. What I did was have the poem as the screen saver on my phone and followed these steps:

  • Look
  • Cover
  • Write
  • Check

It took me about a week, and now I can recite it whenever I want to.

I’ve tried getting students to do this at school with some success. They tell me it’s what they did at primary school and what they do to learn vocab in MFL. It makes sense to me that we as teachers should be making explicit strategies for memorisation with our students: explicit memorisation. In fact, in my role as leader of whole school numeracy I am going to be getting every single student to know a huge number of mathematical facts off-by heart. I will be helping to increase their IQ.

So what is the teacher’s role in this? (with apologies to the huge amount of people who have already advocated knowledge organisers)

  1. Make a list of all the knowledge that you want students to know by heart (I’ve seen Dani Quinn say that anything we as teachers know without having to reason it out should go in the knowledge organiser)
  2. Explicitly teach methods of memorising this knowledge, such as look-cover-write-check and the use of Quizlet
  3. Frequently test the students on this knowledge.




I’m free to be whatever I
Whatever I choose
And I’ll sing the blues if I want

–  Whatever by Oasis *

There are many great freedoms in school.

  • Freedom to have an education
  • Freedom to listen to teachers’ explanations
  • Freedom to be in a calm environment trying one’s best
  • Freedom to learn about the greatest that has been thought and said by humankind
  • Freedom to learn good habits towards hard work

And these freedoms give rise to other freedoms

  • The freedom to choose to study at university
  • The freedom to enter the world of work with good qualifications and habits
  • The freedom to enter the great discussions that are part of what makes us human
  • The freedom to build upon the knowledge of what we have learned at school

School is a truly wonderful place.

There is a price to be paid for these freedoms

  • Discipline so that teachers can teach in the way that is best for all students.

A pretty reasonable price I think.


* In my opinion this is one of the greatest live performances of all time, because of the beauty of the song, the simplicity of the arrangement and the strength of Liam’s voice. The fact that his voice deteriorated so rapidly is one of the world’s enduring misfortunes.

Automaticity: Entering the classroom

The teacher-student three-step.

With thanks to Doug Lemov and the excellent Teach Like A Champion 2.0

Three steps for a teacher to minimise wasted time and disruption at the beginning of the lesson:

  1. Meet and greet the students at the door
  2. Supervise the student three-step
  3. Have a ‘do now’ or ‘recap’ for them to do immediately

The three steps for the students is on a handy poster: three step entry

3 step entry

Implementing this will require me to:

  • explain why and how I want students to do this
  • believe 100% that it is the right thing to do
  • persevere through difficulties
  • guide the students through numerous practices of it until it becomes automatic for them.

Maths: Conceptual understanding first, or procedural fluency?

From Kris Boulton, the best thing I’ve read on maths teaching in a long time the real.

Should you teach conceptual understanding first, or focus on raw procedural fluency?

This question drives endless debate in maths education, but its answer is very straightforward: it depends.

I can demonstrate this quickly and easily with a single example, by teaching you how to multiply logadeons (e.g. 5-:-9,) something you’re probably not familiar with already.

Observe the following examples:

8-:-20     *   2-:-5       = 10-:-25

9-:-20     *   2-:-5       = 11-:-25

100-:-50 *   30-:-7     = 130-:-57

19-:-20    *   5-:-5      = 24-:-25

By this point, you can probably multiply logadeons together quite comfortably.  If you’d like to give it a go, try these two (answers at the end.)

30-:-17  * 4-:-3

17-:-0.5 * 9-:-2

But even if you can evaluate those correctly, you’re probably still not comfortable about all this; you probably don’t…

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Useful bits and pieces

Awesome summary and links for pretty much everything of interest to a teacher from Adam Boxer

A Chemical Orthodoxy

Below is a list of things I have read and found interesting and have helped me develop as a teacher. I’ve been collecting them over the last year or so and tried desperately to keep them in order. This is a work in progress and I’m going to try and update it when I can. I’ve marked everything that I think is super important with a * so you can ctrl+f for it. I’ve tried to keep my summaries as short as possible – the individual pieces will speak for themselves. You will note that I have avoided books too. This is because I don’t really find the time to sit and dedicate time to full books, I prefer to read stuff on the go, in the little snippets of time I find for myself here and there.

This is mainly a list for my own benefit. If anyone else…

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Necessary & Sufficient

What does a good education at school look like?

I think it’s essential to look at this question through the paradigm of what is necessary to a good education and what is sufficient.

Briefly, a necessary condition must be met for a good education, and sufficient conditions guarantee that a good education has been had. My argument here is that Education has been too focused on sufficient conditions, and as a result of this many necessary conditions to a good education are not being met. A good education cannot be had without the foundation of necessary conditions.

My view is that we must look to the necessary conditions and only once these are in place should we layer the sufficient conditions on top of these foundations.

So, what are the necessary conditions of a good education and what are the barriers to them being met? My subject is secondary maths, and this is the lens through which I will be answering the question.

1. Behaviour should be respectful and conducive to learning. The responsibility for behaviour lies with the students and the school as a whole. It is not the responsibility of the classroom teacher.

1b. (added 20-2-16) Homework should be easy to set and mark, ideally with a computer system that automatically marks the work and records the areas of strength and weakness of the students. Ideally, a whole school policy that picks up those students who don’t do their homework would be in place. 

2. Explanations. Teachers should carefully and clearly explain the curriculum to the students. The use of a good set of textbooks would help with this.

3. Hard work and practise. There is no royal road to geometry, and there is no easy way to learn. The responsibility for doing well ultimately rests on the shoulders of the students.

4. Teaching should be responsive. I think that all teachers engage in responsive teaching (a phrase coined by Dylan Wiliam) by which I mean they respond to whether students are learning what is being taught or not and change what they’re doing accordingly. It may be possible to find teachers that don’t but I think they’d be extremely rare. Clearly this can be done to variable levels of success but I think that teachers simply being allowed to exercise their professional judgement in their classes is a necessary condition of a good education.

5. Exam preparation. By this I mean the awareness that students are working towards a high stakes GCSE and referencing this in teaching. The use of past papers and mock exams is, I think, necessary to a good education.

There it is, 5 perhaps obvious necessities to a good education. What are the barriers to this happening?

1. The responsibility for good behaviour is often left with the classroom teacher. Teachers are told to plan ‘engaging’ lessons and to try to have good relationships with students who exhibit bad behaviour. This results in teachers planning for ‘engagement’ and pandering to the troublemakers as opposed to planning for learning and teaching to the top.
2. Teachers have been inculcated into the erroneous belief that they should minimise teacher talk and students should learn by doing and realising things for themselves . Teaching by telling has got a bad reputation even though it is probably the best way for students to learn at school.

3. The responsibility for students doing well is not often enough put on the students themselves. Too often, teachers are blamed for what is simply students not being prepared to graft to do well.

4. Teachers are put off making their own judgements about how to respond to their classes by tick box exercises such as confused whole-school AfL diktats that act as a proxy for teacher judgements.

5. ‘Teaching to the test’ is maligned as being a last resort when other teaching hasn’t worked as opposed to being an integral teaching process. Tests are disparaged as being damaging to students as opposed to a useful learning tool.

So there we go. Let’s start with the necessary conditions for a good education.