Is this the best we can do? Part 2: Diagnosis

‘A correct diagnosis is three-fourths the remedy.’

– Gandhi

This is the second post in a series. In Part 1, I reflected on the vast numbers of 16 year olds who finish 11 years of maths education unable to answer the straightforward questions needed for a grade C. I argued that this is and continues to be a problem, despite rising C-grade pass rates over the past 20 years. This post reflects more deeply on the causes behind why so many 16 year olds are in this state.

Matthew Syed’s ‘Black Box Thinking’ is a great read. He examines extraordinary successes in various fields like medicine, healthcare, aviation safety – like Team Sky, or Google – to find what they have in common. The answer? Such successes ‘harness the power of failure’. They don’t shy away from mistakes, but rather expect them, learn from them, and milk them for all the information they are worth. The aviation industry does this extremely well:

Pilots are generally open and honest about their own mistakes (crash-landings, near misses). The industry has powerful, independent bodies designed to investigate crashes. Failure is not regarded as an indictment of the specific pilot who messes up, but a precious learning opportunity for all pilots, all airlines and all regulators.

Some examples from Syed that bear this culture out: flight accidents are automatically transferred to independent investigators to look into, and those involved in the flight are protected to full disclosure, as whatever they say is inadmissible in court. The report is then circulated and freely available to any pilot in the entire world, ensuring that the entire industry can learn from the accident. (This is a striking comparison to the education sector in this country– but that would require another blogpost in itself…)

As Syed goes on to explain, this culture is particularly useful since the dissemination and circulation of information on failures can quickly reveal ‘signatures’ – particular patterns that keep recurring in various mistakes, problems, and accidents. For example, in one week in 2005, a whole host of reports of near-misses came from pilots landing in Lexington Airport. The investigators quickly cottoned onto the problem from the info from the pilots’ reports: lights had just been installed on an adjacent plot of land, which was then being mistaken for the runway. Within days (apparently, quite slow for the industry – the adjacent land didn’t belong to the airport) the confusing lights were taken down.

In the spirit of this ‘black box thinking’, I’ve thought about what problems and mistakes I frequently observe in my own classroom, together with the mistakes and honest frustrations I hear from colleagues across various schools. I’ve focused particularly on my experience with bottom sets, since it is these groups that usually fail to make grade C (though my conclusions may be more widely true too). From the mistakes with these classes, what are the ‘signatures’ – the common recurring patterns? What are the issues I keep experiencing and hearing about from other teachers? And, in the following post – what way forward can be discerned from these signatures, from these diagnoses?

Signature 1: Behaviour and attitudes

Bottom sets are often known for poor behaviour. They are thus difficult to teach. During my training, I observed a KS3 class, in an Ofsted ranked ‘Good’ school, where the teacher had to spend 20 minutes of the lesson cajoling several pupils into the classroom, while the other 10 or so pupils waited patiently to be taught and given some work. Those pupils seemed extraordinarily difficult. Yet what seemed even trickier was the following: a staff member later mentioned ‘you had to put a student on call in the first 5 minutes of your lesson?!’ – as if such a move by the teacher was too hasty. In my view, it wasn’t hasty enough: it was heartbreaking to see the other pupils waiting patiently for their teacher to come and teach them. If thousands of students spend years of their maths education in classrooms like this, in ‘good’ and worse schools, it is not a surprise that they fail to get grade C passes.

Even in excellent schools, a bottom set is typically where you’ll find the most difficult student in a year group, who can nonetheless have a similar effect upon the entire class.

Disruption is not the only behavioural issue. Bottom sets also feature many students with poor work ethic. Homework is a weekly struggle, and when it is handed in, much of it has been copied from another student. Such students are caught in a paradox: they have the greatest need to work hard outside of lessons, but the least inclinations to do so.

Attitude to learning within lessons is a related issue: many bottom set students do not try particularly hard in lessons, let alone outside of them. Besides general behaviour issues, the main cause I’ve observed is a history of failure in maths assessments: the attitude of ‘why work hard if I will do rubbish in the test anyway?’ seems to be the most debilitating. (Incidentally, I haven’t observed much of the apparent effect of ‘bottom set’ labelling that can’t be explained by the above reasons.) This history of failure is related to the second signature pattern:


Signature 2: They don’t understand it

Explanations and examples which worked perfectly well for other classes often then fall short for low-attaining students. Besides behavioural issues, there seem to be two main reasons behind this phenomenon.

Firstly, the architecture of maths. It’s like a Jenga tower. As Mark McCourt puts it in his excellent post:

At the top, the wooden blocks represent those mathematical concepts that we want the kids to be able to do at the end of their schooling, aged 15 or 16. The GCSE topics. But they are not failing mathematics because they don’t know these topics. They are failing because the blocks much further down, the foundations, are loose, wobbly or completely missing and so the whole tower tumbles…

And you know what, it isn’t rocket science. When kids understand and know the basic grammar of mathematics, they are able to accelerate through the other stuff.

So many students in our bottom sets are being taught high level topics whilst their foundations are shaky, or even completely missing. Foundations like arithmetic, place value, number operations, proportional reasoning. Of course they don’t get it when they’re in year 11 and put through intensive intervention schemes that presuppose some comfort with these skills.

At the same time, I’m not even sure if I know how to teach those basic foundations excellently – they’re the sort of thing I imagine primary school teachers know more about. I wasn’t trained particularly deeply in them. This links to the next reason:

The second reason students don’t understand: teachers’ widespread lack of familiarity with maths pedagogy. Much of GCSE level maths isn’t complicated, once it’s understood. But teaching it can be. And sometimes we simply don’t know that there are better and worse ways to teach certain topics. Sometimes we don’t even know there are options!

Take a look at these various sites, giving different methods for teaching several topics: Ratio; factorising harder quadratics; HCF; addition of negative numbers.

It’s great that these are shared online (and I partly share them because these methods might be new to you too!) – but why weren’t they a compulsory part of my teacher training curriculum? Or during CPD? And why are there no compulsory maths teacher textbooks with titles like ‘the most effective methods and examples for teaching ratio problems’? In comparison, my wife used to work on a database for lawyers which provides up-to-date, comprehensive and practical legal information. Need to clarify a point of law? Look it up there. Need to refresh yourself on the exact steps to take for a legal procedure? Again, head there. Doctors have similar books and websites, judging by my visits to the GP over the years. But for teachers: nothing, as far as I’m aware.

To take another example, I recently asked Twitter for help on this question, which was giving me and some of my bottom-set students grief:

From the replies (click to have a look) it was clear that many teachers also had difficulty teaching this concept. In other words, there’s a definite signature to this difficulty. But it was also clear that many teachers had great methods worth trying out – methods which I’d never seen or even thought of before. But there’s no repository, no go-to resource, for all this accumulated knowledge. Saddest of all, think too of the thousands of trainee maths teachers starting out this September who will again have to teach this topic to struggling students, without even knowing where to look for better methods.

Whether you know how to use the best methods seems to be a matter of luck. There’s no institutional design for it. It’s no wonder that some of our low-attaining students don’t understand various difficult topics, when UK maths teachers enter the classroom without being confidently armed with the vast array of best techniques. Speaking personally, here’s a small sample of topics that I’ve taught to bottom sets where I have a sneaking suspicion that my methods aren’t the best (given how much some of those students struggled): gradient (heck, straight line graphs generally); compound shapes; surface area; written division (heck, even just getting the divisor and dividend the right way around); rounding to significant figures; FDP; using a calculator; transformations; factorising; BIDMAS.

A final point. One possible reason why we don’t pine for the best methods is the fact that most methods are fine for the majority of students. We are lulled into a false sense of satisfaction with them. But this leads to rationalisations like ‘set 3 understood it when I taught it to them. It must just be the bottom set class and their behaviour/intelligence which is the issue’. But if we have that mindset, then it’s no surprise that only a bare majority of students get that coveted grade C. They were the very same students who could cope with our average pedagogy. And so we’ve effectively left the rest of our students behind. I think we can do better than this.


Signature 3: They don’t remember it

The final signature: even where behaviour is not an issue, where students understand what we’re teaching, where students are getting questions right in the classroom… students then fail to remember. So many times I’ve seen teachers (myself included) exasperated about low-attaining students who are completely unable to repeat what they did the week (or even the day) before. Now, assessment for learning within lessons is great and necessary. Yet it seems to be the dominant focus of much CPD. But what about when you assess for learning, you see they’ve got it, your lesson ends, and then the next week they’ve forgotten it? I’ve never received any substantial training on teaching and learning for retention besides ‘make sure you recap’ and ‘use starters’, and judging by the scarcity of blogs on the subject, it doesn’t seem to be a thing many teachers or schools have cracked either.

Instead, we keep on going with our schemes of work where we come back to teaching year 10s how to find the nth term for the nth time since they started secondary school. In other words, we teach according to schemes of work that assume that students will forget, merrily repeating almost exactly the same lessons to the same students, year-in year-out. Again, many students eventually get it by year 11 (and the year 11 intervention cycle does its best to hammer it in), but equally, many students evidently still don’t remember, despite being taught it so many times across the years. (If your experience is anything like mine, just think about fraction operations – a topic students have been learning since they were 9 years old).

Ultimately, we maths teachers don’t really know how, in our daily/weekly classroom practise, to turn our low-attaining students’ 30 minutes of understanding into a reliable long-term memory. Again, with this hurdle, it’s no wonder that so many of our students fail to get grade Cs at GCSE. Those questions are easy; we watched them do them in class; yet they just can’t quite remember them longer term.

Kris Boulton covers this question with much greater depth and eloquence, though mainly to raise the same question I’m raising. In his words:

Why is it that students always seem to understand, but then never remember?  I think many teachers are getting very good at teaching.  I think others are getting even better at helping students understand what they are learning.  I perceive a strong, and desirable push on empowering students with understanding.  I don’t see a similar push towards their remembering what they have understood.  I see an implicit assumption that understanding alone will do the job of memory.  Where people call for a greater focus on memorisation, I see that call being misinterpreted, and fought against. I think we need a dual strategy of building understanding and memory, else what was it all for?



So, to summarise my diagnosis of the signature of the problems behind our poor grade C pass rate/maths performance generally. For students who will struggle to get a grade C:

  1. They aren’t in a position to learn well, due to numerous behavioural issues in their classes.
  2. Even when behaviour is good, they often don’t understand what we’re teaching them.
  3. Even when they understand what we teach them, they don’t remember it long-term.

Once again: is this the best we can do?

This post sounds pessimistic. I don’t mean it to be. My aim is not to beat up on us maths teachers in the UK. But as Syed underlines again and again in Black Box Thinking:

… we have to conceptualise [failure] not as dirty and embarrassing, but as bracing and educative… failure is a part of life and learning, and that the desire to avoid it leads to stagnation… the bigger the fault, the bigger the improvement made possible by its revelation.

My next post, focusing on potential solutions, has a brighter tone. But,as Gandhi says, it wouldn’t be wise to offer solutions without first trying to diagnose the exact problems more accurately. When you look back at the problems and issues of your below-grade-C students and classes, are you seeing the same underlying signatures as I am? Comments very welcome.

Click here for ‘Part 3: success stories’


9 thoughts on “Is this the best we can do? Part 2: Diagnosis

  1. I loved this blog post and I’m sure many teachers will read this and think, “Yep, he’s definitely summed up nicely here.”

    What to do though? I think part of sorting a problem is just acknowledging it in the first place, so a major challenge is just getting teachers to agree with what you have written above. Many will still assume that low achievement and poor behaviour of the bottom set is entirely the fault of the teacher, or perhaps due to some unmet SEND need, or maybe even to be expected if the child comes from a working class background.

    Anyway, as I was reading this I was thinking about maybe secondary schools could try and poach specialist maths teachers (with A levels in maths, backgrounds in finance and maybe a maths-based degree) from primary schools who teach older children and who maybe have personal experience with teenagers, people like me for example. These teachers could then teach up to and including KS4 with ease, leaving the maths grad teachers to teach the A-level students. You could kill two birds with one stone (solves a recruitment issue).

    Also, using a Maths No Problem or Aspire maths textbook based on the Singapore Curriculum could help the children recap and practice to fluency, efficiency and long term memory of those basic maths facts and algorithms. IME, gaps go all the way back to basic number bonds, year 1 etc in those children in year 6, so they’ll be carrying that into their secondary career. I guess if you think about it, these same children currently messing about and not concentrating in your year 10 were doing the same in year 1, 2, 3 etc, therefore they will have gaps in learning dotted the ENTIRE way through the curriculum. In the primary school, leaders and teachers have to make the difficult decision to make this child miss their art/drama/music/singing/ lessons so that they can spend extra time catching up. These children then seem to forget everything, but there will be one or two every cycle who close that gap and then start catching up with peers. Perhaps the same decision needs to be made in secondary school.


  2. Pingback: Is this the best we can do? Part 1: the problem | mathagogy

  3. Pingback: Is this the best we can do? Part 3: success stories | mathagogy

  4. Pingback: Is this the best we can do? Part 4: better pedagogy through better textbooks | mathagogy

  5. Excellent piece.

    One part I’m not so sure about: ‘we teach according to schemes of work that assume that students will forget’. Whilst this may be true in the ‘pre-mastery era’, most schools have adopted ‘mastery’ schemes of work for the new curriculum. Spending more time studying topics in detail to a deeper level of understanding leads to the belief that this understanding will result in long term memory. As a result, I suspect most schools now have schemes of work that assume students will remember. This is equally worrying.

    Liked by 1 person

  6. Pingback: Is this the best we can do? Part 5: hard work & homework | mathagogy

  7. Pingback: Is this the best we can do? Part 6: the problem of forgetting | mathagogy

  8. Pingback: Is this the best we can do? Part 7: the spacing effect | mathagogy

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