JULY 26, 2017
“If you wish to forget anything on the spot, make a note that this thing is to be remembered.”
– Edgar Allen Poe
“The palest ink is better than the best memory.”
– Chinese proverb
This is part of a series offering my views on some problems with UK maths education. The first part looked at the state of affairs with regards to GCSE and PISA results, the second part looked at my attempt at a diagnosis, the third part looked at pre-existing maths education success stories, the fourth part looked at how textbooks offer the largest potential to improve pedagogy across the nation, and the fifth part examined the role of hard work and homework. This part looks at the problem of memory in maths education.
In my career as a maths teacher, I flip regularly between deep job satisfaction and mild despair.
Don’t get me wrong – I absolutely love my job. Most maths lessons I’m teaching pupils some sort of mathematical process. Most of the time pupils then perform that process with aplomb. Whiteboards with correct answers go up, I’m happy, the pupils are happy, I go around and mop up the remaining few pupils who weren’t quite 100%, put up some answers, and by the end of the lesson I and my pupils are feeling pretty pleased with ourselves.
But then comes the test. Even if it’s literally a few days later, and even if pupils face literally identical questions, their answers end up containing every confusion & misconception under the sun.
Whilst marking tests, I’ve often felt like Sisyphus – the mythical Greek figure condemned to roll a boulder up a mountain, watch it roll back down, and repeat forever. We strain and push our pupils up a mountain of mathematical understanding on a particular topic. After hours preparing pedagogy & teaching materials, then 60 minutes of communal graft in the lesson itself, it feels like we might have gotten somewhere with that topic. Then… fast-forward to tomorrow / the next week / the assessment, and those same pupils will literally deny having even heard of that topic before. ‘Gradient? What’s gradient?’
Yet I can’t write this solely as a complaint. After all, throughout my career, pupils have usually followed what I have told them. Lessons have gone well. I’ve explained things; pupils (seem to!) “get it” and show they can do what I just modelled for them; we move on. So, if pupils are doing what I teach them, their forgetfulness must partly stem from a defect in my teaching. (Of course, I’m written previously on how hard work and homework plays a huge role in maths learning, so I can’t take sole responsibility for this failing: I constantly talk about the importance of independent revision, and many of my most forgetful pupils are those who give in shoddy, low-effort homework and never revise.) Nonetheless, there must be more that I can do.
The problem in a nutshell? I can’t think of a better way to put it than in Bodil’s phrase: ‘a lesson is the wrong unit of time‘. Much of our UK educational culture is focused on lessons: lesson objectives, lesson grading, showing progress within a lesson. As a result, we plan topics in lesson-sized chunks (‘addition this lesson, subtraction next lesson’), and we judge our success in lesson-sized chunks (‘the exit tickets show me they all learnt how to answer the key question this lesson’). But if we’re solely focused on lessons, then of course very little gets retained beyond that. Of course pupils forget. After all, as teachers, all we usually aim and plan for is for the pupils to ‘get it’ in the hour itself. To put the problem in reverse: we don’t teach with an eye for long-term memory; therefore pupils naturally don’t remember over the long term.
How big is this problem? And what can we do about it?
I know that my experience of mild despair isn’t unique. I’m sure that the majority of (non-selective school) maths teachers will know exactly what I mean. But in case you haven’t experienced it, choose one of your middle/bottom year 8 or year 9 classes, and set them this quick quiz:
- add 2 fractions
- multiply 2 fractions
- divide 2 fractions
- a) and b) find the area and perimeter of a triangle.
- a) b) and c) find the mean, median and mode of a set of 4 unordered numbers
- simplify 3c + 5c – 4 + 8
- simplify 6w × 3w.
- Round 3.449 to 1 significant figure.
- -3 + 1
The majority of these topics will have been taught again and again since Key Stage 2. These aren’t complicated questions. But if you’ve any experience teaching maths in a non-selective school, I’m sure you can smell all the inevitable misconceptions coming a mile off. My heart sinks simply in compiling this list. Despite the efforts of teachers over nearly a decade of maths education, pupils just don’t solidly remember what they’ve learnt. Therefore, it’s no surprise that UK maths numeracy is low: a 2013 OECD report found that ‘that a quarter of adults in England have the maths skills of a 10-year-old.’ Closer to the classroom, teachers experience the reality of forgetting every time a complex topic comes up. Teaching algebraic fractions to? “Well, let’s do a quick starter on mixed fraction operations… – oh dear. It looks like we’ll have to recap this…”
And this illustrates the first problem of forgetting: because we don’t teach with long-term memory as the goal, there’s a limit to pupils’ confidence in learning and using maths. For example, if pupils aren’t rock solid on the algebra in questions 6 and 7, then they won’t be comfortable in how to use algebra as a tool for modelling scenarios. If pupils aren’t rock solid on the averages in question 5, they won’t ever get how to use and understand statistics – and how not to be hoodwinked by misleading ones. Pupils will instead experience maths as a murky world of half-remembered instructions to follow – not a strong basis for becoming numerate adults.
The crucial cognitive structures of the mind are working memory, a bottleneck that is fixed, limited and easily overloaded, and long-term memory, a storehouse that is almost unlimited. So the aim of all instruction should be to improve long-term memory; if nothing has changed in long-term memory, nothing has been learned. Effective instruction is a simple equation: it minimises the overload of students’ working memories whilst maximizing the retention in their long-term memories.
Yet despite this research, and despite the consequences, the vast importance of memory is neglected. Year after year, maths teachers in staffrooms across the country will grumble that their pupils can’t remember the same fundamental skills that they’re meant to have learnt before. More worryingly, it seems that we often take forgetting for granted. I’ve seen it in comments and attitudes – my own, as well as in others around me: ‘ah, they lost loads of marks on surface area in the test; never mind, we’ll come back to it next year.‘
But this attitude always unnerved me: if pupils would simply come back to the topic afresh next year, why did we even bother spending the curriculum time to teach it this year? After all, the kids who understood it this year would’ve understood it if we’d left it to next year (or even the year after that). And as for the kids who didn’t understand it this year, what (besides getting a better teacher?) makes us think they’ll get lucky and understand it next year? We surely could have planned our schemes of work more effectively. This raises the second problem of forgetting: because we don’t teach with long-term memory as the goal, curriculum time is not used efficiently. This is especially concerning in light of the demands of the new GCSE, the extra content that must be taught, and the increasing complexity of questions. How are we meant to teach complex questions drawing on multiple topics, if pupils barely remember topics by themselves?
Of course, I don’t want to overstate it. Many pupils leave year 11 knowing a lot more maths than they did in year 7. In fairness to the staff room grumble, it is sometimes true that many pupils, who didn’t understand a topic one year, will find themselves grasping that topic in later years. But in my experience, this mainly applies to the pupils who’ve had a good experience of maths in primary school, who have supportive family backgrounds, and have the most highest intrinsic motivation. What about the rest?
As I’ve explored throughout this series, under the old GCSE specification there was a persistent ~40% of 16-year olds each year who couldn’t handle the minimal skills required for a C (the “good ol’ days” where grade C required nothing else but basic procedural recall). Clearly, a sizeable chunk of pupils each year have experienced futility themselves: namely, a secondary maths education where they have remembered very little mathematics. This is little surprise: the weakest pupils are those who have the most limited fluid intelligence and working memory capacity, and if they are to have any chance of succeeding in maths, they must rely heavily on their long-term memories of secure foundational mathematical knowledge to compensate for slower processing speed.
But if we don’t focus on long-term memory? Well, this is the third problem of forgetting: because we don’t teach with long-term memory as the goal, the weakest pupils are adversely affected: they forget (or even totally fail to learn) nearly everything we try to teach them.
So, why don’t we as a profession do more about helping pupils remember? Several reasons come to mind. Firstly, the vast majority of time in my experience of ITT and CPD was allocated to improving one’s day-to-day classroom practice – the craft of behaviour management, explanation, task design – together with broader subject knowledge. Notice what’s missing: in my whole training and CPD, I had a grand total of a 1 hour seminar on memory – and that was an optional seminar that I happened to choose. Aside from that, I had no training on the science of memory, or how to get pupils to remember. We don’t even seem to recognise it as a big problem.
This is reflected in the quality of advice I received early on in my career, when I told tutors and colleagues about my persistently-experienced problem of forgetting: just get pupils doing recap questions as starters, and revision lessons before tests. Yet this didn’t seem enough. Recap starters face these two problems:
- A bias towards recent topics. It’s easy to revisit last lesson’s topic, since it was so fresh in my memory. But topics from the previous week/fortnight/half-term? They were lucky if they came back out ever. And revisiting something just once isn’t a good recipe for remembering it in the long run.
- A lack of systematicity: even when I planned to revise something from a previous half-term, the widening of the time-frame made it very hard to know what to pick to revise. I would pick something that struck me as important, and we would revisit it briefly. But what of the other topics I passed over? Why choose one topic and not another?
Revision lessons before assessments also felt ineffectual: I usually had to re-teach the entire topic from scratch again as if we’d never studied it – so what was the point of studying it in the first place? – and even when I had explicitly ‘revised’ a topic with a weak class the day before their assessment, their seemed to be no consistent effect of that revision on their ability to get that question right on the following day’s assessment. This leads to the fourth problem of forgetting: because we don’t teach with long-term memory as the goal, we’re left with patchwork and ineffectual methods of getting pupils to remember.
To summarise: current maths pedagogy hardly ever focuses on teaching for long-term memory. Yet if this is not the goal of all our teaching, these consequences upon our pupils follow:
- It damages their mathematical foundations, having never secured them in the first place, and thus places limits on their confidence in maths and how high their mathematical understanding can go.
- Teachers are forced to reteach (essentially from scratch) the same topics, year after year, which is a highly inefficient use of curriculum time.
- The weakest pupils, as those who find remembering particularly difficult, are adversely affected and forget most of what they are ever taught in lessons – thus giving them very little hope of securing a decent mathematical education.
- We don’t use or even know proper methods of getting pupils to remember what they have been taught.
These are big problems. As I’ve asked repeatedly in this series, is this the best we can do? Thankfully not. As mentioned at the start, these problems are but the inevitable consequences of an education system focused on individual lessons – lesson observations, lesson grades, lesson plans. Recent years have seen some encouraging changes in this regard, particularly in the fact that Ofsted no longer grade observations. Yet unless we take advantage of this shift and implement a deeper focus on teaching for long term memory, pupil outcomes will not improve. Is this the best we can do? In terms of teaching for memory, I firmly believe no. I long to see a shift of focus towards memory in maths pedagogy; a refusal to accept ‘ah they’ll come back to it next year’; higher expectations of what we can do for all of our pupils.
How? My next post, on the spacing effect, looks at one way of improving memory.