I’ve had a week to think about what direction I’d like to take with this blog, and what questions or issues it should take on. Do I focus just on mathematics? Do I look at systemic issues? Do I dig into the theories of learning? Over the week I’ve lined up maybe seven or eight specific things already that I’d like to talk about, and in general they’re fitting into around three categories:
1) General pedagogy
2) Specific to maths
3) Maths focus, but with a point or conclusion that could be relevant to other subjects (keep an eye out for something involving Diophantine equations and relativistic quantum mechanics in the weeks to come…)
In some cases there are ideas or knowledge that I’ve found useful, and so would like to share. In other cases it’s more about provoking questions and discussion. Since the first post was fairly general in nature, I want to carry on in that vein for the next three, closely related posts. Education theory also underpins a lot of what I do in the classroom, so it will be useful to refer back to in later posts. From there I might detour into some things specific maths for a while. The next three posts will all relate to memory. Preamble over, let’s get to it.
You can’t spell preamble, without ramble.
A few weeks back a fellow teacher posed a question to my form group that I probably should have spotted the answer to much quicker. Having had a difficult time around her own exam years, she didn’t have all the qualifications she might like, and has faced barriers to her progression as a consequence. I’ve known her work hard towards achieving the C grade in maths that she needs to further her career. She told the group that she understands the mathematics, so, given that, what’s her problem, why has she not yet passed the exam? The kids tried a few guesses; it’s too hard, you don’t know what to do, you haven’t seen the questions before – they hadn’t quite picked up on the first point she made: she understood the mathematics, no problem there. My first guess was that the arithmetic might be the barrier. It was much simpler still, and I should have picked up on it more quickly: she understands everything, but then gets to the exam and remembers too little.
Several months ago another colleague told us what a Year 10 girl had said: “What’s the point of learning this?” Fairly typical question unfortunately, but she didn’t mean ‘when am I ever going to use it?’ she was pointing out that in a week we’d be on to a new topic, and she’d forget everything she could now do, so where was the point in learning it?
Why do we keep forgetting so much of what we learn?
There seems to be fervour in education around ‘understanding’, deep understanding, ‘relational’ understanding. ‘Understanding’ has become a much loved buzz word. There’s nothing wrong with that; on the contrary understanding is certainly what we should be aiming for. If I sound disillusioned, if it feels like I’m detracting from its pursuit by referring to understanding as a ‘buzz word’… it’s only because I see so little of that understanding forthcoming, despite the heavy rhetoric and valiant efforts. Perhaps more importantly, I see little that I think would lead to understanding, or worse, a possible institutionalised misconception of where and why understanding is useful. It’s beginning to feel like the rhetoric is in pursuit of understanding at all costs, ironically even at the cost of understanding; blinkers down, off we go! Given how focussed we are on understanding, shouldn’t each child by now be a micro-genius?
Why is understanding important? There are several responses to this. This list is not exhaustive, but if you actually understand something, you’re more likely to be able to abstract and apply it to any new context, rather than just exam questions, say. You’re more likely to be able to twist and manipulate what you understand to form new knowledge, and new understanding, accelerating learning. There’s a further reason, sometimes implicitly understood, other times made explicit: ‘If you understand something, you are more likely to remember it.’
I’d like to set up a straw man: ‘If you understand something, you are certain to remember it.’ This is easily knocked down. Just think of all the lectures you have attended, understood perfectly, and of which you can now consciously remember none. It doesn’t have to be lectures; how about simple TV documentaries? No doubt you’ve seen some, no doubt you had little problem understanding the content, no doubt you’ve forgotten much of it. Books…?
Kris 1, Straw Man 0
It still seems reasonable to say that understanding leads to better memory, but what if it’s not enough? What if memory is a function of understanding and something else? I actually don’t imagine this is a controversial statement. I know several experienced teachers who make a point of revisiting old topics in new lessons wherever they can, though doing so is not part of the curriculum. Who’s going to speak out against the importance of revision? My concern is that all this seems very ad hoc. Individual, experienced teachers, sometimes maybe doing something that might aid memory.
What I’d like to challenge is what I perceive to be the institutional focus on understanding, to the detriment of memory.
Daniel Willingham writes a fascinating article on memory here. He also talks about using narrative to help improve memory in his book, Why don’t students like school? Importantly he talks about the distinction between forming memories, and then later accessing them. In brief, we form memories by thinking about something a lot, but there’s then a separate job to do of building what he calls ‘cues’ to be able to access those memories at a later date (I’ve also heard people refer to this as ‘building pathways.’) He notes that memories rarely fade, but the cues can – so we don’t lose well-formed memories, we don’t ‘forget’ as we might imagine – we instead lose our ability to access our memories.
Stage actors memorise tens of thousands of words through simple rote repetition. The fact that these words ‘make sense’ no doubt aids the process; if they had to memorise a list of ten thousand random words, it would be a much more difficult task. So understanding helps, but it’s not enough; were they to read the script once, they may understand it all, yet remember few or none of their lines! Understanding and practice were both needed.
Stage actors memorise thousands and thousands of words
There are so many techniques available for building stronger memories, and stronger memory cues. Simple repetition is one, mnemonics are another – and they come in several forms – stories are another. Some teachers use these to great effect, others may use them occasionally; some may never focus specifically on building memory at all, and I’ve never seen a check list for ‘Outstanding’ that even suggests they should. I haven’t yet seen any institutional focus on the importance of building memories. Whenever I do see it mentioned in the public arena, I see it derided. It’s tarred with the brush of ‘meaningless facts,’ ‘dry facts,’ ‘rote learning’ and so forth. When one person writes about asking students to memorise things, someone seems to have responded referring to the ‘pub quiz curriculum.’ I’ve often heard the idea of memorisation spoken of as pointless: these days, ‘you can always just Google it.’ As an incidental, rote memorisation is not the same thing as rote knowledge, though the two are frequently conflated.
Here’s the crux of this post:
I suggest that if we put all our thought and effort into building understanding, we do so at the expense of memory, and will nurture students who understood everything, once, rather than understand it, still.
Understanding alone does not = memory; it’s possible to forget what we once understood. I saw the proof that root 2 is irrational on four separate occasions, across the space of a year, before I could reproduce it from memory, despite having fully understood it every single time. …actually as I write this now, I’m not wholly sure whether I still can remember it, or whether I’ve forgotten again. Why does that matter when I can always just Google it? Well for example if in conversation with a student I thought it was appropriate to quickly introduce them to the existence of the proof, then I would do so. If I have to Google it, I’d probably spend those minutes Googling, and have no time left to explain – this has happened before in various forms; opportunity lost. In addition, if I can’t actively recall the proof, then I cannot relate it to any new knowledge I gain, leaving my overall intelligence undermined.
Instead of relying on ‘understanding’ to take all the heavy load of remembering, I would like to suggest that we start to think of building long-term memory retention and recall as a separate concern; that we start to put thought and effort into thinking about how we are going to help students remember what they learn from us, that we ask ourselves at the start of planning a lesson, or a unit ‘How am I going to help ensure my students still remember this six months from now, a year from now, two years from now…?’
Understanding may be an important pillar, helping to support the burden of memory, but is it the only such pillar?
I’d like to offer just one example of where I’ve tried to deliberately focus on helping students remember something that they had already learnt, and understood. I created a story to help students remember the quadratic formula.
Non-maths specialists will appreciate students’ apprehension when presented with this intimidating formula
I’ve heard many teachers in different schools tell students not to try remembering the formula; it’s provided on the formula sheet in the exam paper, so you don’t need to. I’ve also heard one or two teachers insist their students learn it by heart, stating that if they don’t, they haven’t really learnt it. Returning to the philosophical theme of my first post, in The Phaedrus, Plato writes of Socrates’ disdain for the written word. I quite enjoy the following line: “…they will be the hearers of many things and will have learned nothing.” While Socrates arguably takes an extremist position against ‘the gift of letters,’ I think there is a truth to his words – that to truly understand something, completely, you need to have it with you, in you, a part of you, not just symbols on a page you may or may not be able to decipher at some later date.
When I told my Year 11s I wanted them to remember the formula by heart… well you can imagine the reaction, and yes, some of them leapt up to helpfully note that it’s given at the front of the exam paper. I asked them to trust me, and then told them the following story – in the lesson, it was accompanied by my acting out the story a little.
A lady comes home from work, and sees a scratch in her car.
Before she can worry about too much about it, she sees a bee!
She jumps up and ducks down to escape the bee, before running into her house.
She sees the bee through her square window, ‘Phew! I’m safe, but man is it hot in here!’
So she decides to turn the heating down by four degrees, using her air conditioning.
But on her way over to it, she accidentally falls over two apples.
They looked at me like I’d gone mad. I retold the story, and as I did, a few of the quickest in the group clocked what I’d done. As more started to see it, there were more excited cries of ‘Oh I get it!’ For the remainder who still thought we’d all gone a bit strange, I retold the story, and this time drew out the formula as I did – everyone now spotted the link. I told the story one last time, then asked them to try and redraw the formula now, from memory. A wave of excitement swept over the classroom as they eagerly took up pens to test out their new memory. There was almost perfect recall across the whole class – naturally there were still a few minor errors here and there. The students were so proud that they could reproduce, from memory, the most complicated formula they’d ever seen, that the story didn’t stay in my classroom; many of them told it to other students in other classes, and even in other years. As far as I’m aware, to date every student who’s heard it has been excited by, and proud of, what they can remember. That was months ago. How did they do later? Well we never revisited it as we should have done, so as you can imagine, they didn’t do a great job when I asked them to try again maybe three months later. I talked them through the story again, and they got it back – I helped them strengthen the cue. Another month on (last week) I asked them to try again, and nearly everyone now had it back to 100% accuracy from memory, while some no longer need the story to recall the formula.
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?
I’d really like to hear other people’s thoughts and experiences of students and memorisation. Have I failed to make the case that understanding is necessary but not sufficient? Is memorisation a waste of time? Is it simply not as important as understanding? Are my perceptions wrong, and memorisation is a part of all our curricula? Can anyone share stories of how they helped students effortlessly remember what they had learned?
p.s. Here’s the rest of that quote from The Phaedrus
“And in this instance, you who are the father of letters, from a paternal love of your own children have been led to attribute to them a quality which they cannot have; for this discovery of yours will create forgetfulness in the learners’ souls, because they will not use their memories; they will trust to the external written characters and not remember of themselves. The specific which you have discovered is an aid not to memory, but to reminiscence, and you give your disciples not truth, but only the semblance of truth; they will be hearers of many things and will have learned nothing; they will appear to be omniscient and will generally know nothing; they will be tiresome company, having the show of wisdom without the reality.”