Preamble
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.”
...to the real. 
Reblogged this on Scenes From The Battleground and commented:
This is worth thinking about because a lot of dumbing down is done in the name of “understanding”.
Really like this – Kieran Egan writes quite a lot about narrative memory and it’s really powerful. I use this and spacial memory techniques to get pupils to remember key complex vocabulary. They attach a movement to the word or idea – it’s basically what the Jolly Phonics programme works on. When my son was revising for A levels he tried it and said it really worked, but he kept twitching in the exam!! The other thing that helps actors remember lines is making an emotional connection to the text – either through emotion memory or attaching an emotive quality like a super objective to the line. That makes it more memorable, so it’s more complicated than rote learning. Love your bee story idea – will share with our maths dept!
Thanks Debra. I liked what you added about spacial memory techniques – I’ve used something similar in creating sheets to help kids memorise the prime numbers up to 100 and angle facts. The spatial element seems to make a real difference. I borrowed the idea from Sporcle, after using their Africa quiz to learn and memorise its 54 countries last year. That anecdote will feature in full in a future post.
I’ve heard similar tales to that of your son as well; anecdotally I was told once that they heavy use of abacuses in Chinese schools meant kids there were very good at rapid mental arithmetic, but that you’d see their fingers moving in assessments as though using an imaginary abacus! Don’t know if anyone can corroborate that story?
Very much enjoyed and agrees with this piece Kristopher. Thanks indeed. Colin
Thanks Colin.
I think this is an interesting topic and an equally interesting example.
Could you (or someone else) explain what you mean when you say “that they had already learnt, and understood”. What do you mean in this case by “learnt” and what do you mean by “understood”.
I learnt this formula 36 years ago and have not really used it since. I can recall this formula without any problem, without the need for a story like that you present here. Old Andrew, who has reposted your post on his blog is one for talking of dumbing down. I have a feeling that making up a story like this to remember what is a fairly strightforward formula might be an example of dumbing down. I understand it is a memory aid but I have not come across too many examples of kids who couldn’t recall the thing easily. The number who could understand it and actually use it effectively is somewhat less in my experience.
In a situation in which a person does not have access to the forumla externally, the kid who can remember it will clearly have an advantage.
If asked whether is would be better to be able to remember the thing but not use it, or be able to use it but not remember it I would suggest that the latter is more valuable. If asked whether it is better still to be able to both recall it and use when faced with a problem then I would respond “of course”.
A very interesting and thought provoking post. Thanks
It’s fair to say that what we each mean when we use the words ‘learn’ and ‘understand’ can be different. In this case I meant that they had seen the formula, knew its purpose, and could successfully substitute the coefficients of quadratic equations into the formula and determine correct solutions.
Being able to do this, though, does not automatically mean they can recall the formula from memory. If they were to practise using it again and again then over time it might be that they would grow to remember it that way (something that should be touched on in the next blog post). If for any reason that kind of time won’t be dedicated to practice however, the story serves as a kind of memory short cut. As I noted in the post, with additional practice, some students are now quickly writing it down without referring to the story anymore, but the kids I work with don’t always recognise their potential, and so had I not first shown them that they were capable of memorising such a scary formula, they would have forever insisted I provide it for them – which I would have had to do, since they hadn’t memorised it!
Dumbing down in Andrew’s sense refers to expectations. If you’ve largely worked with children who can easily remember a formula like this, then I would hazard we’re perhaps engaged with cohorts in differing circumstances. The students I teach have not necessarily had the most structured and consistent mathematical experience for myriad complex reasons; for example they’re mostly learning to factorise and solve quadratics in Y11 for the first time. In this context, ‘dumbing down’ would be always providing them with the formula, and never expecting them to remember it, which we can get away with since it’s given on the exam paper. By contrast, I expect my students to memorise the formula, but I intend to make that as easy for them to do as I’m able.
Interesting blog post and I quite like the cue to remember the formula. As a non teacher it occurs to me that perhaps you, and other teachers, should go further and aim to develop the ability within pupils to create their own cues to aid recall. Obviously recall is useless if you don’t have understanding as well. I can look at that formula to my heart’s content but I still don’t know/remember what a,b and c stand for. 😦
Thanks Sam, and totally agree, you need to know how to use it as well. Currently though the push as I perceive it would be to get them to *understand* why it works, i.e. derive it from first principles by completing the square of a quadratic in general form. If they understand that, it’s certainly valuable, and then a further argument would suggest that they’d never forget again, since they’d always be able to derive it. However, it takes an awful lot more to learn the derivation than it does to remember a simple story, and actually trying to derive the formula again is incredibly slow!
There is also a public voice that suggests as you say that kids should be asked to come up with these things themselves. There’s nothing wrong with that, and I do think it can have some value, but I hesitate at suggesting it’s a ‘better’ alternative for a number of reasons. First, it means the cohort lose that sense of having shared knowledge, something they all know and value. Second… it’s hard to do! They simply may not succeed, or may not what to try, and in the end it’s rare that a student will individually come up with a mnemonic that’s *so* much better than one we can provide that it’s worth that risk (it does happen, but I’m talking cost-benefit analysis now.) Finally it takes a lot of time. I worked on that little story in my head on and off over several days, maybe even a few weeks before I got it to a place where it would work nicely. Argument can be made for spending that time in class asking students to do it – they’d be spending a lot of time thinking about the formula, and therefore would be *even* more likely to remember it later – but again it’s a cost-benefit that needs to be balanced.
“Obviously recall is useless if you don’t have understanding as well”
As a general statement I would disagree.
In this particular example, one is required to solve a quadratic equation. Being able to recall the formula but not apply it would mean that this had not been achieved, however recalling the formula clearly had.
A reasonable aim here might also be to recognise this as the formula that could be used, and following recgognition to apply it to a problem.
As a first step towards learning how to apply the formula then I think being able to recall it is quite useful.
I am not necessarily one for learning a whole raft of random facts just in case they might be useful one day, but I do think there are many occasions where being able to recall knowledge is actually quite useful. Later, opportunities to apply such knoledge might well present themselves.
When the time comes for you to learn what a,b and c stand for, as you can already redcall the formula you will have an awesome tool at you disposal. Until then, whenever you are asked by someone who is unable to recall the formula, you will be able to tell them without hesitation. That is not an achievement to be sneezed at. After all, it isn’t possible for everyone to know/understand everything.
I have put much more focus on memory with our GCSE sets in History. There is just so much to learn, especially as they they don’t do modules. Learning the material is by far their biggest challenge and yet we had no regular testing. I applied what I had realised from teaching my own children primary maths using direct instruction. I realised that when we did test it was often after half a term’s worth of work and too daunting. I identified the absolute bare bones I wanted all students to know and set regular tests using exactly the same format every two or three weeks. What really made it work well was telling the students exactly what would come up in the test (e.g. learn four reasons there was growing tension at the Potsdam Conference). It meant everyone had a chance of full or nearly full marks. I then set a high pass mark and was ruthless about re-testing those that failed. It has been a really great success and actually well liked by the kids that enjoy the fact they can do well and get a sense of achievement. I am chuffed because they are all remembering the really crucial facts and understanding the ‘next bit’ better because even the week ones know the basic outline of events so far.
My department really like the format too after some concern that we just didn’t have time to fit in regular tests (I was a bit worried too). This said, although I have chatted about the principles behind the format, when I leave making the tests to colleagues the gap of time between tests expands and the guidance becomes much less prescriptive. Its harder than I thought to share the principles behind the method!
Solid stuff! I’d also recommend having a look at Robert Bjork’s stuff on introducing ‘desirable difficulties’ to aid learning & memory: http://bjorklab.psych.ucla.edu/research.html
i blogged about introducing desirable difficulties into the English curriculum here: http://learningspy.co.uk/2013/03/25/redesigning-a-curriculum/
Teaching trig as an NQT to a difficult Intermediate GCSE group I talked about ways to remember which ratio applied to which sides of the right angled triangle. SOHCAHTOA didn’t seem to cut the mustard; Silly Old Hens didn’t trigger anything – I set them off to make up one of their own for homework.
Next lesson we had a wide variety of little stories that all seemed to use “Sex” as the word to represent the S (for sine).
Worked well though – although I did have to read quite a few stories about hilltops and animals; alcohol and holidays etc. when marking the mocks!
Fair enough! Can’t argue with what works. I’m surprised to hear SohCahToa didn’t do the job; I’ve found students to love trigonometry because the acronym’s easy to remember, and it effectively encodes everything they need to know.
Yeah SOHCAHTOA works for most of mine. Thsi class were really boderline between (old) Foundation and Intermediate tiers and this just seemed to stick better for them. Not sure any actually move into the stage of not needing to actually recount the story before tackling the problem though – I wonder if any still remember it?
Sometimes relating to interesting facts about the subject being memorized helps. For example, remembering that for angle in degrees, sin(30) = 1/2, and tan(45) = 1 is a simple substitute for the acronyms. Visualizing a triangle and these values will make it obvious which sides are involved in each case. ‘cos’ is the remaining combination. And then after practice you just remember the combinations.
Really interesting, and sounds like great teaching. I have taught ESL in China, in an education system which apparently prizes memorisation above all else, and even here, no-one actually teaches children how to remember things. I personally hate mnemonics, but they do seem to work for a huge number of people, so this is very valuable stuff.
I also like your analogy with a script. I think part of the reason why there’s such an emphasis on understanding, is that people now want to focus on learning skills, and the idea is that you never forget a skill. You never forget how to ride a bike, for example (so they say!). And I agree with that in many ways: understanding why something works is a qualitatively different experience to memorising how it works. It stays with you, and teachers should aim for that where possible. But it’s not always practical. To this day I struggle to derive the quadratic formula, whereas learning it, with a mnemonic or otherwise, is pretty easy. For children, learning it is definitely the best way to (a) get through exams and (b) get a step closer to the intuitive understanding that will make them able to derive it.
Thanks Phil. I didn’t use to like mnemonics either. I remember picking up a book on memory from my university library during revision time, and it talked about a similar idea of writing a story to remember a complicated formula. For my part, I looked at it and didn’t see the point! I could just rote memorise the formula, and have to remember less than remembering a much longer story! But that works for me; I have the patience to sit and cram things into memory, where others don’t. I think in those cases, being led towards an easy way of memorising something that otherwise seemed insurmountable, is inspiring.
While Willingham has a fair bit to say on different mnemonic techniques, I started to come round to them much more when learning French ‘with Michel Thomas’. His command of mnemonics is so powerful that, having listened to CD1 once, 7 years ago, I can *still* recall that “Venir means ‘to come,’ like the veneer that won’t come off the walls.” To this day I’m not sure what veneer is, but it did the job! Just one of many he uses to outstanding effect.
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We remember the knowledge that we use. The rest is ‘something we did once at school but didn’t use’ (or didn’t use soon enough), so all we remember about it is that we once did it at school. Without practice, knowledge remains passive.
Actors remember their lines because they’ve used them, during rehearsal. I remember equations that I’ve had to use to solve practical problems.
When knowledge it actively applied it becomes something more than mere knowledge: it gets wound up in skill. Interestingly at that point, you don’t have to work at remembering it. You just know it, and you just use it. It’s part of you. It’s there to draw on again and again, when a problem requires it.
The question relating to teachers’ planning is therefore: ‘how am I getting my students to use the knowledge I’ve taught them?’
I think the idea of memory and what students actually remember is an important aspect of teaching and learning. We have an exam system that relies quite alot on what students actually remember and can then recall when sitting their exams and so we do need to pay some attention to getting pupils to think about this aspect of their learning. For some things memory is perhaps more important than the understanding of a concept – I am particularly thinking about times tables. I learned these by mainly by rote at school as that was the favoured technique then. This is not a very fun way to learn times tables but it did help me understand that with repetition my brain was capable of remembering and recalling lots of information. My son found learning times tables pretty hard and I found myself trying to explain to him that he didn’t really have to understand them, just simply memorise sets of numbers and familiarise himself with the patterns.
This video clip demonstrates the use of the counting stick as a simple way to show students how their memory can do pretty impressive things.
I wonder what other educators think about the way the school day is usually organised in high school, and if a change to timetables may result in more embedded learning. I have often wondered what it might be like if high school students spent a whole day on one subject rather than the one hour one subject timetable that most schools follow.
I can imagine there being huge benefits in spending a full day to properly explore a subject or concept, or to complete a painting or a design technology project.
I would suspect that retention is related to the coupling of understanding with the simple matter of living and breathing what has to be understood. The comparison I can make is with Cognitive Behavioural Therapy, where you create a journal of your thoughts and emotions in order to identify those that fail to serve you well.. The process can be understood, but is almost valueless unless undertaken – you must actually do the journalling. So many self-help book readers understand what they need to do to improve their lives, but the understanding is not acted upon.
Maybe not an ideal analogy, but I hope you see my gist.
I was transfixed reading through this. Very stimulating. As a Science teacher, a huge amount of content from key words to equations through to explaining abstract concepts is required to be accessible to students’ memories in the exam. I find that week in week out students leave the classroom demonstrating understanding only to be surprised all over again next time I see them. I particularly enjoyed Willingham’s paper. I feel like starting the year with every year group using his suggested activities.
Roque, have you come across Engelmann’s idea of ‘interleaved learning’ before?
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http://hodteacher.wordpress.com/2013/06/06/sticky-maths/
I have been working on ideas for ‘stickability’ in my Maths Department. I agree that we need to be careful, too much focus on ‘understanding’ could mean throwing the baby out with the bath water. We need to make sure the concepts ‘sticks’!
I like it. Adrian Johnston talks about making things ‘sticky’ in this blog post here:
http://mrreddy.com/blog/2012/12/guest-blog-circle-theorems-and-hula-hoops/
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Asking questions are genuinely fastidious thing if you are not understanding something totally, however this article presents nice understanding even.
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I love your technique here. Very powerful.
Another example: when I was in high school two of my friends decided to give every element in the periodic table a personality based roughly on their properties. They then spent hours talking about how the different elements would react to each other when they met.
You can probably guess the result. Both students aced Year 12 chemistry, a subject that the majority of us found dead boring.
Personification and stories are the original vehicles of knowledge. We’d all do well to bring them into our teaching a lot more.
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The real problem (that you elude to here) is not that they can’t remember but that they don’t want to enough. As you say, they don’t recognise their potential and don’t bother trying to remember it. As human beings we are capable of remembering the strangest things but only if it seems meaningful to us. It is getting the students to want to remember which is the biggest hurdle.
Can we ever achieve that? Is it an inherent flaw in universal education that, since it’s not any kind of opt-in system, we can never guarantee interest across the board? Or could we, if content were delivered in some ideal way? If we can’t, are there alternative ways ensuring retention over time?
The much maligned French system does this with poetry, conjugating verbs and weekly dictations (at Primary School). For a long time the reasoning behind this has been to improve retention. In the UK and US we often look down at the French system, and yet international comparisons such as PISA do not support this supposed superiority.
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Hi Kris, you touch upon so many things here and they are all interconnected! In a chinese cookie once I read the following “To learn, read; to know, write; to master, teach”. Over my years as a student of maths and then practitioner I ve had dozens of peers or students thinking they know something, becuase they read it and it made sense to them at that point in time, but then failing exams miserably because they could not recall it, same as you describe here. Unfailingly, it was becuase they had not practised writing it on their own enough. I asked a friend how he studied for his exams once. He said he read through his notes. I said ‘no silly, you need to solve every single of these problems by hand on your own, not just read through it’. So, yes, as with so many other skills practise is paramount to convert something from short term to long term memory. You also need to make retrieval links in wider context problems. Example: you need to solve a quadratic as part of figuring out the intersection points of a parabola and a line. that’s your context, solving the quadratic is then a means to an end, not the end itself; just an intermediate step on a wider solution process of a more meaningful problem.
If a kid asked me why they need to remeber the formula, or any formula for that matter, I d say, when you are driving from home to school or work, you need to follow a route, right? You need to turn at certain points, and in order to turn, you need to check your mirrors, indicate, and then complete the manouvre. If you had to think hard about completing this task every time you needed to take a turn, your driving would be slow, clumsy and it would take you a long time to reach your destination. You d probably also feel a bit deflated and demotivated to drive further distances.
why do we learn so complicated things like driving to the extend we do them mechanically from memory, on autopilot, but we have difficulty remembering a formula? there is definitely an element of motivation involved. Clearly we want to learn to drive, at the age of 17, probaby because it is cool to do so and shows we are grown ups now. We find it cool because this is the value that has been passed onto us from a very young age, many many people do it very effectively and we want to do it too. this is not the case with maths in this country. other countries do not have this issue. In Greece, Russia, France, India, Singapore it’s not cool not to be good at maths. Not that everyone is good at it, but those who are not recognise it as a stigma, not something to joke about. So the motivation from the society is there, maths is valued for what it is, not for the job you ll get if you are good at it, like you comment on a different post. And even though thats a different discussion, I do feel Kris that unfortunately despite our best intentions as maths educators we cannot beat the societal values we live in. No one would joke about not being good in English but somehow falling behind in maths is fine.
Back to your memory versus understanding questions though. For me, as with every skill a human possesses knowing is the same as doing. You know how to write means you can write, to know how to speak, means you can speak, to know how to play the piano, means you actually play the piano. All these things, you learn through practice, right? You practice your scales, so that then you can play your piece effectively and focus on being musical and interpretation rather than just trying to hit the notes. As you say in your recent TES article re times tables, memory is there to support further understanding, to help us identify patterns, to help the brain draw relations and come to conclusions.
Just a note about the quadratic formula… I prefer to build student understanding of completing the square and the generalization of completing the square. Memorize/remember the formula if that is useful to you but know and understand how to recreate it should you forget and have the need.
So do I, but that is a much more complex task, requiring much greater time.
Being able to memorise the formula is a quick win; I would argue it’s easier to do this first, then join it up to completing the square of a general quadratic later.
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I am not totally convinced such a technique could be used in general. Of course, this will help to memorize one maybe ten formulas, but what about a 2 pages long proof? I cannot imagine I would create a story for every single problem I need to remember because I would not be able to memorize the stories themselves. But on the other hand, I have no clue how to tackle memorization for more complex problems and your technique is the best one I found so far. Do you use any other techniques?
Hi David.
I picked up a book on memorisation from my university library back when I was studying for my exams, and felt exactly the same way.
(A) surely this means remembering more than just memorising the formulae directly?
(B) can I actually create a story for all of it?
For GCSE maths, there aren’t that many complex formulae, so I think it takes care of B… And even for higher level content, and things like two page proofs, it’s certainly possible (far more has been achieved using techniques like this) But it is still a capability that either not everyone has, or that one would need to develop… Not ideal!
For A, I think I was just plain wrong – even if something has more information content, putting it into story form makes it much easier for the brain to recall later.
There are other techniques, but more than I could put into a comment. I’d suggest picking up some books maybe? Like moonwalking with Einstein, and make it stick. There’s a free article you can find online as well by Daniel Willingham, ‘What will improve students memory’
The reason I came to this blog, was due to the same reason. I understand everything currently and when my attention is dislodged, the entire block of understanding goes to ether. Unless as you mentioned the lateral environment to the said understanding is restored, which is a great point. Professor Randy Pausch mentioned learning through head fakes appeals to me. I know my memory exists because given the right circumstances I can at least recall 70% of the original content and meaning.
I also remember things Instantly if my original inquiry was based on my natural curiosity, and this memory just persists without any effort. Especially if it means something to me.
Cognitive science and cognitive load theory have a lot to offer on this topic. I can recommend Daniel Willingham’s Why Don’t Students Like School, in particular.