Beyond the blame game –  the trouble with transfer

It’s a laugh a minute in the Sealy household at dinner as two teachers swap amusing anecdotes about their day while our sons listen on enthralled. Yes, I’m lying. The sons are sticking pins in their eyes in a vain effort to MAKE IT STOP while we drone on to each other about the trials and tribulations of our respective days.  My partner is a maths intervention teacher and trainer who mainly spends his time training other teachers and TA’s how to teach maths to children who are struggling.  The interventions he trains people in are all very effective and have tonnes of evidence to back them up (albeit too expensive to staff for most of us in these cash-strapped times when having a class teacher and the lights on at the same time is considered a luxury). Among his top ten moans[1] is the situation when class teachers fail to recognise that ex-intervention students are now actually quite good at maths, instead seating them in the 7th circle of hell that is ‘orange table’, where there might as well be a sign saying ‘despair all who enter here’ and where the cognitive challenge is low.  When the intervention teacher tries to argue their case, the class teacher, who does not consider their colleague to be a ‘real’ teacher, argues that ‘she might be able to do place value (or whatever) with you, but she can’t do it in the class room where it really matters.’  The unspoken assumption being that intervention teachers – who are not real teachers anyway – don’t really know what they are doing and are easily tricked into thinking that a child has got something because they’ve played a nice game with their not-real teacher who doesn’t understand about important things like Sats and tests and being at the expected level and obviously couldn’t hack it in the classroom. Indeed, a quite senior teacher, worried for her value added, once said to him that he ‘artificially inflated’ pupils learning by teaching them stuff.   To which he countered that all teaching ‘artificially’ inflates learning – that’s what we’re paid to do! We are employed to use artifice to achieve learning.

It occurred to me recently that cognitive science provides an explanation as to why this conflict happens; an explanation that blames neither teacher and also explains equally well why every September, class teachers shake their heads in disbelief at the assessment information provided by their colleague,  the former teacher, a disbelief that is amplified on the transfer from primary to secondary school.

Transferring learning is, quite simply, a bitch.  There are three cognitive hurdles to overcome on the journey from the pupil’s first encounter with an idea to them being able to understand whatever it is in a flexible and adaptable way. First, they need to be presented with the idea in an understandable way that make them think hard[2] about what they are learning. If they think hard about it, it is more likely to make that all important journey from their short term memory to their long term memory. Sometimes teachers try and make ideas memorable by making them exciting in some way. This can backfire if the ‘exciting’ medium becomes more memorable than the actual message the teacher wants to get across. I recall one child who was finding learning to count really tricky, so to engage him we used gold paper plates and toy dinosaurs. He was totally absorbed, but not on the maths, unfortunately – and did much better with plain paper plates and cubes.  But hurdle one is not where the intervention vs class teacher fault line lays.

The second hurdle lies in overcoming the ‘I’ve taught it therefore they know it’ fallacy, particularly common among less experienced teachers.  But even if our panoply of afl strategies tell us that a particular child has grasped a particular concept, it is highly likely that by the next day they will have forgotten most of what we taught them. That is just how our brains work. But that does not mean we labour in vain; the forgetting is an important part of remembering.  The forgotten memory is not really forgotten, it’s floating about somewhere in our long term memory, ready to be reactivated. All it takes is for us to re-teach the information and on second encounter, the material is learned much faster. By the next week it is all mostly forgotten again but with a third presentation, the material is learned very quickly indeed.  And so on.  Each time we forget something, we relearn it more quickly and retain it for longer.

This means that teachers need to build into our lessons routine opportunities to revisit material we taught the day before, the week before, the month before, the term before and the year before.  This is known in the trade as ‘spaced repetition.’  Each time we do so, we enhance the storage strength of memories. Ignorance of this phenomenon accounts for part of the professional friction between colleagues. It wasn’t wishful thinking on behalf of the ‘sending’ teacher.  The pupil genuinely did really know how to partition 2-digit numbers, for example, but has now forgotten. That’s an inevitable part of how our brains work and not some other professional’s ‘fault’.  When faced with a conflict between what it is reported that a student can do and what they appear actually able to do, the most charitable and scientifically probable explanation is that they have forgotten how to do something that they once could do well; with a bit of input it will all come back fairly quickly. If we remind ourselves on this each September and expect to have to cover a lot of ‘old’ ground, that will be better for our students, for our blood pressure and for professional relationships.

However, hurdle number three has, to my mind, the best explanatory power for this aggravating situation.  To understand this, I will have to explain the difference between episodic and semantic memory.  Episodic memory remembers…episodes…events….experiences. It is autobiographical, composed of memories of times, places and emotions and derived from information from our 5 senses.  Semantic memory is memory of facts, concepts, meanings and knowledge, cut free from the spatial/temporal context in which it was acquired.  Generally, especially where teaching is concerned, memories start off as episodic and then with lots of repetition, particularly in different contexts with different sensory cues, the memory becomes semantic and can be recalled in any context. This is the destination we want all learning to arrive at.

So when we learn something new, we remember it episodically at first.   We’ve all had those lessons when we remind our class about the previous lesson and they can recall, in minute detail, that Billy farted, but not what an adverb is.  Or they’ll remember that you spilled your coffee or that Samira was late or even that ‘we used highlighter pens.’  But anything actually important…gone!  Of course, when you recap on yesterday’s lesson, it will all come flooding back.  See hurdle two.  However, the problem for transferring this knowledge beyond working with this teacher in this classroom is that with episodic memories, environmental and emotional cues are all important.  Take these cues away and the memory is hard to recall. We don’t want a situation, for many reasons, where our children can only recall what an adverb is if prompted by the environmental cue provided from Billy’s posterior.  We are a proud profession, we aim a little bit higher than that. We want what we teach to be transferable to any context.  Until that has occurred, how can we say learning has successfully happened?

So, back to our maths intervention teacher. The pupil has learnt a whole heap of maths and made many months of progress in a short space of time.  However, although their teacher has got them to think hard about this material and got them to apply their new knowledge in many different situations, and although the teacher has also used the principles of spaced repetition and revisited previously taught material many times, there is still the very real possibility that the memory of some of this material is still mainly episodic, still mainly dependent on familiar environmental cues for recall.  It is not that the child is emotionally dependent on the familiar adult to boost their confidence – thought that can also happen – but that the academic memory is bundled with the sound and sight (and possibly, the coffee breath of) their intervention teacher and the room in which the intervention happened.  Without these, the memory is inaccessible.

This problem is only exaggerated when the transfer is from one year group to another – with the added difficulty that the student is unlikely to have been doing much hard thinking about either denominators or adverbs over the six weeks summer holiday. It is even more of a barrier when students are transferring to a completely different school, such as at secondary transfer, with all the other attendant changes that brings.

To counter this, when teaching material, we need to try and play about with the environmental conditions to lessen the impact of context cues. So when an intervention teacher asks to come and work in class alongside a pupil as part of their weaning off intervention, that is not some namby pamby special snow flake treatment by a teacher who clearly is too attached to their pupils, but a strategy rooted in cognitive science to help the pupil access episodic memories with most of the familiar context cues removed. Class teachers can try and break the dependence on context cues with material they teach by, at the very least, getting pupils to sit in different seats with different pupils from time to time.[3]  Year 6 teachers, now faced with the post sats quandary of what to teach now, would do well to teach nothing much new and instead ensure over learning of what pupils already know but within as many different  physical contexts as possible  – maths in the playground, or hall or even just by swapping classrooms for the odd lesson.  If pupils are used to sitting next to the same group of pupils in every lesson, now is the time to mix things up, to lessen the dependence on emotional cues (again, episodic) gained from the sense of familiarity of sitting with the same people day in, day out[4].

Transfer can also be facilitated by applying learning in different parts of the curriculum, using maths in DT for example, or in art lessons or maths through drama and also by applying the learning in open ended problem solving.  Indeed, the very sort of ‘progressive’ teaching strategies that card carrying traditionalists usually eschew, are fine for transfer, once the learning is securely understood, but probably still remembered episodically. It’s the use of these methods for the initial teaching of ideas that’s a bad idea – explicit teaching does that job so much better. Whizzy bangy stuff early on – or even in the middle – of a sequence of learning, runs the very real danger of getting children to think hard about the whizz bangs and not the content – so the whizz bangery will be what gets remembered in the episodic memory. See hurdle one. But that’s a whole other blog post.

Accepting the inevitability of the difficulties of transferring learning from one context to another can help us plan better for that and be less frustrated by it both in preparing to say goodbye to pupils in July and when saying hello to students in September.   It’s not that learning slumps as such in September, it’s that it is being reawakened and then transferred from episodic to semantic memory. Once memories have made this journey, they are so much stronger and more flexible, so worth the frustration.  So this September, when your new pupils don’t seem to be able to remember anything their assessment information would indicate they should know, take a deep breath, remember the three hurdles and that is just how learning and memory works. It probably isn’t their former teacher’s fault at all.  Maybe you just don’t smell right.

[1] Just in case a colleague of my partner is reading, he insists I make it abundantly clear this has not happened for a long while where he teaches. It does happen to some of the people he trains (in other schools) though – it is an occupational hazard of being an intervention teacher.

[2] Memory being the residue of thought, as Daniel Willingham explains in this book you really should read.

[3] I am relying heavily on chapter 6 of ‘What every teacher needs to know about psychology’ by David Didau and Nick Rose for all of this. This is also a very good book for teachers to read. If you read both this and the Willingham one above, you would be well set up.

[4] Not that I would recommend this in the first place, but if that is how you do things, shake them up for the last few weeks of term in the interest of better transfer

Beyond the blame game –  the trouble with transfer

Hooray for an array!

Now I maybe wrong, but I was taught that strictly speaking mathematically 3×2 means 3 taken 2 times or in other words 3+3 rather than 3 lots of 2 (2+2+2).  Which means that 3y means y lots of 3 rather than the easier to understand 3 lots of y…. But that’s what I understood the notation to officially mean. Not that it matters with the commutative law and all- but a teacher in America marked some poor kid down for saying 3×5 was 5+5+5 instead of 3+3+3+3+3 (i.e disagreeing with me). The teacher even marked the array the child had drawn for 4×6 wrong because it was orientated the’wrong’ way. The whole point of arrays is that they can be ‘read’ either way and hence are very useful for teaching the commutative law. I love an array! Do you know they are also brilliant for showing the associative and distributive laws, link to the bar model, the grid method, prime factorisation and multiplying by 10!  You just need some spotty wrapping paper……

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Here is 2+2+2 or 2, three times- what I’m going to call 2×3.  Like this it is also a bar model. But like this

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it is an array that could be seen as 2, 3 times or

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3, 2 times.  Or back to the bar model

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3+3

Now I’m sure we’ve all done similar with our children in yr2 and yr3- I used to do it with pegboards and exploring 24. Dotty paper works brilliantly though- get the children to do the cutting and experimenting.

But I’d never thought about using it to teach multiplying by 10, until the new curriculum. The new curriculum stresses multiplicative reasoning. In yr3 they are supposed to understand that 4x12x5=4x5x12=20×12=240. (Associative law and all that).  Which got me thinking. Obviously we NEVER tell children ‘ just add a zero’ when multiplying by 10 and tell then about moving digits one space to the left. My colleague gets the whole class to stand up, jump to the left while chanting ‘multiply, go large, left’ ( for division it’s ‘division, reduce, go right). The local secondary teacher came to see a lesson and finally understood why a quarter of her class would jump up out of their seats and do his routine everytimemshe asked about multiplying or dividing by10! But even though this helps children remember what to do – does it help them understand?  But what if we taught it ( alongside the dance) with arrays? Make an array of 6, then make 10 of these arrays to make an array of arrays eg 2x3x10=6×10=3×20

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Which they can play about with and establish that this also equals

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5×12 and even perhaps after a lot of cutting 2x3x2x5.  Hence prime factorisation. Remembering of course to make the link with factory… The factors are what the 60 factory uses to make 60.

Then photocopy or scan the 10×6 array and by cutting and pasting make another array ten times bigger by putting 10 copies together and establish you know have 600 because 60 can be written as 6×10, so 6x10x10= 600. Then photocopy again to make 6000 and- well maybe leave it at that. When I showed my yr6 that 60×10 could be factored into 6x10x10 they were enthralled and wanted to know if it worked for multiplying by 100 – which they quickly proved it could. This kind of multiplicative reasoning not having been common until recently, they were entranced by it. Last year I showed the level 6 group how you can easily work out eg18x6 by halving the 18 and doubling the 6 and they thought I was teaching them some kind of dark magic- arrays of course to the rescue. Indeed the bottom two photos above show precisely this.  Much experimentation later they got the general rule that you can multiply one factor by any number as long as you divide the other factor by the same eg 96×15=12×120=3×480. By now, they can ditch the arrays and experiment with factorising numbers and then recombining in ways that make multiplying easier. Multiplying even numbers by multiples of 5 being an easy win eg 24x 15= 4x6x5x3=2x2x2x3x5x3=2x5x2x2x9=10x90x4=360. Well,maybe it’s not easier but  I find it kind of satisfying. And they are going to have to understand this if they are going to be able to reason about questions like these from NCETM’s brilliant assessment document. To be honest it did take my yr6 teacher and I a few moments, quite long moments, to get our head around these at first. The last one still gets me every time.

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Have you ever used the dotty wrapping paper method as a prelude to the grid method? It works a dream. The photo below was taken in bright light so it’s hard to see the pen annotations dividing it into 10×10, 10×9, 10×7 and 9×7- try zooming in perhaps.

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Again, get the kids to do the dividing up and cutting, finding the arrays that will make the calculation easiest (the multiples of ten).

This method can form a bridge to the column method. In the class I’d do this with actual paper we had cut up and made into a grid first but I’ve drawn it ( very badly). So put the actual arrays where I’ve put the drawings.

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Finally the distributive law. Easy Peasy when you just split you array into two pieces

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10×6 distributed to make 6×6 and 4×6. Cue investigation- how many ways can we distribute this array. I had to model this to a primary teacher who didn’t believe you were allowed to split numbers up like this in the middle of a multiplication when I did a staff meeting on multiplicative reasoning using the post on Shanghai maths from resourceaholic. This is what we were trying to work out mentally

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He got the bit where you divide 45 by ten and adjust by multiplying 1.58 by 10, but didn’t agree that 4.5x 15.8 + 5.5x 15.8 was the same as 10×15.8. Until I drew this.image

Then the penny dropped. So hooray for arrays and remember CPA all the way . (That’s concrete pictorial abstract, in case you haven’t encountered this yet). If I haven’t converted you yet to the joys of arrays, read this from NRich.

Hooray for an array!