Are ‘knowledge’ and ‘understanding’ really the same thing?

David Didau has just posted a very interesting blog on just this question.  He argues that yes, understanding is actually the very same thing as knowledge; to say that you know something is exactly the same as saying you understand it.

Now I can hear one thousand voices screaming ‘No! No! No!’ and arguing back saying that just because you know that, or can parrot that, 6×7=42 does not mean that you understand it. At all. QED.  Such is the absolute aversion in education to ‘meaningless’ or ‘disembodied’ or ‘dry’ or ‘shallow’ retention of facts as opposed to ‘deep’ understanding, that our automatic rebuttal mode is activated.

But please let me explain.

What does it mean to understand 6×7=42?  I would suggest it means knowing lots of other things too and connecting them together. For example, knowing that 6×7 can be expressed pictorially as an array.

Knowing that 6×7 can be expressed as a bar model.

Knowing that 6×7 is the same as 6+6+6+6+6+6+6

Knowing how to model this on a numberline

Knowing how to model this with cubes

Knowing that all of these examples are equal to the same representations or equations as 7×6

Knowing that 6×7 is the same as 3×6 + 4×6

Knowing how to draw an array to show this

Knowing that 6×7= 7×7-7 and so on.

Every example of ‘knowing’ here could be replace by ‘understanding’ – the two words are interchangeable.

If we know all of these things, we can reasonably be described as knowing what 6×7 means. We understand it. The fact 6×7 is connected to lots of other associated facts.  That’s what understanding is: a load of knowledge connected together. Knowing 6×7 as an isolated fact isn’t very useful; connect it together with all this other information, other knowledge and – voilà – we have our holy grail: deep understanding. AKA knowledge.

So the key here is making sure the individual, ’disembodied’ facts are linked together.  Because teachers have this fear of ‘shallow’ learning, teaching approaches have been drawn up that try and enable ‘deep’ learning from the start.  The problem with this is that the road to deep learning is via shallow learning. You can’t leapfrog over it.  Knowledge becomes ‘deep’ or ‘rich’ or whatever adjective you wish to modify it with when it is connected to other bits of knowledge, but obviously you have to start somewhere. You can’t start with a connection. That would be like trying to build a bridge by building the road before the piers. Shallow knowledge is a necessary step on the way to deep knowledge. We can’t have deep, connected knowledge without remembering a lot of information.

Of course, you could argue that it is useful to reserve the word ‘understanding’ for connected knowledge and retain ‘knowledge’ or ‘knowing’ for shallow knowledge.  The point is though that understanding is not some mystical other thing beyond knowledge; it is lots of associated knowledge stuck together in a schema.

Who does the sticking, the teacher or the student?

Teachers want above all things for knowledge to be connected, for learners to ‘get it’, to understand it, to link different aspects together meaningfully.  So they will try really hard to design sequences of lessons that make the connections as explicit as possible. That’s what variation theory in maths is all about[1] .  The concrete, pictorial, abstract approach is another way maths teachers enable children to make those all-important connections.   Here’s the thing though.  You can’t brute force connections, however hard you try. [2]  And of course we should try (though not the force bit).  We should try really, really hard.  Those connections are much more likely to be made if they are made absolutely explicit, and if teachers really think about all the little in between bits of knowledge we assume children already have, but in fact might not.  For example, to return to 6×7, some children may not have really understood what a ‘group’ of something is: that you can call 6 things, 1 thing. Without this knowledge, 6×7 won’t make sense to them. A vital connection is missing.  They are more likely to think the answer should be 13, making a more familiar connection, albeit to the wrong thing.  Sometimes wrong connections are made.  Students have false knowledge: knowledge that is wrongly connected.  They don’t understand.  The teacher’s job is to be aware of the likely false connections (misconceptions) and explain carefully so that the right connections are made instead.

Learners themselves cannot force themselves to connect disparate bits of information, even if they know the teacher says they are connected.  The brain just does the connecting – or not – independently of either the teacher’s or learner’s desires. Although of course learners can help matters along a great deal by thinking hard about the things they are trying to connect and teachers can plan their lessons to make sure as much hard thinking about the right things as possible takes place.  My suspicion is though that where students just don’t seem to ‘get it’ it is because some intermediate knowledge is missing that just seems so obvious to us relatively ‘expert’ teachers that we fail to teach it. Either that or because the thing we have just taught is hopelessly entangled with the wrong bits of other knowledge in ways we would never guess so can’t untangle. Or probably both.

Any mistakes in this explanation are of course mine rather than David Didau’s.

[1] See appendix 4. Note how this starts with contrasting memorisation with understanding and the expression of surprise that East Asian countries which major on teacher-led didactic approaches outperform Western countries where such methods are seen as inferior, this apparent paradox is explained by reference to the routine use of ‘variation’ in East Asian didactic teaching.  The teacher explicitly draws attention to connected features.  It’s not that didactic teaching per se is ineffective, it’s when it is done badly, in a way that does not exploit every opportunity to make those connections.

[2] I think this is where the suspicion of so called ‘traditionalist’ ways of teaching kicks in. Teachers worry that this means trying to force knowledge into children’s minds by blitzing them remorselessly with facts as if that was a failsafe solution to learning. Whereas the traditionalist would explain their approach as rich in factual information, but factual information presented in a sequence in which inordinate care has been taken to maximise the probability that the right connections are made (variation) and with similar care also taken to ensure it is broken down into small enough steps that essential prior knowledge is not overlooked.

Are ‘knowledge’ and ‘understanding’ really the same thing?

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