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.

I think we “know” whats and we “understand” whys. Every child knows that, if they let a heavy object go at waist height, the object will drop to the ground. That’s a what. And they may understand that they know this fact. But they won’t generally understand why it happens.

Can we build any true sentence that uses “know” into a true sentence that has roughly the same meaning that uses “understand”? Probably. But this strikes me as more of a semantic game than anything revelatory.

If I say I know that 6 x 7 = 42, that doesn’t reveal anything about my understanding about why it’s true. Sure, “I understand that 6 x 7 = 42” is also true, but that’s not really what we normally say. We’re more likely to say that we want students, for instance, to understand multiplication. I can reasonably claim to understand multiplication even if I would never think of putting numbers into an array. Perhaps I understand multiplication using an area model, and never think of numbers in terms of aggregates of things — this is all “why”.

I feel like I get you’re getting at, but I think it jams the typical uses of “know” and “understand” in order to make them full synonyms, which is a counterproductive effort. Why not strive to delineate the terms in a useful manner?

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in a way it is a semantic game but one played in order to point out that the antipathy to ‘teaching knowledge’ as if this were somehow in opposition to teaching understanding, is misplaced. Understanding, in the end, is ‘just’ lots of well connected knowledge. teachers shouldn’t obsess over the supposed dreadfulness of shallow knowledge as it is but a necessary step on the way to ‘deep’ knowledge – aka understanding. in everyday speech I’d agree it is useful to have two verbs, knowing something (having shallow knowledge) and understanding (for when wider connections have been made and the shallow knowledge can usefully be put to work). But it’s good to appreciate that under the bonnet – it is all really knowledge. Realising that also helps with our teaching by helping us really focus on connecting things up explicitly. Really good teachers have always done this.

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I’d agree with all of this.

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Further thoughts: We certainly can’t claim to understand something that we have no knowledge about. Even if understanding is something more complex than “merely” knowing, knowing is still crucial to the process. And knowing things that are technically unnecessary to a simple algorithm can deepen understanding.

For instance, if I’m to add 47 + 53, and 50 + 50, it certainly helps to know about associativity and regrouping (move 3 units from 53 to 47, and it becomes 50 + 50). It even helps to know that regrouping is a technique that leverages the associative, commutative, transitive, and distributive properties. My son (age 8) knows how to regroup and understands the process, but doesn’t know those other terms, and that’s okay, too.

In mathematics, I think this is another place where people balk at “knowledge”: I would rather have a student who understands the concept that right triangles and only right triangles satisfy the property that the square of their largest side is equal to the sum of the squares of the other two sides than a student who knows that a^2 + b^2 = c^2. I understand your point that both the concept and the equation are things that can be “known”, and I agree that “understanding” this theorem involves knowing what a triangle is, what a right angle is, what a right triangle is, what it means to square and add numbers, and so on… there’s nothing to be understood about the Pythagorean Theorem beyond a set of facts and connections to be known. But I know teachers and sources that place far too much emphasis on knowing things: The student who knows the concept of the Pythagorean Theorem (“understands”) but can’t rattle off a^2 + b^2 = c^2 is seen as deficient compared to the student who can rattle off the formula (“knows”) but can’t expand that knowledge to, say, the distance formula.

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Which is why both procedural and conceptual knowledge ( the why) are so important. By conceptual knowledge I suppose we mean – how this bit fits in with all those other bits?

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Knowing it, without knowing all the connects fact is sadly possible

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[…] couple of folk, recently, have written about this. @daviddidau and @claresealey. I only want to add one smallish point to what they have […]

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Knowing it without knowing all the connections is sadly possible (is what I meant to write)

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yes indeed. I hope it’s clear I’m advocating the need to make those connections as clear as possible.

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I agree with all the comments here and OldPrimaryTimer’s analysis of David’s post.

I guess my question is the following: Given that any normal teacher-led, classroom acquired, piece of knowledge is – in itself – just a shallow ‘fact’ in the moment it is delivered/received/stumbled-upon, is the proposal that – with nothing more than a thorough smattering of shallow knowledge, the brain will automatically assimilate these different bits of knowledge into webs of understanding which will – by their own accord – become insights and, indeed, lead to ‘profound’ realisations/appreciations etc…? (whatever ‘profound’ might mean given this discussion).

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Ok… just realised that the above is presented in the form of a single sentence… SORRY!

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‘nothing more than a thorough smattering of shallow knowledge’….i think I’m that the brain will make its own connections so we had better make sure we present knowledge in ways in which maximise the right connections being made. Variation (see the footnote) in the way East Asian maths teachers try to optimise this. I’m objecting to ‘smattering’ as it doesn’t reflect the careful, thoughtful planning I think this needs.

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Yes – you are right – smattering is not the best term. However, I suppose what I’m saying is that there is nothing we can do to inject ‘deep’ knowledge according to David’s hypothesis. Everything we do involves (admittedly skillfully done) injections of surface facts.

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I don’t think you should class facts as surface (shallow is better) as that implies they are deficient in themselves in someway. It is our knowledge that is shallow. the same fact in someone else’s brain may be deep because well connected. But yes, there is an element of this process that lies beyond the teacher’s reach! We all know that though, don’t we – it’s just exasperating not be able to infallibly make kids understand!

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I think I am with your point OPT. What might be a simple drop of knowledge to one child, could end up being a highly ‘deep’ injection for others – being the final piece of the jigsaw in their conceptual framework so to speak. Conversely, the profound ‘Earth-Shattering’ revelation which we think we are revealing to pupils may simply be like an inconsequential spit of drizzle to many of them, for whom the ground simply hasn’t been prepared (or, one way or another, simply isn’t ready).

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Proper juxtaposition and sequencing is key, and I think one place where we’re currently failing. I have three older high school texts on geometry (Heineman’s 1974 ed of “Plane Trigonometry with Tables”, Moise’s 1967 ed of “Geometry”, and Jurgensen’s 1972 ed of “Modern School Mathematics: Geometry”). They all introduce the Distance Formula after the Pythagorean Theorem, and discuss it in terms of the PT (respective page numbers: 3/4, 306/393, 320/452). Heineman is representative: “We shall use the Pythagorean theorem to express the distance P_1P_2 in terms of the coordinates of the points.”

Meanwhile, every current Geometry book I’ve used has placed the Distance Formula well before the Pythagorean Theorem. For instance, in Pearson’s current edition of Geometry, the Distance Formula is in section 1-7, while the Pythagorean Theorem is in section 8-1. And, stunningly, the Distance Formula section STILL REFERENCES the Pythagorean Theorem.

How are students supposed to make connections between facts when the facts aren’t even presented in a logical order?

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I’d go with both sets of ideas, can see why its helpful to join knowledge and understanding, but in the end hold out for the 2 being different things. For exampled, there is a whole bunch of knowledge around the Peasants’ revolt, starch manufacture in leaves and the perspective in art, but understanding the links, the processes and the skills associated requires something different to knowledge, which is in itself measurable. The artificial nature of Bloom’s taxonomy does not make it an unhelpful schema, and the separation of knowledge and understanding is helpful – remembering facts and concepts (knowledge) is not the same as understanding how to use them.

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Or I could write that last bit as ‘ remembering facts and concepts is not the same as knowing how to use them.’ Whatever we decide to call that very last part, that’s the important end game – on that we all agree.

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[…] recently read an extremely enlightening blog by Clare Sealy, in which she investigated knowing and understanding and pointed out that they are more interlinked than people often assume. Traditionally, the […]

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