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 . 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.  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.
 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.
 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.