UNDERSTANDING.

1. Understanding vs knowing.

One of my favourite things of all time is understanding stuff. For example, take one of those retractable click pens. You know the ones where, when you click the top, the tip of the pen is pushed outside the casing and you can write with it and then, when you click the top again, the pen tip retracts. If you take the pen apart you can see how the mechanism works. It’s neat how the same action of clicking the top does two different things each time!

I’ve always been surprised that, in educational institutions, the word “understanding” isn’t used more than it is. Instead the word “knowledge” is more common, where education is then just simply “the getting of knowledge”. (These days we even have something called a “knowledge economy”.) The presumption seems to be that the word “knowledge” can just be taken to include understanding as well. But this is not right. Understanding is not the same as knowing.

The word “understanding” suggests something more sophisticated than “knowing”. Knowing is the result of someone telling you something, while understanding is the result of someone explaining something to you. Explaining is more sophisticated than telling. In the same way that understanding is more sophisticated than “coming to know”.

For example Jack tells Mary that the Eiffel Tower is in Paris. As a result of Jack telling her this she “comes to know” that the Eiffel Tower is in Paris. This is all very straightforward. The whole thing happens quickly with minimal effort from both sides.

But if Jack explains to Mary why it is that two objects of different mass fall at the same rate then both need to exercise a greater degree of mental effort.

It is more natural to say “I UNDERSTAND why two objects of different mass fall at the same rate” than it is to say “I KNOW why two objects of different mass fall at the same rate”.

I suspect that the reason why the word “knowledge” is preferred to “understanding” in educational institutions is because the simpler act of telling is the only one that those institutions are able to do for their students.

Some questions that ask for understanding rather then knowing:

Why do two objects of different mass fall at the same rate?

How does a rainbow happen?

Why don’t hollow objects sink?

What causes the seasons?

How do steam engines produce motion?

Having a good understanding of the world isn’t the same as just knowing lots of facts. Understanding does not simply reduce to lots of factual knowledge. Sometimes I think it might be possible for someone to know all the facts associated with a certain scientific theory and yet still not understand it. For example I might know (be able to tell someone) lots of things about Newton’s Laws of Motion and yet not really understand those laws.

Quiz type tests (the kind you get on TV shows) test for knowledge of facts and people who get high scores on such tests are considered intelligent. But such tests only test for knowledge not understanding.

Sometimes we wrongly use being knowledgable as a false measure of understanding. Suppose Jack knows that laundry detergent contains alyklbenzenesulfonate. When he tells us this it makes him sound clever and makes him sound as if he understands about detergent chemistry. But he might not. He might have just memorised the complicated terminology.

The difference between understanding and knowing lots of facts seems to be that understanding is more about how all those facts relate to each other. How some ‘follow’ from the other. Understanding is therefore something that is more ‘holistic’ than knowing.

Maybe understanding is more about having a skill, a “knowing how” to do something, rather than a “knowing that”. So if I know how to juggle with three balls this doesn’t consist of me knowing any particular set of facts about how those juggling balls move. Like other sorts of “knowing how”, understanding isn’t forgotten as fast as factual “knowledge that” is forgotten. (As per the way people say: “you never forget how to ride a bicycle”.)

Understanding is to do with reasoning. It’s about seeing how one thing follows from another. So from “Jack is either in the garden or he is in the library” and “Jack isn’t in the library” we can see that it follows that “Jack is in the garden”.

More examples. It follows from “when Jack was born Mary was as old as he is today” that “Mary is twice as old as Jack”. Knowing how the latter follows from the former is about understanding. Being good at understanding means being able to see very quickly what follows from something. Similarly it means being able very quickly to pick your way through the logical paradox of a sentence like “this sentence is false”. Rather than just being befuddled by it.

A more complicated example. How you can measure out 4 litres of water using two jugs whose volume is 3 and 5 litres? There are no measuring marks on any of these jugs. (This is one of those famous puzzles that people know.) The answer is fill the 5-jug and pour it into the 3-jug leaving 2 litres in the 5-jug. Then empty the 3-jug and pour into it the two litres from the 5-jug. Now you have 2 litres in the 3-jug. Then fill the 5-jug and from that top up the 3-jug to full. This means you are pouring 1 litre from the 5-jug leaving 4 litres in that jug. Understanding this is all about seeing how each step of the process follows from the previous. And how all the steps together get you to what you want.

We get to an understanding of something the way we get to “knowing our way around” a new town. In a new town you need to wander around a lot. And so become familiar with the territory. Seeing the short cuts. And once you do “know your way around” this sort of knowing is not something that reduces to just knowing a collection of facts about how to get from some places to some other places. In the same way that a map does not reduce to a list of directions from any one place on the map to any other place on the map. (Such a list would be impossibly long anyway!)

Similarly, to understand some content (for example the water jug puzzle just described) you need to go over it in your mind a few times. If you are reading for understanding it is never enough to read just once.

Understanding complicated things is also a bit like coming to see objects when you walk into a dark room. It takes a while for things to emerge out of the darkness.

Understanding is often about ‘seeing’. It’s almost an artistic, non-rational, mystical thing. You have to be in the right frame of mind. (Consider the story of how a scientist got to an understanding of the structure of the benzene molecule, HERE.)

2. Half-understanding.

I hate half-understanding. This is when I understand something but my understanding is clearly lacking. It’s where I have an intuitive, sketchy, gist understanding of something but where I can see this is not enough. Where I know that something is missing from my understanding. (Sometimes I can’t even say what it is that is missing.)

For example I understand that a steam engine (of the sort that pulls trains) uses the force of steam to work. I might have seen steam from a kettle pushing off the lid, so I understand what sort of force is behind a steam engine. It’s not as if I think it’s some magical mysterious force. So on that level, yes I do understand how a steam engine works. But I don’t understand the exact mechanism by which that force of steam is converted into the motion of a steam railway locomotive. So I don’t properly understand how a steam engine works. I don’t have the level of understanding which it would be reasonable for someone to think I had if I said to them that I understand how a steam engine works.

Three other similar examples.

(1) I understand that cell phones work by the transmission of radio waves. But that’s a superficial understanding. It doesn’t say how a cell phone call actually works. Certainly knowing that it’s about radio waves is some level of understanding. But not good enough. Because it’s only understanding compared to somebody who has got no idea and who might therefore think that cell phone calls are done ‘by magic’.

(2) Suppose I have a wooden box which is constructed of five separate pieces of wood stuck together. So I understand the construction. In the sense that I have an understanding greater than just thinking that it’s a box and not even knowing that it’s constructed out of separate pieces of anything. But it’s less understanding than if I knew exactly how the pieces are held together. Is it by glue, pins, screws, or joints or what?

(3) A solar eclipse is when the Moon passes between the Earth and the Sun. I understand that. But then why isn’t there a lunar eclipse every month? If I can’t answer that then my understanding is not complete.

(4) A rainbow is produced by the refraction of light. I understand the gist of the process. I don’t need to invoke fairies or anything. But I don’t understand exactly why it’s coloured the way it is. And, suppose it wasn’t coloured. Even then I don’t understand why it’s curved the way it is.

(5) Suppose someone said to me: “Jack needed to get to an appointment some place but he couldn’t find his car keys so he went to Mary’s”. I would think: I don’t understand why he went to Mary’s. I do understand (so I do have half-understanding) that going to Mary’s is (probably) somehow going to help Jack get to the appointment. But proper understanding would need more. Maybe Mary has a spare set of Jack’s car keys. Or maybe she promised him she’d give him a lift if he ever needed one in an emergency. Or something else.

What level of understanding is sufficient depends on the material being understood. So I understand Pythagoras’ Theorem that the square of the longest side is equal to the sum of the squares of the other two sides. But, as a mathematician, you only really properly understand this once you’ve got a proof.

The difference between half and full understanding is rather like the difference between speaking a language and speaking it fluently.

3. The improvement of understanding.

Is there any way you could teach someone to be good at the kind of reasoning that is involved in understanding? How could someone improve their ability to understand things quicker and more easily? Is that possible?

When I try to understand something and fail my immediate thought is: why? I think: OK, I’m not going to understand it, it’s way beyond me. But if I can’t understand then can’t I at least understand why I don’t understand? Can’t I understand what I am failing to do that other people who do understand this sort of thing do? Maybe if I could figure out what they do that I’m not doing then I could do that and then I would understand!

The work of studying is reading and understanding. And I think to myself: what is the nature of this effort? Exactly. Like if I was digging a hole. I would think about the nature of the effort and understand it so that I could improve my digging technique. Can’t I do the same thing for the nature of the effort involved in studying.

When I finally understand something I look back and think: what’s the difference between now and when I didn’t understand? But it’s hard to see this. I think: if I could be aware of exactly what I did which got me to understanding then I could just do that again, the next time I have something I find difficult to understand. And so then I will be better at understanding things in general. But it’s not possible to be aware of that sort of mental activity, is it? In fact most of our mental activity is beyond our awareness. For example when I recall somebody’s name I’m not aware of the exact mental process involved in that.

There are some methods to help understanding. One is “incubation”. This is where you leave some time between readings of something you are trying to understand. You read it once and try to understand it. Then you leave it a few days and do the same again. Each time it should become clearer. Until eventually you understand fully. During the days when you are not actively trying to understand your mind is somehow working on it.

But that’s a somewhat indirect method. I’m asking: is there any way to directly teach the skill of understanding things? I think: there must be! Because understanding things is just the doing of certain mental actions. People who are good at understanding are just doing these things. So all I need to do is find out what those things are that they are doing and do them too.

Sometimes it seems as if improving your understanding ability might be a quite straightforward thing to do. All you need to do is to find out what mental micro-skills are involved and then master them individually one by one. Thus breaking the problem down.

One of those micro-skills seems to be being able to keep a number of things in your immediate memory at the same time. This is not for the purposes of improving memory as such but to improve thinking and reasoning ability. Because to be able to think about something and see how one part follows from the other you need to have as many parts as possible simultaneously available to you in your mind.

A simple example might be the understanding of why two objects of different masses fall at the same rate. You need to keep two facts in your mind at the same time. These are (1) a heavier object needs more force to move it. And (2) a heavier object has more of a gravity force on it. (I need a better example than this one!)

This kind of “working memory” is not as easy as it sounds. If someone were to read me a sequence of 4 digit numbers and then stop and ask me to recall the number they told me six numbers back I wouldn’t be able to do that! (There’s something called “n-back” training which is supposed to help you with this kind of thing.)

Other mental micro-skills might be:

(a) Noticing things. For example if I gave you a list of eight words and they were in alphabetical order would you notice that fact straight away? What if the last letters of each word were in alphabetical order? Would you notice that?

(b) Concentrating and paying attention without your mind wandering. This is the kind of skill required for proof-reading.

(c) Prospective memory which is remembering to do something.

(d) Thinking about whole and part. Whenever you are thinking about or doing something it’s about a whole big thing. Like if you are following an argument in some text. Or you are assembling a piece of furniture. But at any one instant you are only dealing with a part of it. In an argument you will be reading a particular step explained in a sentence. With the furniture you will be putting a particular screw into a particular hole. What you need to be good at is keeping the part and the whole in your mind at the same time.

(e) At a slightly higher level of abstraction you could break down reasoning into its elements and learn those one by one. Things like the fact “if a then b” does not imply “if not-a then not-b”. If you learnt lots of such rules by rote then that would improve your reasoning abilities.

4. Disagreeing vs not understanding.

Suppose that some text presents an argument X which purports to show that B follows from A. And Jack says “I think that X is wrong (ie invalid). Then Mary might say to him “no, you just don’t understand X. If you understood then you would see that it was right.” And Jack would say: “no, I understand perfectly!”.

Conversely Jack might say that he doesn’t understand X. “I don’t understand how X is saying that B follows from A”. And Mary might say: “that means you don’t think that X shows that B follows from A, which means that you are saying that X is wrong”. And Jack would say: “no I just don’t understand”.

In this latter case it is true that Jack doesn’t think that X shows that B follows from A. But that doesn’t mean he think that X is wrong. For example if someone said something to Jack in a language he didn’t understand. And then Mary asked Jack: “is what they said true?”. It would be true that Jack doesn’t think it’s true. But that doesn’t mean he thinks it’s not true.

Consider this example. Suppose Jack has bought some apples from shop A for £2 per kilo. He tells Mary this and he gives her an apple to eat. She takes a bite and then she says to Jack: “it would have been better for you if you had got your apples from shop B”. Then she asks: “am I right?”. But Jack doesn’t understand what she means. Does she mean that the apples at shop B taste nicer? Or that they are cheaper? At this point Jack can’t answer ‘yes’ to Mary’s question. So she could then say that he thinks she is wrong. But he doesn’t think she’s wrong. He just doesn’t understand what she means.

[12 November 2018]