In the essays Having Understanding Versus Knowing Correct Explanations and Understanding, Shallow Thinking, and Schools, I explain why knowing a teacher's explanation of some phenomenon is not necessarily to understand the phenomenon. In this essay, I want to examine the issue of whether understanding a phenomena or relatively complex subject matter can ever be the same thing as knowing any particular set of propositions about the phenomenon or subject.  That is, is understanding something ever the same thing as knowing various (particular) things about it?

The dilemma is that on the one hand it seems that understanding some subject (say understanding photography, or understanding mechanics in physics, or understanding rate-time-distance problems in math or understanding baseball) means you know some finite set of propositions about it -- propositions of the sort that appear in textbooks to explain that subject along with all the propositions good teachers might give as additional information when students raise questions or make mistakes.  Moreover, it seems that students who do gain understanding have done so by reading or hearing such propositions, perhaps along with some other propositions they have figured out for themselves as they thought about the topic, worked problems, etc.  Surely, there are not an infinite set of such propositions that people have to learn in order to understand the topic.  If a person who understands a topic tells all he knows about it, wouldn't knowing all those things then be the same thing as understanding the topic?

On the other hand, even when students learn all these propositions and can recite them, they do not always understand the topic.  For example, they may make non-trivial mistakes in working problems -- mistakes that are not just calculating errors, but mistakes in the way they think the problem needs to be worked; they do it wrong.  Or they may not know how to do the problem at all even though they have, and in some way know, all the information they would need to work the problem.  They just don't know how to use it; they cannot see how to apply it.  But since even someone who understands a topic may occasionally be stumped about how to work a particular problem or how to resolve a particular issue in the subject, what is it that makes not being able to solve a problem in the field correctly at a particular time a sign of lack of understanding rather than a temporary confusion or lack of insight?

And finally, since many students can work certain kinds of problems by rote, by following recipes mechanically, and yet not understand what they are doing (as can be demonstrated by asking conceptual questions, or as can be demonstrated by modifying the problems in ways that don't quite fit the recipes without some modifications that require some insight), the ability to solve particular problems, by itself, does not seem to be what is meant by understanding the topic. Someone who "understood" the recipes could, generally, still use them to solve the modified problem because s/he would transform the formulas or what they represent as needed, but someone who only knows the recipes and can only use them mechanically, but does not "understand" them, generally will not be able to solve problems that do not simply plug into them.

So what does it mean to understand something?

First, this is a difficult question, and I am not particularly confident of my answer to it.  But I suggest the following for consideration.

It seems to me that understanding something means "seeing" or having insight into how it "works", and that what one knows describes part of that insight, but more importantly, it derives from that insight, and is not the same thing as the insight itself.  Suppose we ask someone who "understands" baseball to tell us all he knows and that he writes down everything he can think of.  It might fill volumes.  Still, we might be sitting next to him in the dugout as he coaches a game, and he leans over and tells the manager, Jones can steal on this pitcher because his high leg kick makes his delivery be too slow to the plate.  If we ask why, if he knows this, he didn't write it down in the book, he might say it was because he never saw this pitcher before and he didn't know about the leg kick.  Plus, he knows of some pitchers with high kicks who can still get the ball to the plate fast enough, but this pitcher does not seem to be one of them.  Or it might turn out that he wrote a chapter on stealing bases and that part of that chapter talked about slow pitcher deliveries and their causes, so that in a sense he did write about this particular thing, but without being able to identify every future pitcher ahead of time who might be too slow.  He might have written about the concept in general or at a level of abstraction that encompasses this particular case.

But suppose while he is watching the game, he comes up with an idea he had never had before and did not write down in the book, but which he thinks of now simply because he sees something in a particular circumstance that, coupled with his knowledge and understanding of the game, he sees will work.  He figures out a new proposition, perhaps a particular one in this case, but perhaps later he will be able to generalize it somehow.  It seems to me that the new proposition stems from his understanding and was neither a previous part of it, nor does it, by itself enlarge his understanding of the game.  However, whatever insight he gets from it that allows him to generalize, may, and probably does, enlarge his understanding of the game.  Notice that it seems quite natural to say that "the insight allows him to generalize" rather than saying "the insight is the generalization."

When one of my children was about six or seven years old, I gave her a scrambled Rubik's Cube, showed her how it turned in various ways, and explained the object was to get its faces each to be one color.  She took it and played with it for some time and then came back to me with one face all in the same color.  I examined it and noticed, however, that along each edge of that face, the colors were all jumbled up, so that though all the yellow faces were on one side, none or only one of the mini-cubes with yellow on them had their adjacent side (e.g., red) on the side it needed to be (e.g., the red side).  I pointed that out to my daughter who took the cube and then thought about what I had said.  She then looked at me and said "You mean all the sides of all these little yellow pieces have to be on the right sides, and in the right places too?"  I said "Yes." She thought about it a moment longer and then handed the cube back to me and said "That would be hard."  She was done with it. She had had all the insight into the cube that she cared about.  She probably could not articulate the problems with doing that, but she could "see" or appreciate them from what she had observed while working on the side she had got done. 

On a more positive note, while I was working with trying to solve the cube, I had progressed to the point of getting all but the last side in place, but I didn't see how to rearrange the pieces in it without messing up the parts I already had in place.  I was pretty stuck.  Then while I was driving somewhere and thinking about the problem, a solution took shape in my mind, but it was neither in pictures nor in words.  It was something of a concept.  I thought of it as moving some of the set pieces to an area where I could sort of squirrel them away while I manipulated the pieces I needed to and then return them from storage.  I could see the concept working in general in my mind, though I could not quite see the details in a picture in my mind.  As soon as I got back to my office I tried what I thought would work, and it did.  It was something like seven steps to move each piece and put all the faces back in place up to that point to make sure and to get ready to move the next piece.  I could neither state how to do it in words, nor could I picture the precise steps in my head but I understood the concept I had figured out.  After I did it enough and became real comfortable with the sequence of manipulations, I could probably have written down the procedure as a set of directions in statement form, but that is not how it came to me. 

A comparable thing on a more mundane level is the sort of situation where someone asks you for directions to some place, and though you easily know how to get there if you were driving yourself, you never thought of spelling it out for someone or figuring out the best way to try to tell them to go.  Sometimes the way you would go would be too difficult for someone to have to follow, so you may give them a route that takes a bit longer but is easier to navigate.  Often when something like this occurs and someone asks you for directions, you might hesitate a moment, and they might then think you don't know and make a comment to that effect.  Your response will be something like "Oh, no, I know how to get there; I am just trying to figure out the best way to tell you to go."  Your knowledge is not the set of propositions you eventually come out with; you figure those out by using your knowledge of how to get there.  You have the knowledge before you have the set of propositions that serve as the set of directions. Morever, if you have driven in any city for a long time, you normally know far more places and how to get to them than any set of statements you may have ever thought about.  Knowing or understanding how to get around in the city is not just having a huge number of sets of directions as statements in your head, nor does it even mean you could tell someone how to get to a certain place because you may not remember street names or landmarks (without seeing them in front of you) or exactly what the cues are you have that let you remember at the time where to turn and where to go next.

At an even more mundane level, suppose you report that..........

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