The Concept of a Variable: Expressing a Pattern

The Basic Problem

One of my problems for students is essentially to define what a circle is. Another problem is to define what a line is. These are not easy problems, but I think the struggle is useful for understanding both a circle and a line. (A line can conventionally be defined as the shortest distance between two points, but to me that is a property of a line, and you would want an explanation of why a line is the shortest distance between two points.)

Should you teach a difficult thinking skill for just two math problems? I think it is more important to teach thinking skills than to teach math, so I use these problems as an excuse to teach this particular thinking skill.

The basic assignment is this: Define what a wall is. (You can use any concept, I will just use wall as an example.) According to my informal research, there are very few words that English speakers can adequately define. I sat with one college student for 40 minutes while she tried to define "table". When she was done, a chair fit her definition. The problem, in a nutshell, is that your unconscious knows exactly what these words mean, but your consciousness does not.


I once had a student from China. She said that when she asked a native English speaker the meaning of an English word, she usually could not get a good answer. Her example was asking people the meaning of the word "wall". I start my "lecture" with this, to motivate the students that this is an important type of problem that deserves an accurate answer. When I gave the problem of defining a circle to my students, they resented it whenever I showed them that their answers were not correct. Thier goal was to get the problem marked correct, so I was being an obstruction, not a help. They of course should have the goal of constructing a correct answer, and they should see my complaints as helping them. In fact, they should be making their own efforts to find the problems in their answers.

My student's observation was more than just a complaint about the difficulties of learning English. One of the projects in computer science is building "expert systems", which are computer programs that capture the expertise of experts. As my student pointed out to me, the main obstacle in making these systems is getting the information out of the experts. To give a well-known example from psychology, there are people with the job of separating male from female chicks when they are first born. They are very accurate, but they cannot say how they do it. To give another example, it was found that doctors were teaching their students one way of reading X-rays but then were using a different method when they themselves actually read an X-ray. A third-grader has spent about 7 years immersed in learning the English language, which is enough time to make them experts, and they are experts. And they have the same problem as any other expert -- they are not very good at saying what they know. For example, they probably cannot construct a good definition of wall.

One more example, though I did not use it for my lecture to my students. When I was a graduate student in psychology, we once had one of the expert teachers from the sociology department come and give a lecture on how to be a good teacher. It was a big deal -- most of the psychology faculty came to listen and there was a big crowd. One of the points he kept emphasizing was this -- First you tell 'em what you are going to tell 'em; then you tell 'em; then you tell 'em what you told 'em. In other words, introduction, main body, conclusion. Okay. During the questions after his presentation, one of the graduate students, Diane Morrison, said this to him: "When I start to tell my class what I told 'em, they start packing their bags to leave." The expert teacher answered, "When you tell 'em what you told 'em, you lift it to the highest level of general and importance."

Whoa. That's completely different. If you tell 'em what you just told 'em, they are likely to be bored. If you lift your message to it's highest level of generality and importance, you have a great conclusion. That advice has served me well. But why did he leave it out of his presentation? He never thought of it, until Diane asked her question. (As I later realized, no one then asked him what you really do for an introduction.)

Phase 1: The First Answer

The first thing you ask students to do is provide definition. They usually do. It is usually inadequate. However, you do not want your students leaving with a bottom-line impression that there first impulse is bad. It is good. It is very important. First, the subsequent steps don't work unless you already have a defnition to word with. Second, the first impulse always contains good information. It might not be a complete answer. But it's a step, and maybe even a giant step, towards a good definition.

Eugene Gendlin has developed a technique called Focusing, designed to help people know what their feelings are. The problem of defining a word is not the same as the problem of describing a feeling, but the two are very close. One of my students was a poet, so I likened the problem of defining a word to the problem of finding what you wanted to say in poetry. Anway, both processes start with blurting out an answer. This answer comes from the unconscious, and it contains useful information. It's just not the end of the story.

So the bottom line message should be that the student's first answer is good and desirable, and they are making good progress. Then on to the next step.

Phase 2: Finding the Flaws

Unless you are very good at this, and even if you are very good at this, you should probably not just choose a random word and do this exercise along with your students. If you do, you will have the same problems as them, and it might difficult for you to find the problems in their definition. So you should have some words, like "wall" and "bottle", that you are prepared for.

An adquate definition of a wall has two properties. First, it should fit all walls. Second, it should not fit things that are not a wall. Finding the flaws means finding walls that don't fit the definition and non-walls that do fit the definition.

You do not just look at a definition and know that it is inadequate. To do that, you would have to consciously know the correct definition. The whole point is that you are trying to find the correct definition even though you consciously do not know the definition. Instead, you are looking for counterexamples to the definition. You are using the fact that your unconscious will tell you what is and what is not a wall. The problem is, how do you find these counter-examples?

Phase 2a: Do Other Walls Fit the Definition?

When someone constructs a definition of a wall, they usually look at a wall and report the features of that wall. There is more to it than that; the point is just that they probably have a specific wall in mind. So the first point of attack on a definition is to think of other walls, then see if they fit the definition. For example, one student defined a wall as something separating one room from another. That seemed like a good definition to me, but then I looked at the other wall of the room, which separated the room from outside of the building.

Phase 2b: Do Other things Fit the Definition?

Second, you look at features in the definition, think of other things that fit those features, then ask if they fit the whole definition. Suppose a student defines a wall as something that is flat and vertical. You need to think of things that are flat and vertical. If they are walls, fine. But if you happen to think of a window or a door, you will realize that flat and vertical is not enough to define wall.

Phase 3: Correcting the Definition

Once you find a wall that doesn't fit the definition, or a not-wall that does, the definition needs correcting. The process of correcting usually happens easily and spontaneously, which is to say, the student changes the definition without prompting. This may be a completely new definition, but usually it is just a change to the existing definition.

It is important to realize how this process occurs. When you see a wall that doesn't fit your definition, the problem with your definition immediately comes to consciousness. Your unconscious brain won't tell you the correct definition, but it will tell you the difference between your definition and the wall you are looking at. Same thing for not-walls that accidentally fit your definition. You can look at the thing that is not a wall, such as a window or door, and your unconscious will usually tell you how it is different from a wall. For example, mobility might come to mind.

The point is this. The counterexamples don't merely show you that your definition is wrong. They also help you improve your definition. Essentially they are the keys for extracting more information from your unconscious.

A second point to realize about this phase is that any change in a definition undoes half of your past checking. If you make your definition of a wall more lax, to accomodate some wall that didn't fit your definition, you have to worry that some not-walls fit this new definition, even if they fit the old definition. Similarly, if you make your definition more strick, to rule out some not-walls that were fitting your definition, you have to worry about walls that don't fit the new definition, even if they fit the old definition.

Avoiding Circularity

It is well-known that a definition cannot be circular. For example, you cannot define a bottle as being something that looks like a bottle. The problem is hidden circularity. Suppose one part of the definition of a dog is that it barks. The question then needs to be asked, what does the student mean by "bark"? The student cannot define this as the sound a dog makes, because then the definition of dog is circular. Similarly, if a wall was defined as the things surrounding a room, you might want to check the student's definition of room, just to make sure the student can define "room" without using the concept of wall.

On the other hand, a person cannot be forced to define all of the words used in his/her definition. The whole point of this exercise is the difficult making definitions.

Winning the Game

The actual conclusion to the game depends on what word you are trying to define. Most words have "fuzzy" definitions and cannot be defined completely accurately. For example, if you wanted a very good definition of "dog", you would end up with a long list of features that dogs have. Things that were not dogs would have only a few of these features. However, something with all the features but one would probably still be called a dog. For example, dogs have four legs, but something with all the features of a dog but with 3 legs would be called a 3-legged dog. I call this the three-legged dog problem, but you also have problems with a miniature dog, a hairless dog, a dog that can't bark, a dead dog, and a plastic dog.

Mathematical concepts are not fuzzy, so you can construct definitions that work perfectly. For example, you can make a definition of circle such that all circles fit the definition and all non-circles do not. However, this ending too is not completely satisfactory. The problem is that there are many such definitions of a circle. From a mathematical/philosophical point of view they are all equal, though psychologically we might feel that some are better than others. Students will rarely appreciate that, but the point is, there is little reason here to stop with just one definition.

The real winning is understanding the concept better. When you reach a reasonable definition of a fuzzy concept, that's good enough. When you have found one property of a circle that serves as a definition, you can break out the champaigne, and you can also go on to look for other distinctive properties. But you are a winner when you better understand the thing you were trying to define.

Other Examples

As noted, you can probably do this with any concept, you just have to be prepared. "Bottle" is a good word to try to define, because most people forget the bottleneck. For "wall", the usual problem is constructing a definition that does not include windows and doors.

This exercise can also be done about the perception of things. For example, just looking at a line drawing of a face, how could you tell that it was a drawing of a cat? In my research, I have found that almost everyone will use the almond shape of the eyes, but that feature is very difficult for people to report. Or, standing a small distance from a lake, how could you tell that it was a lake? I don't know the answer.