How are mental models taught? And -- really the same question -- how are existing mental models added to and improved upon? This chapter presents several methods. They all might use the same underlying mechanism, but they look different from a teachering perspective.

Achieving Goals

My idea of different types of knowledge was inspired by psychological research. In that research, mental models are learned when people try to achieve a goal and receive feedback after each effort. So according to that research, mental models can be taught by giving students a goal to accomplish. I call this goal-oriented learning.

By itself, that research adds up to just an idea to try. I have tried goal-oriented learning. It works. And because the human brain was designed to achieve goals, this is a natural way for human beings to learn. Putting this all together, the natural technique for teaching mental models is goal-oriented learning.

However, to use goal-oriented learning, you cannot first teach your students a method of accomplishing their goal. If you do, they will try to store your instructions in memory. That creates memories, not mental models. To achieve that goal, they will consciously recall the instructions, and they will consciously use the information. That's using memory, not mental models. To develop a mental model, students must be given a goal that they do not know how to accomplish. I imagine them stuck on one side of the river, and their goal on the other side of the river. When they don't know how to cross, they have to construct a bridge. That bridge is a new piece in their mental model.

Problem Solving. In problem solving, the goal is solving the problem. So problem solving fits goal-oriented learning perfectly. Again, students cannot first be taught a method of solving the problem. If they are, they just form and use memories. For example, the conventional method of teaching mathematics is to first tell students exactly how to solve a problem, then give them a problem. Most students just memorize and consciously apply the method they are taught. So they form memories, not mental models. So even though the conventional method of teaching mathematics centers around solving problems, the learning process is not goal-oriented learning.

The discovery of a solution is called an insight. The Gestalt psychologists said these insights occur when there is a reorganization in the brain. This reorganization is a change in a mental model. You can think of the mental model as wiggling around and changing itself until it morphs into something that can solve the problem.

In studying and talking about insights, it is natural to focus on difficult insights requiring a major reorganization. But problems can also require just a minor reorganization. Sometimes, a problem just strengthens a mental model. For example, suppose you want your students to be able to solve problems like 5x + 7 = 22. The first time they solve this problem, they are building a mental model. But it is a weak mental model. Solving problems like this again and again strengthens the mental model. As will be described in the next chapter, as long as the student is not bored, the student is probably strengthening a mental model; when the student is bored, then no learning is occurring.

Feedback is an important part of goal-oriented learning. Fortunately, problem-solving usually has clear and immediate feedback.

Specific Skills. If you are trying to teach some specific skill, just put your students in a situation where they need to use that skill. For example, there are two standard methods of teaching college psychology majors to be good experimenters. The first is to lecture. That doesn't work. The second is to have them experience examples of good experiments. That doesn't work either. What does work, amazingly well, is simply to give them goals to accomplish via experiments. This forces them to develop and use the skills of experimenting. I would give my students a research goal to accomplish. They had to design an experiment to accomplish this goal, then run the experiment using their classmates as subjects. In the course of trying to achieve their goals, they learned.

I also wanted to teach them how to analyze data. So I had them analyze their data. To teach the skill of thinking of ideas for experiments, students were given the assignment of constructing ideas for experiments. What's the pattern here? Have students do what you want them to learn.

General Skills. There are some general skills you want students to learn -- problem solving skills, social skills, and communication skills are probably the big three. General skills are taught by putting the students in situations where they need to use those skills -- giving them problems, having them interact, and having them communicate. You won't know exactly what they are learning, because you have little control over what actually happens. But that might be for the best -- psychological research suggests that you have poor conscious knowledge of your skills, so you probably could not lay out what your students need to learn even if that was an effective way of teaching. For example, if you want students to learn social skills, let them work together. Obstacles and difficulties will naturally occur. In trying to resolve these problems, your students will receive feedback and learn mental models.

Projects. Giving students projects to do is unstructured goal-oriented learning. If they become invested in a project, they will form goals; when they try to achieve their goals, they will develop mental models. You don't have much control over which goals they form, but they are learning. The feedback usually occurs naturally.

Resolving Conflict

Once I asked students, "If you went to the mall and selected 20 people at random, what do you think the average age would be?" and "If you went to the mall and selected 40 people at random, what do you think the average age would be?" One student answered 20 to the first question and 30 to the next question. That reveals an incorrect mental model of averages -- whatever the expected average is, it doesn't change just because more people are sampled.

What should I have done? Most teachers would tell the student that she is wrong and explain the correct concept to her. Maybe that would have worked. But this student had already heard the correct answer many times. (She had taken a college level statistics course.) Why should one more presentation of that same information have any different effect? I asked her what the expected age would be if 100 people were sampled. She quickly said 50. I then asked her what the expected age would be if 1000 people were sampled. She paused.

I could hear the parts of her brain gnashing together. Her mental model of averages said that the answer was 500, but she knew that was ridiculous. She said she wanted to think about it, then she went off by herself. She came back a little while later and confidently said that the expected average age is the same no matter how many people are asked. Exactly right.

Magic! On her own, she learned something that she had not learned in her previous classes. I provided no information. I did not tell her that her answers were wrong. All I did was ask her questions, questions that she could have asked herself.

But she hadn't asked herself those questions. My questions were constructed so that she could experience herself the problem in her mental model. It was important that my feedback to the student wasn't simply "your answer is wrong." The student needed to experience that her answer was wrong.

People can happily go through life with a wrong mental model, as long as they don't have to experience any of the wrong predictions. But an experience of wrongness creates conflict. When there is conflict, sometimes the person is just confused, and sometimes the mental model is right and something else is wrong. But sometimes, people can realize their mental model is wrong and fix it. So this is a second method of teaching mental models.

These conflicts usually happen spontaneously, when a mental model makes a wrong prediction. In my teaching, I induced the conflict with a careful question.

Another example. To diagnose and repair errors in my students mental model of standard deviation. I gave them problems such as "Which has more standard deviation, the set <0,1,2,3,4> or the set <0,10,20,30,40>?" They had once memorized the formula for calculating standard deviation, but there wasn't time to do these calculations, not to mention that most of them had forgotten the formula. So they had to use their mental model of standard deviation to answer my question. Some students concluded that these two sets had the same standard deviation, which is incorrect. Next I had to lead them to a conflict. I asked them to compare the sets <0,1,2,3> and <0,10,20,30>. Of course, they gave the same answer. I continued to drop a number from each set until I got to <0,1> and <0,10>. For this comparison, the students knew that the second set has the larger standard deviation, even though it follows the same pattern. This was their conflict. They resolved this conflict in the appropriate way, improving their mental model of standard deviation.

Another example. When I tried to teach third-graders to calculate area, I ran into the problem that they didn't seem to have a concept of area. How do you teach the concept of area? If you have a memory approach to learning, you try to define it. If you have a mental model approach to learning, you try to think of a problem that area is the answer to. So I showed them two shapes, both rectangles. I told them they were chocolate bars, and asked them which one they would choose. That's a question about area, and they knew it.

One of the shapes was a 1 x 8 rectangle, and the other was a 3 x 3 square. They choose the 1 x 8 rectangle -- because of a perceptual illusion, it looks larger. Then I asked them why it was larger. They explained about length. So I showed them a 1/4 x 8 rectangle, and asked if they wanted that over the 3 x 3 square. The illusion is still there but it isn't powerful enough to overcome the fact that the square had more than four times the area of the rectangle. So that created a conflict. I am not sure exactly how they resolved that conflict, but it had to be in my favor.

Recall that they choose the 1 x 8 rectangle over the 3 x 3 square. I now cut up the 1 x 8 rectangle and showed how it could fit inside the 3 x 3 rectangle with some left over. Another conflict, because they new this meant the 3 x 3 square was larger. Again, this conflict had to be resolved in my favor -- area wasn't just length.


Presenting information to students has been the bad guy so far in this discussion. But there is nothing wrong with information per se. When students practice goal-oriented learning, their feedback is information. When students experience conflict, that conflict is information. So sometimes presenting information works simply because it is in the format of goal-oriented learning, or in the format of creating a conflict. For example, in this book I have tried to tell stories with outcomes different from what your mental models would predict. This conflict, if you experience it, hopefully opens you up to changing your mental model.

Hints and answers. Once your student is in a goal-oriented situation, hints and answers aren't so bad. The idea is that you have set up a situation where your student needs to build a mental model. Ideally, your student builds the mental model, but sometimes you can help in the construction. Again imagine the metaphor of the student on one side of a river and needed to build a bridge to get across. If the student really is on one side of the river, and really wants to get to the other side, and can see how to use the bridge, it's no so bad if you help the student build the bridge. But if the student is in the desert, handing him a bridge does not build a mental model.

So it's okay to help occasionally when needed. But if you find you are doing all the work, then the situation is wrong for building mental models. Simlarly, if the situation is right, your student will ignore any advice doesn't help him/her produce the solution or achieve the goal. That's good. If your student is ignoring your good advice, give it up.

The bottom line is that if a student can't solve a problem, you can always give an easier problem, or you can always wait for another day. If a student is working on a problem, you can wait to see if the student solves it, and even if the student doesn't solve it, the student is probably learning a lot. If a student gets the wrong answer, you can always ignore it, or wait for the student to get feedback that it is wrong, or provide further experience that creates feedback or conflict. Students do not need to correctly solve every problem they are given. So you can avoid giving a lot of information, and in goal-oriented learning, you probably should not be providing a lot of information. But when the student is stuck, or going in circles, it might work to give a hint, or maybe even the answer on occasion; when a student is not noticing an error, you might need to give feedback. But that help should be "light". If you feel like you are trying to force your student to learn, give it up.

Adding to existing mental models. Ironically, in the course of natural learning, presenting information often adds to a mental model. First, sometimes your mental model has what I call a hole. A hole is a piece of information that you know is missing from your mental model. Your emotional experience is being interested to learn that piece of information. It will be exactly like when you have heard most of a story and desperately want to learn the ending. When someone tells you the information that fills up the hole, your mental model will suck up the information. (This is unlike memory, which requires rehearsal.)

Mental models can also have rough edges where you can hang new information. You won't be interested to learn the information prior to learning it, because it isn't a hole, but the information will be added to your mental model. Your emotional experience is finding the information interesting. You will consciously dwell on this information, while at the unconscious level your mental model will reorganize itself. This is unlike the learning of conscious knowledge, in which there is no interest in the rehearsal. For example, suppose you hear something new and interesting about your friend. You think about it and change your mental model of your friend.

So information can be added to mental models. However, the mental model has to be there in the first place, and it has to be complex enough to have holes and rough edges. The problem with today's educational system is that it doesn't create these complex mental models. The students are given information to memorize. They memorize it. That creates no mental model. So when the next piece of information is presented, there is nothing they can do with it except memorize it. So the educational system fails.

Sometimes the students have mental models, but the mental models are not activated by the class. Instead, the students just memorize the new information and ignore their mental models. I found in my research that students walk into a college-level statistics class with a good mental model of dispersion. Then they are taught the formula for standard deviation. Standard deviation is just a measure of dispersion, but the formula is never connected to their mental model of dispersion. So students often leave a statistics class with a poor mental model of standard deviation. Similarly, physics students usually walk into class with their own mental model of how the world works, and the class creates new information. Students then can report the conflict between their old mental model and the information they have learned. Teaching without engaging the students' mental models is like wallpapering a room without first taking down the pictures on the wall.

Remembering Conscious Knowledge

My student Beth Friedman did this experiment on mental models. One group of subjects read an encyclopedia article about how pens worked. Then, a week later, they were testedfor their memory of information in the encyclopedia article. Think of these subjects as receiving the traditional style of education.

The second group of subjects first tried to explain how a pen worked. Then they were asked questions about their explanation, such as, "Does your explanation predict that the ink will leak out of the pen when it touches the paper?" Then these subjects read the same encyclopedia article and took the same test a week later. The questions created conflict, and the overall process probably deepened their mental model of pens. We would have loved to test which group of subjects had a better mental model of pens,. Alas, we didn't know how to do that. So we tested their knowledge of facts. Which group of subjects will remember more facts?

The second group did. Why? Obviously, they spent more time thinking about pens. But, to understand this experiment, you must recall the experiment by Helene, described in the last chapter. In that experiment, one group of subjects spent a lot longer with word pronunciations, but they had worse memory for the word pronunciations. The reason was because there is no mental modelling to learning a word pronunciation, that's just memory. And they spent their extra time on wrong pronunciations, so that interferred with their memory.

In this experiment, the subjects who had to say how a pen worked never said the correct answer. So they too were spending their time with the wrong answer. The questions didn't reveal how a pen worked, they just revealed the problems with their explanations. But the critical difference was that the students could develop a mental model for pens. The questions had them do this. So even though students produced the wrong answer about pens, and even though the questions to them did not provide any information, these activities helped them remember facts from the encyclopedia article.

Still, we have a mystery -- why did having a mental model of pens, or reading the information to improve a mental model, improve their memory for the facts of the encyclopedia article? The answer, I am fairly sure, is this. Experiment after experiment has shown that people have poor memory for things that are not attached to mental models, and good memory for mental models. This experiment suggests that people have good memory for the facts that are a part of their mental model.

Now I can make the final point about why our educational system fails. The goal in education should be to teach mental models, not facts. Our educational system usually fails at this. But suppose the goal was just to teach facts. Then would our educational system be perfect? No, because people forget facts when they are not embedded in a mental model. So students are given tests, and they can pass these tests if they are given right after the student studies. But a while later, the students forget what they learned.


The natural technique for teaching mental models is to give your students a goal to solve, then let them develop mental models in the context of trying to reach their goal. The goal can be solving a problem, or doing something that requires a skill, or just some general project the student is invested in accomplishing.

However, it is very important that you not first teach students a formula or method of solving the problem. Then they will just memorize what you have said and try to consciously apply it. That forms memories, not mental models. (As Lao Tsu said many years ago, the path that can be laid out to follow is not the true path.)

A second method is to have the student experience a conflict created by the student's wrong prediction based on his/her mental model. This happens naturally in real life, when a mental model makes a wrong prediction. Sometimes a teacher can intentionally craft a conflict to help the student.

So the process of learning mental models is very different from the process of learning memories. Ironically, the presentation of information can be useful once the students have a complex mental model, the problem is that education today does not lead students to forming mental models or even using their existing mental models. Hints and other information can also play a role in learning mental models, as long as all the elements of goal-oriented learning are in place. But the presentation of information should be "light" -- students should be mostly thinking for themselves and working on their own solutions. They should not be memorizing what the teacher says.