*In this section, Wit Busza discusses common sources of confusion for students during problem solving.*

### Problems Cannot Actually be Solved

One element of problem solving that often confuses students is that when we give students a problem, as a rule, it cannot be solved. There is no solution to the question, because in the real world, things are extremely complicated. When I drop a ball, it doesn’t go through the air without resistance. There is also wind. When I drop it, I slightly rotate it, etc. In reality, there are a million things that determine how the ball falls. But when we set up the problem, we are not interested in those complications. We are interested in something specific: the concept that relates to the chapter we are teaching.

Now the poor students don’t know this. So they may be preoccupied or worried about the very things that they’re supposed to ignore. In reality, some of the factors they are asked to ignore are more important—because they have a bigger effect on the answer—than the question that is being asked.

For the professor, this is obvious. He or she has assigned the same problem many times before. The professor knows what he or she wants. But this is not explained to the students in the problem because it’s impossible. You would have to write a book on every problem. Instead, students are asked to ignore some effects and approximate others. Students justifiably find the problem very hard because they don’t have the experience needed to discern what you can approximate and ignore and what you cannot. This is confusing because the students may actually be correct: the very effects they are asked to ignore may, in the real world, greatly impact the solution to the problem.

This is particularly true in electricity and magnetism. I can find, in every book on the market, problems where the solutions are wrong, because the part that the students are supposed to ignore is more important than the part which they are expected to take into account. How are the students to know that?

But from a professor’s perspective, it is very difficult to write a problem—to tell students what they need to ignore, without giving away the answer. By the time the professor finishes telling students everything that is an approximation or can be ignored, she’s more or less told them the answer!

To address this challenge, professors need to spend lots and lots of time writing problems. I once gave a rule of thumb, when I was advising younger faculty, that they should spend as much time preparing the assignments as they do preparing the lectures. Assignments have to be incredibly carefully prepared because it is through them that students learn. You have to ask questions that really teach students the essence of a problem without confusing them.

### The Famous Equal Sign

Something in physics that confuses students to no end is the famous equal sign—those two parallel lines! The problem is that those two lines have many meanings. In some cases, the equal sign indicates a definition. For example, density = mass / volume. That’s a definition. You cannot prove it. I cannot prove that density is mass divided by volume. I cannot derive it.

Another meaning of the equal sign is the equality of two mathematical expressions. For instance, 2 + 3 = 5. That’s not a definition. I can’t prove it to you. I’m writing the same thing twice. It’s an identity. The two expressions are logically the same—this is a completely different meaning than a definition.

The next thing an equal sign can mean is a physical law. An example would be the change of potential energy is equal to the change of kinetic energy. You cannot prove that, but it’s also not a definition. It follows from the law of conservation of energy.

These three meanings of the equal sign are completely different. When the student sees a derivation by the professor, he or she may get confused when trying to understand how the professor got from line 1 to line 2, thinking it was a mathematical maneuver, when in reality, the professor brought in a definition. Or perhaps the professor writes the famous F = ma. The student might say “prove that F = ma.” This is not possible, and so the student may wonder, “Where did she get that from?” Well, the professor brought in a law of nature—and if the student could prove it, he or she would be as famous as Newton!

A lot of this confusion arises because we use the same symbol to mean different things without alerting students to the fact that we wrote the same symbol for three completely different meanings. We should explain this. Like any other professor, I am not perfect. I don’t always make it explicit when the meaning of the equal sign changes in my work, but in the problem solving videos for *8.03 Physics III: Vibrations and Waves* on OCW, I’ve tried very hard to follow my own advice.