Monday, October 03, 2011

You Can Do It, But Did You Understand It?

Chatting with a bunch of people about our times in college brought up something that had crossed my mind a few times. Was there a course or subject area in which you could pass or get through, or even got good grades in it, but you thought that you didn't actually had a good grasp on it?

I certainly did. And the subject area was Thermodynamics. For some odd reason, even though I got good grades in it, I just didn't think I actually GET IT when I was an undergraduate. At that time, I thought that the subject was disjointed, and I see a lot of "starting points", where such-and-such an equation or description goes with such-and-such a problem. I didn't see any underlying uniform idea through the whole thing. Yet, I could still do well in exams.

I didn't get that feeling in other subjects, such as Classical Mechanics, E&M, QM, etc. I'm not saying that I find those easy, or easier, but at least I had a clear "view" of the material and able to figure out where I was at any given time. In Thermo, I had a pieces of the puzzle, and I can work with that puzzle, but I never had a clear idea of the whole picture.

That, of course, changed very quickly in graduate school where I finally had to 'apply' stuff that I learned in Thermo, and finally, many parts of it started to sink in. Now, I think, if I were to retake my undergraduate class in that subject, I can see the bigger  picture that applies to that particular problem or area. But I found it rather strange and disconcerting that I didn't have that level of understanding at that time.

I'm guessing that this is not unusual, and that a lot of students, especially in the intro physics courses, are able to get through by simply knowing how to work out a problem, rather than having a profound understanding of what they are dealing with.



Ryan Dickherber said...

I had the same experience. For me, the problem was the concept of probability. This term was left undefined in every textbook, and I had no intuitive grasp of it like I had of charge, mass, space, time, etc. The key, for me any way, was the information theoretic interpretation. But it's not as simple as saying that probabilities (and entropy) represent knowledge (and uncertainty). So here's my interpretation which makes perfect sense to me, and now all the pieces fit.

There are three kinds of probabilities that are relevant in statistical mechanics. One is Bayesian probability. These represent your knowledge or uncertainty. Another is frequency. These represent the fraction of the time a system has a certain state. Then there are quantum probabilities. These represent the information in the environment with respect to some basis (a basis is an extension of the concept of a frame of reference---just like a particle can have a different location in a different frame of reference, the information in the environment can be different and, bizarrely, have different amounts of entropy in different bases).

In statistical mechanics, you might be tempted to ask whether probabilities are Bayesian or frequency. This is a false dichotomy. In your model, the probabilities are Bayesian. In the physical system of interest, they are frequencies. If your Bayesian probabilities are the same as the frequencies, then your model is accurate. Being accurate does not necessarily mean being certain.

Same thing with quantum mechanics. Your model has Bayesian probabilities which are accurate if they match the real probabilities in the system of interest. (Note that the probabilities of your model need to be the same in every basis, not just one, if your model is to be accurate.) Being accurate does not necessarily mean being certain (although, actually, entropy is always zero in a basis where the state is a basis element).

I have never seen a textbook that distinguishes between the different kinds of probability. If I wrote such a book, I would put it front and center. Instead, textbooks treat the different probabilities as different interpretations of the same thing. This isn't just misleading, but is actually false. No wonder students are confused.

(I've copied my response to my blog here: )

Pi-Guy said...

I went to a colloquium last spring on this exact issue. The speaker was concerned that his students were getting good grades in undergrad physics by memorizing equations, but without cultivating any internalized intuition. It was a lot like in Surely You're Joking, where the Brazilian kids could solve any textbook problem, but made no connection with the real world.

It's a legitimate issue, but I had mixed thoughts about this particular prof's take on it, because his methods of illustrating the rift even left the other attending seasoned physics profs at a loss to understand what he was trying to demonstrate.

Eventually he did get his point across and the other teachers were in a position to swap their own cases-in-point. Apparently it's quite common.