I am not sure if this is the right part of the forums to post this, but I was reading the book Surely you're joking Mr Feynman when i reached this part:

This got me thinking. How does one "understand" physics and mathematics? It's certainly more than remembering some formulas. What do you think?

For me understanding comes in cycles. First, I achieve is bare familiarity by knowing the definitions and a rough idea of how it fits together. Second, I achieve the ability to solve simple problems following recipes and methods fairly close to those used in the textbook or course explanations. Third, I grow into the ability to cook up my own recipes and solve problems that I have not encountered before using the same important principles.

This is as far as I usually get with my first encounter trying to understand a new idea in math or physics. Sometime later, I circle back around and begin to appreciate things better, usually how the particular idea or techniques fit into the broader picture, how they were developped, and what later developments depend on them. One of the higher levels of "understanding" that I achieve is using a given set of knowledge in math or physics as a paradigm to pattern potential approaches for completely different problems in other fields.

How does one 'understand' anything? There have been a lot of studies about how experts vs. novices categorize and draw on information from their field (try a google search and a number of resources will pop up).

I think Dr. Courtney summed up the process well. I sort of feel like (to use another Feynman analogy) that understanding is sort of like an onion in that there are many layers that you'll progress through over time. I find this true even for some of the most rudimentary aspects of physics and math.

Some old (Hindu perhaps) proverb was on a tea bag that I was drinking before teaching class one day and it said something like "To learn, read. To know, write. To master, teach." I think this is a good recipe for developing understanding. For all of my classes I prepare (almost) all of my own materials. This involves a significant amount of writing and allows me to really find the best way to present clear arguments for each topic and also to understand the subtleties that might be involved in each argument. Also, having students ask questions forces one to think about things in a way that might not have occurred to them. I can't think of specific instances off hand, but I know there have been many times students have asked questions on topics that I felt I understood and I didn't have a satisfactory answer to give them. Whenever this happens I tell them that I'll get back to them and it forces me to do further research or put together my own thoughts in a more logical way.

So. If you want to develop the best understanding of a subject I would suggest that after reading (doing exercises, etc), you write about it at the very least (explain it in your own words, write your own exercises).

When I was in school, I wouldn't really understand the material of one class until I took a more advanced class in that subject. So I learned very basic E&M by taking Griffiths, and I learned Griffiths by taking Jackson, etc. And then teaching has helped me understand physics all the more, as brainpushups' proverb suggests, because it reveals all the weak places in my understanding that I glossed over the first time. It's so easy to confuse "That makes sense" with real understanding.

When students ask me how to study, one thing I often recommend is that they try to explain the current subject to someone else. A roommate or classmate is best, someone who can ask questions. But even talking to a teddy bear, or making a fake video blog, can help I think.

The cycles idea, and how understanding is rather a quantity with levels is something I'd totally go with. In some cases, it takes many many years.

The point I in particular identify here is Feynman's implication that learning by rote is NOT part of understanding. So Perry's statement "It's certainly more than remembering some formulas" almost contradicts this. I.e. the question is whether learning off formulas, however useful - necessary, even - is for example, can it be even called a first stage understanding of something? My feeling is Feynman is saying no.

However, I myself have often recalled formulas I've learnt off and never understood, mainly during boring bus, train or plane journeys, and started to write them down and try to understand them - this time - properly. Helps pass the time, so I wouldn't totally condemn memorizing stuff in the face of understanding it. Maybe I would admit it as a first step in a much longer journey of understanding.

Perhaps, the chief problem is that learning by rote, gives both you and others the idea that you do understand it, so it's a little insidious there.

Finally just to point out that the Hindu proverb, and Scott Hill's Teddy bear lecture are totally in key. After going through the cycles, get up and lecture to an imaginary audience. If you manage that, you advanced to "mastering" level!

To me, it appears that Feynman is trying to communicate how little understanding one has when learning mathematics and physics without ever having to apply it to real world situations. Making the connection between concepts and real world demonstrations of concepts is the first step towards understanding mathematics and physics. Why is this? Because mathematics and physics are used to describe the world we live in, and on a broader scope, everything real and imaginary.

I think a lot of this failure to recognize and apply physics and mathematics to real world situations stems from these rarely being taught in an applicable way. Too often classes are taught under ideal conditions so that nicely cooked problems can be solved. What if, instead of placing so much emphasis on getting an exact answer, high school teachers and professors spent more time focusing on what's really happening in our world?

One example is projectile motion. In high school physics courses, it's always assumed that air resistance is negligible. If that were the case, then sustained flights would not be possible. Then, in Physics 1, the formula to calculate approximate air resistance for specific shapes is shown and problems start to include air resistance within them. I would be willing to bet that a lot of students never realize what terminal velocity actually means and its causation by air resistance. It's things like that that should be taught alongside projectile motion to help students understand what is actually happening on earth, not in a place where air resistance is negligible.

The problem of mathematics getting too complicated pops up. Yet at higher level almost everything is approximated, so why can't things be approximated earlier on? To return to the previous example, there is no reason why numbers cannot just be given in the initial conditions. There seems to be no point in requiring students to have their knowledge "dumbed down", so they miss out on using air resistance in their problems. Providing students with it and teaching them how to apply it within problems will yield answers that are more realistic to what you would find on earth.

Physics and mathematics knowledge builds up incrementally. However, students will have a much easier time and be better off if they are able to see the inherent connections with the world they live in every single day. This also promotes thinking about mathematics and physics, which will most likely increase their understanding (and perhaps interest?) and help let students see what they are learning.

Obviously for a subject such as linear algebra dealing with "n" dimensions, such visualizations and recognitions in the real world are not possible. However, I believe that for a lot of physics and mathematics courses, teaching students to do more than just "plug and chug" and be able to make connections to the real world is far more important than solving something nicely cooked and under ideal conditions.