I ran across an excellent passage in one of Feynman's "extra" lectures about the need to develop physical intuition in learning physics:
Now, all these things you can feel. You don't have to feel them; you can work them out by making diagrams and calculations, but as problems get more and more difficult, and as you try to understand nature in more and more complicated situations, the more you can guess at, feel, and understand without actually calculating, the much better off you are! So that’s what you should practice doing on the various problems: when you have time somewhere, and you’re not worried about getting the answer for a quiz or something, look the problem over and see if you can understand the way it behaves, roughly, when you change some of the numbers.
Now, how to explain how to do that, I don’t know. I remember once trying to teach somebody who was having a great deal of trouble taking the physics course, even though he did well in mathematics. A good example of a problem that he found impossible to solve was this: “There’s a round table on three legs. Where should you lean on it, so the table will be the most unstable?”
The student’s solution was, “Probably on top of one of the legs, but let me see: I’ll calculate how much force will produce what lift, and so on, at different places.”
Then I said, “Never mind calculating. Can you imagine a real table?”
“But that’s not the way you’re supposed to do it!”
“Never mind how you’re supposed to do it; you’ve got a real table here with the various legs, you see? Now, where do you think you’d lean? What would happen if you pushed down directly over a leg?”
I say, “That’s right; and what happens if you push down near the edge, halfway between two of the legs?”
“It flips over!”
I say, “OK! That’s better!”
The point is that the student had not realized that these were not just mathematical problems; they described a real table with legs. Actually, it wasn’t a real table, because it was perfectly circular, the legs were straight up and down, and so on. But it nearly described, roughly speaking, a real table, and from knowing what a real table does, you can get a very good idea of what this table does without having to calculate anything—you know darn well where you have to lean to make the table flip over.
So, how to explain that, I don’t know! But once you get the idea that the problems are not mathematical problems but physical problems, it helps a lot.
This passage makes a point similar to the one in Glen Coughlin's introduction to his translation of Aristotle's Physics: that knowledge and thoughts about the physical world are prior to the abstract knowledge of modern mathematical physics:
To understand Newton's argument for universal gravitation, one must have experience of weight in things and in oneself, of the motion of the stars and planets and moons. Knowing calculus is not enough. This hybrid science [mathematical physics], then, comes after the consideration of nature through non-mathematical means.
Richard P. Feynman, Michael A. Gottlieb, Ralph Leighton, Feynman's Tips on Physics: A Problem-Solving Supplement to the Feynman Lectures on Physics (Boston: Pearson, 2006), 52-53.
Aristotle, Physics, or Natural Hearing, trans. Glen Coughlin (South Bend, IN: St. Augustine’s Press, 2005), xii.