Last revised 1999/08/20 |
If you have already read the first section, you may want to jump directly to "How to Learn Physics" |
In the evolution of western, scientific human thought the step out of the "ocean" onto the "shore" was the very revolutionary idea that we should seek some sort of understanding of phenomena by looking at the numbers generated by (freely designed) measuring devices. I think this began with Boyle, et al, in the 1600's. Prior to this, such a mode of understanding was not even a part of human thinking. Since then it has become the essential feature of every human invention which goes by the name of quantitative science .
Bob Sciamanda Physics, Edinboro Univ of PA (retired) PHYS-L mailing list, 1998/05/13 |
Physics is a logical structure composed of quantitative statements, and the language it is written in is mathematics. Historically mathematical expressions have often proven to be applicable far beyond the phenomena that first gave rise to the expression. As a result I believe that the mathematical nature of physics is a reflection of reality itself being mathematical. However, pure mathematics is not physics; physics must correspond to observable and measurable events. In physics there are always pictures behind the mathematics. If there are no pictures there is no physics, and if you don't understand the pictures you don't understand the physics. In essence, mathematics is most useful as a compact description of the pictures.
Physics has levels of understanding, and in fact physics is more deep than broad. If the mathematics at the first level is properly understood, that mathematics acts as one of the pictures for the next level. Thus full understanding at deeper levels cannot be achieved without first having obtained reasonable understanding of the mathematics of simpler levels. At each stage you must translate the pictures into a mathematical understanding that you can use to construct your qualitative understanding of the next stage. The interplay between qualitative understanding and quantitative calculation is vital. Neither alone is physics.
An example of a group of equations that are always stored together are the equations for motion under constant acceleration. Incidentally, experts always store a label of this type with the equations. The label ensures that the equations are not misused in situations to which they don't apply. The equations for constant acceleration are
x = x_{0} + v_{0}t + (1/2) a t^{2}
v = v_{0} + a t
x is the position of a moving object v is the velocity of the object a is the acceleration of the object (a constant, remember) t is the time at which the position and velocity are measured A subscript of 0 means the value of that quantity when t=0. |
There are some additional things I have observed in myself and other trained physicists. I carry a picture of the graph of the equations in my head which I recall whenever I see an equation of reasonably simple form. For complicated equations I generate a mental graph of individual terms or even of factors within the terms. Given v = (1/2) a t^{2}, my mind reacts with "parabolic in t, it will rise quickly." However, the reaction is conceptual, not in terms of a statement in English.
Another form of visualization is providing interpretations for the symbols. I read "x", I say "x", I write "x", but my mind always reacts "position." I may even translate an equation mentally, just as I would with a difficult phrase in a article in French. Hence
is read mentally as "the change in velocity is given by the acceleration times the elapsed time."
Experts always keep the physical meaning of equations in mind. The equation itself is just a sequence of nonsense syllables. What is important is what the symbols mean. I have a mental image of my moving around a fixed starting point, changing velocity or not according to the situation. Sometimes I am the starting point and it is a car moving. Often I visualize a checkerboard pattern on the ground as a kind of graph paper. I do similar things with other equations, whenever possible.
Physics must be "understood." You will hear this from me and other physics teachers again and again, only typically we don't explain what we mean. "Understanding" refers to making the connections between phenomena we know about and the logical/mathematical descriptions of the phenomena that make up physics theory. For each principle and equation, find something you are familiar with or at least have seen demonstrated that is an application of the principle. Figure out how changes in the principle that seem otherwise plausible to you would conflict with the familiar events. Use these conflicts to "understand" why the principle is expressed the way it is rather than in some alternate form. Practice making these connections. Ideally, after you are done, you find it difficult to understand how a piece of physics could be any different than it is. If you have to memorize physics by rote, you don't understand it yet.
I do not mean to say that memorization is unnecessary. Definitions are essentially arbitrary, and at least some will have to be memorized. Moreover, memorization provides fast recall of equations for tests, is a familiar process you already know how to do, and by now you have a pretty good feel for when you have something reliably memorized. It is also a boring, time-consuming task, and memorized information is prone to disappearance in the middle of a test when you need it most. So you want to memorize judiciously and have backup ways of recalling information if necessary. If you know you can derive or reconstruct an equation, you are far less likely to forget it anyway.
So here are some tips on what to memorize and how to organize your memory work: