(c) Fernando Caracena, 19 July 2012

*Why students struggle with physics*

I taught physics after receiving a PhD in theoretical physics to a variety of college students for several years. Generally, most students find physics to be a difficult subject. Physics is a course that weeds out pre-med students, and they take the course with much trepidation. In graduate school, I tutored pre-med students in physics. It was a painful process all the way around. The students wanted to hide their ignorance, and I searched for the weak spots in their understanding. In this area, a missing 20% of understanding can render the other 80% noneffective. By concentrating on improving that 20%, the students were able to improve their grades to A-level. So, some pre-med students gladly paid me a rather expensive tutoring fee, which they made reasonable by getting others to tutor-pool; but, I found one-on-one gave the best results.

*Mathematics is the language of physics*

Perhaps one of the biggest blocks that many students have in understanding physics is that they learned to use mathematics the wrong way. Mathematics deals with patterns that can be described with the precision of numbers, which are often expressed abstractly using symbols. Physics is concerned with patterns in nature and how nature operates. For this reason, the laws of physics are expressed as equations. Students think 'formulas and rules', which for them means memorization, whereas they should really be thinking 'patterns' that the equations describe. Richard Feynman talks about how his cousin had trouble with algebra using this approach. Although is possible to teach almost all of the math-side of physics using nothing but high school algebra, students have to learn to use that algebra in a different way than they learned it. In fact, it could be taught entirely with advanced algebra, but it is easier to teach advanced physics using calculus.

*So what is wrong with the student's conception of algebra that makes physics hard?*

In high school, students learn algebra as a series of arithmetic-deferred problems. For example the following one: Tim is three years older that his younger brother, Sam, and three years younger that his older brother Bob; Bob being twice Sam's age, what is the age of each of the three brothers? Had you not taken algebra, you might do this problem in your head by trial and error and come up with the solution: Tim is 9, Sam is 6 and Bob is 12. However, if you were a high school student, the teacher would probably say, "I want to see how you reasoned it out." He would not accept an an answer that you simply, grokked it. What your teacher would be looking for would be that you assigned three distinct symbols for each of the brothers' ages [such as, Sam(S), Tim(T) and Bob (B)] and by applying the rules of simple algebra arrived at the numerical value corresponding to each symbol.

OK, maybe you should be able to do that, but the bigger issue is that you should be able to see the pattern. That is what grokking does. Actually, the algebra problem presented above is a simple example of a general class of solving a set of linear equations. It is easy to solve this problem using the simple rules of algebra for equations. This class of problems can be solved once and for all by employing a powerful methods of linear algebra that use matrices, vectors and determinants. The methods of linear algebra allow one to solve such problems numerically in a flash--very little thinking required, just enter in the numbers correctly! In this case, the issue is not how to handle the numbers and arithmetic operations (unless you are designing the computer routines), but rather, knowing when and how to to use the numerical tools, and what the results mean.

*Experiments give the symbols of physics meaning*

For physicist, the symbols and equations of physics acquire meaning the way the words describing everything acquire meaning for everybody: by experience. A series of staged events, called experiments, provide the experience behind physics. Even a theoretical physicist acquires laboratory experience. It is part of a graduate preparation in physics to assist students in the laboratory. Later, experience is acquired by sharing, such as from from other experimenters. Without experiment, the equations become a mass of symbols that are easy to forget.

Richard P. Feynman, the famous Nobel laureate who shared a Nobel prize with Sin-Itiro Tomonaga and Julian Schwinger for work on Quantum Electro-Dynamics (QED), devised a method of organizing calculations that involved diagrams, which were easier to follow than equations needed to do rather involved calculations. The diagrams, representing the patterns visually, were easier to look at and organize into a more symbolic calculation. In the end, these patterns converted into various abstruse equations, which were solved for the answers of interest.

If you are studying physics, try to understand the equations and symbols in terms of experiments and the underlying patterns. remember what Einstein said:

“Imagination is more important than knowledge. For knowledge is limited to all we now know and understand, while imagination embraces the entire world, and all there ever will be to know and understand.”