Paul G. Hewitt makes a noteworthy point in emphasizing the meaning behind symbols in a physics equation, but there is even more to his story. While he focuses on classical physics, the climax of revelation through equations is rooted in modern quantum theory. Newtonian physics “taught” many to lean on the idea of a purely deterministic world whereas the quantum implications touch free will and the basis of human consciousness. These ideas are all touched on by simply considering the implications of an equation. It is one thing to know how to utilize an equation, but mastery can only come through also fully understanding the origins and conceptual roots of a mathematical framework.
While an incredible amount of information can be taught and learned through analyzing the message in any equation, simply understanding the underpinnings of the formulas still leaves individuals mentally separated from the origins of the equations, as if the creation of them is some esoteric work. It almost feels as if, when we work with equations, we are borrowing someone else’s tools. Even students with much experience in physics or mathematics are often left guessing as to how an equation was born.
The key to full understanding lies in derivation — derivation in the sense of determining the inertia of some shape when rotated about an axis as compared to “deriving” Newton’s second law. The derivation process begins with a question. “We have some system here, and we need to describe it using the tools [equations] we have. How do we describe [x aspect] about the system?” Keep in mind that this is derivation as a form of teaching and not experimentation or theory development. Following the question comes the adaptation of one’s current knowledge to the problem at hand. Known foundational equations are rearranged and substituted into each other, the method of the calculus is probably employed, and the end result is tested against the aspect of a system to be described in order to confirm its validity.
Many students today cannot visualize or describe relationships between equations dealing with similar aspects of systems (for example, F=ma and K=(1/2)mv^2). Some do not even know that relationships exist! Mentally, this creates a disconnectedness between mathematical descriptions and deceptively leads students to believe that physics is ultimately a system of memorization. Students do not see the harmony and rationale behind many equations, how from several foundational concepts, today’s physical framework can largely be purely derived. This limits students’ ability to adapt a formula to fit a problem (see Hewitt’s mention of part (a) of the tennis problem in the aforementioned article). Equations become associated with a particular problem type and then are neglected at first sight of a (seemingly) unfamiliar problem.
Notice how the process of derivation mirrors that of experimentation (here I am focusing on experimentation for purpose of teaching and not the modern trend of theory giving rise to experimentation). First, a question is posed or an interest is piqued. Then, by using and modifying foundational information, one can come up with some sort of mathematical description of the experimental subject. Finally, the mathematical model is tested against experiment and observation.
The derivation process reveals to students the purpose and origins of an equation. The need for such an equation is presented and students are then exposed to the actual use of mathematics in producing the equation. This naturally demonstrates the deep harmony behind many equations in physics and reinforces the meaning behind each concept in the equations. Thus it is not enough to understand even the meaning of the symbols in each equation but it is necessary that students understand the connectedness of the physical mathematical framework.