Imagine you're in high school and are just starting to learn about physics.

"s = vt"

If v = 100 cm/s and it's 3:00 in the afternoon, how far have you traveled? Impossible to say.

If v = 100 cm/s and t = 20 s, how far have you traveled? Impossible to say.

If a = 0 cm/s^2, and v = 100 cm/s, how far will you travel in 20 seconds? You'll travel 20 m.

Sometimes "t" means "duration" and sometimes "t" means "instant in time (a point on a graph)". Without careful teaching, it all gets mixed together. And then when it comes time to discussing differentials and limits, it can all just become a mishmash of mathematics.

I've mentioned an excellent book on physics teaching here before - but I can't find the name of it at the moment. But an on-line copy of an essay that makes similar points is here - http://web.physics.u...oy/challenge.html

A similar confusion arises concerning the work-energy theorem, Eq. (2). When students are asked to explain what kinetic energy means, the most common response is that it is "one-half the mass times the speed squared." By fixating on the mathematical definition, they fail to grasp the essence of the work-energy theorem: that the kinetic energy of a particle is equal to the total work that was done to accelerate it from rest to its present speed, and equal to the total work that the particle can do in the process of being brought to rest.

This tendency to focus on a mathematical definition rather than physical meaning was shown convincingly by Lawson and McDermott. [10] They presented students with a simple question concerning the work-energy theorem. As depicted in Fig. 2, an object of mass m and another object of mass 2m are initially at rest on a frictionless horizontal surface. The same constant force of magnitude F is then applied to each object. The question to be answered is "Which object crosses the finish line with greater kinetic energy?"

Using the work-energy theorem, and keeping in mind the physical meaning of kinetic energy, it can easily be seen that each object has the same kinetic energy upon reaching the finish line. Yet in interviews with 28 students taken from two classes at the University of Washington, an honors section of calculus-based physics and a regular section of algebra-based physics, Lawson and McDermott found that only a few honors students were able to supply the correct answer and the correct reasoning without coaching. While most of the remaining honors students were able to eventually achieve success with guidance from the interviewer, almost none of the students from the algebra-based course were able to do so. No less disappointing results were obtained with a written version of the question presented to a regular section of calculus-based physics. I have had similar experiences with my own students: Their performance on conventional homework-type problems shows that they can compute quantities such as work and kinetic energy, but their performance on conceptual questions shows that they have much more difficulty explaining or interpreting their results.

This example shows again that emphasis on numerical problem-solving can obscure major conceptual deficiencies in students. It underscores the importance of requiring students to apply the fundamental concepts of physics in a variety of different situations, as well as requiring them to explain the logic that they use in solving physics problems of all kinds.

This confusion and lack of understanding can have real consequences. People become great at memorizing equations and doing "plug and chug" but they have no understanding. For example, I heard a story about someone at their physics PhD defense being asked whether "the wavelength of light was bigger than a breadbox." They couldn't answer correctly.

Teaching the basics carefully and clearly is really important if you want to impart understanding. If someone makes things "easy" in the wrong way, it is damaging.

My $0.02.

Cheers,

Scott.