In all of my physics classes (high school, undergrad, grad), we were allowed to have a formula sheet with whatever we wanted written on it. Furthermore, important equations would just be given to us on the test, in case you forgot to write it. Even so, test scores were usually below 50% (corrected by curving after). Memorizing formulas to study would have been a laughable waste of time, the better approach is to practice a wide variety of problems.
If one has to look up a formula, can we say that they have a good understanding of it? If one had to look up a multiplication table each time they calculated something, how far would they have gone in learning?
I think you have it backward. All that time/energy wasted memorizing multiplications could have been used to understand more complicated topics. It's a cool party trick but it won't make you better at solving problems past high school level. I would prefer to work with someone who understand the concepts well and how to apply them than someone who can recall 32x9 or 10 digits of pi from memory.
When people complain about the difficulty of their college math courses or the math problems they solve at work, I can assure that it's not because they had a hard time remembering the answer to a multiplication. In a lot of cases, the equations they are trying to solve don't even have numbers in them. And when there are numbers, you won't be able to calculate them mentally because they won't simplify like they do in a no-calculator math exam.
You might not have realized, but a lot of the work when designing a no-calculator exam is to make sure people can compute the numbers mentally. When you step out in the real world, 32x9 becomes into 32.091x9.1^1.1. It turns ugly real fast when you have to deal with real numbers instead of a carefully crafted exam question.
I would have to look up Maxwell's equations right now if I needed them. Nevertheless, I am confident I have a good understanding of them. What I have in my head are important relationships (conservation laws, symmetries, wave solutions, etc.), not the exact mathematical formulas.
