Throughout the duration of my K-12 education I’ve come across many methods that are trusted ways to solve a given problem. The more important ideas are taught rigorously so that they are ingrained within the minds of students permanently. Occasionally, I’ve come across a method that I know is valid, but I do not know why. In my experience most of these kinds of methods (that are valid, but unproven to myself) appear in math classes for some reason.

My most recent experience with this situation is as follows:

One day while sitting in math class my teacher was explaining derivatives. He then proceeded to present a list of rules that must be followed when finding the derivative of and equation (e.g. the chain rule, product rule, etc.) No one in the class, not even myself, questioned the rules. We all just took notes and followed the rules.

A couple of days later while sitting in physics class my physics teacher presented the class with a sample problem that required the use of derivatives. When he solved the problem he pointed out something that caught my attention. He explained that any constant in a derivative is reduced to zero. Why? My math teacher was supposed to be the authority on the subject, and all he gave us was a list of rules. The "constant rule," as it turned out to be called, was on that list my math teacher gave me, but there was no explanation for it. It took my physics teacher all of twenty seconds to explain that the power of the constant (5*x0 for example) is being multiplied by the whole term, rendering any result to be zero. (See below for a full explanation.)

If it took so little time to explain it, why not just explain it instead of make a rule for it? It sometimes seems to me that the teacher is missing the point. The true lesson here is that process is more important than product.

Some other examples include methods to find nth roots, converting decimals to fractions, and so on. I’ve always used a calculator to find an nth root. Heck, I don’t even know how to find the approximate square root of a prime number. I’ve been told that some countries generally don’t let students use calculators to solve problems like these. In hindsight I think I would have appreciated not having a calculator. Shortcuts eliminate the use of methods. Perhaps, with more methods and fewer shortcuts, the question "When are we going to use this in life?" might be eliminated.

Derivative: the slope of a function.
e.g. The derivative (or slope) of b*xn is (b*n)xn-1.
Derivative of a constant: c = c*x0. So the derivative of c is 0*c-1, which equals 0 because anything times zero is zero.