I was wondering how we actually learn math and science (physics). Some people say that it is important to "understand" the formulas/equations. However, if anyone were asked what 5 divided by 5 is, they would immediately respond 1. Most people probably aren't picturing a group of 5 apples (for instance), making groups of 5 out of those apples, and counting how many groups they have. This suggests to me that we do not really understand math/physics -- it seems to me like it is memorization or pattern recognition.
Yet, if I asked someone how many groups of 5 apples they could make from 5 apples, they would know to do 5 divided by 5 and end up with 1. Here, it seems like the person has a true understanding of the concept of division.
Do we understand math/physics or just memorize it? How are we able to seemingly do both? Of course, the example I gave was very simplistic, but I can find many more. For example, I could ask someone with basic physics knowledge to calculate the power in a circuit given that there is 5 volts across it and 1 amp running through it. Most people will quickly calculate that there is 5 watts consumed without even thinking about the "meaning" of the equation they are using (power = voltage × current). Yet, if I asked them to explain to me why they used this formula, it would not be too hard for them to prove that it is correct. In other words, they have an understanding of the formula, but did not apply this understanding when asked to solve a simple problem.
I was wondering if this has a logical explanation. Is our knowledge understanding or memorization? Is one form of knowledge more advantageous than the other? Why do it look like we have both forms? Thank you!