I don't know if this is the right place to post this or not, but I will go ahead anyway (sorry if it ain't the right place)
Yesterday I was discussing a particular theorem of geometry with my brother which he just learnt in the school. I had asked him if he knew the proof for it, he replied saying his teacher has said that wouldn't be necessary.
Then, I asked him to sit and try to prove the theorem. He said that knowing the theorem counts for more than knowing the proof. How do I explain to him that knowing the proof is more important and how it can even help expand his thinking?
I know this question doesn't have a single pointed answer as is pre-requisite for questions posted here, but I would appreciate any replies
Best Answer
An idea on why learning proofs (not just theorems) is important: One could say that it is unimportant to know how to prepare food because there is a McDonald's down the street. But, if a person becomes strictly reliant on McDonald's for preparing food, then we can be assured that (s)he will never be able to produce a (worthwhile) dish of their own creation.
Likewise with proofs--one could say it is unimportant to know how to "prepare" the "food" of a theorem via proof because there is the "McDonalds" of the math book nearby. But, after years of just relying on memorizing theorems, a person will never be able to come up with a sound theorem of their own.
Being able to prove something makes it much more solidified in one's mind, and gives you a tool that is applicable to many circumstances, not just a single instance. For example, my double angle formula may not be useful when I need a triple angle formula, but I could use the proof/derivation of the double angle formula to find my own triple angle! This is where proof is much more powerful than memorization.