It's a great question. my only ( not so great ) answer is that, often, if you don't understand the proof of the concept or statement, then this means that you don't truly understand the concept or statement So, for me, understanding or coming up with a proof is a testament to the deep understanding of what is being proven.

Great question, but very likely a duplicate…

In the opposite direction: there are many students on this site taking a first course with proofs, or perhaps they are in contests that require proofs. They don't seem ever to try examples. I suggest having your child learn an easy computer language (maybe python, it is popular) and and begin to check problems to look for patterns... after that, a proof just catalogues the reasons a pattern must hold.

@rschwieb thank you for posting that link, I think that's an interesting question, I'm not sure this is a duplicate of that specific one as this is far broader?

@WillJagy thank you Will. That's a good idea, he knows python well so that might be a good entry point into a conversation about this.

You could maybe read this: www2.math.uconn.edu/~hurley/math315/proofgoldberger.pdf and try to get the main points across

You can also see this post: math.stackexchange.com/questions/111440/… . You could tell him how patterns which are apparent at first may not be true. (Top answers maybe a little heavy, but Pólya’ conjecture is one of the most famous ones where a counterexample was found. Maybe this answer and this) Math can also sometimes be counterintuitive, such as the Banach-Tarski paradox.

Agent, see projecteuler.net . Very well done. Beginning of the problems at projecteuler.net/archives

Tell your child that without a proof, you don't actually know. And that math, like bicycling, is learned by doing it. Doing math does not mean having a list of previously proven items. You study proofs to learn how to use and understand logic and reason. I recommend the books by G. Polya.

"An argument is something that convinces a reasonable person, a proof will convince an unreasonable one". (Marc Kac)

@user619894 love that! :)

I like to tell students that In Mathematics, Nothing Is True — ( unless there’s a proof that it’s true ).

@Cornman: Gödel showed that there are true statements in the theory of arithmetic that cannot be proved from any given set of axioms for arithmetic (satisfying certain technical conditions). Your ill-informed simplification of the incompleteness theorem doesn't bear on Lubin's excellent comment and doesn't help the OP or his child.

@RobArthan I think what you are trying to say is that there are true statements that can not be proven.

@Cornman: no, that is not what I was trying to say. Please read what I wrote.

You can also try asking your question on the Mathematics Educators Stack Exchange, which is for teachers but welcomes this kind of question. Presumably there will be many people there who've experienced just this issue and tried different ideas.

I think we should emphasise more the serious nature of proof. Should your son decide to continue maths at the higher level, every textbook they will read and almost every exercise in those textbooks will be full of proofs; this is for good reason. I don’t think I have the right words to explain why

"true" and "false" depend on the interpretation. Goedel showed that a statement can be proven if and only if it is true under every possible interpretation. But for this question, bringing Goedel into play is off-topic anyway.

@user619894 There are people so unreasonable that even a proof does not convince them.

@Peter it's fun tho :D :)

Sorry @Peter, I am not convinced...

Note: Basically the same question is cross-posted on the Math Educators SE site at Math Proofs - why are they important and how are they useful?.