When we first started discussing the values of proofs in class a few days ago, my first thought was, "they're more trouble than they're worth." Of course I was thinking about my high school geometry class in which everyone complained about them and how this reaction hasn't changed since I've come to college. But I didn't truly understand the purpose and implications until a couple days ago.
An important implication that we discussed in class was that proofs establish critical and logical thinking. They are essentially arguments that, if written well enough, will never fail. That is a very powerful skill, to write effectively enough that what you've said can never be contradicted. So although proofs seem like they're very specific to whatever math conjecture you're trying to prove and require some foundation work to learn applicable theorems, they really establish writing skills and logical thinking that can be applied to many areas of life. Thus they're very important for students to learn.
One of the biggest problems I have with proofs is that the purpose of the proof is lost in the details of memorizing theorems and definitions. I find this particularly relevant in high school geometry where it's expected that the student will memorize all of the relationships between the angles and such. As teachers, I think it's important that we practice the names of the relationships and theorems and make sure students understand what they mean before they attempt to write a proof. I believe that if we have them try to piece things together too soon, the task will seem too daunting and students will become too frustrated, which will lead to them writing proofs off as “too hard.”
On his blog, Daniel Schneider emphasizes the maze-like structure of a proof. That is, given a beginning and an end, find the path between those points. I think this is a very important connection to establish because it takes the the bewildering nature of a proof and simplifies it to a well understood maze. So with this simple concept in mind, and with full understanding of applicable theorems, I think students can easily tackle the daunting task of writing a proof and be able to realize their importance.