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So, I'm wondering whether there is a way to get that into a curriculum.
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<p>There is a way to get it into a curriculum. The problem is that relatively few high school math teachers (and even fewer middle school math teachers!) feel comfortable teaching students how to write proofs.</p>
<p>And, it is labor-intensive to grade and comment on proofs. It's easy just to check short answers. Part of the reason the standard school math courses don't require students to do much in the way of proofs (except possibly for very pro forma, routine, cookie-cutter type proofs in geometry) is that it takes SO much time for teachers to read the proofs and provide helpful feedback. (This is similar to the reason that many high school courses don't provide students with many essay writing assignments. Easier to grade multiple-choice and short-answer format.)</p>
<p>So getting it into the curriculum isn't going to be easy. Most of the students who get involved in doing proofs in middle-school or high-school have taken the initative to read books on their own, to participate in things like the the USAMTS mail-in competition I mentioned above, to take some college courses that require proof-writing, and/or to participate in summer programs like Ross, PROMYS, SuMAC, the Hampshire summer math program, etc. which have critical mass of instructors who can help students develop the ability to write proofs.</p>
<p>The team-based proof competitions like ARML's Power Round and Power Contest are very nice gradual introductions to proofs, because they have multiple related parts of varying difficulty and complexity. The idea is that the less experienced members of the team will tackle the easier problems and the more experienced members will tackle the harder problems, which often build on the results established by the earlier, easier parts.</p>
<p>There are plenty of books of old proof-based problems out there. One thing to do is to encourage a student to try a proof-based problem. If he's stumped (which often happens in the early stages), look at the model solution in the back of the book. Then, a couple days later, retry the problem and see if you can recreate the steps of the proof from memory. Eventually, you just get the "hang" of proof-writing and take off from there.</p>
<p>Edit: One more thing---learning to do challenging problem-solving math and proof-writing math is kind of like learning to ride a bicycle. It seems pretty hopeless at first, you think it's impossible, you'll never get the hang of it, you just don't see how other people manage to do it. But you keep trying and somehow, magically, it happens. Some people can manage to teach themselves how to ride a bike by just watching other people, followed by trial and error experimentation until they "get" it. Other people need encouragement and perhaps someone running alongside next to them holding the handlebars a bit at first.</p>
<p>The most important thing that kids need is a sense that it's OKAY to be stumped on a math problem, that it's okay not to know right away how to proceed, that it's okay to try several things that don't work before hopefully finding something that does work, that you can learn an awful lot from carefully examining why your "false starts" didn't work. </p>
<p>Too much of standard garden-variety math is about giving kids formulas and recipes to plug and chug. AOPS (the website recommended above by token) is great because it encourages a totally different mindset. And you can see the problem-solving process evolve as kids discuss challenging problems in their forums and in the transcripts of their "math jams."</p>