Yeah, philosophy teaches you counterarguments to help along discourse. Too bad most teenagers go into rage-mode instead of debating.
I'm not a fan of the formal debate structure. It's not focused on who's closer to the truth; instead, it's about who presents their argument with more style. This is why we see creationists desperately trying to "destroy" evolution using formal debate -- techniques like the Gish gallop play on cognitive biases and attempt to force the opponent to focus on rapid-fire non-truths and a formal debate does nothing to address this. The only formats I've seen that address the Gish gallop are asymmetrical in nature (e.g. a call-in show) where a hopefully rational host can easily cut off the caller and force them to go one point at a time, not getting to present their whole argument if anything in it is blatantly false.
Science is the best tool we presently have to find the truths of nature, but it's hardly ever a performance art. The evidence and theories needed to overturn a consensus can't be presented in one debate but instead move along at the glacial pace of peer review, figuring out the math, and repeating experiments.
Too true, but I'm sorry to say that you misinterpreted what I meant to say and thus your (correct) analysis came off as somewhat non-sequitur (goddammit, must not drink, else latin). In philosophy class, the normal argument comes off as this: thesis-antithesis-synthesis.
Step 1: you announce your point of view. (The thesis, in which you state your initial claim)
Step 2: you present the counterargument. (The antithesis, in which you offer a rebuttal to your initial claim)
Step 3: you conclude whichever way you feel like. (The synthesis, in which you conclude your argument)
It's all so prim, proper, and codified it should feel like playing mental chess. My fondest memory of the lycée was one where I had to play devil's advocate to an avowed communist. Toughest mental exercise I'd gotten 'til that point. It was delicious. What I was trying to say, and that makes what you said a tangent, is that nowadays you get either reactionnaries who are yelling (and I don't exaggerate) that the old ways were better, or the young guard who are yelling (louder) that the reactionnaries are fascists. We're pretty far from civil discourse, let alone Socrates' methods.
as an aside, we study René Descartes in class, which bridges the gap between math and philosophy. By the way, I was never a fan of philosophy class, even if I come off biased. Thinking about people thinking about how people think is too much of a "yo dawg" moment for me, unfortunately, the teacher assumed that I was trolling whereas I was willingly testing the philosophy (or way of thinking) du jour to the breaking point. That's why I was the teacher's favorite devil's advocate when we talked about it: I am a crash-tester. To this day I do not regret having sunk 8 hours weekly into this course for a school-year. It's just that the lesson was learned as I matured, but I'm not a philosopher by any stretch of the imagination. Hell, LeTipex has a degree, I'm a historian. He philosophizes, I study people who philosophized.
(I love the english language, you can totally create a term like "philosophize" and make it sound like it's not awkward at all...) 
@Sigma: The discourse exercise is "convincing others" and nothing else. That's why it's so tough on the nerves and on the students. But to the gish gallop, I counter personnally stenographied notes and a point-by-point rebuttal (whatever works, I guess).
