One thing I think math teachers struggle with (and I am one of them) is "proofs." These can be student-generated ones on the level of basic geometry (use SAS to prove these two triangles are congruent) or teacher-driven ones (where does the Power Rule come from?).
As you saw in my recent post about the card game "Mao," I tried to get students to recognize patterns by looking at various problems. They needed a lot of coaching to get things going in the right direction, but once they got the hang of it, they recognized the patterns pretty well.
The problem then became--what's my motivation for proving the rule in a general case? I usually fall back on the concept of "Yeah, but how do you KNOW it works for all functions and all numbers? Maybe you just randomly chose the right ones by luck."
You can also see this when coming from the other side. I asked my students in calculus to show that the equation of the tangent line to a given line at any point turns out to be the original line itself. A lot of them just picked a couple numbers for slope and intercept and showed that it was the same for those, then "since I chose random numbers, this should work for all of them."
Clearly that's not how math proofs work, but how do we motivate it beyond a lot of "what if"s?
I suppose I could pull out some of the less intuitive formulas or give some examples of sequences where the simple patterns don't fit the values for large indexes, but those always seem somewhat contrived.
I think there's power in working with general formulas and showing that we can say definitively that such-and-such is true. On the other hand, the students seem very used to just believing the teacher when she says, "The formula for circumference is pi times the diameter." So, they just want to take my word for it.
Any other ideas to motivate proofs?
I try to get them withe the ol' "what, you're just going to take my word for it?" appeal to social norms of independent critical thinking. also, getting philosophical about what makes math so special yadda yadda epistemology-wise hooks the strongest students. finally, I model this attitude of awe and excitement, jump up and down over elegance (sincerely) and that goes a long way too. But ultimately you are asking how do we create the wish/need for deductive reasoning instead of other ways of convincing ourselves, and there really is no easy answer to that question. I'll ask out theory of knowledge teacher and get back to you though.
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