I'm not happy with the way I've taught graphs of Trig. Functions this year (the last few years, honestly). My students inevitably get confused by the details (adding fractions of pi is extremely difficult for them, and therefore extremely distracting from what is actually going on), and I haven't really provided any real world motivation... how hard would it have been to throw some simple harmonic motion at them and ask for a model? Why didn't I start that way and let them play around with the functions to really feel what's going on?
Ok, now that I've vented: What this really shows me is that way to often the "natural flow" of material in a textbook is completely bass akwards. A whole bunch of math, neatly divided into six or seven sections, followed by a section dedicated to "applications." By the time they get around to the part where we (speaking as a typical math student) finally find out why we're supposed to care, it's too late: we already don't.
I know that is far from a groundbreaking or original statement, and if you had said to me yesterday, or even last year, I would have given you a smug smile and congratulated you for seeing things the way I always have... but now I'm really starting to become aware of the allure of the textbook approach: the math builds so nicely upon itself... initial problems are easy to understand, and we can apply the concepts to more difficult problems when we finally encounter them. Its so perfect... what could go wrong?
Here's what goes wrong: no 17 year old in their right mind cares about anything you want to tell them unless its the answer to a question they've asked (go find a teenager on the street and regale them with some interesting facts, I dare you). If you don't get them to ask the question first, they'll never care about (or remember) the answer.
Here's the interesting part: I'm exactly the same way. I've been re-inspired as a teacher this year by the likes of Dan Meyer and Shawn Cornally (two of many, but two that were the doorway to the many), who scream the last paragraph I wrote at me in every post or tweet. I read, nod, agree, give an "Amen" to myself, and then start a class talking about how the graphs for the sine and cosine functions are a natural extension of their relationship to the unit circle. Wow... feeling inspired, kids?
I've been told the solution to this over and over again: make them want to know before you give them answers. Then you won't need to give them anything, they'll just go get it. Just like them, however, I hadn't run into the wall yet. I didn't need that solution to satisfy me. I don't know why it took so long (probably the same reason it takes them so long), but now I need them to ask the questions. I need them to need the answers.
Why have I been driving the buggy this whole time? Time to drop the spurs, hang a few carrots out there, and enjoy the ride.