I remember my 'ah-ha' moment in high school (there were actually numerous moments that kept reinforcing each other). In pre-calculus, I remember my teacher showing me how to derive the quadratic formula. Wait, what? This isn't just a random formula that some old dude came up with that just-so-happens to work? *Mind Blown* Then again, later in the course, we started doing trig. identities. Wow! I think I'm the only student in the class that figured out the awesomeness behind trig. identities. Everything in pre-calc made sense after that point, because you really can derive anything. Wow!
Within my department we set a goal for the year to create lessons that require higher-level thinking/questioning from our students and allow them to dive deeper into in the material. I know when I started teaching, I was part of the 'I teach what's in the book, how its in the book' crowd. Now that I'm comfortable with the material, I've drastically changed my style. My goal is to teach the student, not the curriculum. Sure, I have a set up standards and topics that I have to teach, but guess what? I'm going to cater to the students. If they ask me a question that's related to what we're talking about, but would normally come up later in the semester, I'm teaching it now. Why should I turn down students' interest in math? They've had enough math teachers in their past destroy their possible love of the subject already; I want to rebuild that.
Anyway, all of that to say the following... In my geometry class yesterday we started the area/perimeter unit. Its obviously something that high school students have seen and mastered already, so I posted this problem to put a spin on it and give a little challenge:
Since we just finished working on some right triangle trig, most students were able to calculate the height and go from there. However, I had a student set it up incorrectly, but get the correct answer. Here's what she did:
Oooooooooo. She recognized her mistake in copying it down wrong, but was still curious if her still correct answer was a fluke or would it happen every time. Now, there are some teachers that would simply say "it was a coincidence" or "you set it up wrong so therefore you're wrong" or ignore it all together. However, I saw this as an opportunity to teach and expand minds. Why is the sine of 105 equal to the sine of 75? Long story short, we got into a discussion about the unit circle and a more detailed reasoning as to where these trig. values come from, and made the connection among the sine function and supplementary angles. The students were HOOKED! They learned some pre-calc in geometry! Holy crap, is that allowed? It's encouraged.
Throughout this team-teaching that I've been doing with my colleague, one of the areas we've been focusing on in making those connections among math concepts. We've both noticed that when we're relating topics to other topics to other topics to other topics and they're all from different math areas/subjects, the students are ridiculously engaged and focused. They Learn the material instead of ignoring it or memorizing it. Its awesome. Even students who don't really care about math can see value in where it comes from. There is something perplexing and fascinating about knowing where it all comes from and how it all works together.
The 'why' in math needs to be focused on more in classrooms. Students appreciate the subject when they can make those connections and the subject has meaning.
So why don't more teachers do this? I'm not quite sure. Maybe its a lack of confidence in the material? Maybe its laziness? Maybe its because the textbook doesn't go in that order? Maybe its one of a hundred other excuses/reasons. No matter what, its something that needs to happen to increase our students' knowledge.