Quadratic equations seem to lend themselves particularly well to math history lessons (or "math commercials" as my principal calls them--love that). For example, when we talked about the Square Root Principle, I gave a mini-lesson on Christoph Rudolf (names that rhyme are the best, aren't they?) and the introduction of the square root symbol and how it's supposed to resemble a lowercase r, etc.
I asked them innocently here, "So, do you think mathematics is invented or discovered?"
If they said, "invented," I said something like, "So, the square root of two didn't exist until Rudolf came up with a name for it?"
If they said, "discovered," I said something like, "So, you're just going to ignore the contributions people like Rudolf made to mathematics?"
I let them hash it out a little, playing devil's advocate all the way. And then I ended with, "Well, interesting conversation, guys," and proceeded to my next slide. Which, inevitably had the effect of "WAIT! Aren't you going to tell us?"
Which then had the effect of, "I'm going to Google it!"
"Go for it."
The next day, Day 2, we continued with the square root principle, but now we tried to solve equations like x^2=-1. I let them try to convince each other that there is no real solution to this equation (though their multiplication skills are still lacking, so...sigh). Here's where we talked about imaginary numbers and a mini-lesson on Euler ensued. I told them Euler couldn't stand not having an answer to this problem, as it--along with other problems like it--had been appearing in mathematics for nearly two thousand years. So, Euler made his own solution, and called one of the solutions i.
"Now do you think mathematics is invented or discovered?" We took a poll:
At the end of class, I had them write a letter to me defending their answer. They were instructed to choose only one (invented or discovered). Here are two really great letters, one from each point of view:
Dear Mrs. Peterson,
Mathematics was discovered, because just because a human didn't know the answer to something doesn't mean it doesn't exist. Before the Pythagorean Theorem was invented, a right triangle still had an area. Humans simply put words/letters/numbers and theorems to help us find the answer, and explain math, but the problem they solve, and answers they find, were always there. Some species of animals haven't been discovered yet, but when someone finds them, they didn't invent the animal, they discovered it.
Dear Mrs. Peterson,
I believe mathematics is invented. I believe this because invented means to create or design something that has not existed before, or make up an idea, name, story, etc. You have to create a name for mathematics to exist. One apple is not one apple unless you give a name to the number or quantity of the apple.
The next day, something happened that I think will go down as one of my favorite teaching moments of all time. A student, who has said from Day 1 that she's not good at math, came up to me before class and looked at me with her precious, sincere, huge brown eyes:
"Mrs. Peterson? I really need to ask you something."
"Go for it. What's up?"
"Can you PLEASE tell me--is mathematics invented or discovered? I can't stop thinking about it."
Cue burst of emotion and huge cheesy grin on my face. Why was this so wonderful? Because she just experienced what makes mathematics so addictive: the deep longing to solve or to prove, and the pleasure that follows the accomplishment.
Luckily (or maybe unluckily) for her, on this day, Day 3 of our discussion, I wrote a letter to my classes, defending my point of view. Now, I didn't give myself the same restrictions I gave them, and I typed this up the night before (I know, bad Rebecka), so it's pretty rough around the edges (hey, that's the great thing about teaching--now I have a whole year to make it better). But, here's what I wrote:
Discovered or Invented
What was so, so cool about this whole discussion is that it really appealed to most of my students. They were hooked. They kept asking about it. They wouldn't let it go.
What more could I ask for?
So, what do you say: Is mathematics invented or discovered? I'd really like to know your input...so I can make my letter better for next year.
 When I checked in on these students, they seemed more confused than when they started. Let's hear it for UnGoogleable Problems!
 Don't let the total numbers fool you. Only 1/2-2/3 of my classes participated (don't want anyone thinking I have a class of 17!). Not sure if the rest didn't want to commit to a single answer, if they didn't have access to a phone, or if I just didn't quite hook 'em...