The above image was a problem that I presented to my class a few weeks ago. We were concluding our unit on area and perimeter, so I decided to kick it up a notch and ask them to find the area of this figure. We had practiced irregular figures such as this one already (actually about 90% of what we did was irregular figures due to the fact that they are literally everywhere in the world), but I don't think anything this abstract or complex. I just made it up on the spot, purposely giving them little information. Below is what occurred when a student volunteered to solve it on their own in front of the class.
Naturally, I did not turn this brave individual down. I encouraged the class to copy down the problem and try it on their own while this brave young soul gave it a shot. The picture above does not due the entire situation justice. As a geometry teacher, I try to teach my students the curriculum (obviously) but more importantly I teach them how to think, a skill that many lack. The conversations that went on throughout this problem (I'd say it took them a good 20-25 minutes until they collaborated, asked questions, fixed mistakes, etc.) were absolutely amazing. At one point, I looked at the clock and realized how long they'd been working; I considered stopping them but I let them go. Why should I stop them from working and thinking original thoughts? I come back to a previous point I made: as a teacher its my job to make sure that they are thinking and problem solving; nothing should stand in the way of that, its in their best interest (but more on that later). Just look at the difference in pictures; I love when students write all over diagrams. It sheds light on their thought process and how they internalize information. As I said, I intentionally didn't draw this to scale and left certain parts unlabeled. I actually tried to give them as little information as possible so they would be forced to think and ask questions. Ultimately, I could care less about their answer, I want to know what's going on in their minds. They had to ask if the 'triangle' on top was isosceles or if the bottom was a rectangle or if both sectors were ninety degrees or if those two vertices were 'midpointish.' And most of the time, I replied with vague answers and forced them to come up with ways to verify their own questions (in the beginning of the year, they hated this, but now they're used to it and rise to the challenge). The majority of the students came to an answer, and most were correct. Those that weren't understood where they went wrong when we discussed it as a class. Its amazing what happens when you just let your students experiment even if it means taking extra time.
I try to do stuff like this all the time. Some of my colleagues have argued with me because its not real-world, or because it takes 25 minutes for one problem, or because its not entirely written 100% into the curriculum and there is no time. All of these are bogus (while relevancy is an important aspect of teaching, especially math, it is not the only aspect that should be focused on). In the end, if its good for the students, I'm going to do it. I didn't have one student complain about this problem and I had 100% participation without fighting with them. In talking with them, I could tell that many of them learned from it. Students love a challenge when they know they have a chance at success, and that's what I gave them.
I could go on and on about this experience, but ultimately it comes back to a question I pose to my department members on a regular basis: If it's good for the students, why would I not do it?