As I've discussed previously, I've been trying to incorporate much more higher level thinking and processing into my lessons this year (as I think most teachers have been). My style has become one where I present a question or scenario to the class and together we build the necessary concepts from it. Five years ago, this style would've scared me to death; I would not have been able to give my students that much freedom. I was not comfortable enough with the material nor was I confident enough in my teaching abilities. However, now that I've taught geometry multiple times per year for six years, I'm pretty much at the place where I could walk in every morning and just go without missing a beat. As a student teacher I remember of dreaming about getting to this point in my career so I wouldn't have to plan much, but now I'm realizing this is not a place in my career that I want to be in. I'm getting tired of teaching the same topics over and over again. I wouldn't say that I'm getting burnt out, but its getting more and more difficult to come up with some new ways of teaching these same topics.
I've been reading some recent blog posts lately about challenging myself as a teacher, and while I don't think I'm at the peak of my career or abilities by any means (there's a ton that I could improve upon), I was able to relate to these posts in terms of my scenario. Teaching these same topics is challenging to me because I've taught them all in multiple ways and I can't think of anything new. I'm starting to get bored but I do not want to be. This new style has forced me to think outside the box and caused me to come up with some really neat things for my students.
Every year when I get to the area/perimeter topics, I wonder if it is really worth exploring. I mean, really? Teaching honors high school students area/perimeter seems unnecessary. For this year, I started with Pizza Doubler and just let the students discuss. It's an easy problem to get into, but one that gets real in-depth real quick. Without any prompting, just with me projecting the picture, my students immediately started talking to each other about it. Listening to their conversations was awesome; so many great, in-depth thoughts and arguments. From their talks, I asked for their estimations/reasoning and we went right into the calculations. They gave me different ideas as to how to calculate the area of the sector and arc length and we tested each one. Their explanations allowed the rest of the class to see their thought processes and we figured out which way seemed to make the most sense. Both of my classes came up with four or five different ways of making the calculations, and still a week later, I have different students doing different things (which is awesome!). They are all doing whatever makes sense to them. They are not begging me for formulas or getting mixed up with the algebra - they fully understand what's going and they have internalized it well. Seeing this makes me excited on so many levels. To be honest, just to see what would happen, after all of this occurred I gave them the formulas that were in the book. I wanted to know if my students would go back to their old ways of regurgitating with formulas or if they'd work with what they'd developed. They did not fall into the trap. Actually, some of them told me that the formula didn't make a lot of sense to them and the way they thought of it worked better. Obviously I didn't fight them on this and had a little victory party after class was over.
Anyway, as we went through the calculations for area of a sector and then for arc length, the students were instantly intrigued as to why the area multiplied by four and the arc length only doubled. They were shocked and perplexed and instantly began asking questions and hypothesizing. By answering their questions, we turned a discussion on area/perimeter into a discussion on similarity. They were listening out of pure curiosity, not because it was going to be on the test. I had them hooked and they wanted more. When I finished my explanation, they sat quiet for a few seconds just taking in everything that they had seen. Breaking the silence was a student with the quote of the year: "We went up the banana tree and found oranges." Absolutely, yes we did! I don't think students are seeing these connections being made within their other math classes, and that bothers me. Looking at their faces throughout this lesson, I could tell that they were captivated with how all of these things tied together, and how predictions could be made. And all of this came from just a (seemingly) simple pizza problem!
I've seen and worked with teachers that are not able to give up that much freedom in their lessons. Had students asked them about similarity during area, they would've said, "We'll come back to that in a few weeks, don't forget your question." Or, I've been with teachers that start with vocabulary, then example one, example two, example three, then their students do twenty identical problems that are carefully made to look just like what they did in class and nothing more. I need my students to have ownership. I need my students to be constantly thinking about what they can do with the math in front of them. I need my students to take the basics to the next level. I need them to understand that all of this 'math stuff' is interconnected in a beautiful way. This lesson achieved that, and I desperately want to develop more lessons that do it too.
I do not know what courses I'll be teaching next year, and even though I've been able to find a new avenue to explore geometry, I would like to try my hand at something new. Regardless of whatever I teach, I'm going to approach it in this style from day one. I want my entire department to approach their lessons in this style so the students in our school develop mathematical thinking abilities and critical problem solving skills. I want to hear meaningful discussions among our students as I walk down the math wing and I want them to get excited about being successful because they've done something important to get there. I don't think that this is too much to ask. As I've stated before, it might be an adjustment for some teachers; it might be way out of their comfort zone. But, if you're not doing something that challenges you, are you really doing everything that you could be?