Today, I was asked to teach a calculus lesson for another teacher. For the past few days we have been having a discussion on how to best present “area under the curve.” Being my favorite topic in all of mathematics, I agreed to teach it. The teacher was unsure how to connect the sigma notation with the sum of the areas of the rectangles. In fact, I was having trouble developing a method of breaking the concepts down to a level that these students would be able to understand. Earlier in the day I had a mathematical epiphany while taking a leak. I was having a problem relating i into the sigma notation. The solution was simple. The i represents which rectangle of which you are attempting to find the area in the summation.
I observed the class before and I knew some of the students. Since this is an AP Calculus class, the students were all highly motivated and easily engaged. They also like to have a lot of fun which made my job extremely easy (the classroom management part anyway). I was surprised that I was actually a little nervous right before I started the lesson but as soon as I started speaking I remembered how teaching felt over the summer and I immediately became acclimated. The lesson went very well and it was fun interacting with students who knew how to both have fun in the classroom but doing it while participating in the lesson.
A few criticisms:
In general, I have noticed that students in this school do not know how to take notes. Their teachers never required it so they never developed the habits. This calculus class was filled with intelligent students that didn’t know how to take good notes. I knew this going in and tried to implement a strategy. One thing I noticed while I taught in the summer is that when my notes on the board are organized and accurate, that has a direct positive effect on the students’ performance. The first step in teaching how to taking notes is to model good note taking behaviors and force the students to write what you write. I attempted to do this. I put together organized notes on my clipboard and intended to given them an example of good notes. The quality of my notes quickly degraded. One reason is because I need more chalkboard practice. Notes on loose-leaf don’t translate the same on a chalkboard. The other reason is the organic style in which I try to teach. You can’t always plan on how to reach different learning styles. Sometimes you need to think on your feet to be able to break a concept down. The Fundamental Theorem of Calculus is a complicated, multitiered concept that is difficult to see the big picture. Just because you can follow each step doesn’t mean you can see the whole thing when it’s complete. Throughout the lesson I needed to explain a step by making related notes. I so became disorganized. I gave good explanations and was able to relate ideas but I was unable to model good notes.
This topic is neat because you see so many different ideas (i.e. summations, limits, factoring, simplifying, graphs, etc) and building this extremely elegant math concept which eventually because a simple operation (integration). If you can’t neatly show these concepts flow together on the chalkboard, then the lesson is not nearly as effective as it could be. I need to keep this in mind. It is an easy tangible improvement that can be made with practice and reflection.
3 comments:
hi tommy - these are fun to read and have helped me reflect on my own teaching.
also, teaching students to take good notes is difficult, but modeling is an excellent idea. one of the things i do is directly state some of my strategies during my lessons. so i might say, "it is really important for you all to organize your notes in a ways that make most sense to you. what works for me is . . ." many students take your suggestions immediately, as they might not even have a model to begin with.
the other thing i did when i taught 8th grade (we focused on note-taking a lot), is complimented students' note-taking skills by using their models of note-taking to teach the class. i remember one class saying, "chris g. organizes his notes in this way, and so and so does this." a lot of students then tried to model these behaviors and adjust so the models suited their own learning styles.
sorry if you've thought of this already.
It is frustrating to try to address this to students that have had a complete educational experience without someone caring how they take notes. I struggle with the idea of grading notebooks or demanding a certain organization because everyone is different. In many classes I never took notes because I don't believe they help me. I also couldn't focus on the teacher while trying to take notes. But they need a starting point, right? If you can establish certain basic skills, structure and discipline then and only then can they move on to something more specific or individualized.
And when am I going to have time to grade notebooks?
Great work.
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