Making Connections and Learning Something New

Today I observed an AP Calculus class for an hour and thirty-five minutes with Dr. R. (different from my mentor teacher). They were covering the second derivative test, which tells the students whether critical values are minima, maxima, or neither. I had never heard of this test, so it was interesting for me. In general, I was very sad to find that the teacher spent the entire class going over homework problems (which I've heard in both methods classes is not at all how we should teach). He didn't collect the homework until the end of class, so the students were copying the work onto their homework the entire class. Also, the students have solutions manuals with the odd numbered problems, but they only went over odd numbered problems in class. The teacher admitted to me that many of the students copy homework straight out of the solutions manual, but he didn't seem concerned about it. I had to wonder how much of the homework the students actually understand. I was especially surprised that the class did not cover any new information, especially since it is an AP class.

Another thing that stuck out to me was that while they were discussing the second derivative, one student asked a question about inflection points (which can be found using the second derivative). The teacher said, "No, forget what you know about inflection points and only think about the second derivative test." I was really surprised by his reply because I have had a conversation with my mentor teacher and have heard from Dr. Manizade's class how important it is to make connections between concepts in math. Had I been asked that question, I would have praised the student for making that connection but explained that looking at the inflection point would not help us in determining maxima and minima.

From this experience, I have learned a few things:
First, it is extremely important to make sure that homework is completed at home. It should be something that builds on student understanding rather than a way to teach students because there was not time to cover new material in class. I feel very strongly against spending much class time to go over homework, so I would have simply asked them what problems they did not understand. Also, I would make sure to assign even problems similar to odd problems so that students actually had to complete the homework rather than just copy the solutions manual. By doing that, I would provide students the opportunity to look at examples similar to the problems they are doing and look at their assigned problem so that they could get help on how to solve their problems. In addition, I will make sure that I collect the homework at the beginning of class so that students won't be able to complete their homework in class.

Another thing I learned about is the importance of reinforcing connections in mathematics. In Dr. Manizade's class today we learned that most high school students will have jobs in 2010 that do not exist right now. This means that no one has done what they will be doing, so they will have to be able to think on their own and make connections between concepts to solve problems. When I am a teacher, I will encourage my students to try to discover how each topic we cover relates to previous topics (and will make a point to show them that myself). I will certainly encourage students for trying to make connections so that they will be able to see how new information builds on previous material. That will encourage my students to take learning into their own hands.