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.

Proofs gone wrong

This past Tuesday, I finished teaching the lesson I began last Friday in Geometry CP on proving isosceles triangle conjectures for the hour and a half class. Part of the lesson that I had planned involved completing two flowchart proofs together, both from the textbook. In both of the proofs in the book a lot of information was already given to the students, leaving them to only have to complete some of the remaining information themselves. I decided to take out some of the information to challenge the students to think. I went through them and took out about one piece of information from every pair given. I used the think-pair-share method of problem solving so that they students would have to think by themselves, work with each other, and share their reasoning with the class.

While we were going over the proof, I discovered that I should have been more careful and strategic about the information that I omitted. There were two adjacent boxes dealing with linear pairs: one of which had the reason of the definition of a linear pair and the other of which had the reason of the linear pair conjecture. There was one student who absolutely could not understand why the reasons for the two boxes could not be interchanged. I tried having the students explain first, followed by myself explaining that the definition was in one box and the conjecture was in the other box, so the reasons could not be changed. We must have spent at least ten minutes going over just that part of the proof. I didn't know what to do because it was only one student who was really struggling with the concept while the others seemed to have grasped it. I tried to explain a couple of times and then had to move on, but I went over to her individually afterward to ask if she had any other questions. The entire experience was a bit frustrating for both me and the students as we found it very difficult to communicate with each other.

Looking back, I think that I perhaps should have given them both reasons so that the students would not have been as confused. From this entire experience, I have learned the importance of really thinking through each problem to detect any potential problems that students may have and to create/edit problems accordingly. From now on, I will try to look at each problem in the students' eyes to see how they will approach the problem and think about explanations that address these ahead of time.

Finding Balance

Yesterday I was at school for nearly seven hours, an hour and a half of which I taught Geometry CP. I was teaching on Proving Isosceles Triangle Conjectures and, overall, it went very well. I was a lot more comfortable this time and the students all behaved very well. During the lesson, we somehow got into a discussion about different dimensions and planes. The students were all asking very good questions and it was a mathematically rich discussion. It did not really have anything to do with what we were talking about, but it is an interesting topic and most of the students were actively engaged in the discussion, trying to grasp the concepts. That diversion went on for probably about 20 minutes, and I did not get to finish my lesson as a result. To me, it was more important that the students were thinking mathematically and asking lots of questions that it was for us to steam through the topics that we were supposed to cover.

During lunch, I talked with my mentor teacher about the lesson. He told me that he thought it went very well and that he thought I handled things well. At one point, they asked me a question that I didn't know, so I told them that I didn't know. One of the students commented that he really appreciated my ability to admit that; my teacher later told me that he agreed. I was glad that went over well with the students. However, I know that students would not like it if a teacher said that every day, so it's something that I'd have to use sparingly. My teacher reminded me that with the Promethean Board, you can access the internet and said that would be a very useful resource in the future--something I'll have to keep in mind.

We also discussed how to balance between rich mathematical discussions and covering standards that must be met. Obviously this issue is important to me because I love when students ask questions stemming from topics covered in class that shows their curiosity and interest in math beyond what's being covered, but we still have to cover certain material. He asked me if the tangent we discussed addressed any of the standards covered in the lesson I taught and pointed out that if side topics build a stronger foundation of understanding, they are worth it. Otherwise, it will not be as beneficial to the students and they could use that to try to get the teacher sidetracked so that they never really have to do any work (kind of like in Mr. McCourt's classroom in his book). I know that students are smart about trying to get out of work, so I will have to work to see when that is the case. As a teacher, I will try to be open to talk about other topics to the extent that it will be helpful to the students. Otherwise, I need to limit those conversations to ensure that my students will know the material they need to know for End of Course exams and other tests.