The Finite Element Method and Problem Solving in Geometry

The last two weeks have gone well in Ms. Tran's classroom, and I am starting to feel like the students appreciate my presence. Yesterday, there were students in both of the first two classes who looked around for me at the beginning of the day to see if I was there. A highlight occurred yesterday in fourth hour just before my talk, when a student came up to me and said: "You would make a good teacher." That made me feel like I am doing good things in this classroom, and that I am getting through to at least some of the students. Last week I came in with a talk about nodal networks and finite element models, for which I recycled much of a powerpoint slideshow that I used last year, with an iteration of improvement. This week I came in with a new discussion on problem-solving, showing them an example of a real-life problem that I encountered just a few weeks ago. This will be the basis of the students' first project of the year, which I helped Ms. Tran formulate.

My talk last week was pretty successful. I had been talking with Ms. Tran the week before and asked her what topics would be good to cover. She actually remembered the talk that I gave last year around this time, which discussed nodal networks, which was something that they covered in class last year. This year, the curriculum changed, so they did not talk about nodal networks, so that was my opportunity to give them their only exposure that to the topic that they'll likely get during high school. My title slide included a figure from their assignment the previous week, which had a bunch of points connected by lines, and I told them that it was a network of nodes. I then asked them what they knew about networks of nodes, and most of them knew very little, so I broke it down into the words "node" and "network." In all of the classes, at least one person knew what a node was or could guess based on my intro slide; most of the students were able to define what a network is, and I asked them for examples of networks. Many of them came up with the phone company's definition of your "in" network, the internet, and facebook. This was great, so in each case I asked them what the nodes would be in their network; respectively, those were the individual cell phones, the computers and servers on the web, and people's profiles on facebook. I then showed them a few more examples of what I view as nodal networks, including rollercoaster truss structures, the human circulatory system, and an ipod's circuitry. I then introduced the students to finite element models, where we again broke the phrase down into individual words and then brought them together to define the phrase. I showed them an example of a Ford Taurus model with over a million elements, or nodes, and showed them how the network of nodes is useful for crash test simulations with a quick video animation. We wrapped up with a discussion on the purpose of models and why we simulate them. I thought this went fairly well - most of the students were attentive, and I noticed more students answering questions that had in the previous talk.

During Ms. Tran's lesson after my talk, there was some time for in-class problems, and for most of the classes I was able to go around and help students work through the problems. I was rather disappointed that some of the students (particularly in the non-accelerated classes) don't seem to know basic algebra, which is technically a prerequisite for this course. Not knowing how to solve problems of the form (3x+1)/2 = 5 is a real problem, and I have a feeling that the students who don't get it now will have a hard time getting it in the future, when they're fully expected to know it. Being able to get a variable by itself is so important, and it seems to me that the root of the problem is not understanding multiplication and division. There were at least two students that I was trying to help, and when time was up I still didn't feel like they understood what we were doing. I want to help them, but I get such little time with them that it is difficult to make any lasting impact. I spoke with Ms. Tran about this yesterday during the planning period, and she said that some students in the regular geometry classes actually failed algebra, and there weren't enough spots in the algebra classes, so they were placed in her geometry class. I told her that I will let her know who these students are so that she can push them to seek help outside of class, but I am still concerned about these students.

Yesterday I came in expecting time in class to help the students with in-class problems, but there was a change in plans and Ms. Tran gave a test yesterday. She told me when I came in yesterday morning that I could present for the last 15-20 minutes of each class, so I prepared my slides while the first class took their test. We had discussed this topic beforehand, when at the beginning of the year Ms. Tran asked me to help her come up with a project for the first half of the first term. I thought about it for a while, and I came up with an idea while I was helping a friend move into her new apartment. After hanging a curtain rod, we spent maybe 20 minutes trying to solve a problem that we had with a protruding bracket, and when we finally fixed it, I had a real sense of satisfaction from solving the problem. We went through several steps before finally fixing it, and I thought of this as a good example of real-life problem solving that students don't often think about during math class. I also thought that there was a good way to link this with geometry "logic," which mostly revolves around conditional statements. So I proposed to Ms. Tran that the project be to find an example of a real-life problem and discuss the steps and methods for solving it; then, the students could write 5-10 "if-then" conditional statements about the problem solving process, such as "IF this method doesn't work, THEN try this other idea."

So it wasn't a huge surprise when she asked me to do it yesterday morning, and I already had a bunch of photos prepared for it, so I sat in the back of the room and put together my powerpoint while the first class took the test. The slide show started out by defining what problem solving is and asking the students if they've ever had a problem or ever solved a problem (I may or may not have mentioned that Jay-Z had 99 problems). I also posed a question here about how problem solving relates to math, and I had an opportunity to discuss how math is really just a language that we can translate real problems into so that we can solve them. I then presented the curtain-hanging scenario, and walked the students through the three ideas that we tried before finally solving the problem, at each stage asking them for suggestions on how to fix it - they almost always came up with the same ideas that we did. I concluded by asking them to come up with some if-then statements about what I just did, and then I put a few examples up on the board. The classes got a little bit rowdy because many of my pictures were cheesy and kind of goofy, but I think because of this, more people paid attention. My first time through it, in second hour (accelerated), took a lot longer than I expected (maybe twenty minutes), but the students seemed engaged for the most part. One of the students actually gave a really good conditional statement that I hadn't thought of, and I later added to the slides: "If I don't solve the problem, then my friend is unhappy." Third hour (non-accelerated) is shorter than the others, and I only had 10 minutes to breeze through it, which rushed me quite a bit. The bell rang right before I got to the conditional statements part, but we told them to hang on one minute so I could get through it - the downside of being rushed was that I wasn't able to ask as many questions and give them time to respond. Fourth hour (non-accelerated) seemed to really get it, and I was impressed that most of them were engaged throughout and had really good answers to my questions. This was the only class where a student guessed where I was going with it before I got to the "if-then" slide, which made it go a lot quicker (too bad this didn't happen in third hour when I was rushed!). Sixth hour hadn't yet been exposed to conditional statements, so I felt like the part at the end didn't get as much feedback or response. Overall, I think this was an excellent exercise, and I'm looking forward to seeing what kinds of projects the students come up with.