Summary of Video:
I watched the video "Using Dynamic Geometry". It starts with a female teacher. She asks the students to create a hexagon by rotating lines. They created tessellated patterns by rotating the hexagons with patterns inside of the hexagons. Teacher liked the technology because she could get instant results, and she could move faster to the reasoning and the proof, which is a higher level of thinking and learning, rather than just spending the day drawing. She asked students why the pattern fit so well with the hexagons and they talked about the 120 degree exterior angle of the hexagon. A student said that using the technology is easy and quick and you can see how angles fit better than if you were drawing, because drawing isn't perfect. Even with the software students have to know the math to know what they're doing like when they looked at rotations of the hexagons. Then they looked at pentagons and saw that they didn't work like the hexagon. The students could explore the math of the exterior angles fast because of the technology.
The second teacher was a male. He had his class find shapes in the pattern of circles that he gave them. They had to find the shape, justify it to themselves, to a friend, and then to the teacher. By doing this he let the students look at the properties of circles (radius, 360 degrees) and the properties of the shapes they're finding. GSP makes it so the students don't have to worry about drawing correctly, the program does it for them, but they have to give the program the correct instructions, which requires students to understand the math. The technology gives students the opportunity to prove why they know they see certain shapes. It also allows students to more easily point out properties once they have discovered them. The teacher says that writing it on paper is good, but you loose the interaction of talking about it and explaining it. GSP then allowed students to reproduce the proofs for the class without the effort of drawing.
Summary of Class Discussion:
In class our discussion started with the question, "Was that a public school?" which referred to the two schools that were shown in the video, and they both were. This question was asked because in both of the videos there are a lot of computers available for students, so my classmate was wondering if it was a public school computer lab, or a wealthy private school that had computers for every student. Because of that the question was asked, "what would happen if we only had half as many computers?" If this was the case in our classroom, then there would be two people per computer, which might make it so one person gets more experience than the other. This can create conflict because the two people may want to explore different things, but the person with the mouse is pretty much in control. Having people paired up at computers can have positive and negative effects. Some studies have shown that when you have to share you're more cooperative, but at the same time, students don't like it. If you're going to use computers with your students you need to make sure you have enough for what you want to do. You can have meaningful learning experiences even if you don't have a computer for every student.
We then spent a little bit of time discussing the two different teachers and their teaching styles. The female teacher talked to the students about tessellations, and it seemed like she was telling them what to do and what to see. Later in the discussion we also noticed that because she talked a lot that can take away chances students have to understand something, but for this class, the students are still engaged in the mathematics, which is what really matters. On the other hand, the male teacher had a very open ended task, and because of how free the students were to explore, some got more out of the exploration than others. During the task the students were supposed to find shapes and prove that those shapes were correct. He asked the students to "convince yourself, convince a friend, then convince me". This is probably a good method to use because the students have to understand what they are doing so that they can explain it to a friend, and they have to make sure they're doing it right so they can explain it to the teacher. This lead us into a discussion about proofs. Asking why is looking at a concept from a different perspective than just proving it. Proofs don't always play the roles we wish they would because a student can do a proof but still not understand anything. There are different types of proofs/ reasons to prove: establishing something for the mathematical community, convince yourself, understand why. Then we went on a small tangent about proving the angle sum of a triangle.
After talking about teaching styles we talked about the technology that was used. Technology can be precise, which would be hard to get from a drawing. Then someone asked "What is it about doing this on a computer that makes this different than doing it on paper?" A few people pointed out that students don't want to draw and erase and they can communicate properties, but there are some potential problems: no one was taking notes, and the students and teacher seemed to keep the figure static. They didn't make good use of the dynamic aspect of the geometry. Also, programs take training which takes time out of the mathematical learning.
We concluded our discussion by talking about whether or not the SMART board (interactive white board) is really worth it if it's only used as a projector or i clicker. One student suggested that if you're going to spend the time and money to get the technology, then spend the time and money to learn how to use it. Dr. Leatham pointed out that you're not going to use a part of the technology if you don't know about it. He then said that we go through stages with learning about technology. We get it, play with it, learn more about it, realize why it's cool (or not). A student then said that we've made this assumption that if we really did put the time in to learn about it, and he asked, would it really make a difference? So far Dr. Leatham doesn't think so, but then again he said that he might have thought that about using a computer in the classroom, and now he wouldn't teach without the computer. The SMART board is designed to do what we do on a white board, but then the work can be saved and it's available to everybody.
Critique:
I enjoyed watching the video, and actually didn't have any complaints about it the first time that I saw it. It gave me some good ideas about how to use the technology. What surprised me were the things pointed out to me during our class discussion. I didn't notice that the first teacher was telling the students what to do all the time. Once that was pointed out to me I realized that although the technology can be very useful, without an affective teacher it's useless. At the same time, although the other teacher didn't tell the students exactly what to do, he also left it so open that some students didn't get as far as other students because they were content to stop at the easy shapes. When designing the tasks the two teachers should have paid more attention to how they thought the students would react. But I think that using the technology was a great way to teach about geometry. I liked that the teachers in the video and in class we pointed out that this is so much easier and more effective than trying to draw the shapes, or just teach the concepts without the pictures.
Connection:
This experience has made me realize that although something may look good at first (like the activities that were presented on the videos) I still need to go through the activity and make sure it will work. I also need to make sure that the way I am orchestrating the task is appropriate for the type of learning that I want my students to have. Watching this video and having this discussion I hope that I will have the opportunity to use some of this technology to teach my students. Because the second teacher left so much up to exploration, some students just explored easy shapes, while others explored more complex shapes. If I have the opportunity to teach with technology I'll need to remember this for my tasks. If I see that students are doing this, then maybe I can point out some shapes and have my students explore them so that they all get that experience. (This would happen after my students had already explored the shapes a little bit.) While my students are exploring what is important is that they are engaged in the mathematics. So, like when the first teacher was giving the students all the answers, the students were still talking, the student who the teacher was talking to was still able to answer all the questions, so obviously her students were still getting something from the activity. But if her students weren't so interactive, then the teaching method would need to be changed.