Journal Article 2

Goodman, T., (2010). Shooting free throws, probability, and the golden ratio. Mathematical Teacher, 103 (7), 482-487.

I think the author is trying to get across the point that contextual problems can give students the opportunity to use the math that they already know to model that problem as well as others like it. He spends the article explaining one problem in particular. He gives a probability problem with shooting free throws. He showed the different ways that the students solved the problem. This is a very flexible problem so it allows for many different variations on the contextual problem. So he then showed some different variations and each of the ways that students could solve them. He doesn't spend much time tying the problem to his main idea other than to simply say that the ways students chose to solve the problems depended on what their previous knowledge was.

I agree with the authors main point, but I am confused as to what the purpose of the article was. I thought the task he presented to his students was good for learning probablity; however, it seemed like the students already understood probablity, and thus they were not learning so much as testing already aquired knowledge. Students will use previously learned techniques for contextual problems, but understanding this should only be the first step in how to teach students (I would have liked him to explain how this connected to the way teachers should teach).

Journal Article 1

D'Ambrosio, B.S., Kastberg, S.E., Viola dos Santos, J.R., (2010). Learning from student approaches to algebraic proofs. Mathematics Teacher, 103 (7), 489-495.

I think one of the main points that authors are trying to make is that analyzing students work is not only a way, but an important way, to learn the depth of students understanding. They use the example of 12 grade math students and their understanding of algebraic proofs. They go into detail with one question in particular. Prove that if a number ends with a 5 that the number squared will result in an answer that ends with 25. They then give a hint that should start students on an algebraic proof of this topic. They show the work of many different students and they analyze the work of each of the students, and they show what you can learn about each students understanding of how to do an algebraic proof.

I agree that students work should be analyzed in order to ensure that what the teachers are teaching is getting through to the students. It is obvious from the paper that students are having a difficult time understanding specific parts of algebraic proofs. They are having a hard time with understanding the purpose of a variable in an algebraic proof, and they are giving explanations in some places where they could be showing arithmetic. Analyzing the mistakes they make will make a large difference in the teachers ability to help correct the mistakes that the students are making.