“The very function of proof is not to prove the obvious in order to meet some sophisticated and purely professional standards and goals, but to prove what is not obvious.”-Morris Kline
This is how most people are inclined to think, at least in my opinion. But, the question here lies in what is considered to be obvious. What assumptions are we allowed to make? Obvious varies from person to person, from audience to audience. What is obvious to a nuclear physicist might not be obvious to your everyday lawyer.
Most students, especially early on, are not typically strong with proof, (if I’m honest, I’m not strong with proofs either). This might be due to levels of exposure, or the fact that they might think certain things don’t require proving. Also, throughout the article, many of them believe proof is by example. They show an example, and there is the proof.
I think that might be a start. It’s like the basis step. For example, is what they are giving me even true for a few cases? If it is, then they can move on to generally applying what is given by using prior/existing knowledge and assumptions. I would have started by looking at a few examples and seeing some kind of pattern. Since the proof gave a hint, I would have started looking at the hint first, expanding it and seeing what it is telling me. Then I would have proceeded from there. These students all seem to have the same idea, but they don’t have a strong background in proof or even the basic concepts. (At least not enough to make the connections.)
As teachers, we should work on providing an exploratory environment where students can think and make connections. We need to give students more tools that they may be able to use in coming up with ideas. Create an open classroom focused on thought an interpretation and making the rules of the game clear. Explain what proof is. Explain different types of proof. Explain that math is connected and not this distorted group of topics that have no relation to one another. The goal is to make deeper connections, but the question is how, and I think more experimentation and research is needed on that front.
More focus needs to be on the student and their understanding and the article seems to fall short on explaining itself. Though, it does make some useful points and brings up an issue that students are having. Yes, we should analyze student work, but we also need to expand student knowledge and encourage deeper understanding through inquiry and study. With that being said, we might want to try to discourage teaching strictly for the test. Students will never learn if they study for the one grade and never go back to the material. Maybe we need a strong use of cycling within the classroom. So, go back to old topics and show how they might be connected to the new ones. Topics in the classroom are not isolated. There is a connection. And students need to see that. It’s like history. dates are chained together, and one event may be connected to another event resulting in this domino effect. Math is the same way. There are chains, links, between topics.
In all, analyzing students work is helpful and seeing where students need help, but applying that knowledge to helping students root their understanding in the best manner is the issue that needs more research. What ways can we help our students?