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This is the viewpoint of an engineer. The salient difference in thought is that the goal is rarely to be right (initially), but to get insight about mathematica
by j2kun 13y ago
This is the viewpoint of an engineer. The salient difference in thought is that the goal is rarely to be right (initially), but to get insight about mathematical stuff.
Mathematicians do (and students should) focus on proofs over computations, and there is an extreme amount of subjectivity there. If you produce a false proof, it can nevertheless be beautiful and yield fantastic insights. Likewise, a correct proof can be unsatisfying and ugly. Your goal then is to revise it (or completely rewrite it) to make it more beautiful and insightful.
Here is an example: the 7 Bridges of Konigsberg problem is one of the most famous (solved) problems in all of mathematics. One can easily give a proof by exhaustion, but that is wholly unsatisfying and perhaps the ugliest possible proof. A much better proof involves some insight into graph theory, and a satisfying and elegant proof would give you a characterization of these kinds of trails in graphs.
You don't need a mechanical background to start trying to solve such puzzles (I know because I've taught it that way), and you don't need any mathematical background to appreciate the very elementary proof. But as you let students flounder with puzzles like these, you can slowly introduce technical matter. The point is that it has a context they care about (people like working on puzzles!).
- ghaff 13y agoI guess I have an engineering viewpoint as well--which may not be surprising because I, well, was one. I never especially connected to geometry with its proofs in high school. What I did learn was lots of basic trig and algebra and the beginnings of calculus (which I took more of in college). The result is that while I've never had much of an interest or flair for advanced math, I did develop most of the math skills I needed both for school and, more importantly, through career and life. I've had stats as well. As I was just discussing with someone in a different context, I think stats, accounting, and similar applied math would be far more useful to most students than either most calculus (past simple differentiation which is used in a lot of contexts), proofs, or set/group theory. And, arguably, could be linked a lot better to real-world applications.
- j2kun 13y agoThe question is what are you trying to develop in your students: factual knowledge or mathematical thinking skills? I think it's the latter, and if that's the case then proofs are by far the best way to do it. And do you really think students would be any more motivated to learn about accounting than they are about current math? I mean, come on, nobody wants to do that stuff. It's a chore even for adults.
- rayiner 13y ago> The point is that it has a context they care about (people like working on puzzles!). I'd guess most kids hate puzzles.