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April 5th, 2010, 10:31 PM   #11
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Re: Abstract Algebra

What I meant by overuse of terminology is when a fellow mathematics student throws in technical terms specific to their expertise that they know I don't know, instead of trying to offer possibly longer explanations in terms that are appropriate to my background. In my experience it is the algebra whiz kids that are the most likely to do this, but perhaps it's only a mistake of not realizing that they at one point didn't know these words. I wish I could remove this statement altogether though as it's a gross generalization fueled by finitely many cases of personal frustration.

Originally Posted by pseudonym
With regards the terminology, on an undergrad course it can seem like its just there for its own sake. You prove a lot of stuff that seems like busywork. But this is just because even relatively advanced undergrad/beginning grad courses are really only introductions. They're trying to give you an overview of the tools that are available but they rarely have time to motivate them by going into the problems from which the definitions emerged.
Yes this is an excellent point, and a topic that should be explored in its own thread (but not on the algebra forum of course). It's interesting to look back at high school books, and early undergrad books at the problems to see how they were really setting you up for later material. Like integral convergence questions in my calc book use for the exponent the power p, as a primer to showing the difference between convergence in the different Lp spaces. That's a bad example perhaps, but you know what I mean.

I think a good professor will tell the students why something will be important later. The downfall to this, is that it can lead to students ignoring other "less relevant" parts of the course material. But if only I knew how important Taylor's theorem was when I was learning integral calculus as a freshman. Something I know consider to be the most important tool in applied mathematics is something I used to think was busy work to fill the end of the semester.

Originally Posted by cknapp
Of course, problem sets will always be important-- I never expect to understand a book until I work the problems, and never expect to understand a lecture or paper without working out the proofs on my own
This is an excellent point as well. In the transition to theoretical mathematics I foolishly began overlooking the importance of "drill work". However, in studying for the GRE math subject test I've seen an amazing improvement in my problem solving skills as a whole, that are no doubt a result of repeated drill work. Tools I knew about but in practice never thought to use are now actively surfacing in my consciousness , and I feel so much more empowered.

Above all the foundations of your knowledge base need to be practiced over and over again as you progress (i.e. algebra, geometry, trig. , calculus calculus calculus). A building is only ever as strong as its foundation, and an A grade almost never implies true mastery. I can't tell you how many kids who get A's in algebra can't apply the same tricks in the calculus setting or beyond.

Originally Posted by cknapp
...even when they are "trivial"-- so you'll be able to comfortably hide inside a cozy problem set for a bit whenever you get too afraid of the wilderness.
This can help build confidence and help alleviate some of the fear of mathematics, it's important for the student to realize 'hey, I CAN do this stuff'. Fear of mathematics is such a powerfully negative force for some people. In tutoring calculus I have seen near brilliant people fail to answer the simplest of questions, only because of the fear and preconceptions of calculus. If I had asked the same questions without calculus floating in the air, they would have thought I was belittling them. So in learning mathematics an air of confidence (but not over confidence or self importance) is powerful and necessary. Maybe I should really say an understanding of one's own potential. I have a saying I made up about this:

Knowledge is only useful if you know you have it
But only a fool thinks himself otherwise
So praise not what you think you know
And embrace only the potential to grow
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