Teaching styles
In math, there are two extreme styles of teaching: the prof can either develop the entirety of the subject, so essentially the students’ only job is to take notes, or the prof can give just definitions and ask guiding questions, allowing the students to develop the subject almost independently.
The usual compromise struck is that the teacher gives lectures that cover the bones of the subject– the named theorems and other big ticket items– while homework is assigned that allows the student to put the meat on the bones themselves. So ideally, homework covers the little indispensable tricks, helps the students rediscover the most useful (at least judging from history) paradigms for approaching problems that apply the theory, and if designed particularly well, even helps the students learn to spot problems where the theory can be applied.
Then of course, there’s the path that seems to be taken in teaching CDS202: motivate the hell out of the students in class, pump them up by making incisive analogies to the finite dimensional or linear cases… then relegate the raw meat of the course, the not quite as beautiful true face of the subject, to the reading.
Possibly relevant posts:
- Fluid Mechanics Films (2/17/2008)
- ACM11: Introduction to Matlab and Mathematica (9/26/2008)
- A buddy system (7/14/2008)
The first method invariably leads to rote learning. The second however is most likely to blow up in the professors, and the student’s, face.
A rote course will be isomorphic to random scribblings. However, a “guiding questions” approach can be an immensely frustrating experience if the guiding questions are so vague or open ended that people are left with no idea what they are even supposed to be proving.
The drawback in both approaches I think is they inevitably leave gaps in the topic. Gaps which will go unnoticed or are not cared about by those learning the topic. A good mathematics course proves every relevant statement, or at least constructs appropriate examples. A great mathematics course tells the students what they need to prove, and then gets them to prove it themselves (but you must be through).
An excellent example of a great mathematics course is “The Abel Theorem in Problems”. Every single example, motivation, derivation and theorem is set as a problem for the student to solve, and everything is set out in impeccable logical order. Apparently the author taught the Abel theorem concerning the impossibility of root formulae for fifth and higher order algebraic equations to second level students in six months using this method.
I’m in favour of getting students to put the meat on the bones themselves, but only if they are given the right bones, in the right order and with the right tools to put the meat on those bones. People who claim that this is dumbing down mathematics or making things too easy are missing the point. Mathematics should be simple! The best mathematics is so clear that even a child could follow it.