Anonymous wrote: > David: > >>Actually from what you say here it seems pretty likely that >>you do know what linear independence _is_, but the >>way you're stating the definition is totally wrong. > > Right. That's my point. I *do* know the definition, but the phrasing always > gets me. It's like this, probably, for a bunch of the main terms. Still, you > say the number one problem is that students don't understand all the > definitions. Is my understanding of "independence" then not good enough? Or > is it?
A definition may be phrased differently in different texts. But these variatons should be eqivalent. Starting with one of these variants with the *exact* phrasing you will usually build an internal mental representation. You should be able to reproduce the original definition from this internal representation. If this does not work (i.e. "the phrasing gets you"), this indicates that something went wrong along the way.
The problem is that you will confuse yourself each time you need to know the original concept, e.g. in a proof.
I suggest the following steps when confronted with the definition of a new concept X:
1. While reading the definition of X for the first time (a) ignore previous encounters with anything that just "looks like" X. (b) if the definition of X uses concept Y then lookup the definition of Y unless you are able to reproduce it yourself. for example, when you encounter the definition of "vector space", ignore that you might met the term "vector" before.
2. after having read the definition of X look for (non)examples of X. at this stage, _do_not_hesitate_to_look_at_the_definition_of_X_again. Do not count on being able to memorize the definition of X just from reading it once or twice. It is far worse to memorize a distorted version of the original definition. Working on examples this way will eventually help you to memorize the definition.
3. look at proofs of statements that involve concept X. Try it yourself. This will help you to memorize X.
4. After a while check if you can reproduce the original definition of X.
The main point in steps 2 and 3 is to avoid the fatal sequence "read X once , remember X' instead, use X' until it sinks in, never really understand X"
For this it is crucial that you check constantly whether you really are really using the correct definition.
> >> Next time you take the class try actually >>_learning_ the definitions of all the important terms, >>_precisely_. > > OK, there's my answer, I guess. I've always had courses where professors > emphasized that we don't have to memorize definitions and terms word for word, > whether it's a foreign language course, a social studies course, an economics > course, whatever.
They probably just meant to stress, that you should not _stop_ at memorizing word for word. Also they may have counted on previous knowledge about the concepts taught. But I guess, you learned at least foreign vocabulary word for word.
> It seems like definitions are usually just things that you > have to get the basic jist of. It's sort of unfair that math is different and > no one warns us about it.