>Ullrich: > >>Some of us are saying that memorizing the definitions by rote >>is stupid, instead the student should learn the definitions by >>using them, the way we do when we learn a new topic. This >>is missing the point: _regardless_ of _how_ one learns those >>definitions, "we" all agree that one _does_ need to _know_ >>the definitions eventually in order to be able to do the >>things one is supposed to be able to do. > >Right. But I still think it's possible to know the definition of something >*without* being able to state or explain the definition clearly.
I've already acknowledged this, at least twice, in reference to your garbled attempt at a definition of "independent". If we're talking about being able to prove things (which by the way _is_ what we're talking about) then this sort of understanding is useless - to be able to give a coherent proof of something involving independence you need to be able to state things clearly.
What I don't understand is why you asked for advice but continue to insist that the advice people are giving is wrong. You wanted to know why you're flunking linear algebra. Several people conjectured is was because you simply don't know the material, probably starting with the definitions. When you gave a "definition" of "independent" that _confirmed_ people's conjectures about the fact that you simply don't know the definitions.
If you had any sense you'd stop insisting that you do know the definitions in some sense, and instead simply go back and _learn_ the definitions, in the sense that's _required_. ("Required" in the sense that it's necessary in order to succeed in what you're trying to do, whether or not explicitly required by the instructor.) Because the advice you've been given by many people is _correct_, whether you believe it or not. (The advice has been essentially unanimous - essentially nobody disputed the fact that you needed to know the definitions precisely, the quibbles were over whether you should memorize them by rote or learn them through using them. When you showed us your incoherent version of the definition of "independent" people stopped quibbling about that.)
You're reminding me of the student who asked me yesterday for help getting some graphics from Mathematica into Word. I had a plan for how to do that. It took a little while because instead of doing what I suggested first he wanted to try this and then try that - I really didn't understand why he wasn't willing to just try what I wanted to suggest, given that the things he'd been trying had not been giving satisfactory results, and he'd asked me to help. When I finally got him to try the things I wanted to try it worked out just fine.
>For example, >the word "the". We all know what it means and when it can and can't be used. >But, can any of us actually define it? Probably not. In fact, I'd be willing >to argue that we actually understand things *better* when we know what they're >used for and how to use them, and we can do that without defining them.