>>>Memorize? I can't remember ever memorizing anything. Better just to >>>practice until you understand. Discuss, ask questions, apply. That way >>>you memorize, of course, but that's just a side-effect. >> >>No, for proof classes, which linear algebra is in many places, >>it is crucial to *memorize* the definitions. This holds for >>all the proof classes at higher levels also.
>I disagree. As I said, you memorize (of course). But not by way of >"cramming, swotting" (or "bÃÂÃÂ¼ffeln" - German). Worked for me.
Hmmmm...Memorizing a definition is not the same as, say, memorizing digits of pi. At the very least, you should have the quantifiers in the right place and have something logically equivalent to what is given in the book. The problem comes when a student comes to me and I ask what the definition of 'independence' is. The student says something like 'c_1 v_1 +...+c_n v_n=0 and c_1=...c_n=0'. I'm sorry, this is not the definition, it is not equivalent to the definition and a student that uses this as the definition will not be able to either get the proofs done that are assigned nor will they understand the proofs in the book, nor will they really understand what independence is until they get it right. I do see there as being a bit of 'cramming' in getting it right.
I agree with Ullrich. Learning mathematics is like learning a language. You have to memorize some things before you can even get off the ground. After a while, you can pick up meanings quicker and see the subtleties sooner, but memorization at some level remains crucial for understanding.
>>There is simply no >>way of giving rigorous proofs if you don't know the actual >>definitions.
>Absolutely correct. But that doesn't disprove my assertion.
And they can't understand the theorems or the proofs in the book until they have the correct definitions. The quickest way, at least at first, for a student to get the correct definitions is to memorize them.
>> All too often, students have some very vague ideas >>of what is going on and then can't even get started on a proof >>because they don't know the *exact* definiton used in the course.
>When I'm not sure, I go back to my book and look it up (Happened >recently, when I wasn't sure what the exact definition of a refinement >is (in the context of paracompact top. spaces)). You forget, of >course, unless you solve problems or apply in a regular way.
Yes, we all forget things over time. But if you are taking a course in topology that covers paracompactness, I would certainly hope you have the definition of 'refinement' memorized (by whatever method) come time for the exam. Or even time to read the next theorem. If you don't, the definition of paracompactness will be very hard to understand.
>>The student should be able to give a rendition that is at least >>equivalent to the one given in the book and that uses precise language.
>Absolutely right. But it should go above Parroting, that's what I >meant.
Of course it should. But I am lucky to get anything close to an equivalent statement of the definition from my students. Simply getting them to understand the difference between 'if...then' and 'and' has been a struggle.