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Topic: Failing Linear Algebra:
Replies: 91   Last Post: Jan 10, 2007 12:56 PM

 Messages: [ Previous | Next ]
 Daniel Grubb Posts: 526 Registered: 12/10/04
Re: Failing Linear Algebra:
Posted: Apr 24, 2004 11:02 AM

>>>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 "bÃÂÃÂ¼ffeln" - 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.

--Dan Grubb

Date Subject Author
4/24/04 Daniel Grubb
4/24/04 Marc Olschok
4/24/04 Daniel Grubb
4/24/04 Marc Olschok
4/24/04 Daniel Grubb
4/24/04 Thomas Nordhaus
4/24/04 Dave Rusin
4/25/04 Jonathan Miller
4/25/04 Felix Goldberg
4/24/04 Daniel Grubb
4/28/04 Tim Mellor
4/28/04 James Dolan
4/28/04 Daniel Grubb
4/28/04 James Dolan
4/28/04 Daniel Grubb
4/28/04 gersh@bialer.com
4/29/04 Daniel Grubb
4/29/04 Dave Rusin
4/28/04 Guest
4/29/04 Guest
4/28/04 Guest
1/10/07 David C. Ullrich
4/29/04 Dave Rusin
4/28/04 Guest
1/10/07 Law Hiu Chung
1/10/07 Dave Seaman
1/10/07 Marc Olschok
1/10/07 George Cox
4/28/04 Guest
1/10/07 Dave Rusin
4/28/04 Lee Rudolph
4/28/04 Guest
4/28/04 Guest
1/10/07 Marc Olschok
1/10/07 Toni Lassila
4/29/04 Guest
1/10/07 M L
1/10/07 Thomas Nordhaus
4/30/04 Guest
1/10/07 David C. Ullrich
1/10/07 Toni Lassila
4/30/04 Guest
1/10/07 George Cox
1/10/07 Marc Olschok
4/30/04 Guest
4/30/04 Guest
4/27/04 Guest
1/10/07 Thomas Nordhaus
1/10/07 David C. Ullrich
1/10/07 Dave Rusin
1/10/07 David C. Ullrich
5/9/04 James Dolan
5/10/04 David C. Ullrich
5/10/04 James Dolan
5/10/04 David C. Ullrich
5/10/04 Marc Olschok
5/10/04 David C. Ullrich
4/27/04 Guest
1/10/07 Thomas Nordhaus
4/27/04 Guest
1/10/07 magidin@math.berkeley.edu
1/10/07 David C. Ullrich
1/10/07 Marc Olschok
1/10/07 David C. Ullrich
1/10/07 Tim Mellor
4/28/04 Daniel Grubb
4/28/04 Daniel Grubb
4/27/04 Guest
1/10/07 David C. Ullrich
4/28/04 Dave Rusin
4/28/04 Daniel Grubb
4/27/04 Guest
1/10/07 Marc Olschok
4/24/04 Wayne Brown
4/24/04 Thomas Nordhaus
4/24/04 David Ames