Replying oh, so late, to a message Dan Triplett wrote on 12-31-96:
-> I can understand this at a higher level.  Do we really need to
-> "understand" how all algorithms work or is it enough to understand
-> how to manipulate the numbers?  Some of that higher math stuff is
-> really weird (at least to a math novice it can appear to be weird).
All high school and below math should teach the underlying
reasons behind the algorithms. Otherwise it's just a bunch of black
magic, hocus-pocus to the students. Students who understand the reasons
know better when to apply them. They also remember them better and feel
more secure using them.
If a student goes along through their entire math education, at each
step learning all the underlying concepts and reasons, then the "math
stuff" won't look "wierd" to them. They will be progressing through a
developmental process, and as each new "wierd" topic or algorithm comes
along, they will be ready to learn it, having previously understood all
the prerequisite material.
You say "higher math stuff looks wierd" (to a novice). Well, Japanese
writing looks wierd to me. I can't understand any of it. Oh, well, I
never learned it.
-> I used to give guitar lessons and I would tell my students that I
-> cannot "teach" them anything.  I can show _them_ how to learn
-> something.  If they took the lesson home, and worked at the ideas I
-> presented them, then and only then would they learn it.  I think this
-> idea is true about everything we learn.  Only when we as the learner
-> work with a
-> concept/skill will we advance in our knowledge and understanding
-> about it.
This is a good point, and the same with math (and comp sci...I'm tearing
my hair out lately over the lazy-asses--excuse me--in my comp sci class
who expend NO effort outside of class). "higher math stuff" looks wierd
initially, but if the student has learned and UNDERSTOOD the
prerequisite material, then they can learn the current "wierd" stuff.
They just have to spend some time thinking about it on their own,
digesting the facts. Just like you can show them how to play the guitar,
but they must work on it on their own at home, also I can show them
wierd math stuff, but they must work on it at home until it becomes
comfortable.
Sheila
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