On Jun 29, 2007, at 7:18 PM, Andrew Lentvorski wrote:
Christopher Smith wrote:
That is purely anecdotal. I work with a bunch of young math whiz's
who are dripping with insights, and never had to memorize
multiplication tables.
And you have selection bias because you are working with "whiz's".
That doesn't invalidate either set of observations.
That was exactly my point. Absent some hard data, it would seem to
suggest that mathematical insight is independent of memorization of
multiplication tables.
I work with a bunch of college students who can't estimate for crap
because their basic arithmetic is so slow. And these are at least
"above average" students because they have had at least a year of a
CS degree.
Rough estimation merely requires counting zeros. I would agree that
counting is still necessary even in a world with machines. You often
do not have a counting machine around, and on top of that it really
does help to know that 6 is less than 7 without having to check. ;-)
Memorization of things like multiplications tables does not
inherently provide insight. It is an enabler of arithmetic speed
which leads to algebraic speed which leads to calculus speed.
Oddly, many great mathematicians are pretty lousy at arithmetic (some
are practically human calculators, but it doesn't seem to necessarily
be an impediment).
It is the ability to calculate "with speed" that enables the
ability to have insights.
...and with few exceptions, calculators allow you to calculate with
speed better than most people who have memorized tables.
Ask folks when the difference between O(n), O(n log n), and O(n^2)
is perceptible on modern machines. The folks who don't use a
calculator invariably finish first and give better answers.
Arithmetic skill is not necessary for that (literally no arithmetic
is done). Calculus helps. Some understanding of graphs helps. As I've
said, based on my observations calculators actually seem to help with
both of those.
--Chris
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