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numerical methods (was: banning calculators)



Justin Parke wrote:

... He wanted to know if long division is still taught in the elementary schools and how long was typically spent on it. (i.e. do they do division of a 5 digit number by a 4 digit number or only 3 by 2, for example.) His thought was to drastically decrease the amount of time spent practicing an algorithm and use that time to practice estimating what the result of the division *should* be and then confirming that with the calculator. This is similar to the idea that students can do integrals while having little idea of what they mean.

What are your thoughts on cutting instruction on long division in favor of estimation/prediction?

Tim O'Donnell wrote:

Unfortunately, a student's idea of estimating is doing the
calculation on a calculator then rounding the answer to
the near 1 or 10 or 100, etc.

Then it is the instructor's job to assign problems where
the thoughtless approach doesn't work.

Here is a modest contribution along this line:

Questions:
a) What is 24/25 (to 2 sig digs)?
b) What is 23/24 (to 2 sig digs)?
c) What is 24/25 - 23/24 (to 2 sig digs)? Show your work.
How is this related to answers (a) and (b)?
d) What is 24/25 - 23/24 (to 4 sig digs)?
How is this related to answers (a), (b), and (c)?
e) What is 24*25?
How is this related to answers (c) and (d)?
f) Find a number x such that your calculator cannot accurately
evaluate the expression x/(x+1) - (x-1)/x directly, but
explain how you can use your brain (with or without the
aid of a calculator) to obtain a very accurate result.