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Re: Algebraic Manipulation
Posted:
Jan 22, 1999 1:55 PM
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In a message dated 01/16/99 12:55:19 AM Eastern Standard Time,
paisj@MEDICINE.WUSTL.EDU writes:
<< My 9 year old daughter is learning and *understanding* by doing precisely
what Michael describes. At each step in the process she does an heuristic
multiplication approximation problem followed by a subtraction problem. Over
the holidays we worked on showing whether or not 1999 is prime and it is
clear to me that the by-hand work is precisely what she needed to do (i.e. a
lot of thinking and by-hand calculating) to feel comfortable about and
ultimately own the concepts. A while ago I asked her what she thought about
the idea of kids always doing their "math" using a calculator. Without any
prompting from me (ever) she answered that "they might be able to get the
right answer, but they won't *know* anything." >>
I remember one day introducing the concept of convergent vs divergent Series
based on an Infinite Sequence of Constants. I defined such a seq and series.
I showed how to graph them on the (then) TI-81 (or 82?). We first compared
and contrasted the seqs 2^n and 2^(-n) and then the series for these
sequences.
Motivation was given by the Penny a Day question:
How much money do you get in one month if your employer starts paying you 1
penny the first day and then doubles your pay each day after. What was your
last paycheck? How much did you have in your piggy bank if you saved it all?
and Zeno's Paradox:
Does the arrow ever really reach the bullseye? Zeno would say no, as the
arrow has to travel 1/2 the way there, then another 1/2 of the remaining
distance and so on. Zeno could not fathom the notion of limit. He argued
that the sum of an infinite number of constant lengths would have to be
infinite!. So much for a geometric approach to problem solving...
BTW, an excellent book that starts with the history of i from a geometric
perspective is The Story of i, An Imaginary Tale by Nahin (Princeton
publishing).
That night, I happened to be watching The Cage (the second pilot for the
original Star Trek series) with my son. Sulu comes on as the Science/Math
officer (no, helmsman was latter after being Spock's protege and then came
Chekov, but I digress) and posing the same Penny problem to Kirk (to explain
his friend's exponential growth in dangerous ESP abilities) and I asked my son
if he could solve the problem with pencil and paper (not even a 4 function
calculuator, mind you). He came back to me the next day with correct answers
(on ton's of scribbled paper) and a better grasp of The Calculus I was trying
to impart the day before than most of my seniors did! That was acheived all
by-hand. My son, BTW, was then only in the 4th or 5th grade (9 or 10 years
old?). What sons won't do for their mathematically crazed dads....
Regards,
A. Jorge Garcia
http://members.aol.com/calcpage/
The Calculus Archive Project!
Applied Math & CS
Baldwin SHS & Nassau CC
Long Island, New York, USA
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