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....