I have added the following piece by Don, and the video, to the growing Flow Channel collection about balances in mathematics:
http://naturalmath.wikispaces.com/Flow+Channels
Simple exercises and complex problems
by
Don Cohen
From a local parent whose 2 children came to work with Don for a number of years:
"Typical approach studies one simple concept at a time- boring-
isolated, irrelevant. Instead- have a more interesting, complicated
problem, that uses these concepts in finding the answer. This leads the
student through math concepts, seeing them in their natural context and
usefulness. Also, when the problem is finally solved, the "Look what I
can do!" feeling spurs further exploration of math."
In trying to solve the complicated problem, the student needs to use the simple concepts to understand the whole thing.
An example
I give a 4th grader a quadratic equation like x^2 - 5x + 6 = 0
Before I do this I make sure they know 5^2=25 and 3-4 = -1
Then we guess a number (need the rule for substitution and order of operations). So they give me a guess number, say 1.
1->x 1^2 - 5*1 + 6 =? 0
So they are squaring a number, multiplying, subtracting,adding negative
and positive numbers, and deciding if the sentence is true - all without
thinking about it. When they find the 2 numbers that work, they really
feel like they have accomplished a great feat! And when they see a
pattern of how the 2 numbers relate to the adding number and coefficient
of x WOW - this is not busy work! Then I give them a few others to
solve... Then they have to make a quadratic equation for Mom and Dad and
one for me. And they make up really hard ones for me! "Look what I can
do!" feeling spurs further exploration of math." That's the real payoff.