Marsha's comment, "We need to change the debate to "How can we teach
mathematical reasoning and number sense to students who use calculators?",
deserves a couple of responses. First, the next debate may be about people
that can't manipulate symbols because they have become overly dependent on
symbolic math programs. Second, the method that I use to teach number
sense is to talk about numbers occasionally with no notes, no paper, no
calculator, no computer allowed.
A recent example will help. A homework problem was, ".....the sum of
their ages is 23, the product of their ages is 132. How old are they?" We
set the problem up in Excel with tabular data and graph adjacent and my 7
year old noted that there appeared to be two solutions. Hours later, I
asked him if he had ever seen a problem with two solutions before. It now
struck him that it was unusual and he asked if some problems might have a
hundred solutions. I said yes and proceeded to construct one.
R: "If x-1 = 0, what's x?
G: "1"
R: "If I multiply two things and either one is zero, the result is zero.
So, if I have (x-1)*(x-2)=0 for what values of x will the whole thing go to
zero?"
G: after a longer silence, "1 and 2"
R: "And if I have (x-1)*(x-2)*(x-3)=0?"
I really don't remember now if it was at x-5 or x-6, that he finally said,
"Oh, I see, you just keep going all the way to a hundred." It doesn't
matter. The point is that we talked about the problem with no paper, no
pencil until he could see the pieces in his head.
Speaking of talking of things, my friend Kent pointed out to me the other
day that almost all kindergarten kids have an intuitive understanding of
derivative and integral from several everyday experiences. How much water
is in a bucket depends both on how fast the hose runs and how long it runs
at each flow rate. How far you've gone in your car depends both on how
fast and how long. In fact the odometer/ speedometer and dash clock
provide the source for a very nice introductory data set. From that
perspective, it seems like a big mistake to avoid using the words integral
and derivative in kindergarten. Kids can learn 10 words a day and they
aren't particularly scary words. I think that calculus was a nightmare for
many education majors, so they think that it is intrinsically difficult.
In their zeal to shield young minds from the details of "impossibly"
difficult subjects, they avoid using the terminology and fail to plant the
seeds of concepts at a young age. In later life, those children don't have
the anchors to hang all the details on.
If I had to try to summarize this message, it would be that much of
mathematics is jargon. The ability to learn new languages begins to
decline rapidly at age 5. The earlier you begin using the terminology
correctly, the more fluent you will be.
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