🚀 go-pugleaf

In response to the interesting questions brought up by rww taylor
regarding how technology affects our perceptions about what
problems are "difficult" (and perhaps bends our interests
towards solving certain types of problems)...

I think that certainly we perceive some problems as newly-"doable"
with technology that earlier we'd never have thought of
tackling.  Imagine having students calculate Riemann sums
with partitions of size 5, 10, 20, 60, etc to see the
convergence even with a *scientific* calculator.  Never.
Or finding arclengths that before were unyielding because
we ran head-on into integrals for which there is no closed-
form antiderivative.  Today, for instance, I asked my BC
calculus students to determine which of y=x^3, y=2^x-1,
y=sin(pi*x/2), y=tan(pi*x/4) or y=log3(2*x+1) traced the
shortest path from (0,0) to (1,1).  They had a lab
assignment recently to define a function on [0,1] whose
actual arclength differed from that produced by adding the
lengths of 100 line segments by at least a factor of 10.

None of these activities would've been possible without
some fancy tools.  The other isue is whether some other
types of problems are supplanted by technology.  Again, I'd say
yes.  And probably, the loss of some of that is a shame.

What we need are clever teachers to devise ways to bring out some of
those clever and beautiful methods of great mathematicians, with
or without the use of technology.

I've often wondered what calculus would be like today if Newton or Leibniz
had had a graphing calculator.

Mark Howell
mhowell@jacks.gsfc.nasa.gov

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