[From Chris Cherpas (981018.0200 PT)]
Chris Cherpas (981016.2340 PT)--
...asked if it's more probable to go to the beach and
see women or tanned women, kids say tanned women...
Bruce Gregory (981017.0825 EDT)--
My experience is that the mistake is often ours and not
the students. We assume that if we present the students
with an abstract representation that they can manipulate
and repeat back to us, we have done our part and they
will do their part by "internalizing" the model and
"applying" it to new and unfamiliar situations. Great
idea, but we have a mountain of evidence that
it never happens.
I couldn't agree with you more. Besides Tversky and
the whole gang of decision theorists who look for
deviations from the rational/normative models by
asking a few trick questions and publishing what are
perfectly reasonable answers (given interpretations
which are not the official ones, but are not unwarranted
either) as "biases," I get pretty annoyed with the reverence,
particularly in the education community, for Piaget.
If you follow Piaget (instead of, say, Fischbein), kids
should be clueless about probability until they're just
about ready to start having sex (maybe there's an
evolutionary argument here I've overlooked...). Although
people like Hugh Petrie or Gary Cziko seem to be able to
interpret Piaget's notions about learning appropriately,
I'm not too sure about the rest of the world, including
Jean himself.
It turns out that the kinds of tutorials and practice
exercises that would counter these supposed biases can
actually be effective because of their general properties.
For example, in the case above, involving supposed violations
of monotonicity, it is pretty easy to set up classic
colored-balls-in-urns problems which represent inclusive
subsets of the sample spaces in Euler (Venn-ish) diagrams
and work through some basic properties of measurement that
you'd want to get to anyway. And, to boot, if some
sneaky standard test item is tossed at them which plays
these kinds of games, they will think twice about getting
duped by the evil experimenter syndrome.
One more cheap shot at Piaget: some good insight about
the relationship between mathematical structures/algebras
and psychology, but before going overboard, at least
study your Bourbaki a little more closely to get the
math right before building a theory of psych on algebraic
structures. And one not so cheap: one can't say one is
evaluating an educational practice, qua education, unless
one is at least converging towards being able to say
something to the effect that doing X, instead of Y, in
the First Grade works better for doing Z in the Eighth
Grade...and one will never get to that level of
analysis unless one has controlled a longitudinal,
multi-year, individualized instructional process,
not just asked a few questions every year or so and
publish the group averages.
Best regards,
cc, a different kind of evil experimenter