Bruce, you ask:
>This is your learning style. What about the X-1 students of the class?
>How many of them learn in the same manner? Would another approach have
>been better? How many passed the class?
I suspect you meant these questions to be rhetorical. I'm going to answer
them anyway, because I think you fundamentally missed the point of Piaget
and other stage theorists. [I'm not an expert in stage theories, but I
did study them and will try again to do them justice so you can make up
your own mind.]
Learning "style" is not very relevant to the point about physical
manipulation ("concrete operations.") Perhaps some learned from the
manipulation by observing (visually) what happened; others learned by
listening to descriptions; others from the tactile sense of the turning
cube. The point of stage theories is that early learning must derive from
performing "operations" (in whatever learning style) on physical, concrete
objects.
So, to answer your questions:
If there were any X-1 students, they learned just fine.
I saw several other approaches that worked much more poorly.
Everyone passed the class -- with B- or better -- but that had a lot to do
with who was in the class. We were not from the "heart" of the bell curve
on mathematical ability (he immodestly claims.)
--"John Gunkler" <jgunkler@sprintmail.com>
Learning-org -- Hosted by Rick Karash <rkarash@karash.com> Public Dialog on Learning Organizations -- <http://www.learning-org.com>