Re: Meaningless problems in algebra texts

[John Denker <jsd@AV8N.COM>]

> John C wrote in part:
> |
> | Consider that less than 10% of HS graduates can apply formal
> | logic according to a paper by Lawson et al.  This means that
> | students will have great difficulty really understanding
> | proof.  Now it is possible to get them to do proof, but it is
> | not possible to do it by most of the conventional methods.
> | Consider that the majority of middle school children are
> | concrete operational so that the conventional methods of
> | teaching algebra are doomed to failure.  This means they are
> | incapable of hypothetico-deductive logic, but they can use
> | empirical inductive logic.  Actually only about 20% of HS
> | graduates are fully able to use hypothetico-deductive
> | deductive logic and some are able to use it part of the time.

That's pretty much meaningless.
 -- First, it switches back and forth between high-school and
  middle-school.  Alas what's true for one is not true for the other.
 -- Also, it switches back and forth between Piagetian theory (which
  speaks about what _can_ be learned) and numerically-precise yet
  vaguely-defined assertions about _has_ been learned.

Piaget's theories are sometimes grossly mischaracterized in postings
to this list;  I recommend people be skeptical.  More conventional
introductions are readily available, e.g.
  http://www.google.com/search?q=concrete-operational+age


RAUBER, JOEL responded:
> How does one push a student from piagetian level to the next.  I suspect
> that one has to do tasks that are above the level that the student is
> currently occupying.  (gently and patiently of course)

Here's where it is important to distinguish between
 a) Piagetian stages, and
 b) what _has_ been learned.

Specifically:

a) There's no way you can teach a normal two-year-old to ride
a bicycle.  No amount of pushing (gentle or otherwise, patient
or otherwise) will make any difference;  you'll just have to
wait until the kid is more developed.  In the meantime, stick
to tricycles and rocking-horses.

b) There are plenty of seven-year-olds that can't ride a
bicycle.  It's not because they _can't_ learn it;  it's just
that they haven't yet had the opportunity to learn it.  They
just need a bike and a teacher (and some suitable terrain).

> Meaning even if only 10% of high school students can do a proof; it
> probably is a good idea to introduce them to the idea of a proof.

Agreed!

I would go farther and expect them to be able to "get it" i.e.
to learn how to do a proof ... maybe not up to J. Am. Math. Soc.
standards, but still a proof.