Re: Meaningless problems in algebra texts

[John Clement <clement@HAL-PC.ORG>]

>  From a mathematician's point of view, the physics approach is
> at once too broad and too narrow.  Mathematics is supposed to
> be strictly logical.  For example, we might know "A implies B"
> because we have derived it, step by step, starting from the
> axioms.  To repeat: math is validated by formality and rigor.

Yes, it is certainly true that math requires rigor and proof, but math
teaching, especially at the lower level does not.  It requires connections
to what the students already know, and to physical reality.

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.

Essentially math teaching is stuck with the Dykstra "folk theory of
education".  It is nearly 100% concentrating on educating future theoretical
mathematicians.  So only a few percent actually understand the abstract
math.

The real problem is that mathematicians approach the teaching of topics as
if it were a math problem, but getting students to understand and apply
ideas is an experimental science.  The result is very much like the Van
Heuvelin drawing which shows that there is an extreme impedance mismatch
between the teacher and the learner.  This is now being alleviated in
science, but math is a bit behind.

Incidentally if you wish to find some math books which were designed by a
mathematician, but with help from PER, look at Workshop precalculus and
Workshop calculus by Nancy Baxter Hastings.  Of course the collaborator was
Priscilla Laws who has copyrighted Workshop xxx.

John M. Clement
Houston, TX