On 4/25/2010 at 10:10 am, Dom Rosa wrote:
> The truly superb article, "What Is Mathematics For?,"
> by Underwood Dudley has been published in the May
> 2010 issue of the AMS Notices.
If mathematics is taught well and the students learn it, then mathematics
can help train the mind. Other subjects can as well. But the key is
that teachers encourage critical thinking and not just mere recitation
of facts and mere regurgitation of solutions to drill and standard
problems (for instance, the kinds of problems we often see as worked-
out examples in textbooks). But it is best that we are exposed to
a variety of subjects if we are to learn general critical thinking skills
and not just critical thinking for a specific subject.
Much of mathematical reasoning is inductive for trying to discover
patterns and discover theorems, but then only deductive reasoning is used
to prove theorems. The catch is that deductive reasoning is rarely used
outside of mathematics. But I would think that adding critical thinking
to any subject--whether mathematics or not--can help students learn to
think. But most courses in school today focus on memorizing a bunch of
facts rather than on learning to think. Reducing any subject to rote--
whether math or not--destroys the higher purposes of education.
Teaching students to think should be our main goal as teachers.
Perhaps much of the thinking behind mathematics does not apply directly
to real life, but I do wonder if that thinking behind mathematics can
still complement these goals.
In fact, reducing education to all job training also destroys the higher
purposes of education. That does not mean that it is necessarily a bad
idea to try to motivate students about the uses of various subjects in
careers and in everyday life, but I think we get too carried away about
this. As Underwood Dudley says--and I think he is right--those drawn
to mathematics are drawn to the subject for reasons that go beyond
getting a good job. Of course, such people are most likely thankful
for the good jobs they did get with their mathematical knowledge but
also find pleasure in mathematics for additional reasons as well.
Furthermore, that does not mean that I oppose career-oriented schools
such as Kaplan University or Argosy University or other similar schools;
we still need them. Employers do want potential employees who can think
but also want them to have certain career-specific skills as well.
And we must face reality: Many students do want to go to college to
train for a specific career. Some of them do want to learn to think,
and others can be convinced that this is a good goal to acheive, but
they also want to learn career-specific skills as well: Most of them are
not going to college to become pure scholars.
I do question Dudley's claim that the public wants more mathematics taught
since most people in our culture fear and hate mathematics. But it is
possible that many of these same people wish they understood mathematics
better and might support more mathematics being taught in schools if
mathematics were taught well in schools, which is often not the case
right now. I don't know since I don't recall reading anything that tells us
what the general public thinks about whether math should be taught in
schools and how much should be taught (this article is the only exception
I can recall).
I do not think it is reasonable to conclude that kids turned off by the
traditional curricula of math cannot be interested in any kind of
mathematics. Mathematics is often taught as a boring, uncreative,
uninspiring subject, so it should be clear why so many kids do not like
math. If we were to fix these problems with math teaching and work
harder at helping students find something enjoyable about math, then I
believe we would see far more students liking math or not seeing math
as such a burdensome or tortorous subject. We should shed the notion
of the "one-size-fits-all" approach to teaching because students
are not clones of each other: What works well for one student
may not work well for another student. Maybe some of those turned off
by traditional curricula might like math better because they have more
options that now appeal to them or simply because the traditional curricula
was taught to them in these bad ways. In short, our definition of "school
mathematics" is too narrow, so I think it is a good idea to consider expanding
students' choices of which math courses to take in middle and high school
and college. Kirby Urner on the math-teach list has plenty of good ideas
worth considering for expanding these options: He proposes including
more discrete and digital and computer mathematics in school. Courses
on mathematical modeling are worth considering. Berea College in
Berea, KY, has a freshman mathematics course on mathematical modeling
using computers (called Math 101). Case Western Reserve University has
an interesting freshman mathematics course (Math 150) called
"Mathematics from a Mathematician's Perspective."
Does Dudley prove in this article that mathematics is not useful to
most people or that mathematics applies only to a few careers? No.
First, he focuses just on algebra, not on mathematics as a whole.
Second, his article does not say that mathematics is not important to
these various professions but instead argues that the math can be
done without going through all the algebra because formulas and other
rote rules and tables have been developed to help professionals get
the necessary information. For example, problems requiring a system
of linear equations are done by using formulas that give the solution
to the system of equations; all we need to do is plug in the given
data and crank out the solution. But mathematics lies behind these
rules and procedures and other principles, and I find it at least
a bit distressing that many people apply these rules and formulas
and use these tables without having at least some idea of what
justifies what they are doing.