The discussion has been interesting. I would like to offer 3 points.
1. I would expect both mathematicians AND lawyers to be practised in supposing "What if....?"!
2. Experience counts in maths perhaps even more than in other areas. Think what happens if you pick up a graduate text NOT in your area
(i) in maths (ii) in history.
3. I once sought out metaphors used to describe the doing of mathematics and easily found more than 50. No surprise that we dont find a one line answer to the question.
There is an opposite kind of question that might clarify some ideas: What harm can mathematics do?
Bob Burn
University Fellow, Exeter University
Sunnyside
Barrack Road
Exeter EX2 6AB
01392-430028
________________________________________
From: Post-calculus mathematics education [[log in to unmask]] On Behalf Of 英史 [[log in to unmask]]
Sent: 20 May 2010 07:03
To: [log in to unmask]
Subject: Re: What Is Mathematics For?
A short argument about mathematical thinking
One of the reasons that mathematical thinking is a better training of critical thinking is:
Mathematics is not so close to life compared to other subjests and daily life experience would not help much when doing math thinking. That is to say that in math you have to really think and purely think to get the conclusion, not like in other subjests the conclusion might come from what you already knew or heard.
In these context, the usual advocate of math teaching that emphasizes on making things more practical and avoiding abstractness might not be so right as it seems to be.
2010/5/18 Murray Eisenberg <[log in to unmask]<mailto:[log in to unmask]>>
I read Underwood Dudley's article and am skeptical about his thesis that one learns mathematics in order to train the mind, to learn to think, etc.
What evidence is there that particular skills taught and learned in mathematics generalize to other areas of thinking and reasoning? Or is it just that people who are particularly good at learning and doing the kind of reasoning employed in math happen to be good, too, at reasoning in other subjects.
Dudley's thesis reminds me, in a way, of the claim that one learns Latin in order to better understand English grammar. But surely learning English grammar is the best way to better understand English grammar (without superimposing upon it some artificial and inappropriate Latin structure).
I do agree with Dudley that many of the purported "applications" foisted upon students (and teachers) in math books is so much nonsense. In many such applied problems, you are given what you could not possibly already know and are asked to determine what you already know. (Exercise for the reader: find 10 such examples in the first three chapters of a current calculus text.) Then there are the ridiculous problems, again foisted off as having real-world import, that ignore critical real-world constraints. E.g., finding the dimensions of a window consisting of a rectangle surmounted by a semicircle that maximizes the area given a fixed perimeter (when you really need to take into account architectural limitations not to mention aesthetic considerations); or to minimize the material used to make a circular can given the volume it will hold (without taking into account odd shapes that don't fit shelves or shipping containers, or again without considering appeal to the buyer).
On the other hand, Dudley may be underemphasizing genuine real-world applications which are often not taught because they are too messy. Again from calculus, there's the old chestnut about the lifeguard running along the beach and swimming in the water in order to reach the drowning man; or the "smart" dog who knows how to do the same sort of minimization. Such problems are posed, typically, because the numbers work out tractably. But too often a significant, real-world application is ignored -- the behavior of light rays in different media, e.g., in passing from air to water, where the numbers are not so nice. (My one-time colleague Frank Wattenberg taught me to use that application.)
On 5/18/2010 8:41 AM, Jonathan Groves wrote:
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.
http://www.ams.org/notices/201005/rtx100500608p.pdf
Dear All,
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.
Jonathan Groves
--
Murray Eisenberg [log in to unmask]<mailto:[log in to unmask]>
Mathematics & Statistics Dept.
Lederle Graduate Research Tower phone 413 549-1020 (H)
University of Massachusetts 413 545-2859 (W)
710 North Pleasant Street fax 413 545-1801
Amherst, MA 01003-9305
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