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On 5/18/2010 at 10:30 am, Murray Eisenberg wrote:

> 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.




I'm also struggling with this question as well.  I can believe that
Dudley is correct but not completely correct about this conclusion.
Dom Rosa, the one who started this discussion thread, had posted this
article in several math education discussion lists, two of them being
mathedcc and math-teach.  And arguments arose on those lists regarding
this question and regarding the question as to how useful math is.
One of them had claimed that mathematics helps people become better
mathematicians and nothing else and had also claimed that even if
math does help teach people to think (which he doubts), are there not
other ways to do that as well--that mathematics is not the sole way
to do that?  I think so, and I think it most likely requires that one study
a variety of subjects rather than just one to learn to think so that one 
does not become deadlocked into the kind of thinking required in that one 
subject.  I'm sure it is a safe bet to conclude that most (probably nearly 
all) people can think well in certain disciplines and not so well in others.  

In short, I still believe that mathematics can help teach one to think but 
also that one must study other subjects and everyday logical reasoning
to learn to become a good critical thinker (in the everyday sense).  




> 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).




That same person had mentioned this same idea, too.  Studying Latin
can probably help one to understand English grammar better, but I don't
think it is necessary to study Latin to understand English grammar.
Many people are very good with English grammar who also do not know
any other languages.    



 
> 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.)




Too messy in that these problems would take far more work and understanding
behind them than typical homework and exam problems and examples presented
in class and because too many students--especially those who are worried more
about their GPA than whether they have actually learned anything--would complain 
that these genuine applications problems are too hard.  According to 
school and college administrators, something like that is inexcusable and so 
cannot be tolerated.  Unfortunately, the obsession with grades in K-12 and
massive grade inflation and "rewarding" student laziness by giving everyone
A's and B's continues to encourage massive numbers of college students to
dread challenges rather than to take pleasure in tackling challenging
problems.  I can't blame them for thinking this way because their K-12 
schools have encouraged such thinking.    



Jonathan Groves          




> 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]
> Mathematics & Statistics Dept.
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