Bob,

Unfortunately one on another discussion list has called me a fascist at
least several times already for insisting on mathematics beyond arithmetic
and a statistical literacy course as general education requirements.
And I must get under his skin a lot more than anyone else on math-teach
because I am the only one he continues to reference to others as
wanting "to shove math down people's throats" though several others
take similar views on mathematics as a general education requirement.

I know the question "What harm can math do?" would not work with him
because he cites the harm math has done to lots of students: massive
math anxiety and massive hatred of math among lots of students.  So he
now believes that no math beyond sixth grade should be required.


Jonathan Groves        



On 5/21/2010 at 8:45 am, Bob Burn wrote:

> 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
> red 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]>
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