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On May 18, 2010, at 9:30 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.
>
> 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]
> Mathematics & Statistics Dept.
> Lederle Graduate Research Tower phone 413 549-1020 (H)
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