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Topic:
What Is Mathematics For?
Replies:
29
Last Post:
Aug 18, 2010 7:38 AM




Re: What Is Mathematics For?
Posted:
Jun 3, 2010 8:11 AM


Jonathan I dont doubt the reality of the harm which can be done in maths classes. Presumably that harm is not what mathematics is for. Should the fact temper our idealism, or enhance our awareness?
Bob Burn University Fellow, Exeter University Sunnyside Barrack Road Exeter EX2 6AB 01392430028 ________________________________________ From: Postcalculus mathematics education [MATHEDU@JISCMAIL.AC.UK] On Behalf Of Jonathan Groves [JGroves@KAPLAN.EDU] Sent: 22 May 2010 03:03 To: MATHEDU@JISCMAIL.AC.UK Subject: Re: What Is Mathematics For?
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 mathteach 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 > 01392430028 > ________________________________________ > From: Postcalculus mathematics education > [MATHEDU@JISCMAIL.AC.UK] On Behalf Of ?? > [eaglle@HEF.ORG.TW] > Sent: 20 May 2010 07:03 > To: MATHEDU@JISCMAIL.AC.UK > 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 > <murray@math.umass.edu<mailto:murray@math.umass.edu>> > 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 > realworld import, that ignore critical realworld > 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 realworld 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, realworld 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 onetime 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 subjectwhether mathematics or notcan 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 notdestroys 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 saysand I think he is > rightthose 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 > careeroriented 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 careerspecific > 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 careerspecific 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 "onesizefitsall" 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 mathteach 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 > > > > > > > > > > > > > > > > > > > murray@math.umass.edu<mailto:murray@math.umass.edu> > Mathematics & Statistics Dept. > Lederle Graduate Research Tower phone 413 > 5491020 (H) > University of Massachusetts 413 > 5452859 (W) > 710 North Pleasant Street fax 413 > 5451801 > Amherst, MA 010039305



