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Re: Why study math?

Why study algebra. A good starting point is the Rhind Mathematical
Papyrus of 1500 (or thereabouts) b.c.e., as described in Gillings,
<Mathematics in the Time of the Pharaohs> (Dover 1972). It describes how
to calculate (without using algebra) the payment due priests who were paid
in loaves of bread and jugs of beer (apportonment problems). The point is
that the calculations are incredibly complicated, and are greatly
simplified by starting with the statement; "let x=the unknown quantity"
(suggesting an approach that is absent from most modern-day algebra
texts). Algebra, in other words, in important because it lets one solve a
whole set of problem at once by thinking symbolically. This approach is
particularly important in our present computer age, when the algebraic
formulation can be placed on a computer that generates a solution for
every specific input.


Algebra is also important for general thinking, such as (taken from an
early work of Hake):
If gravity provides that, for a body of mass m:
GMm/r^{2) =ma, then we expect that motion of bodies
wubjected only to gravitational forces (including frictional forces that
are proportional to the body's weight) should be independent of the body's
mass.
Thsee are, in our present educational system, extremely difficult
propositions for student because they do not want to reason with symbols.


Regards,
Jack


On Fri, 26 Mar 2004, Richard Hake wrote:

In his Math-Teach post of 19 Mar 04 10:33:57-0500 (EST) titled "Re:
Saxon Math," math student "Elrohir" wrote:

". . . . show us where math is used to build bridges and roads, how
it's used in balancing a checkbook, and how it's used in everyday
things. . . . My prime example is Trig. We were given formulas to
memorize, situations to memorize, with no explanation as to why . . ."

In response, on 19 Mar 04 12:43:53-0500 (EST), Timotha Trigg wrote:

"Is it my imagination, or have bridge building activities enjoyed
almost as much of a boost in popularity as bar graphs in recent
years? Since not all that many students have bridge building in
their future, I find it curious that this has become such a popular
'real world' example. If students need real world examples, WHY NOT
SHOW THEM SOME PROBLEMS FROM SERIOUS PHYSICS AND CALCULUS TEXTBOOKS
and show that algebra/trig knowledge is a useful prerequisite for
solving these problems? Students need not be able to understand the
physics and calculus problems to appreciate that learning
algebra/trig will help keep the doors open to whatever they
eventually decide they want to do with their lives." [My CAPS.]

Better yet, why integrate science problems seamlessly into math
texts, as suggested to deaf ears almost 4 decades ago by that
bug-a-boo of direct instructionists Morris Kline (1967, 1970). In the
preface to his calculus book Kline (1967) wrote:

"The second essential respect in which this book differs from current
ones is that the relationship of mathematics to science is taken
seriously. The present trend to separate mathematics and science is
tragic. There are chapters in mathematics that have value in and for
themselves. However, the calculus divorced from applications is
meaningless."

My colleagues in the Indiana University math department used to take
great delight in telling me that Kline's calculus books were a
disaster because they had bombed in the market place!

Responding to Timotha Trigg's post, Ralph Raimi on 19 Mar 2004
13:54:37-0500 (EST) submitted to Math-Teach a post titled "Why Study
Math?":

"Textbooks have always been big on bridges, and invariably feature
large colorful photographs of bridges with captions explaining that
geometry (sometimes algebra) is used in building a bridge such as the
one pictured. Actually, they should mention trigonometry, and
perhaps some books do. But the bridge is *de rigeur*. For a less
colorful explanation of why one should study trigonometry let me
suggest the essay [by Ralph Raimi] found at
<http://www.math.rochester.edu/people/faculty/rarm/trig.html>.

In his essay, Ralph likens the question "Why study trig?" to the
fabled query "How would you use a barometer to measure the height of
a building?" [For copious references and commentary on the "Old
Barometer Story" see Hake (2000).] Ralph writes: "The true answer is
that trigonometry is part of our culture, and should be as visible in
daily life as anything in Dickens or Shakespeare." For other answers
see Snyder (1997).

While Ralph's answer is, of course, correct, I doubt that it will
provide much motivation to the average grumbling math student.
Perhaps physicist Al Bartlett's aphorism:

"THE GREATEST SHORTCOMING OF THE HUMAN RACE IS OUR INABILITY TO
UNDERSTAND THE EXPONENTIAL FUNCTION"

might jolt some students into an appreciation of math.

Another answer to the question "Why study math" was given by Timothy
Keller (2004) in his Math-Teach Post of 26 Mar titled "Re: Why study
math?" Keller wrote:

"Well, of course one could [give Snyder's (1997) answers] but the
general question [is]: 'Why study anything ???' Daily life in the
modern world requires very little knowledge of science, history,
literature, mathematics or anything else for most of us to function
as economic or social creatures. On the other hand, part of being
human is an innate need to understand the world around us. Satisfying
this need may or may not have a personal and/or immediate practical
utility. . . . I find all this business where some educational
administrator says 'This is what the kids need to learn to get a good
job in today's Information Economy (or some such catch-phrase )..', a
really stilted view of what being human is about."

With the same bottom-line mentality as Timothy's educational administrator,
Business Week (2004) states: "Because the quality of a nation's
workforce has such a huge influence on productivity, effective school
reform could easily stimulate the economy more than conventional
strategies, such as the Bush tax cuts. CONSIDER WHAT WOULD HAPPEN IF
THE U.S. COULD RAISE THE PERFORMANCE OF ITS HIGH SCHOOL STUDENTS ON
MATH AND SCIENCE TO THE LEVELS OF WESTERN EUROPE WITHIN A DECADE.
According to Eric A. Hunushek . . .
.[<http://edpro.stanford.edu/eah/eah.htm>]. . . . , a senior fellow
at the Hoover Institution <http://www-hoover.stanford.edu/> at
Stanford University, U.S. GROSS DOMESTIC PRODUCT GROWTH WOULD THEN BE
4% HIGHER THAN OTHERWISE BY 2025 AND 10% HIGHER IN 30 YEARS." [My
CAPS.]

For a Bartlettian protest to Business Week's (2004) myopic
concentration on the Gross Domestic Product, see Hake (2004).

Richard Hake, Emeritus Professor of Physics, Indiana University
24245 Hatteras Street, Woodland Hills, CA 91367
<rrhake@earthlink.net>
<http://www.physics.indiana.edu/~hake>
<http://www.physics.indiana.edu/~sdi>


REFERENCES
Bartlett, A.A. 1976. "The exponential function - Part I," Phys.
Today, October, pp. 393-401. For a related online paper see Bartlett
(1996).

Bartlett, A.A. 1996. A slight revision of "The Exponential Function,
XI: The New Flat Earth Society," The Physics Teacher 34(6): 3432-343;
online at
<http://csf.colorado.edu/authors/Bartlett.Albert/flat-earth.htm>.

Business Week. 2004. "America's Failure in Science Education," 16
March; online at
<http://www.businessweek.com/technology/content/mar2004/tc20040316_0601_tc166.htm>

Ferguson, R. 1998. "Theodolites, Barometers, and Laptops," Math-Teach
post of 10 Sep 1998 12:38:22-0500; online at
<http://mathforum.org/epigone/math-teach/heldpheldder/3.0.1.32.19980910123822.00699fe8@accd.edu>.

Hake, R.R. 2000. "The Old Barometer Story (was Problem Solving in
Physics)" online at
<http://listserv.nd.edu/cgi-bin/wa?A2=ind0007&L=pod&P=R15115>. Post
of 22 Jul 2000 15:03:43-0700 to Math-Teach, PhysLrnR, and POD: I
wrote (see that post for the references): "Despite its fictional
nature, the Old Barometer Story (OBS), in its various guises, can
serve as a reminder of the boring algorithmic nature of many
physics/math 'problems' and the rebellion of many good students (ref.
4,5) against the rote learning required in many [usually ineffective
(ref. 6,7)] traditional passive-student physics courses and in much
[usually ineffective (ref. 8-10)] K-12 science/math instruction (ref.
11). Fortunately, more interesting and challenging physics problems
are now available (ref. 12)." For an amusing offshoot of the OBS see
Ferguson (1998).

Hake, R.R. 2004. "Re: SPECIAL REPORT: America's Failure in Science
Education (Business Week)," online at
<http://lists.nau.edu/cgi-bin/wa?A2=ind0403&L=phys-l&O=D&P=28211>.
Post of 23 Mar 2004 15:55:46-0800 to Phys-L, Physhare, Chemed-L, and
PHYSOC.

Keller, T. 2004. "Re: Why study math?" Math-Teach post of 26 Mar 04
11:18:13-0500 (EST); online at
<http://mathforum.org/epigone/math-teach/broxbrangbre/2ywxu5e8ob5n@legacy>.

Kline, M. 1967. "Calculus, Part 1." John Wiley.

Kline, M. 1977. "Why the Professor Can't Teach," St. Martins Press.
Kline writes: "The writing in mathematics text is not only laconic to
a fault; it is cold, monotonous, dry, dull, and even ungrammatical. .
. . The books are not only printed by machines; they are written by
machines."

Snyder, M. "Why Study Math," Math-Teach post of 31 Mar 1997
23:42:53-0500; online at
<http://mathforum.org/epigone/math-teach/broxbrangbre/v01530501af661fb3cfcd@%5b206.119.237.3%5d>.


--
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General Custer's unremembered message to his men,
just before leading them into the Little Big Horn Valley