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Topic: [ME] Math Myopia (John Allen Paulos)
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Jerry P. Becker

Posts: 15,675
Registered: 12/3/04
[ME] Math Myopia (John Allen Paulos)
Posted: Jan 27, 2000 12:03 PM
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This is an interesting column you may want to share with a lot of your
colleagues ... from Lynn Steen and Ken Ross, with thanks: Forbes
[magazine], January 24, 2000

Math Myopia

By John Allen Paulos

An uproar over soft teaching methods shouldn't blind us to the fact that
mathematics does not have to be boring.

"Ambition, Distraction, Uglification, and Derision" is how Lewis Carroll
referred to addition, subtraction, multiplication and division. Although
most people resonate with this repugnance toward computation, most would
also grant its frequent necessity.

This tension underlies the latest skirmish in the simmering Math War. The
issue is the proper place of computation and algorithms (step-by-step
procedures) in the school curriculum. What, in particular, is their
relation to such often neglected skills as understanding graphs,
interpreting probability, modeling situations, applying mathematical
concepts in other domains or estimating and comparing magnitudes?

Textbooks and curriculums that attempt to foster the skills mentioned
above have been criticized as insufficiently rigorous. When the Department
of Education recently endorsed some of these new curriculums as
"exemplary," a group of prominent mathematicians published a letter to
Education Secretary Richard Riley claiming that many of the recommended
books and programs neglect basic algorithms.

This might seem a parochial controversy were it not for the social cost of
our arithmetical failings--clerks who are perplexed by discounts and sales
taxes, medical personnel who have difficulty reckoning correct dosages,
quality control managers who don't understand simple statistics, voters
who can't recognize trade-offs between contrary desiderata and journalists
who are sometimes oblivious to serious risks but apoplectic over trivial

Although there is no real opposition between understanding concepts and
mastering algorithms, extreme positions are easy to parody. Assigning 500
long-division problems to elementary school students is a sure way to
stultify them. So is requiring older students to factor 500 polynomials in
algebra class or to differentiate 500 functions in calculus class.

On the other hand, the reformist endeavor (with which I've been
associated) to tell stories, describe applications, play games and
naturally embed mathematical insights and ideas into everyday life can
also be mocked. Thoughtlessly implemented, it can lead to a feel-good,
wishy-washy ineffectiveness. A new "aha" experience and engaging vignette
can't be required for every problem, and a mere glimmer of the idea
generally isn't sufficient to secure numerical answers.

The proper balance depends on the student's age and background, and the
specific algorithm. There is no royal road to mathematical education,
certainly not one capable of being reduced to a column. Despite common
belief, arithmetic is not easy (see the problem at the bottom of this
page); nor are "higher-level" subjects necessarily difficult. Some
"elementary" algorithms, such as those for dealing with fractions, may be
drudgery if they are not presented well, but they are mathematically
significant and essential to real understanding. No stories about
combining parts of pies or salaries, for example, can replace the formal
rules for finding 2/7+3/11.

Acknowledging that there are glaring weaknesses in some of the new
recommended programs, I'm pleased that they stress applications and
concepts and do not place an undue emphasis on rote repetition. We should
no more be teaching our children to try to compete with $5 calculators
than we should be training them to dig ditches with hand shovels.

In arithmetic the stories and applications should set the stage and
provide motivation for understanding the algorithms. The many good people
on opposite sides of the Math War should recompute their strategies.

Problem: Imagine buying 100 pounds of potatoes and being told that
they're 99% water. After the potatoes have been left outdoors for a day,
you're told that they're now 98% water. What is the weight of these
slightly dehydrated potatoes?

Answer: 50 pounds. Since 99% of the original 100 pounds of potatoes is
water, only 1 pound is "pure potato stuff." Hence this 1 pound must
constitute 2% of the P pounds of partially dehydrated potatoes remaining,
which means P equals 50 pounds.


Jerry P. Becker
Dept. of Curriculum & Instruction
Southern Illinois University
Carbondale, IL 62901-4610 USA
Fax: (618) 453-4244
Phone: (618) 453-4241 (office)
(618) 457-8903 (home)


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