[magazine], January 24, 2000
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 ones.
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
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
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.