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04.07
Backscatter: More About Math
By Donald Christiansen
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What Good Is It?" |
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Late last year I wrote a column
called “Math . . . What Good Is It?,” in which I
noted the difficulty in interesting U.S. K-12
students in mathematics. The reason often given
is that students are not made aware of the
practical uses of math. They find it an abstract
concept that bores them.
I am still getting e-mail
responses to the column — as I write this, number
560 has just arrived. There is little doubt the
math quiz included at the end of the column
helped trigger the unusual amount of feedback.
And the article appeared both online and in
print. That helped, too.
Many readers expressed sympathy
for the plight of the students, blaming a
dysfunctional K-12 education system and, in some
cases, teachers who are not adequately prepared
to teach math. While he agreed it is important
to imbue students with the notion that math is
useful in helping solve many real life problems,
Edgar Bristol, Fellow Emeritus of the Foxboro
Co., observed that teachers are not likely to
know anything about the kinds of problems that
any particular student may encounter in real
life ten years hence. Vadim Braginsky, a patent
attorney, wondered whether enough of the best
and brightest minds are motivated to choose
teaching as a career. Could it be, he
speculated, that the K-12 teaching profession
tends to attract candidates who are themselves
weaker at the “meaning stage” of math than would
be desirable. Perhaps, he noted, the
compensation/reward system for teachers may be
at fault.
Several readers disabused me of
the notion that all engineers had been intrigued
by numbers and enthusiastic upon their earliest
encounters with them. Some noted that
mathematics in the early stages of elementary
school is taught in such an abstract way as to
turn off students who later, perhaps through
their own devices, learn to find math useful and
even fascinating. Justin Faulk, a recent EE
graduate of Oklahoma State University, found
formal math pointless until his “revelation in
Calculus 2.” Now he relishes discussions of
Fourier Series or other mathematical subjects
and believes more practical examples of applying
math in the public school system would have
hastened his appreciation of math as a
“beautiful thing.” Sid Gilligan, who earned his
BSEE from Purdue in 1943, remembers how he
disliked arithmetic in grade school, but by the
time he encountered algebra and geometry was “in
love with it all.” The only motivation for those
who were inept or inattentive during Gilligan’s
grade school days was extra sessions with the
teacher, a visit to the principal’s office,
detention, a call to one’s parents, or “staying
back for another run at it.” Arthur Kalb thinks
we scare the average teens away from math by
psyching them out by assuring them that it is
difficult, then further deter them by failing to
relate math to any physical situation. Bob Boyd,
a math instructor, concludes there is too much
emphasis on teaching theoretical rather than
applied math at the high school level. Boyd has
also practiced electrical engineering for nearly
three decades, and so is sensitive to how much
math was useful in his job and how much was not.
He thinks algebra should be taught minus
subjects he defines as unnecessary — set theory,
proofs, lemmas, theorems — which would be left for
the university math major. Geometry would be
taught with practical examples. A high school
student ought to learn how to use the
Pythagorean Theorem, not how to prove it, he
believes.
Other readers questioned why it
is even necessary for students who intend to
become engineers to learn math skills they very
likely might never use in their day-to-day
assignments. Yet many of their colleagues
disagree, noting that even though many engineers
may use but a fraction of the math they learned
in high school and college, they see math as the
best training for abstract thinking. Patent
attorney Braginsky views the study of math as a
process that more generally develops useful
cognitive abilities.
The culture as culprit
The environment in which
youngsters find themselves was faulted by
several readers as contributory to their lack of
interest in math. Gary Polhemus noted that his
14-year-old son enjoys math and appreciates its
many possible uses, but feels that most kids are
not inspired to learn math, or even many other
subjects. He observes: “Our children in the U.S.
seem to be far too interested in looking cool,
which for some reason in our current culture
equates to not being smart. Academic achievement
is just not widely respected in the pop culture
that controls most teens today.” Polhemus
commends those teachers who can reach a student
and then convert him or her into a success
story. Tien Pham agrees that having too many
uninterested students has its root in the value
system in the United States. “How many
mathematicians/engineers get the same respect as
a rock star or football star?” he asks. In
countries like China, engineers are well
respected, not considered nerds as they are in
the United States, he notes, suggesting that the
big question for the average U.S. high school
student is “Do I get respect, and, most of all,
fame, money and sex by being good in math?” (In
the television show “Numb3rs,” the math-genius
hero gets them all. But, of course, this
contrary role model is pure fiction. If the
series survives through several seasons, a trend
may be in the making.)
Peer pressures notwithstanding,
the kids were not completely exonerated from
exercising personal responsibility. “Isn’t it a
shame that students demand a practical use for
anything you are attempting to teach them,”
wrote one reader. Bob Maynard emphasized a
student’s need for mental discipline and a
willingness to persevere through uninspiring
course material. Michael Hammer wrote “Most kids
today seem to have a ‘learn as little as
possible’ mentality, as though everything could
always be looked up if really needed. There is
no concept of knowledge dependencies, where
ideas build upon one another. They think
knowledge is flat, and never venture beyond the
first rung.”
The quiz
The problem I chose to include
in the article was simple: The square of 24 in
base b equals 554 in base b. What is base b?
I suspected it might be
difficult for most teenagers, but easy for those
who were members of a high school math club. I
thought it would be a piece of cake for
practicing engineers. My expectations were
generally confirmed. Interestingly, though I had
hoped readers would try the quiz on their kids,
most could not resist solving it themselves,
even as they remarked on its simplicity or
mocked it as trivial. Evidently only a few
readers challenged their sons or daughters to
tackle the problem. One gave the problem to his
two sons, neither of whom wanted to attempt a
solution. The 16-year-old granddaughter of
another easily solved it, but she would not be
deemed an average teenager, as she is an
accomplished pianist, violinist, and music
teacher, and is not the product of the public
school system but is home-schooled. Dahe Chen of
Cadence Design Systems gave the quiz to his
daughter, an eighth-grade student. He first
refreshed her memory about the concept of place
digits and explained how to convert numbers from
any base to base ten. She was then readily able
to solve it.
By far the majority of those who
attempted the problem first converted it to a
base ten equivalent: (2b + 4)(2b + 4) = 5b^2 +
5b + 4. Solving the resulting quadratic equation
yields roots of 12 and -1. But a few to whom
base conversions are second nature solved it
essentially by inspection. John Smaardyk
reasoned as follows: Multiplying the two least
significant digits yields 16 in base 10. Since
the remainder of the multiplication is a 4, the
base must be 12 (16 – 12 = 4).
Others used a trial and error
approach, while a few enlisted the aid of an
Excel program. Fewer still proposed a
logarithmic approach, but could not recall what
the process would be. (Sorry, I cannot help!)
While most teenagers might find
a problem of this type of little value, several
readers found it pedagogically useful. Philip
Falcone observed “The great thing about this
problem is that it ties quadratic equations with
base arithmetic, whereas each subject most
likely is taught in a vacuum. This type of
problem gives an opportunity to generate an even
higher order polynomial both to impress and
empower the students . . . not a big deal for
experienced electrical engineers, but a major
conceptual hurdle for anyone encountering such a
polynomial for the first time and possibly
trying to solve it.” James Morse observed that
while students may not be aware of any practical
application for a problem of this type, such
problems are a good exercise to make the student
think about how to solve problems.
Resources
For more on helping K-12
students appreciate math and science:
-
Christiansen, D., “Math . .
. What Good Is It?,” Today’s Engineer
Online, October 2006 (includes eleven
additional resources on math history,
learning issues, puzzles and problems).
Donald Christiansen is the former editor and
publisher of IEEE Spectrum and an independent publishing
consultant. He can be reached at
donchristiansen@ieee.org.
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