Talk About Teaching
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2 2
2 2 -x - y
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(A graph of the equation above accompanied this piece.)
By Dennis DeTurck
What is mathematical literacy? How is it achieved? Why bother with
advanced mathematical education? The immediacy of these questions is
emphasized by the nature of the debate over the book The Bell Curve, by
Murray and Herrnstein. The book brims with data, statistical analyses
and interpretation. Its conclusions concerning race and IQ have been
criticized widely. Unfortunately, much of the commentary is emotional
rather than rational, even though there is plenty of room for analytical
criticism of the work.
There is a story about the great eighteenth-century Swiss
mathematician Leonhard Euler. He was summoned to the court to debate an
esteemed but nameless philosopher about the existence of God. The
philosopher offered a long, eloquently-worded argument to refute the
existence of a deity. Then, Euler stepped up to a blackboard and wrote
some complicated mathematical equation (perhaps the title of this
essay), stepped back, and intoned, "Therefore, God exists." The
philosopher was speechless in the face of the mathematics because he was
not mathematically literate enough to recognize its irrelevance.
Similarly, when we hear that "Housing starts were up 3% in
October," how many of us know what this statistic means? How is it
measured? Up 3% from what? From September housing starts? From October
1993 housing starts? Is this good news?
How does all this relate to freshman Calculus? It all comes back
to the issue of why one learns mathematics. Probably, if a student
intends to be a physician, an engineer, a lawyer, a businesswoman or a
humanist, she will never need to calculate derivatives and integrals.
But she must understand that mathematics provides useful tools and
language for describing, measuring and predicting all manner of natural
and human phenomena. Moreover, she should know how problems particular
to her discipline get translated into the language of mathematics, to
what extent realism is lost in the translation, and how accurate,
reliable and relevant the results of mathematical analyses are to the
original problems.
College students arrive having taken mathematics for many years.
Most Penn students who need Calculus for their majors have studied some
in high school. They say they "know Calculus." But anyone who has taught
Calculus has heard the student lament: "I understand the math, it's
those word problems!" These students have missed the whole point of
studying mathematics in general, and Calculus in particular. Their
mathematical education consisted primarily of drill on problems that
were stated in mathematical terms ("Solve the equation...") and ended
with "the answer in the back of the book." It's as though one learned
the conjugations of all the regular and irregular French verbs, but
didn't bother to understand the meaning of any of them.
Developing facility with symbols and getting correct answers to
drill problems is important, but is only a small part of learning
Calculus (or any mathematics) for two reasons. One is described above:
for all but a few of us, mathematics is pointless unless it is done in a
non-mathematical context. We must be using it to solve real problems.
Thus, we must be able to translate something from the real world into
mathematical language, solve the resulting mathematics problem, and
interpret the answer in the language of the real world. We might do this
to compare mortgages or investments, to decide to elect a risky surgical
procedure to avoid not-quite-certain dire consequences, or to form
opinions about public policy (should medical professionals be tested
randomly for AIDS?).
The other reason that "symbol-pushing" is such a small aspect of
mathematics education is that we can now relegate most of it to a
machine. This causes controversy at every level. In elementary school,
one wonders whether students should use calculators (although, in the
words of Peter Drucker, "Life is too short for long division"). For
Calculus students, the issue is the use of computers and programs like
Maple.
Computing has evoked many essays that begin like this one: What is
mathematical literacy? etc. This question is only half-answered in much
that is said and written around the University. Many students come to
Penn having used word-processing and publishing programs, e-mail,
perhaps a spreadsheet, and games. They say they "know computers".
Similarly, many faculty (see the Provost's essay "The electronic
environment" in October's Penn Printout) view the computer primarily as
a tool for communication: we use it to "talk" to each other more, and to
gain access to more information. This is a limited view, which ignores
the fact that computers are also used for computing. What do we do with
information once we have it? What kinds of thought take place before we
post the message to the newsgroup? What are the tools for organizing,
analyzing and interpreting textual, graphical and numerical data?
Against this background, the use of Maple in Calculus courses can
be discussed. Maple is a programming language that can solve most of the
"rote" problems in a typical Calculus book: Without much thought, the
user types in the problems with only minor notational adjustments and a
few decorative commas and semi-colons, and the computer types back the
answers. Maple can draw two and three-dimensional graphs (the one on
this page [in the print version of Almanac] is the graph of the title),
and can solve problems in many parts of advanced mathematics. Like any
programming language, Maple has a fairly rigid syntax, and demands of
the user a high degree of precision of thought. So using Maple comes
with a cost. What is the return for paying this cost?
Many students will attest that the return is not that Maple makes
Calculus easier to learn. Calculus is a profound achievement of human
thought, and takes a great effort to master. Maple's graphics can help
students visualize some difficult concepts, but making Calculus "easier"
or helping students finish their homework faster is not the point.
Rather, we return again to the theme of this essay: why we learn
mathematics. Tedious details of mathematical calculations should be
relegated to the computer, so that humans can focus on the intellectual
and difficult part of the enterprise: the translation of the problem
into mathematical language and the interpretation of the results of the
mathematical analysis. In fact, this has been one of the biggest sources
of student frustration. Since the machine can do the routine
calculations, courses now place more emphasis precisely where the
students are the least comfortable: translating word problems into math
problems. The return for using the computer comes as students develop a
deeper appreciation for the power and usefulness of the mathematics they
are studying, and a greater ability to apply mathematics and
quantitative thought when it is appropriate. This is evidenced as
students and faculty in Biology, Chemistry, Physics, Engineering, and
Economics classes begin to use Maple (or one of its cousins) to do
mathematical work.
Another effect of the use of Maple in Calculus is the experience
students gain in dealing with the software. Although most of our
students will not be using this software in the future (indeed, it will
undoubtedly be obsolete by the time this year's freshmen graduate), many
will be faced with computer languages of one form or another (whether
for the operating system of a machine, or statistical analysis, or
specialized no-tation for music, etc.). Basic programming skills learned
in one environment transfer to others, and are becoming more and more
valuable.
Penn's Mathematics Department continues to look to the future.
Mathematicians, together with faculty from the natural sciences,
engineering, medicine, and the social sciences are working to develop a
collection of classroom modules and interdisciplinary courses that
combine the teaching of mathematical techniques with applications drawn
from the research of the Penn faculty and from the work of users of
mathematics in industry. The Society for Industrial and Applied
Mathematics (whose national headquarters is in Philadelphia) is
assisting with industry links, and is part of a Penn-led consortium of
institutions pursuing this "holistic" approach to mathematics and its
applications throughout the curriculum. Our new administration has
committed itself to a significant new initiative to position Penn as a
national leader in undergraduate education. The goal of this
interdisciplinary effort is to make Penn a national leader in innovative
undergraduate mathematics education.
Dr. DeTurck is professor and undergraduate chair of mathematics.
His column is the second in a new Almanac series developed by
the Lindback Society and the College of Arts and Sciences.
Almanac
Volume 41, Number 11
November 8, 1994
Return to Almanac's homepage.