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Mario G. Salvadori
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Over 20 years ago I started visiting New York City public schools to evaluate the widespread
belief that the city's schools were terrible. To my great satisfaction, I discovered that most of
our schools were teaching correctly -- and often enthusiastically -- the courses assigned by the
New York Board of Education. My
complaint was that state educational authorities compelled the teachers to use curricula that
represented the state of knowledge at the time of the ancient Greeks, ignoring the progress
made in math and science during the last two millennia.
Upon inquiring about this
strange freezing of education for some 2,000 years, I discovered that the teachers were not to
blame; school officials believed such a program would satisfy the needs of future graduates in
nearly all theoretical and practical activities. Shocked by this state of affairs, I abandoned my
decades-long tenure in engineering, architecture, and the graduate faculties at Columbia and
began teaching in the New York City schools, with the help of some of my graduate students.
My approach to the teaching of mathematics and science, based on the idea that students
between 9 and 13 were perfectly ready for some of the complex subjects I had taught at
Columbia, soon acquired a name, the "Salvadori method," and produced unexpectedly good
results.
I taught topics that were useful in everyday life but had been ignored as
too complex for the young. The more complicated these were, the more they interested my
students, whose apparent deficiencies were due more to the approach their teachers used than
to incapacity on either side. The program's assumptions made it inevitable for the students to
believe that topics they were not taught were incomprehensible. The solution to this circular
misunderstanding was to teach both groups concepts that would satisfy the needs of our time,
not just those of the ancient Greeks.
Clarifying the idea of gravity, for example, I
would guide a student toward the recognition that he or she attracted the Earth as much as it
attracted him or her -- or, as Newton stated, that equilibrium
depends on equal and opposite forces -- and that any attraction, between a person and the
Earth or between two people, depends on the distance between these bodies. Explaining
Newton's idea that attraction varies inversely with the square of the distance, I would add that
he could not prove this assumption; intuition suggested this relationship, and he accepted it
because experiments corroborated that it worked. Students were fascinated to find that 200
years after Newton's death, Einstein proved
mathematically that if Newton had made any other assumption except that of the inverse
square, he would have been wrong, and his entire theory would have collapsed. Einstein's
proof did not suggest that one should or could have guessed this result, but that rare geniuses
like Newton could intuit it and, of course, then show by experiment that it was correct. It is
most important to explain to students how science advances by a combination of intuition,
experimentation, and mathematics, rather than describe discoveries as the result of intuition
or chance alone. Nine times out of 10, one can explain in simple words the mental processes
leading to a discovery, allowing the students to participate in the surprise and joy of
discovery.
Once the students understand the basic ideas, they can appreciate other
concepts that may appear surprising at first: for example, the Greeks' important assumption
that given a straight line and a point outside it, only one line can be drawn parallel to that line
passing through that point. One must state that this is an axiom, i.e., a statement to be
accepted without proof, and that it is 2,000 years old. It may be useful, but it is not
necessarily true. To the surprise of most students, I state that it certainly does not apply to
straight lines drawn on the Earth's surface, since the surface is round. It becomes obvious to
the students that the concept of a line is pure abstraction. We can make statements about
straight lines that cannot be made on the Earth's curved surface unless these lines are
infinitesimally short.
Students are not amazed that practically at the same time,
but unbeknown to each other, the Hungarian mathematician J. Bolyai and the
Russian N. I.
Lobachevsky proposed a geometry in which two different lines could be drawn parallel to
a given line from a point outside that line. Then at the beginning of the 20th century a
German mathematician, G.F.
Riemann, proposed a geometry in which an infinite number of lines could be drawn
parallel to the given line. Students may feel at first that these assumptions do not make sense,
but they easily change their minds when we ask them to consider that the Greek assumption
is just as arbitrary as the newer proposed assumptions. A bright student may ask whether
these newer "abstract" lines can be used to practical purposes since, strictly speaking, they do
not exist on the Earth's surface. The answer is yes: These more recently invented non-Euclidian geometries have been extremely useful in
many engineering applications -- for example, the design of electrical circuits. Rather than
being amazed at the utility of the non-Euclidian geometries, students are often amazed that
the old geometry remains useful.
Since the mathematics we use every day can be
extended to concepts as abstract as those of the above geometries, we are showing students
that mathematics is as useful as it is because of its abstract concepts. It would be useless
without them. We can dispel the fear of mathematics in all its aspects by showing that each of
us has been using abstractions, unaware that we should have been afraid of them.
I
hope to have shown that our students are perfectly ready to tackle problems grounded in
advanced research -- and to enjoy solving problems considered "difficult" by those who have
not learned yet how bright youngsters can be.
MARIO G.
SALVADORI, Ph.D. (1907-1997), was the James Renwick Professor
Emeritus of Civil Engineering and
Applied Science and Professor of Architecture Emeritus at Columbia. Among many other
achievements, he founded the Salvadori Educational Center on the Built Environment. We
are indebted to his colleague at the center, Dr. Lorraine Whitman, for assistance with this
manuscript.
Photo Credits:
Photo: Columbia Office of Public Affairs