
Mario G. Salvadori
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 decadeslong 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 nonEuclidian 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 nonEuclidian 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.Related links...
Mario Salvadori, "A Life in Education," The Bridge vol. 27 no. 2 (summer 1997), National Academy of Engineering
Mario Salvadori, "Can there be any relationships between mathematics and architecture?" Nexus 96 conference on architecture and mathematics, Fucecchio, Italy
Tom Lehrer, "Nikolai Ivanovich Lobachevsky" (satiric song lyric)
MARIO G. SALVADORI, Ph.D. (19071997), 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.
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Photo: Columbia Office of Public Affairs