An engineer's personal experiences form the basis for an innovative educational program that uses concrete, visual problems to teach abstract math concepts to students in inner-city public schools.
Many are the paths to a successful educational career, and they vary from country to country. In Italy, my native land, it is not unusual for an elementary or high school teacher to publish learned papers or win a national competition and by so doing earn an academic appointment within the public-university system. The most famous example, perhaps, is that of the young Roman barber who, having mastered 46 languages and started to open the doors of the mysterious Etruscan tongue, was elected to the prestigious chair of linguistics at the University of Rome, without ever having attended elementary school. Others, like Maria Montessori, were less fortunate; although recognized today worldwide as a seminal thinker in the field of elementary education, she never obtained a university position in Italy.
My path along the road to an academic career was never consciously concerned with educational methodologies. It never occurred to me that I might become interested, much less active, in the philosophy and practice of education. My purpose from the very beginning was to teach almost any subject I was interested in and not be concerned with how I would teach it. Moreover, by chance, my career in education has followed a downhill path opposite that of some Italian school teachers: It started in an Italian university and is ending in an American kindergarten.
Because of this downward evolution, and because of my rather unusual early education, it may be of some interest to the reader to find out how I was led, on the one hand, to a particularly happy educational career and, on the other, to an approach to education labeled by some American educators "the Salvadori educational methodology."
There is no denying that the way educators teach is deeply influenced by the way they themselves were taught. My education started with a single year in an elementary public school in Genoa, Italy, a year I remember to this day for two funny reasons: the cold, home-cooked omelet I had to eat for lunch, and the chanting of the multiplication table in unison with the other kids in the class.
To my unconfessed relief, my father took our family to Spain at the end of that year, and I began attending the French Lycee in Madrid. This second educational experience lasted all of 2 months, at which time I informed my parents that I refused to attend a school where the teacher hit the students' fingers with a ruler in order to elicit correct answers. Although I was only 7 years old at the time, my parents took my firm decision at face value and decided to educate me at home themselves. Mother, who according to the customs of the time had not attended school but had been tutored privately, taught me the humanities, and father, an engineer with an amazing background in the sciences and a few years as professor of electrical engineering at the University of Rome, opened my mind to the glories of science.
As a consequence of this decision, and for purely psychological reasons, I became progressively more enamored with the humanities and developed a growing hatred for the sciences. Mother was sweet and protective, and under her warmth it became natural for me to love literature (at the time I spoke three languages: Italian, French, and Spanish) and even to enjoy Latin declensions as if they were an amusing game. On the other hand, father may have been a first-rate teacher (he was), but he was also, or at least pretended to be, a strict disciplinarian and a severe taskmaster. My fear of him as the unchallenged master of the household became sublimated into a hate-fear of math and science, particularly because father overvalued my capacity to grasp the abstract concepts of these two subjects and, above all, their application to what he referred to as "practical problems."
I still have nightmares about the problem he asked me, with great glee, to solve at the very beginning of our scientific-mathematical intercourse: "The hands of the watch are one on top of the other at 12 noon. At what other times during the day are they one on top of the other again?" Besides not seeing any practical value in the determination of the watch hands' superposition, I was totally unable to determine these unfathomable times by mathematical manipulations. I got lost on square one and hated math and science for the next 12 years of my educational life.
Upon our return to Italy after an absence of 5 years, my middle and high school education followed the so-called classical curriculum, established for those students who by money and/or brains were destined to attend the university. The curriculum was particularly heavy on the humanities, but was also demanding in math, physics, and chemistry. Yet, I found it altogether fascinating, enjoyed each day of school and passed the terrifying final exam, consisting of 5 written and 7 oral examinations, with high honors.
The time had come for me to choose a career. This was not an easy decision because, meanwhile, I had become totally enamored with music. By then I was a fairly good pianist, spending more time playing and concertizing with professional string players and singers than studying either the humanities or the sciences. I dreamed of becoming an orchestra conductor. Unfortunately, most if not all of the male members of my family were engineers and since my early childhood I had been conditioned to say, "I want to be an engineer." It did not take much pressure from my loving parents to convince me of the total impracticality of my musical aspirations. I entered engineering school.
I finished first in my class, with a deep distaste for our "family" profession, declaring to my amazed parents that I would never practice engineering. Father may have been strict, but he was also understanding: "Why don't you try a Ph.D. in mathematics?" he said. I thought he was kidding; he knew that the main reason I did not like engineering was that I couldn't understand the math in it. Yet, between starting an unpleasant career and trying a new one, I unhesitatingly chose the latter. After a month of attending lectures at the mathematics school delivered by some of the greatest mathematicians in Europe, I fell in love with the subject I had previously hated.
I remember to this day the words with which my calculus professor began his course: "Let M objects be chosen N by N . . ." I was totally mystified by this seemingly meaningless statement, but within a week I became enamored with the abstraction and particularly the beauty of pure mathematics, purely taught by the Cauchy approach. My pleasure increased when I discovered that the math curriculum would allow me to attend courses in physics. I quickly signed up for the course in quantum physics taught by the young Italian Enrico Fermi. I was, thus, painlessly introduced to the mysteries of quantum mechanics, relativity theory, and the other revolutionary ideas about the physical world that changed the way we look at it.
I started my own career in teaching as an "instructor in the theory of structures" in the faculty of architecture of the University of Rome. It was there that I discovered, almost by accident, how to coax unwilling students (of the kind I myself had been) into a fascination with a new subject and, thus, to excite their will to learn.
The circumstances of my first teaching experience could not have been more propitious. Architectural students chose their career because they were attracted to the creative aspects of architecture while being almost totally uninterested in its technological aspects. I, too, was fascinated by the creativity of architectural design but was also well aware of its technological aspects and, particularly, of the usefulness of mathematics in the solution of structural problems.
In teaching architectural technology, I decided on a different approach than the German theoretical methodology favored by most professors. I decided to have the students look first at problems from a physical, intuitive point of view, after which I would show them how simple the mathematical solution was when dictated by a clear understanding of the underlying physical problem. (Who could have predicted then that 60 years later I would adopt the same approach in dealing with students in the elementary and junior high schools of New York City?)
Architecture was ideal for my purpose, because the numerical answers to structural problems always have physical meaning. They represent lengths, areas, volumes, weights, among other physical entities. My approach worked like a charm. The students understood the physical meaning of what they were being asked to calculate, appreciated the importance of such an understanding to their future professional activities, and enjoyed what they were learning.
A few years later, I was assigned the additional task of assisting the professor of structures at the school of engineering. One morning he called me from his sick bed, at a quarter of nine in the morning, asking me to teach a 9 o'clock class of 150 students! On the spur of the moment, I decided to introduce them to a brilliant American method for the solution of structural problems by successive approximation, which presented the advantage of giving a physical meaning to each successive numerical move toward the correct solution. With this unexpected lecture, I gained the admiration of the students and, of course, the hatred of the professor. (The method was known for years in Italy, not by the name of its inventor, a professor at Yale University, but as the "Salvadori American method.") The enthusiasm of the engineering students confirmed my belief in a physical approach to the mathematical solution of engineering and architectural problems.
My tenure at the University of Rome lasted only 7 years because of my decision to leave Italy rather than fight in the second World War on the side of the Nazi-Fascists. My arrival in the United States in January of 1939 was made possible by the unexpected and never admitted help of my mentor and by then good friend Enrico Fermi.
In January of 1940, after a sad year spent working in a factory while looking for a job, I was offered a 4-month position in Columbia University's school of engineering as a substitute lecturer in elementary mechanics. I delightedly accepted this temporary assignment and, to the surprise of my colleagues, kept going to the campus after the 4 months had expired. At times madness knows its hidden purpose and, sure enough, within a month, I was offered a lectureship to teach elementary mechanics for an unspecified length of time. Shortly thereafter, when a famous professor of civil engineering was called to Washington by the war effort, I was assigned to teach his courses, and a tenured career at Columbia became a goal I could dream of reaching.
I could not have found myself in a better spot at a better time. The school of engineering had always relied on the mathematics department to teach the one course in engineering mathematics required of all its students. The mathematicians, having become too busy with the war effort to waste their time on engineers, gave up the course. I was put in charge of engineering mathematics. For the next 20 years, under the impetus of emerging technologies, I developed 17 different courses in engineering mathematics and published a number of textbooks with titles ending in "Engineering Problems," an indication of the importance of the physical approach to the solution of such problems.
At first, my books were not widely accepted in the mathematics community. Despite this initial high-brow disapproval, however, my books were adopted by an increasing number of engineering schools. Again and again, I proved to myself and to my students that the understanding of a physical problem should precede the adoption of a mathematical solution. Thus, my conviction was reinforced that the difficulties stemming from the abstract nature of mathematics could be overcome through the understanding of concrete problems.
A renewed interest in this approach to structural theory was awakened in me when, starting in the 1950's, I began designing architectural structures in the engineering office of Weidlinger Associates, meeting architects not as students but as professionals. In addition, I was lucky enough to be offered a position by Princeton University as a professor of architecture. My 5 years of experience at Princeton reinforced those at the school of architecture of Rome and inspired me to write my first book on structures for architects. To my pleasant surprise, it was adopted by almost all American schools of architecture and was translated into 14 foreign languages. The book dealt with some of the most refined concepts of architectural structures, but it did not contain a single mathematical formula. I found that I could explain such concepts with words and pictures as clearly as with mathematical formulas, and I was almost as surprised by this discovery as my readers were. This experience made my collaboration with some of the greatest architects in the world a wonderful experience.
The 1970s saw me relinquish some of my structural design activities in favor of work as a "forensic expert" in court cases involving structural damage or collapse. I thus had the opportunity to explain the causes of collapses to members of a jury, most of whom were totally untrained in the intricacies of structural design.
It is with apologies for this extravagant (but, I believe, relevant) preamble that I can now explain my approach to education in the lower and middle grades of our schools, a cause to which I have dedicated a significant amount of my energies and time over the past 20 years, as founder of the nonprofit Salvadori Educational Center on the Built Environment (SECBE).
In all of this work, I have looked at education from the point of view of the student. None of us ever learned anything unless we were interested in what we were supposed to learn; a bored child is not only unwilling but unable to learn. I, for one, hated mathematics throughout high school because I neither understood it nor saw any purpose in it. This occurred because, at that time, math was taught as a series of rules to be accepted without question, in contrast to the approach used in the humanistic subjects. Since mathematics is the only scientific subject in which there are no rules whatsoever but only agreed-upon assumptions, which we call axioms, the inanity of this approach is obvious to anyone who has been lucky enough to discover what mathematics really is: a pure, abstract game to be played according to accepted axioms, which can be changed whenever suggested by a greater applicability to practical problems or upon pure creative whim.
I choose to discuss mathematics first, because of the totally abstract nature of that field, its only justification stemming from the logic of its development, and the unfortunate fact of life that abstraction does not come naturally to most of us. Moreover, there are two different kinds of mathematical abstraction: the geometrical kind and the analytical kind. (Most of the really outstanding mathematicians and scientists have an extraordinary, natural tendency to think geometrically rather than analytically.)
The difficulties that arise from the "game" quality and abstract nature of mathematics can be remedied in one of two ways: 1) by imposing an authoritarian acceptance of nonexistent mathematical "rules" (i.e., by teaching mathematics by rote) or 2) by explaining mathematics as it really is, a serious but pure game, and by showing students how to play the game through the application of math to concrete problems that interest them.
Although it might be acceptable in countries with values different from ours, I believe that the rote approach is unacceptable in the United States, both from an educational and a political standpoint. How then should we teach math? In the SECBE "approach," we adopt two basic principles.
First, we introduce students to any and all mathematical concepts and techniques by means of problems that are relevant to their daily lives and are at their level of maturity. The concreteness of the problems and the students' interest in their practicality do away with the fear of abstraction and thus naturally motivate.
Second, we do what I call "opening windows on the future." We allow the student to apply a less intuitive mathematical technique to a practical problem that may seem to be so abstract as to be useless in practice but that can be shown to be of great practical value. To make this point, sometimes we have to resort to imaginative examples or even metaphors. But once these two principles are adopted, their impact on students is often amazing.
Let me illustrate this approach with an anecdote. A seventh grader once asked me, "Mario, why is 2 x 3 equal to 3 x 2 ?" The correct answer to this question (though a meaningless one to the student) is: "Because this is an axiom of real number theory." Instead, I showed the student that putting two apples on each of three plates or putting three apples on each of two plates resulted in the same number of apples. His satisfied reaction was, "I understand."
I followed up by explaining that the axiom used in the example did not apply to certain numerical entities (usually labeled by capital letters) in a different kind of algebra, called matrix algebra, in which A x B is different from B x A most of the time. I then told the students that without matrix algebra (and computers) I could not have designed high-rise buildings with 100 or more floors.
Thus, the students simultaneously encountered the arbitrariness of the operations of elementary algebra and were given a hint of the usefulness of matrix algebra. I had both answered the student's question and opened his mind to the practical advantages of a more unusual, more abstract area of mathematics. Of course, I had not taught him matrix algebra, but I had opened a window on a topic he might run into later on in his educational career, and I might even have stimulated his curiosity about such a career.
Because the axioms of mathematics are so well adapted to the solution of problems, it is hard for young students to believe that different axioms may be as "real" as those of number theory or Euclidian geometry. I consider it essential to the understanding of mathematics to have students realize that the well-known axioms of elementary mathematics are as abstract as those of more advanced mathematics.
For this purpose, I use the Euclidian axiom concerning how many lines can be drawn parallel to a line from a point outside the line. I have the student state Euclid's axiom, and then I play dumb and ask, "How long is that line we are talking about?" "Forever," answers the student. I appear to be confused and say, "I, like you, live on the Earth and if I keep going forever in the same direction I move on a spherical surface and go along a circle. I do not walk on your line, I walk along a circle." The student is now confused, and it is my turn to point out that an infinite straight line is a pure abstraction, although it represents our reality if we work "in the small." I similarly point out that the Euclidian notion of a "point" is a pure abstraction, as is that of an integer.
I go on to explain that 20,000 years ago, when a human being first noticed that three stones, three people, and three stars had something in common--the property of "threeness"--that was a demonstration of an amazing capability for abstract thought. I then clinch the story of the Euclidian axiom by pointing out that toward the middle of the last century, at the same time but unbeknownst to one another, two geometricians, one Hungarian (F. Bolyai) and one Russian (N.I. Lobachevsky), proposed a geometry in which not one but two parallel lines could be drawn from a point outside a line. And that at the end of the 19th century, a German geometrician (F.F.B. Riemann) proposed a geometry in which an infinite number of lines could be drawn parallel to a given line from a point outside the line.
I finally explain that the first two non-Euclidian geometries have been found to be very useful in the design of electrical circuits, and that without the Riemannian geometry, Einstein could not have constructed his general theory of relativity. In this way, I hope to convince students that not only are numerical and Euclidian axioms pure abstractions which do not represent our "reality," but that even more abstract axioms, like those of matrix algebra or non-Euclidian geometry, can be just as useful in solving "real" problems as apparently "more real," yet still abstract, axioms.
To my surprise, I have discovered during my years of teaching school that not only students, but also teachers, including teachers of science, often believe that science explains natural phenomena. I use a question-and-answer session to do away with this belief. I start by asking a student how much he or she weighs and, upon being given a answer of so many pounds, I ask, "How come?" Usually, the same student or sometimes another states with great authority, "I know, I know. It is because of gravity." On hearing the magic word "gravity," I play dumb and ask: "Did you say 'gravity'? What do you mean by gravity?" Seldom do seventh graders know the answer, but once in a while one states: "By gravity we mean that the Earth pulls on us." I agree with this answer but then ask the same student: "The earth pulls on you. And do you pull on the earth?" My question is usually met with general laughter, and the student says, "Who, me? I don't think so. No."
At this point, I state what is usually called Newton's gravitational law (which I prefer to call "Newton's gravitational hypothesis") and write the corresponding equation on the board. It is proudly recognized by a few students. Then I ask, "Is this what Newton said, that two bodies attract each other in proportion to the product of their weights and in inverse proportion to the square of the distance between them?" Everybody agrees that this is what Newton stated and it is then for me to clarify that what he said was: "Two bodies behave as if they attracted each other . . . " where the "as if" clearly indicates that Newton did not know why the two bodies attracted each other but only described how they attracted each other.
(I am aware that because the Newtonian gravitational law explains much more than the attraction between two bodies, it is said by some scientists to "explain" gravity. I cannot disagree with their statement, if by the word explain they mean that it enables us to better understand how certain natural phenomena occur. An assumption that the force of gravity acts in inverse proportion to the cube of the distance between two bodies would be just as logical as that it acts in inverse proportion to the square of the bodies' distance, but it would not check with our experiments. In fact, the inverse-square law is necessary to Einstein's general relativity theory. It makes his theory more comprehensive, just as correct in limiting cases, and certainly more elegant. There is no question in my mind that physics does not explain the cause of gravitation and never will.)
I attribute this lack of understanding of abstract concepts and of the limitations of science to the absence of science history in our curriculum. While we learn in school the names and contributions of the creative men and women responsible for progress in the humanities, math and science are taught as if they had miraculously descended on us from a celestial sphere. This lack of historical context not only leads to the kind of misunderstandings I mentioned above, but also ignores the dramatic sequence of discoveries that have slowly brought us to our present knowledge in the sciences. To focus on one minutia of math history, students are amazed to learn that the equal sign we use in mathematics today was only suggested and adopted at the end of the 1700s. More importantly, our students are taught physics as if its development had stopped 300 years ago. Our schools still live mostly in the Newtonian world, while the scientific world has gone through the first revolution of special relativity, the second of general relativity, the third of quantum mechanics, and the fourth of "string theories."
I am not suggesting that our middle school students should be taught still-evolving theories, but I wonder whether it is satisfactory to talk only about particles at a time when the very concept of a particle has vanished from physics. It would seem logical, practical, and useful to introduce the elements of the history of science as we move along its path so as not to contribute to the basic ignorance and fear that science elicits in so many of our citizens. This, then, is an essential part of the SECBE approach to the teaching of science--to mention science's history at every step of its evolution so as to uncover its human dimension and to reduce its abstract nature.
The second essential part of the SECBE approach is to introduce abstract concepts through concrete, visual problems. For example, using just a piece of paper, ruler, and scissors, it is possible to construct a geometrical proof of Pythagoras' theorem, perhaps the most memorable theorem in the history of geometry. Similarly, the common yet extreme interest of students of all ages in the behavior of structures (including the influence of shape on the strength and stiffness of structural elements) can lead to an understanding of the variation of "strength through form" by means of simple experiments, requiring inexpensive and easily available materials. (See, for example, Salvadori, 1994.) Students of all ages love to participate in hands-on activities, and they learn social skills along with their mathematical and physical concepts.
There is no doubt among educators that, once everything is said and done, the entire educational system revolves about one pivot: the teacher. Yet, in times as dynamic as ours, teachers are confronted with a serious dilemma: If they keep teaching the way they were taught, they will be left behind by the rapid progress in subject matter and in pedagogy. Yet, seldom is time or compensation provided for the additional effort required. Hence, the continuing education of the teacher remains one of the most difficult issues in modern education.
Our center has tried to address this problem through a modest, direct approach--essentially the same strategy we use with students. We offer teacher-training graduate courses in math and science based on concrete problems of interest to teachers that, not surprisingly, do not differ much from those that interest students. Of course, the center can only do this on a limited scale and in schools in and around New York City. Still, by delving deeply into the basics of math and science, we succeed in generating in teachers new enthusiasm for their work.
It is often said that educational problems cannot be solved by throwing money at them. If interpreted literally, this statement is obviously correct. No significant human activity has ever been motivated by money alone. But if the statement is meant to imply that once the needs and solutions to educational problems have been identified, they cannot by their very nature be solved through the use of appropriate sums of money, they could not be more wrong.
So, I cannot end without expressing my hopes for the future of the U.S. education system. I believe deeply that teachers, administrators, boards of education, and politicians have all become aware of the need for change, both in terms of what children are taught and how they are taught it. At stake is our position as a world leader in science and technology as well as our economic well-being in the 21st century. But, while I am hopeful, I remain realistic. The types of changes contemplated can take place only over long periods of time and with the focused efforts of many individuals and organizations. We have reason to expect success, but not immediate miracles.
Salvadori, M. 1994. Strength Through Shape: Paper Bridges. New York: Salvadori Educational Center on the Built Environment.