a life in education

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A Life in

Education

Mario Salvadori

The Changing Nature of Engineering

The Bridge Vol. 27 No. 2, Summer 1997

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.


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any are the paths to a successful educational career,and they vary from country to country. In Italy, mynative land, it is not unusual for an elementary or high

school teacher to publish learned papers or win a nationalcompetition and by so doing earn an academic appointmentwithin the public-university system. The most famousexample, perhaps, is that of the young Roman barber who,having mastered 46 languages and started to open the doorsof the mysterious Etruscan tongue, was elected to theprestigious chair of linguistics at the University of Rome,

without ever having attended elementary school. Others, likeMaria Montessori, were less fortunate; although recognizedtoday worldwide as a seminal thinker in the field of elementaryeducation, she never obtained a university position in Italy.

My path along the road to an academic career was neverconsciously concerned with educational methodologies. Itnever occurred to me that I might become interested, much

less active, in the philosophy and practice of education. Mypurpose from the very beginning was to teach almost anysubject I was interested in and not be concerned with how Iwould teach it. Moreover, by chance, my career in educationhas followed a downhill path opposite that of some Italianschool teachers: It started in an Italian university and is endingin an American kindergarten.

Because of this downward evolution, and because of myrather unusual early education, it may be of some interest tothe reader to find out how I was led, on the one hand, to aparticularly happy educational career and, on the other, to anapproach to education labeled by some American educators"the Salvadori educational methodology."

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There is no denying that the way educators teach is deeplyinfluenced by the way they themselves were taught. Myeducation started with a single year in an elementary publicschool in Genoa, Italy, a year I remember to this day for twofunny reasons: the cold, home-cooked omelet I had to eat forlunch, and the chanting of the multiplication table in unisonwith the other kids in the class.

To my unconfessed relief, my father took our family to Spainat the end of that year, and I began attending the French

Lycee in Madrid. This second educational experience lasted allof 2 months, at which time I informed my parents that Irefused to attend a school where the teacher hit the students'fingers with a ruler in order to elicit correct answers. AlthoughI was only 7 years old at the time, my parents took my firmdecision at face value and decided to educate me at homethemselves. Mother, who according to the customs of thetime had not attended school but had been tutored privately,

taught me the humanities, and father, an engineer with anamazing background in the sciences and a few years asprofessor of electrical engineering at the University of Rome,opened my mind to the glories of science.

As a consequence of this decision, and for purelypsychological reasons, I became progressively more enamoredwith the humanities and developed a growing hatred for the

sciences. Mother was sweet and protective, and under herwarmth it became natural for me to love literature (at the timeI spoke three languages: Italian, French, and Spanish) andeven to enjoy Latin declensions as if they were an amusinggame. On the other hand, father may have been a first-rateteacher (he was), but he was also, or at least pretended to be, astrict disciplinarian and a severe taskmaster. My fear of him as

the unchallenged master of the household became sublimated

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into a hate-fear of math and science, particularly becausefather overvalued my capacity to grasp the abstract conceptsof these two subjects and, above all, their application to whathe referred to as "practical problems."

I still have nightmares about the problem he asked me, withgreat glee, to solve at the very beginning of our scientific-mathematical intercourse: "The hands of the watch are one ontop of the other at 12 noon. At what other times during theday are they one on top of the other again?" Besides not

seeing any practical value in the determination of the watchhands' superposition, I was totally unable to determine theseunfathomable times by mathematical manipulations. I got loston square one and hated math and science for the next 12years of my educational life.

Upon our return to Italy after an absence of 5 years, mymiddle and high school education followed the so-called

classical curriculum, established for those students who bymoney and/or brains were destined to attend the university.The curriculum was particularly heavy on the humanities, butwas also demanding in math, physics, and chemistry. Yet, Ifound it altogether fascinating, enjoyed each day of school andpassed the terrifying final exam, consisting of 5 written and 7oral examinations, with high honors.

The time had come for me to choose a career. This was notan easy decision because, meanwhile, I had become totallyenamored with music. By then I was a fairly good pianist,spending more time playing and concertizing withprofessional string players and singers than studying either thehumanities or the sciences. I dreamed of becoming anorchestra conductor. Unfortunately, most if not all of the male

members of my family were engineers and since my early

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childhood I had been conditioned to say, "I want to be anengineer." It did not take much pressure from my lovingparents to convince me of the total impracticality of mymusical 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 Iwould neverpractice engineering. Father may have been strict,but he was also understanding: "Why don't you try a Ph.D. inmathematics?" he said. I thought he was kidding; he knew that

the main reason I did not like engineering was that I couldn'tunderstand the math in it. Yet, between starting an unpleasantcareer and trying a new one, I unhesitatingly chose the latter.After a month of attending lectures at the mathematics schooldelivered by some of the greatest mathematicians in Europe, Ifell 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 meaninglessstatement, but within a week I became enamored with theabstraction and particularly the beauty of pure mathematics,purely taught by the Cauchy approach. My pleasure increasedwhen I discovered that the math curriculum would allow meto attend courses in physics. I quickly signed up for the coursein quantum physics taught by the young Italian Enrico Fermi.

I was, thus, painlessly introduced to the mysteries of quantummechanics, relativity theory, and the other revolutionary ideasabout the physical world that changed the way we look at it.

I started my own career in teaching as an "instructor in thetheory of structures" in the faculty of architecture of theUniversity of Rome. It was there that I discovered, almost by

accident, how to coax unwilling students (of the kind I myself

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had been) into a fascination with a new subject and, thus, toexcite their will to learn.

The circumstances of my first teaching experience could nothave been more propitious. Architectural students chose theircareer because they were attracted to the creative aspects ofarchitecture while being almost totally uninterested in itstechnological aspects. I, too, was fascinated by the creativityof architectural design but was also well aware of itstechnological aspects and, particularly, of the usefulness of

mathematics in the solution of structural problems.

In teaching architectural technology, I decided on a differentapproach than the German theoretical methodology favoredby most professors. I decided to have the students look first atproblems from a physical, intuitive point of view, after whichI would show them how simple the mathematical solution waswhen dictated by a clear understanding of the underlying

physical problem. (Who could have predicted then that 60years later I would adopt the same approach in dealing withstudents in the elementary and junior high schools of NewYork City?)

Architecture was ideal for my purpose, because the numericalanswers to structural problems alwayshave physical meaning.They represent lengths, areas, volumes, weights, among otherphysical entities. My approach worked like a charm. Thestudents understood the physical meaning of what they werebeing asked to calculate, appreciated the importance of suchan understanding to their future professional activities, andenjoyed what they were learning.

A few years later, I was assigned the additional task of

assisting the professor of structures at the school of

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assigned to teach his courses, and a tenured career atColumbia 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 themathematics department to teach the one course inengineering mathematics required of all its students. Themathematicians, having become too busy with the war effortto waste their time on engineers, gave up the course. I was putin charge of engineering mathematics. For the next 20 years,

under the impetus of emerging technologies, I developed 17different courses in engineering mathematics and published anumber of textbooks with titles ending in "EngineeringProblems," an indication of the importance of the physicalapproach to the solution of such problems.

At first, my books were not widely accepted in themathematics community. Despite this initial high-brow

disapproval, however, my books were adopted by anincreasing number of engineering schools. Again and again, Iproved to myself and to my students that the understandingof a physical problem should precede the adoption of amathematical solution. Thus, my conviction was reinforcedthat the difficulties stemming from the abstract nature ofmathematics could be overcome through the understanding ofconcrete problems.

A renewed interest in this approach to structural theory wasawakened in me when, starting in the 1950's, I begandesigning architectural structures in the engineering office ofWeidlinger Associates, meeting architects not as students butas professionals. In addition, I was lucky enough to be offereda position by Princeton University as a professor of

architecture. My 5 years of experience at Princeton reinforced

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because, at that time, math was taught as a series of rules to beaccepted without question, in contrast to the approach used inthe humanistic subjects. Since mathematics is the onlyscientific subject in which there are no rules whatsoeverbut onlyagreed-upon assumptions, which we call axioms, the inanity ofthis approach is obvious to anyone who has been luckyenough to discover what mathematics really is: a pure, abstractgame to be played according to accepted axioms, which canbe changed whenever suggested by a greater applicability topractical problems or upon pure creative whim.

I choose to discuss mathematics first, because of the totallyabstract nature of that field, its only justification stemmingfrom the logic of its development, and the unfortunate fact oflife that abstraction does not come naturally to most of us.Moreover, there are two different kinds of mathematicalabstraction: the geometrical kind and the analytical kind.(Most of the really outstanding mathematicians and scientists

have an extraordinary, natural tendency to think geometricallyrather than analytically.)

The difficulties that arise from the "game" quality and abstractnature of mathematics can be remedied in one of two ways: 1)by imposing an authoritarian acceptance of nonexistentmathematical "rules" (i.e., by teaching mathematics by rote) or2) by explaining mathematics as it really is, a serious but pure

game, and by showing students how to play the game throughthe application of math to concrete problems that interestthem.

Although it might be acceptable in countries with valuesdifferent from ours, I believe that the rote approach isunacceptable in the United States, both from an educational

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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 mathematicalconcepts and techniques by means of problems that arerelevant to their daily lives and are at their level of maturity.The concreteness of the problems and the students' interest intheir practicality do away with the fear of abstraction and thusnaturally motivate.

Second, we do what I call "opening windows on the future."We allow the student to apply a less intuitive mathematicaltechnique to a practical problem that may seem to be soabstract as to be useless in practice but that can be shown tobe of great practical value. To make this point, sometimes wehave to resort to imaginative examples or even metaphors.But once these two principles are adopted, their impact onstudents is often amazing.

Let me illustrate this approach with an anecdote. A seventhgrader once asked me, "Mario, why is 2 x 3 equal to 3 x 2 ?"The correct answer to this question (though a meaninglessone to the student) is: "Because this is an axiom of realnumber theory." Instead, I showed the student that puttingtwo apples on each of three plates or putting three apples oneach of two plates resulted in the same number of apples. Hissatisfied reaction was, "I understand."

I followed up by explaining that the axiom used in theexample did not apply to certain numerical entities (usuallylabeled by capital letters) in a different kind of algebra, calledmatrix algebra, in which A x B is different from B x A mostof the time. I then told the students that without matrix

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algebra (and computers) I could not have designed high-risebuildings with 100 or more floors.

Thus, the students simultaneously encountered thearbitrariness of the operations of elementary algebra and weregiven a hint of the usefulness of matrix algebra. I had bothanswered the student's question and opened his mind to thepractical advantages of a more unusual, more abstract area ofmathematics. 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 stimulatedhis curiosity about such a career.

Because the axioms of mathematics are so well adapted to thesolution of problems, it is hard for young students to believethat different axioms may be as "real" as those of numbertheory or Euclidian geometry. I consider it essential to theunderstanding of mathematics to have students realize that the

well-known axioms of elementary mathematics are as abstractas those of more advanced mathematics.

For this purpose, I use the Euclidian axiom concerning howmany lines can be drawn parallel to a line from a point outsidethe line. I have the student state Euclid's axiom, and then Iplay dumb and ask, "How long is that line we are talkingabout?" "Forever," answers the student. I appear to beconfused and say, "I, like you, live on the Earth and if I keepgoing forever in the same direction I move on a sphericalsurface and go along a circle. I do not walk on your line, Iwalk along a circle." The student is now confused, and it is myturn to point out that an infinite straight line is a pureabstraction, although it represents our reality if we work "inthe small." I similarly point out that the Euclidian notion of a

"point" is a pure abstraction, as is that of an integer.

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I go on to explain that 20,000 years ago, when a human beingfirst noticed that three stones, three people, and three starshad something in common--the property of "threeness"--thatwas a demonstration of an amazing capability for abstractthought. I then clinch the story of the Euclidian axiom bypointing out that toward the middle of the last century, at thesame time but unbeknownst to one another, twogeometricians, one Hungarian (F. Bolyai) and one Russian(N.I. Lobachevsky), proposed a geometry in which not onebut two parallel lines could be drawn from a point outside a

line. And that at the end of the 19th century, a Germangeometrician (F.F.B. Riemann) proposed a geometry in whichan infinite number of lines could be drawn parallel to a givenline from a point outside the line.

I finally explain that the first two non-Euclidian geometrieshave been found to be very useful in the design of electricalcircuits, and that without the Riemannian geometry, Einstein

could not have constructed his general theory of relativity. Inthis way, I hope to convince students that not only arenumerical and Euclidian axioms pure abstractions which donot represent our "reality," but that even more abstractaxioms, like those of matrix algebra or non-Euclidiangeometry, can be just as useful in solving "real" problems asapparently "more real," yet still abstract, axioms.

To my surprise, I have discovered during my years of teachingschool that not only students, but also teachers, includingteachers of science, often believe that science explains naturalphenomena. I use a question-and-answer session to do awaywith this belief. I start by asking a student how much he orshe weighs and, upon being given a answer of so manypounds, I ask, "How come?" Usually, the same student or

sometimes another states with great authority, "I know, I

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know. It is because of gravity." On hearing the magic word"gravity," I play dumb and ask: "Did you say 'gravity'? Whatdo you mean by gravity?" Seldom do seventh graders knowthe answer, but once in a while one states: "By gravity wemean that the Earth pulls on us." I agree with this answer butthen ask the same student: "The earth pulls on you. And doyou pull on the earth?" My question is usually met withgeneral laughter, and the student says, "Who, me? I don'tthink so. No."

At this point, I state what is usually called Newton'sgravitational law (which I prefer to call "Newton'sgravitational hypothesis") and write the correspondingequation on the board. It is proudly recognized by a fewstudents. Then I ask, "Is this what Newton said, that twobodies attract each other in proportion to the product of theirweights and in inverse proportion to the square of thedistance between them?" Everybody agrees that this is what

Newton stated and it is then for me to clarify that what hesaid was: "Two bodies behave as ifthey attracted each other . .. " where the "as if" clearly indicates that Newton did notknow why the two bodies attracted each other but onlydescribed howthey attracted each other.

(I am aware that because the Newtonian gravitational lawexplains much more than the attraction between two bodies, it

is said by some scientists to "explain" gravity. I cannotdisagree with their statement, if by the word explain theymean that it enables us to better understand how certainnatural phenomena occur. An assumption that the force ofgravity acts in inverse proportion to the cube of the distancebetween two bodies would be just as logical as that it acts ininverse proportion to the square of the bodies' distance, but it

would not check with our experiments. In fact, the inverse-

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square law is necessary to Einstein's general relativity theory.It makes his theory more comprehensive, just as correct inlimiting cases, and certainly more elegant. There is no questionin my mind that physics does not explain the cause ofgravitation and never will.)

I attribute this lack of understanding of abstract concepts andof the limitations of science to the absence of science historyin our curriculum. While we learn in school the names andcontributions of the creative men and women responsible for

progress in the humanities, math and science are taught as ifthey had miraculously descended on us from a celestial sphere.This lack of historical context not only leads to the kind ofmisunderstandings I mentioned above, but also ignores thedramatic sequence of discoveries that have slowly brought usto our present knowledge in the sciences. To focus on oneminutia of math history, students are amazed to learn that theequal sign we use in mathematics today was only suggested

and adopted at the end of the 1700s. More importantly, ourstudents are taught physics as if its development had stopped300 years ago. Our schools still live mostly in the Newtonianworld, while the scientific world has gone through the firstrevolution of special relativity, the second of general relativity,the third of quantum mechanics, and the fourth of "stringtheories."

I am not suggesting that our middle school students should betaught still-evolving theories, but I wonder whether it issatisfactory to talk only about particles at a time when the veryconcept of a particle has vanished from physics. It wouldseem logical, practical, and useful to introduce the elements ofthe history of science as we move along its path so as not tocontribute to the basic ignorance and fear that science elicits

in so many of our citizens. This, then, is an essential part of

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math and science based on concrete problems of interest toteachers that, not surprisingly, do not differ much from thosethat interest students. Of course, the center can only do thison a limited scale and in schools in and around New YorkCity. Still, by delving deeply into the basics of math andscience, we succeed in generating in teachers new enthusiasmfor their work.

It is often said that educational problems cannot be solved bythrowing money at them. If interpreted literally, this statement

is obviously correct. No significant human activity has everbeen motivated by money alone. But if the statement is meantto imply that once the needs and solutions to educationalproblems have been identified, they cannot by their verynature be solved through the use of appropriate sums ofmoney, 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 allbecome aware of the need for change, both in terms of whatchildren are taught and how they are taught it. At stake is ourposition as a world leader in science and technology as well asour economic well-being in the 21st century. But, while I amhopeful, I remain realistic. The types of changes contemplatedcan take place only over long periods of time and with the

focused efforts of many individuals and organizations. Wehave reason to expect success, but not immediate miracles.

ReferencesSalvadori, M. 1994. Strength Through Shape: Paper Bridges.New York: Salvadori Educational Center on the Built

Environment.

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About the Author:Editor's Note: Mario Salvadori, author of thisarticle and posthumous recipient of the 1997 National Academy ofEngineering Founders Award, died 25 June 1997. Mario Salvadori, amember of the National Academy of Engineering, is honorary chairman,Weidlinger Associates, Consulting Engineers, P.C. and founder,Salvadori Educational Center on the Built Environment (SECBE). Alonger version of this paper, containing examples of hands-on classroomactivities, is available from SECBE, (212) 650-5497.

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a life in education

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