UC Santa Cruz - Center for Teaching Excellence

Anthony Tromba — Teaching Statement 2001-02
Professor of Mathematics

Teaching, A Philosophical Snapshot

My teaching career began, as an instructor, almost forty years ago while I was still an undergraduate at Cornell. It was the 1960s, when we had almost six times the number of mathematics majors in the United States that we have today, a matter of immediate national concern. Mathematics is the pathway to the sciences, medicine, technology, finance and economics. If students in large numbers cannot proceed down this pathway they are denied the opportunities to explore the most exciting, financially rewarding, and diverse career paths in human history and the nation is denied the human resources required for an advanced technological society. Yet just one bad classroom experience in mathematics can send students rushing for the exits.

What has happened over the last forty years? For one, there are serious problems with mathematics education in K –12. For another, universities, in their insatiable appetite for research prestige, have sacrificed commitments to the critical need for teaching excellence in mathematics education, and have also failed to address the K –12 fiasco. I am proud to be a faculty member of Cowell College where the old UCSC values cling to life and honored that my students have nominated me for a teaching award in a field that I loved as a child and still love today.

Mathematics is almost entirely an abstract discipline: its ultimate purpose, applicability or relevance, even at the graduate level, strikes many as opaque. The subject has evolved over thousands of years as the principle second language of science and it has had an enormous impact upon the development of civilization. However, students, as well as teachers of mathematics, are unaware of how significant areas of mathematics have evolved and how profoundly it has influenced the modern world. On the contrary, there are contemporary educators who argue that most of modern mathematics is simply the manipulation of abstract symbols according to rules “made up” by university professors. How could these educators arrive at such a cynical point of view?

Much of mathematics is presented “cookbook” style with instructors following prescribed texts that usually skimp on context and motivation in an effort to get through the material. For the upper division this material is selected mostly for the purpose of incubating future mathematicians. Today many students in large classes find themselves in a situation like French geese, buried up to their necks, force-fed until their livers explode, only to be transformed into mathematical pate de foie gras. Calculus texts now weight upwards of seven pounds. The connection of mathematics to the humanities and to its very own roots has been severed.

The situation was not always like this.

Before the early Greeks, the world seemed almost incomprehensible. For some inexplicable reason, the Greeks came to the conclusion that the world was “rational,” that it could be understood. They believed that the means through which this understanding could be achieved was through the mathematics they had developed, in particular through the study of geometry and number. Plato went as far as to state that the only real truths were those attainable through the study of pure form, in particular through mathematics: that the universe was designed mathematically. This connection between mathematics and philosophy was necessary to sustain the discipline itself.

It is almost breathtaking to contemplate that the Platonists understood, 2,400 years before Newton explained the motion of the planets, the tides, and even the shape of the Earth; before Maxwell’s equations predicted radio waves; before Einstein’s field equations implicitly predicted the expansion of the universe; and before the advent of quantum mechanics empowered us to predict the existence of new elementary particles, that mathematics lies at the heart of the comprehension and prediction of natural phenomena.

My main goal in both graduate and undergraduate classes is to teach not only content, but also process and context: to show that many concepts, abstract as they may be, are historical necessities of their antecedents. These abstract concepts were born as a consequence of the necessity to explore or to understand other significant scientific or mathematical problems. The cynical educators are wrong: important ideas rarely develop in a vacuum; if they do, they are usually doomed to oblivion.

As much as is possible I like to place students into the mindset of a “creator” of mathematics. In this mindset they can see the often insurmountable obstacles, logical and philosophical, that mathematicians themselves struggled to overcome, the orderly, pico bello textbook presentation being a total deception. One hopes to demystify the subject, to reconnect it with its roots, with the history of science and with philosophy. One should try to make clear what has driven developments of the best areas of mathematics. Finally, one dreams of initiating students into a true mathematical experience. Going up the road towards higher mathematics is an incredible adventure in exploring the possibilities, power, and incomparable beauty of abstract thought and structures, a beauty which has intoxicated and addicted many of those who have managed to touch it. For some, like Leibniz, mathematics is the language of the Creator. Others are drawn by its beauty, its logical self-containment, its challenging problems and by its awesome power to predict.

Simply by making calculations following from a formal theory, Isaac Newton stunned the 17th century. One of his predictions was the onion shape of the Earth. It was Maupertuis, the first President of the Prussian Academy of Sciences under Fredrick the Great, who went to Lapland to make the measurements to determine if Newton was indeed correct. Voltaire, himself awed byNewton’s achievements, exclaimed: “Maupertuis confirmed in places boring and muddy, what Newton knew without ever having to leave his study.”

To attain my educational goals students must never feel intimidated. They must do more than just solve problems, they are encouraged to engage in dialogue and, unconventionally, to intellectualize about mathematics itself. We educators must all be prepared to continually re-examine our lecture material as well as our pedagogical approaches; we must never be afraid to experiment! One must, if possible, approach every lesson, no matter how trivial, with enthusiasm and humility, almost as if one is discovering the material for the first time, full of wonder and a sense of mystery.

Naturally, no class will be successful if mathematics is not presented neatly, in a logically compelling manner, in a voice that tends to wake rather than to lull. Keep eye contact to see when students are drifting; when they do break stride, tell a joke or story just get them back!
Hold onto a strong sense of humor and never, never take yourself too seriously.