As described in subsequent sections, I have had some success in teaching mathematics. I attribute this to a style of teaching in which, although without a textbook and a prepared lesson plan, I try to teach exactly to the students’ understanding of the material. Firstly, I answer every question asked of me in the classroom, or on my home telephone, which is available seven days a week. In order to encourage questions which might ordinarily be embarrassingly simple, I purposefully do not learn the names of any of my students until the end of the term. Secondly, on a given subject, I give an introductory lecture, followed by worked-out examples; then mountains of homework. The next class session is a problem workshop. By answering all of the students’ questions, until all of them are able to understand the jugular vein issues, I am able to provide complete explanations to students who may not be thinking along the lines I am with my professional   training and aptitude in theoretical mathematics. The syllabus is almost always above the assigned material in the departmental syllabus, but is accessible to any undergraduate student. Even in service courses, I teach the subject hoping to obtain a bimodal distribution of grades with one peak about 88% and the other peak in the twenties. This approach is probably criticizable for its difficulty in raising an inattentive F student to a D grade, but my experience has been that many, many F students raise their grade to A or B with this involving method of teaching


1997-98 was a very bountiful year. At the Spring commencement, I received the coveted and unique St. Vincent de Paul Teacher/Scholar Award from St. John’s University for my teaching and research. The University also awarded me the Faculty Outstanding Achievement Medal at a later awards convocation. As a corollary to all this, they nominated me for the Carnegie Foundation for the Advancement of Teaching/ Council for Advancement and Support of Education’s Award as the U. S. Professor of the Year(1998). In the Fall of 1998, I addressed the St. John’s Univ. Faculty on my teaching methods (the script of my speech appears below). It was a wonderful opportunity to explain the influence of Jerry Garcia on effective teaching. In 1997, I was recognized by St. John’s University for exemplary teaching of mathematics. In April 1999, the Student Government of St. John’s Univ. named me Professor of the Year for the St. John’s College of Liberal Arts and Sciences. In 1978, I received the Carl F. Wittke Award for Distinguished Undergraduate Teaching from Case Western Reserve University in Cleveland, OH. In 1976, I was appointed Fulbright Senior Lecturer to Colombia. There, I taught a graduate course in differential geometry at La Universidad del Valle in Cali, from which a textbook emerged. I also gave a series of faculty discussion groups and public lectures on advanced mathematics education in Medellín, Cucutá and Bogotá. I have been a reviewer on a grant review panel in Washington, DC, for the Education and Human resources Directorate of the National Science  Foundation.


In 1973, I was appointed by the United States Agency for International Development (U.S.A.I.D.) and the Venezuelan government to create a graduate school of mathematics at La Universidad de Oriente   in Cumaná, Venezuela. Under their aegis, I established a curriculum, stocked a library, hired a faculty of Spanish-speaking mathematics professors, and taught many of the courses, myself. The endeavor was evaluated by the School of International Education at the University of Pittsburgh by having the first graduating class of the mathematics graduate school at La Univ. de Oriente take their last semester  of studies at the Univ. of Pittsburgh, then pass the Masters degree examinations from the Mathematics Dept. of the Univ. of Pittsburgh. It was an unqualified  success. I also have been professor of mathematics at the Univ. of Washington (1971-73) and La Universidad Simón Bolívar in Caracas, Venezuela (1974-75). In addition, I have often taught courses in the History of Mathematics based   on the textbook which I wrote (in Spanish) at La Univ. Simón Bolívar.

TEXT of SPEECH Upon Receiving the 1998 St. Vincent de Paul Scholar/Teacher Award

In 1971, I took my first position   as a doctorate holding professor of mathematics at the University of Washington, in Seattle, not far from Eugene, Oregon, where I got my degree. One of my friends, Jerry Garcia and his friend, Robert Hunter, had just written and recorded a song with their band, the Grateful Dead. Jerry and the band had often visited  me in Eugene to decompress from the rigors of touring. They could not know it  then, but the lyrics they wrote were predestined to precisely apply to me in 1998.

Sometimes the light’s all shining on me.            Other times I can barely see.                                       Lately, it occurs to me.                                           What a long, strange trip it’s been.


Today, completely cured after a long bout of ill health, I want to thank Dr. Julie Upton and Father Harrington for all the support they have given me over the years and for this award, the St. Vincent de Paul Scholar/ Teacher Award. But most of all, I wish to thank my students, many of whom are sitting before me today as members of the President’s ociety. Although I have won such a teaching award before, in 1978, at Case  Western Reserve Univ., this award feels all the better because of the distinct  possibility, a few years ago, that I might not be here to receive it. You may well ask: What is behind  my teaching? The ex-hippie with the long hair and the earring is not the obvious   role model of a formal, straight-laced teacher. My teaching method is based   on my empirical observation that there are two completely different types of human memory. One, called the working memory, is associated with cold   memorization. This is where songs and poems go to be later recited, without thinking. We all remember how, in college or high school, we would memorize all the bones in the body or all the names of the state capitals. Then we would vomit them down on an exam paper, and go to the local bar to download them forever   from our mind. This is certainly not to be confused with education. The other type of memory is long-term memory. This memory stays intact for over thirty years, and it clearly the one a good professor is aiming for in the education   process. Later conversations with other educators showed me that introduction of ideas and processes into a student’s long-term memory carries with it a strong emotional component. (Damasio, A. R., Descartes’ Error: Emotion, Reason, and the Human Brain. G. P. Putnam’s Sons, New York. 1994.) All along, I had been calling this emotional component,   the “OH, WOW !” effect. I had been trying to impress on my students, at the moment of their analysis of a given problem, how exciting and   impressive is the ease and power of the procedure. And it works. They remember   my analyses for years and years. In particular, I usually follow the following pattern in my teaching:

  1. I present an overview lecture of the foundational concepts;
  2. Several example problems are worked out in great detail, exposing the jugular veins of the concepts;
  3. The students receive my own typewritten  notes;
  4. Tons of homework is assigned on the material; and
  5. A Problem Workshop is scheduled for the next class period.

Obviously, some comments are in order about each of the items in this list. (1) When I walk in for my overview lecture, I never bring notes, nor 3 x 5 cards, nor neat, worked-out problems. I just show up in my levis and tee shirt to discuss the topic. If one has to dress up in order to convey expertise, there must be something missing in one’s own self-evaluation. I strongly feel that teaching in a suit and tie only increases the gap between student and teacher. There is also the well-known contribution of relaxation into the creative process. I lecture in response to the questions of my students as well as the looks on their faces. If they look panicky, I slow down and extract simpler explanations from my experience. If they look like they understand the concepts, I will go to a higher level and introduce stronger material. (2) There is no more effective way  of teaching than for the teacher to distill down the material to its jugular veins. This approach requires that the professor be an expert in the material. She or he must stay at the forefront of the field, but not necessarily conduct   and publish research. The latter path is my method, but it is certainly not a requirement. The best teachers I know are either researchers or are dedicated teachers who maintain their mastery of their field(s) by reading, studying and   attending conferences. Many teachers I know maintain that their field stopped growing when they left school, and that their only responsibility now is to read old, out-dated textbooks to their students. Certainly theirs is not a well-chosen career. The secret is that you must love your material. I love mathematics. (3) Your classnotes should contain your own distillation of the course material , expressed in a rigorous fashion, but in a flowing , familiar tone. Sadly, most textbooks follow the time-honored   dictum of the incompetent pedagogue, “I had to do it this way, so you have to do it this way.” There are some excellent textbooks available. But the textbook selection committees of most departments are bound to the “let’s not make waves” dictum which precludes their adoption. Therefore, I rarely use textbooks. Your notes should include, as mine do, some extremely difficult   looking examples with accompanying “piece of cake” solutions. These impressive solutions allow you to recall how the previously seen methods from high school or previous courses took hours to accomplish what just took a few minutes. This is the “Use the Cannon to Shoot the Fly” technique which is the source of the “OH, WOW !” component of this teaching approach. (4) Assign tons of homework. There is no more effective way of instilling what you are teaching than the solution of many, many problems by the student. But the problems cannot be just meaningless calculations and drudgework.. Surely, the long calculations must be eventually  done. I have done them myself on my way up the ladder. But the impetus to do them must come internally after the student is seduced into believing that the topic is beautiful and powerful. The impetus must come from the student’s   own love of the material, not from a professor’s order. I never give any  credit for the homework, but I grade every problem turned in to me. I explicitly tell each student that they can never hope to pass my course without working out every problem they can get their hands on. The reason becomes clear when we discuss topic number. (5). Give a problem session immediately after the homework assignment. The heart of the problem session is my out front promise to answer every question asked of me, no matter how stupid, at any time in the course. Actually, most of the questions are very incisive and not stupid at all. This promise gives the student complete freedom to express their own lack of understanding of what I am saying. In responding to their questions, I generate the remaining notes for the class. I hang additional topics on my  own responses at a time when the students are open to ideas. For instance, I  allow them to postpone the date of an exam by asking a sufficient number of  questions. The gain in transference of knowledge completely outweighs any issues of scheduling. Although many teachers will tell you it is most important to  learn each student’s name and to address them by their first name in order  to increase familiarity, I, contrariwise, use no roster and run an anonymous   class. In that way, each student can ask any question they have without fear that they will be penalized later for a stupid question. My style is already   loose enough without my worrying that they are not my friends. In order to accomplish   my insertion of the analytic procedure into the students’ long-term memory,  I make myself freely available to them for questions. I give all my students  my home telephone number (631-924-9420). They can call me from noon until 10:00 PM, seven days a week or e-mail me to: The idea is that if they can get over the hump on a problem at the moment they are tackling it, the analysis will go into the long-term memory, along with the accompanying “OH WOW !” factor. The next issue to be addressed is my exams. Since I am teaching processes to students in mathematics classes, there is no credit given for the right answer. All exam questions are graded completely through partial credit. An arithmetic error, early on, means just a few points off if the incorrect calculation is used correctly in the remainder  of the answer. Correct answers without analysis receive no credit. For this  reason, the students may not use hand calculators during the exams. I still haven’t figured out how to respond to the assertion that the mistake lies in keying in the numbers. Also, I have grave reservation about giving higher grades to rich students who can afford more expensive graphing calculators. I am certainly not against technology, and assign laboratory problems on the PC computer, using computational languages such as MAPLE. On exams, each student  receives a different exam. This is easy in mathematics where the problems are   based on numbers. I simply ask the student to write down their social security   number; then take, say, the third digit and use it in the problem. This procedure   ensures that the students study for the process rather than the answer. Also, the decrease in cheating is profound. If at all possible, I give exams with no time limits. When all the grades are assembled, my students usually achieve a bimodal frequency distribution of grades: one large peak about 85% and another, smaller peak around 20%. I do not plan this distribution, but am very happy that it occurs. Every professor who espouses the transference of knowledge should seek a bimodal distribution of grades as just described. The traditional goal of a bell-shaped curve centered at 70% means that your aim is teaching the majority of your students at the C level. Certainly, we should aim higher than that. Sadly, my method does not work for all students. It usually fails dramatically for the students who are not motivated   enough to keep up with each day’s assignments so that the next day’s   question and answer session is meaningful to them. For this reason, my teaching   method includes an almost daily pep rally, enthusing the students with the power   and ease of use of the mathematics I am presenting. I have taught many years in South  America, and currently, I, and my wife, Collette Fortin, are working on translating Maya glyphs from the ruins in central America. This work has led me to integrate  into the Committee on Latin American and Caribbean Studies here at St. John’s. I also enjoy teaching the Beauty of Mathematics and Art to non-science majors. Both of these endeavors have shown me the importance of realizing that different   people think in different ways. If I am to be effective as a teacher, of anything, I have to shape my presentation to the thought process of my audience, my students. I am a theoretical mathematician by trade. There is no reason whatsoever to   suppose that anyone thinks the way I do. That is why my procedure of creating   class notes from the student’s questions is so effective. Recently, when I was in Berlin to   participate in the International Congress of Mathematicians, a very effective   mathematics educator, Prof. David Smith, of Duke Univ., gave me a reference to The Seven Principles of Good Practice in Undergraduate Education, enunciated in 1987 by Profs. A. W. Chickering and Z. F Gamson Chickering,   A. W. and Z. F. Gamson, eds., Applying the Seven principles of Good Practice in Undergraduate Education, New Directions for Teaching and Learning No. 47. Jossey-Bass Publs., San Francisco, 1991). They are: 

  1. Encourage student-faculty contact.
  2. Encourage cooperation among students.
  3. Encourage active learning.
  4. Give prompt feedback.
  5. Emphasize time on task.
  6. Communicate high expectations.
  7. Respect diverse talents and ways of learning.

Without realizing it, I had been doing exactly that for, lo, these many years.

I want to leave you today with the words of another Grateful Dead song, “Ripple.” It is a somewhat religious song about ripples in still waters when there is no pebble tossed, nor wind to blow. And the last line is:

If I knew the way, I would take you home.



 My Bumper Sticker


CALCULUS: The Agony and dx/dt