Alumni newsletter

Spring '97

In this issue:

The WeBWorK Project

For the past year Professors Gage and Pizer have been developing a software package called WeBWorK for the automatic grading of math homework. Last summer two undergraduate programmers, Basem Moussa '97 and Scott Douglass '97, were hired to help with this effort. Last semester Pizer used the package in Math 140A, the first semester of our new calculus with foundations sequence. The students have responded favorably and they say the program motivates them to work harder in the course.

With this software an instructor can define a type of problem and the program will vary it slightly (say by randomly changing the numerical or symbolic input, subject to predefined constraints) for each student. The student can use the program on any computer with internet access, for example from a personal computer in a dorm room. He/she gets a personalized assignment, and then learns immediately if her answer is correct. The student is allowed to try again subject to a deadline.

Experience has shown that the instanteous feedback is very helpful, far more so than the traditional method where a student has often forgotten about the homework by the time he/she gets it back from the instructor. In the words of one student, "I can fix my mistakes while [the] problem is fresh in my mind." The student can then think about the problem, find the correct answer, and come to a better understanding of the mathematical principles involved. Students have a strong incentive to do so and the WeBWorK experience is that they do do so. In contrast, when home work is graded by hand, often students will not receive back corrected assignments until a week or more after the assignment was completed. By then, the problems are no longer fresh in the mind of the student and the student does not have a strong incentive to understand his or her errors. The immediate response provided by WeBWork has stimulated students to ask more questions about mathematics, thus creating an excellent environment for learning.

Aftermath of the Renaissance Plan

We are happy to report that, after a one year hiatus, we are now accepting applications for the entering graduate class of 1997. The negotiations that led to the reinstatement of our doctoral program last spring have also improved our relations with the rest of the University. We suggested then that we would meet individually with every department that has a math requirement for its majors to discuss curricular issues. So far we have had about half of those meetings, and they have all been very constructive. Some have led to ideas for new courses. The Computer Science Department has agreed to make our discrete math course a requirement for its majors.

We have especially close ties with the Physics and Astronomy Department. They have asked us to teach a graduate level math course that they require all of their graduate students to take. There is a new mathematical physics seminar that meets twice a month. The two departments together are seeking a federal grant that would enrich both of our graduate programs.

We are also pleased to report that the Renaissance Plan has been successful in recruiting a very talented freshmen class. The Mathematics Department has for many years offered a very challenging honors calculus sequence. This year the freshman enrollment in that sequence is extraordinarily large.

Meliora Mathematica

This past Fall Semester the Mathematics Department began its "Meliora Mathematica" program. As all who are familiar with the Latin motto of the university will know, meliora means "to make better." The goal is to improve mathematics education and research in all it aspects, with a strong focus on undergraduate courses and graduate training.

We have some ongoing projects which could use financial help.

Computing

Academia is becoming more compute intensive every year. We now use computers to communicate with our students, write their exams, grade their homework, and illustrate mathematical concepts in the classroom. The computing needs of the department are expanding, and our equipment needs to be upgraded as technology improves. Contributions in cash or kind are welcome. If you work for a high tech company, equipment that you consider obsolete could be useful to us. Please think of us before discarding it. For more information please contact Professors Arnold Pizer at <apizer@math.rochester.edu> or Michael Gage at <gage@math.rochester.edu>.

Graduate Student Expenses

Most of our graduate students receive modest stipends from the University, but we would like to supplement these in various ways. In particular we want to help our advanced students travel to conferences and other scientific meetings. For more information please contact our Director of Graduate Studies, Professor Michael Cranston, <cran@db1.cc.rochester.edu>.

Meliora Mathematica Speakers Fund

The department has a busy seminar schedule. In addition to the applied math and mathematical physics seminars mentioned above, we have seminars in topology, number theory, probability, geometry and analysis, as well as our colloquium, a series of lectures aimed at a general mathematical audience. All of these enrich the intellectual life of the department and improve the educational experience for our students.

Quest

Quest Calculus

Undergraduate calculus is a focus of experimentation with new ways of teaching. The department has instituted a new course sequence, Quest Calculus, which demands that the students make extraordinary efforts to learn both concepts and techniques of calculus. In Quest Calculus, students are given two kinds of problems, the first kind developing a mastery of the standard calculus techniques like differentiation and integration and the second kind, which are called quest problems, requiring much more original and conceptual thinking. These quest problems are harder and the students work together in small groups to solve them. The traditional calculus course is set up so that students attend lectures and recitations. In recitations students have the opportunity to see problems solved by graduate and undergraduate teaching assistants. In quest calculus, recitations are replaced by work sessions where the students work together under supervision of teaching assistants and professors. Work sessions require more work from students, teaching assistants, and faculty. They are very active sessions and, after one semester of trial, they seem to be both successful and popular with the students.

Quest is a college wide program, now in its second year, for enriching the academic experiences of selected freshmen and sophomores. Our two year honors calculus sequence (the only two year course in the College) has been fulfilling this function for many years. This year 27% of all Quest enrollments are in Mathematics.

Quest Calculus as a Laboratory for Modifications of Traditional Calculus Courses

The most significant difference between Quest Calculus and traditional calculus is that the students who enrolled in the Quest version selected themselves by registering for a calculus sequence which would demand more effort but which would pay off with a deeper understanding of mathematics. This means that the setup of Quest Calculus cannot be adopted without change by the traditional calculus course. But the department is much encouraged by the success of the work sessions and it is entirely possible that these sessions can be made part of the traditional calculus. These work sessions have the advantage that students can have fun working together.

Sample problems from Quest Calculus

To get the real flavor of what we are doing in quest calculus, there is nothing like trying some sample assignments from the course. Here are two of them. Can you answer these questions?

Assignment 1

An urn contains an infinite number of billiard balls labelled 1,2,3,... . At 1/2 minute before midnight, balls 1-10 are removed from the urn and ball 1 is replaced. At time 1/3 minute before midnight, balls 11-20 are removed and ball 2 is replaces. At time 1/4 minute before midnight, balls 21-30 are removed and ball 3 is replaced. And so on.

At 1/n+1 minute before midnight, which balls are removed and which one is replaced? Which balls are in the urn at midnight?

Assignment 2

A light projects a slide through a lens onto a screen. Suppose this arrangement projects the point with coordinate x on the slide to the point with coordinate x3 on the screen. For example, 2 is projected to 8 and 3 is projected to 27. The interval [2,3] is projected to the interval [8,27] which is 19 times as long. For large x the lens magnifies a great deal; for x near 0 the lens shrinks a great deal.

How would you define magnification at a point x? Explain why this makes sense. What is the magnification at x = 2?

Answers are available at the end of the newsletter.

New advising system for math majors

The math department has instituted a new system in which all math faculty are involved as advisors for individual majors. This will allow each student to develop a close relationship with a professor that will last for his/her entire undergaduate career. It is modeled on the way in which graduate students have always been advised, namely, a one on one mentor-student relationship.

Applied Mathematics Seminars

At the graduate and research level, the department has instituted a new applied math seminar. This seminar takes mathematics beyond its traditional boundaries and has been very successful. The audience and lecturers are a diverse group consisting of faculty and graduate students from the Mathematics Department and related departments in science and engineering. For example, one of the lectures presented the work of Professor Barbara Shipman of the Mathematics Department on the dance that honey bees do to tell their hivemates where they have found a good food source. The bees change the form of the dance acording to the location of the flowers that constitute the source. The surprising thing is that there may be a deep mathematical reason for how the dance changes form. The reason is related to a space in symplectic geometry known as a "flag manifold." Although no one is suggesting that honey bees understand flag manifolds, it is possible that the instincts which control their behavior are wired in such a way that the principles of this kind of geometry apply.

Financial Mathematics

The Mathematics Department has been studying the possibility of starting a Masters program in Financial Mathematics, similar to ones offered by Carnegie-Mellon University and the University of Chicago. This semester we are offering two new courses in this subject, one at the graduate and one at the undergraduate level. Financial mathematics is the study of mathematical models for the pricing of financial derivatives such as options to buy or sell stocks, bonds, commodities, foreign currencies, etc. It is estimated that the current market for derivatives exceeds $10 trillion, and there is a growing demand for mathematical expertise in this area.

Alumni: We want you!

Let Us Hear From You

Students and parents often ask us, "What can you do with a math degree?" We want to know what you are doing with yours. How has mathematics helped you in your career?

Knowing what you know now, would you have pursued your undergraduate career differently? Are there courses you wish now that you had taken then or not taken then? Are there other courses that you wish we had offered? Which of the math courses you took have been the most or least useful to you?

Hearing the answers to these questions from as many alumni as possible will help us give better advice to our students. We are especially interested in hearing from you if you live in the Rochester area or if you are planning to visit here soon. We would like to arrange meetings (either formal or informal) between students and successful math alumni. Please contact us if you are interested in doing this.

The Math Alumni Career Resource Network

We want to encourage all alumni to give our students advice about careers in mathematics. There are many different forms these contacts could take:

We want to create a list of math alumni that our students can call upon for career advice. We will soon have a mechanism on our web site for collecting this information. Please contact us if you are interested in helping our students in this way.

Perspectives on Mathematics: Historical Quotes from our "Mathematics Survival Guide"

In addition to being a practical guide to learning mathematics at the University of Rochester, the "Mathematics Survival Guide" contains selected quotes which provide a bird's eye view of the understanding of mathematics down through the ages. We think you will find these quotes interesting.

From Ancient Egypt, mathematics passed to Ancient Greece where it played a central role in philosophy and science. When asked by his pupil, Alexander the Great, for a short cut to learning geometry, the tutor Menaechmus replied: "O King, for traveling over the country, there are royal roads and roads for common citizens; but in geometry there is one road for all." The Greeks valued mathematics as part of their understanding of reality and as training for the mind. Many years later in 1860, Abraham Lincoln expressed similar views in his Short Autobiography:

He studied and nearly mastered the six books of Euclid since he was a member of Congress. He began a course of rigid mental discipline with the intent to improve his faculties, especially his power of logic and language. Hence his fondness for Euclid, which he carried with him on the circuit till he could demonstrate with ease all the propositions in the six books; often studying far into the night, with a candle near his pillow, while his fellow-lawyers, half a dozen in a room, filled the air with interminable snoring.

The mathematician Augustus De Morgan had a different view: The moving power of mathematical invention is not reasoning but imagination.

The Greeks put their emphasis on geometry and measurement. In these terms, they developed a good understanding of, among other things, musical scales and harmony. The other great source of mathematics is, of course, the concepts of numbers and counting. In the Enlightenment, the cofounder of calculus, Leibnitz expressed his understanding of rhythm in music: Music is the pleasure the human soul experiences from counting without being aware that it is counting.

When Descartes invented analytic geometry, he made it possible to derive geometry from numbers. At the beginning of the 20th Century, this was forcefully expressed by one of the great developers of the theory of complex variables, Magnus Gustaf Mittag-Leffler: Numbers are the beginning and end of thinking. With thoughts were numbers born. Beyond numbers thought does not reach.

But most mathematicians would not accept such a restrictive definition of mathematics, much less of all thought. In the 19th Century, the founder of set theory, Georg Cantor, said: The essence of mathematics is its freedom. A little later, the English mathematician, G. H. Hardy, was more specific: A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

Scientists have always used mathematics and, in doing so, have developed distinctive views on its nature. Compare the views of two great 19th Century physicists. Ernst Mach (whose name is connected with the speed of sound) said: Strange as it may sound, the power of mathematics rests on its evasion of all unnecessary thought and on its wonderful saving of mental operations. Heinrich Hertz, the discoverer of radio waves, said: One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.

Answers to sample Quest problems

Assignment 1

At 1/n+1 minutes before midnight, balls 10(n-1) to 10n are removed and ball n is replaced. Since every ball is replaced before midnight, at midnight all the balls are in the urn.

Assignment 2

In general, magnification is the ratio of the size of the image divided by the size of the object. So, magnification at a point is the size of the image of a small length at x divided by the small length. Magnification at x is the limit as the change in x (called delta x) approaches 0 of the length from x3 to (x + delta x)3 divided by delta x. This is the definition of the derivative of x3. So magnification at x is given by 3 times x2, which for x = 2, is 12.