# Background and Context

## Instructor

Stephen J. GreenfieldDepartment of Mathematics

Rutgers, The State University of New Jersey

greenfie@math.rutgers.edu

www.math.rutgers.edu/~greenfie

Stephen Greenfield is a Professor of Mathematics at Rutgers. He is a member of DIMACS, a national center for discrete mathematics and theoretical computer science. His research interests are complex analysis and partial differential equations. He has also worked in some areas of discrete mathematics and cryptography.

He is interested in educational innovation and sustaining improvements to instruction. He has taught about 35 different math courses in his career, ranging from precalculus to advanced graduate courses. He jointly taught an undergraduate course with a physicist. He was in charge of changing various undergraduate calculus sequences at Rutgers. These changes included appropriate use of technology (involving both graphing calculators and Maple), the use of workshops with peer mentors (small groups of students working on nonroutine problems in class with written reports graded for both content and presentation), and Web support for instruction. He has advised undergraduate research efforts, and led seminars for undergraduates.

He has served as Undergraduate Vice-Chair of the Mathematics Department and as Graduate Director. He obtained significant internal and external funding for educational activities affecting both undergraduate and graduate students.

He has worked with high school teachers, both as part of an educational program which was a direct ancestor of the course presented here and as an enthusiastic participant over the last decade in the Advanced Placement Calculus program. He learned a lot about teaching from these activities. For the last four summers, he has given series of 14 lectures to 100 high school students in the New Jersey Governor's School of Engineering and Technology.

His efforts have been recognized by awards including the Rutgers
University Minority Advancement Program alumni award for
encouraging minority graduate study in mathematics, the FAS Award
for Distinguished Contribution to Undergraduate Education, the
Warren I. Sussman Award for Excellence in Teaching at Rutgers (the
university's most important award for teaching), and, recently, the
2003 Mathematical Association of America-New Jersey Section's Award
for Distinguished College or University Teaching.

Mathematics is part of human culture. Its applications to engineering and the "hard" sciences have been assumed. It is interesting to learn and to teach about how some parts of mathematics have been used to support other needs of society.

## Course History

### Where is the course taught?

The course is taught on the New Brunswick/Piscataway campus of Rutgers University. Here is the official statement of the mission of Rutgers University:

As the sole comprehensive public research university in the New Jersey system of higher education and the state's land-grant institution, Rutgers University has the mission of instruction, research, and service. Among the principles the university recognizes in carrying out this three-fold mission are the following:

Rutgers has the prime responsibility in the state to conduct fundamental and applied research, to train scholars, researchers, and professionals, and to make knowledge available to students, scholars, and the general public.Rutgers should maintain its traditional strength in arts and sciences, while at the same time developing such new professional and career-oriented programs as are warranted by public interest, social need, and employment opportunities.Rutgers will continually seek to make its educational programs accessible to an appropriately broad student body.Rutgers is committed to extending its resources and knowledge to a variety of publics, and bringing special expertise and competence to bear on the solution of public problems.

The university itself has a unique history, as a combination of
colonial college (founded in 1766), land-grant university (1864),
and, finally, as the "flagship" public research university of New
Jersey (1956). Several colleges and other schools with intricate
histories of their own were consolidated into Rutgers at various
times in the last century. The university has more than 50,000
students at campuses in Camden, Newark, and New
Brunswick/Piscataway. The last-named campus, in the center of the
state, is by far the largest, with more than 35,000 students. Most
of the university's Centers and Graduate Programs have their
principal presences in New Brunswick/Piscataway. The university is
a member of the Association of American Universities, "an
association of 62 leading research universities in the United
States and Canada," and is also a member of the National
Association of State Universities and Land-Grant Colleges an
"association of public universities, land-grant institutions and
many of the nation's public university systems."

New Brunswick/Piscataway has more than 25,000 undergraduates, affiliated with a wide variety of schools and colleges, with many majors. The diversity of the undergraduate population is remarkable. Slightly more than half of the students are female. Barely half of the undergraduates are classified as "White," with significant percentages of African Americans (10%), Latinos (10%), and Asian Americans (20%). About 90% of the undergraduates are New Jersey residents. 30% of New Jersey's population is African American, Latino, or Asian American. The first two segments are slightly underrepresented in the Rutgers student body, while the Asian American portion has a much higher proportion of undergraduate enrollment than its percentage of state population, which is about 7%. The mean SAT scores for "registered first-year, regular-admit enrolled students" in fall 2003 were 588 (verbal) and 615 (math). The most recent six-year graduation rate for Rutgers undergraduates is 70%. Many undergraduates do not have English as their family language, and many come from the first generation in their families to attend college.

In December, 2003, the Rutgers New Brunswick/Piscataway Mathematics Department wrote a self-study documents, primarily authored by its chair, Professor Richard Falk. Professor Falk further particularized the mission of the department in the following way:

The Department of Mathematics supports the University's mission of instruction, research, and service in the following ways. The Undergraduate Program of the Mathematics Department provides professional training and liberal education, both to its majors and also to the wide variety of students in other disciplines who enroll in mathematics courses. The Graduate Program provides high quality training to produce the next generation of scholars, researchers, college and university teachers of mathematics, and mathematicians for government and industry. The Department seeks to hire outstanding scholars, whose creation of new knowledge in mathematics will continue the development of the University as a national and international resource. The faculty of the Department is also committed to providing high quality service to the University, the state and federal government, and the mathematics profession.

The Rutgers Department of Mathematics is usually ranked among the top 15 departments nationally for scholarly excellence. Faculty hiring and promotions are primarily based on scholarly promise and achievement, with appropriate attention given to teaching and service. Managing the mathematical enterprise is not simple. About 20,000 students take mathematics courses during the calendar year (even the summer school runs more than 55 math courses!). Almost all students select majors which require some mathematics courses. Also, most undergraduate colleges have quantitative skills requirements which students must satisfy. These requirements are often satisfied by taking mathematics courses.

### What is the course's role in the undergraduate curriculum?

The course was directed at students with minimal college mathematics preparation who wanted to satisfy the quantitative skills graduation requirement of various colleges.

Rutgers is one of many U.S. institutions of higher education which
have graduation requirements involving *quantitative
reasoning.* The ability to think quantitatively and to have a
certification that such is possible (a passing grade in an
appropriate course!) is probably good. Calculus or precalculus
courses certainly fulfill that requirement. Therefore the majority
of our undergraduates are now covered, since most undergraduate
majors require one of those courses. Unfortunately, the learning
communities fostered by many of these courses are not ideal.

At many schools the courses intended to satisfy the quantitative reasoning graduation requirement are not pleasant experiences for the instructors and the students. The students emphatically view these courses as an evil and irrelevant hindrance to their academic journey. The instructors may think that courses discuss esoteric and possibly uninteresting topics taught to an unwilling clientele.

At Rutgers-New Brunswick, the academic niche for a mathematics
course directed at non-majors with modest mathematical background
is Math 103, called *Topics in Mathematics for the Liberal
Arts.* This 3 credit course has two 80-minute meetings each week
for 14 weeks.

I realized that a certain portion of these students are frightened
of mathematics. Even after I got to know each class, I saw that
when I turned to the board and wrote 10100 or (worse)
*A ^{B}* that some of those in the room would blink
nervously or laugh or look blank. Ingrained in these intelligent
students was a fear or distaste for math as an object of
intellectual interest. I don't know whether this was inborn or if
it resulted from previous educational experiences, but great care
and gentle persistence must be used. The students must be enticed
to study math. By contrast, I've taught engineering students for
much of my instructional career. Many of these students are
correctly sure that math is a principal key to success in an
engineering curriculum. Some students in Math 103 are almost as
mournfully sure that math is their personal fate, with unpleasant
connotations.

The math background I requested was good knowledge of Algebra 2, and some knowledge of analytical geometry. I remarked that some involvement with computers (use of the Web for research on papers and elementary use of Maple) would be part of the course.

### Funding Source

Partial support for the creation of this course was provided by the
National Science Foundation under grant DUE-9850071.