Directions For Mathematics Research Experience For Undergraduates

Mark A Peterson, Yanir A Rubinstein

• Publisher: WSPC
• Year: 2015
• ISBN:
  9789814630337
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

“The US National Science Foundation (NSF) Research Experiences for Undergraduates (REU) program in mathematics is now 25 years old, and it is a good time to think about what it has achieved, how it has changed, and where this idea will go next.”

This was the premise of the conference held at Mt. Holyoke College during 21–22 June, 2013, and this circle of ideas is brought forward in this volume. The conference brought together diverse points of view, from NSF administrators, leaders of university-wide honors programs, to faculty who had led REUs, recent PhDs who are expected to lead them soon, and students currently in an REU themselves. The conversation was so varied that it justifies a book-length attempt to capture all that was suggested, reported, and said. Among the contributors are Ravi Vakil (Stanford), Haynes Miller (MIT), and Carlos Castillo-Chavez (Arizona, President's Obama Committee on the National Medal of Science 2010–2012).

This book should serve not only as a collection of speakers' notes, but also as a source book for anyone interested in teaching mathematics and in the possibility of incorporating research-like experiences in mathematics classes at any level, as well as designing research experiences for undergraduates outside of the classroom.

“The US National Science Foundation (NSF) Research Experiences for Undergraduates (REU) program in mathematics is now 25 years old, and it is a good time to think about what it has achieved, how it has changed, and where this idea will go next.”

This was the premise of the conference held at Mt. Holyoke College during 21–22 June, 2013, and this circle of ideas is brought forward in this volume. The conference brought together diverse points of view, from NSF administrators, leaders of university-wide honors programs, to faculty who had led REUs, recent PhDs who are expected to lead them soon, and students currently in an REU themselves. The conversation was so varied that it justifies a book-length attempt to capture all that was suggested, reported, and said. Among the contributors are Ravi Vakil (Stanford), Haynes Miller (MIT), and Carlos Castillo-Chavez (Arizona, President's Obama Committee on the National Medal of Science 2010–2012).

This book should serve not only as a collection of speakers' notes, but also as a source book for anyone interested in teaching mathematics and in the possibility of incorporating research-like experiences in mathematics classes at any level, as well as designing research experiences for undergraduates outside of the classroom.

Readership: Undergraduates, practitioners and teachers in mathematics.

Chapter 1

Undergraduate Research and the Mathematics Profession

Donal O’Shea

President’s Office, New College of Florida
5800 Bayshore Road, Sarasota, FL 34243, USA
[email protected]

1.Introduction

Much has been said and written about undergraduate research in mathematics: about its possibility, about its intrinsic quality (or lack of), about different models for facilitating it, and about its value to students, mentors, and even departments. Very little has been written about the systemic effect of undergraduate research on the mathematics profession and the development of mathematics as a whole. What follows is a first, necessarily speculative, attempt to begin to fill this gap.

2.Historical Sketch

I begin by remarking that mathematics as a discipline, and as a profession, in the US is comparatively young. Although higher education dates back to before the American Revolution, it was difficult to get a good graduate education in mathematics in 1900 in the US (and virtually impossible in 1890). A student with a serious interest in mathematics had little choice but to go to Europe, especially Germany or Paris. This would change in the beginning of the twentieth century as some of the educational, social, and technological ferment that followed the Civil War took root. The nation needed professionals and postsecondary education grew rapidly. The Morrill Act established the land grant institutions and in the three decades from 1869 to 1899, the number of institutions of higher education increased from 563 to 977, enrollments quintupled to just under a quarter of a million students and bachelor degrees awarded tripled to over 27,000.¹ The new universities needed professors. Some older, more established, American institutions, notably Harvard and Yale, responded by establishing programs to train university professors with disciplinary expertise. Johns Hopkins University, Clark University, and the University of Chicago were set up to emulate the best German practices. The emergence of the American mathematical research community parallels the emergence of American research university out of the German research model. This is a fascinating story, well documented in [22].

The new mathematics research community soon began to experiment with better methods of graduate education. R.L. Moore began teaching topology in 1911 to beginning graduate students using a method that basically mimicked research, and others followed suit. Efforts to encourage undergraduate research followed shortly after. In 1934–35, the Michigan Section of the MAA appointed a committee to create a wider interest in undergraduate research, and subsequently opened space for undergraduates to present at their annual meetings (see [26]). In [10], Frank Griffin surveyed college and university mathematics departments to find out which were involving undergraduates in research, where he defined “research” as “exploring some new question or re-exploring some old question and getting results previously unfamiliar to specialists in the field.” He documented an impressive array of student research achievements, including those at his home institution, Reed College, and noted that colleges that encouraged their undergraduates to do research produced more students who went on to receive graduate degrees in mathematics.

The point I make here is that undergraduate research in mathematics, although not common in the early decades of the twentieth century, is nearly as old as mathematical research in the US. A few individuals had noted its efficacy, observing that students who had been involved in research as undergraduates were more likely to continue in mathematics and were, in turn, more likely to involve undergraduates in their research. Some sectional MAA student meetings encouraged student presentation. Some mathematics departments required student theses. Others, however, had a comprehensive exam, a practice almost orthogonal to student research.

The late 1950s to mid-1970s and the 1960s in particular, mark the greatest period of change in higher education in the United States since the post-civil war period, save perhaps the present. The exuberance of the times coupled with the large increase in college enrollments as a result of the GI bill and the post-World War II baby boom fueled huge growth and diversification in American higher education. Enrollments in postsecondary institutions more than doubled in the 1960s, from 3.6 million enrolled in 1959 to more than 8 million 1969, and tripled from 1960 to 1975, reaching 11.2 million in 1975. See [17, Table 313.10]. The number of institutions grew more slowly, but still dramatically, increasing 25% in the 1960s (from 2004 in 1959–60 to 2556 in 1969–70) and an additional 10% in the first half of the 1970s (reaching 2765 in 1975–75). See [17, Table 317.10]. The Sputnik launch by the Soviet Union in 1957 and the subsequent press, not to mention the 1960 presidential campaign, produced a sense that the US needed to catch up by producing more scientists, engineers and mathematicians. Federal research dollars flowed into the system and the American university morphed into what Clark Kerr, first chancellor of UC-Berkeley and president of the University of California system, called the multiversity: not one community, but a community of communities [14].

From the point of view of mathematics and undergraduate research, this era was more notable for what did not happen than what did. Federal research agencies, especially the then recently established (1950) National Science Foundation funded a number of experiments to complement mathematical and science education. Among them, was the URP (undergraduate research program), which funded undergraduate research through the 60s and 70s. This funding, however, was primarily focused on biology, chemistry and physics, not mathematics, and allowed students to spend time, typically ten weeks, in active research laboratories, at research universities and some liberal arts colleges. The program did not impose the requirement that departments accept students from other institutions. Chemists at liberal arts colleges started incorporating more summer research into undergraduate programs.

In mathematics, the projects funded by the National Science Foundation tended to involve enrichment activities aimed at complementing formal mathematical education. Most involved accelerated instruction, coupled with a format that encouraged students to work together and competitively on challenging problems. Perhaps the best known such program is the Ross Summer Mathematics program for high school students which originally began at Notre Dame in 1957 and which moved to Ohio State University in 1964 where it has been ever since. A number of government labs provided research opportunities in which mathematics students could participate, but there seems to have been little pressure from the mathematical community to create summer research opportunities similar to those in the other sciences. There were a number of projects to institute academic year programs and to improve the major. In particular, a program directed by Kenneth May at Carleton College was explicitly aimed at producing undergraduate research. The project involved selecting particularly strong entering undergraduates and supplying them with a rich set of problems and a social environment to encourage research in their junior and senior years. It set up an undergraduate colloquium joint with St. Olaf [15].

The first conference on undergraduate research in mathematics of which I am aware took place at Carleton College in 1961. The Summary and Resolutions of the conference committee made it clear that the prevailing assumption was that ordinarily undergraduates could not be expected to do significant research in mathematics (see [27]). “[The aims of undergraduate research] are the training and stimulation of the student, not the attainment of new results, though such bonuses will come occasionally.” The document goes on to state that “Undergraduate research should be judged by standards different from those now employed by mathematicians” and notes that using the word “research” may in fact inhibit the establishment of undergraduate research programs at colleges and universities, suggesting that one might broaden the appeal by calling undergraduate research “independent study.” Nonetheless, the resolutions of the conference carried the clear sense that “student activity of the research type was a part of good educational practice” and were aimed at the best means for achieving it.

So, there were disputes about what precisely “undergraduate mathematical research” meant and others who questioned its value altogether, asserting that it came at the expense of time that would be better devoted to “mastery of established topics.” Nonetheless, some faculty members in mathematics departments, both in liberal arts colleges and in research universities, were encouraging undergraduates to do research, usually but not exclusively as part of a senior thesis. (See [10], [11] and [26], as well as the articles [28], [23] and [16], reprinted in [25].) There was considerable anecdotal evidence that undergraduate research enhanced the likelihood that students would continue to graduate studies in mathematics. It was clear, however, that even in a time of relative plenty, there was neither sufficient appetite nor consensus to use research dollars to fund undergraduate research. In retrospect, this seems to have been a missed opportunity for the mathematics community.²

In 1977, Joseph Gallian established a summer research program in mathematics at the University of Minnesota Duluth. See [6]. That program, which is still running today, brought a small number of undergraduates from across the country for ten weeks to work on unsolved problems, with a goal of obtaining and publishing new results. In other sciences, of course, it had already been accepted that undergraduates could contribute usefully to the research enterprise.

The 1980 presidential election brought a new ethos to the country as the growth of preceding decades tapered off. Perceptions of an oversupply of scientists abounded and the URP program was terminated in 1981. By the mid-1980s, alarm had arisen at the decrease in the number of young people going into science and mathematics careers. The David Report [18] documented a critical shortage of young people entering mathematical careers. Data gathered by the American Chemical Society showed a 25% decrease in the number of B.S. level chemists graduated during the early 1980s. Congress gave the National Science Foundation a mandate to create a plan to counter the trend.

One result was the Research Experiences for Undergraduates (or REU) program which began to solicit proposals in 1987 for the so-called summer research sites. Sites were encouraged to tailor programs to suit their own calendars, and to make efforts to increase populations of women, Native Americans, minorities and other under-represented groups (see [24]). Mathematicians with regular NSF research grants were also allowed to tack on REU supplements that would allow funding one or two undergraduates as part of an investigator’s research program. This program resulted in the establishment of a number of REU sites in mathematics, one organized by an alumnus of the Duluth program. Some of these sites still exist — in fact, it is the twenty-fifth anniversary of the REU site at Mount Holyoke College (see [21]) that has occasioned the present conference.

In July 1988, a conference on undergraduate research in mathematics was organized at Carleton College in connection with the Second National Conference of the Council on Undergraduate Research.³ The proceedings, entitled Models for Undergraduate Research in Mathematics and edited by Lester Senechal, surveyed some older, as well as some new, summer programs. Although the issue of whether undergraduates could do research had not entirely been laid aside, a number of the new sites focused on having undergraduates working together to create new mathematics. Some of the sites exploited the fact that advances in algorithms and computing speeds had made it possible for undergraduates to explore examples that were at the forefront of what was known theoretically.

This time around, enough mathematicians embraced undergraduate research to ensure that it would flourish. A mathematical and computer science division was added to the Council on Undergraduate Research in 1989. In addition, research institutes such as MSRI that had been established by the National Science Foundation as an attempt to enhance the infrastructure for the mathematical sciences began to incorporate undergraduate research. Shortly thereafter, the National Science Foundation started funding sites and departments to vertically integrate mathematical education and research. Although NSF subsequently discontinued the formal VIGRE program, undergraduate research played an important role in these projects. Undergraduate research sites also became an important component of a number of programs to enhance the participation and retention of women and under-represented minorities in mathematics.

Today, there can be no question that undergraduate research is firmly established. Many colleges and universities have found ways to institutionalize programs, either with gifts or by convincing university administrators to all...

Table of contents

• Cover page
• Title Page
• Copyright Page
• Contents
• Introduction
• Chapter 1: Undergraduate Research and the Mathematics Profession
• Chapter 2: FURST — A Symbiotic Approach to Research at Primarily Undergraduate Institutions
• Chapter 3: A Laboratory Course in Mathematics
• Chapter 4: REUs with Limited Faculty Involvement, “Underrepresented” Subjects in the Undergraduate Curriculum, and the Culture of Mathematics
• Chapter 5: The Berkeley Summer Research Program for Undergraduates: One Model for an Undergraduate Summer Research Program at a Doctorate-Granting University
• Chapter 6: Fifteen Years of the REU and DRP at the University of Chicago
• Chapter 7: Why REUs Matter
• Chapter 8: Integrating Mathematics Majors into the Scientific Life of the Country
• Chapter 9: The Gemstone Honors Program: Maximizing Learning Through Team-based Interdisciplinary Research
• Chapter 10: The Freshman Research Initiative as a Model for Addressing Shortages and Disparities in STEM Engagement
• Chapter 11: Determining the Impact of REU Sites in the Mathematical Sciences
• Chapter 12: The Mathematics REU, New Directions: A Conference at Mount Holyoke College, June, 2013