Preface to the First Edition

In 1989, after much discussion, we added a new sophomore-level “bridge” course to our curriculum. Just as it is difficult to learn to write unless you have something you want to say, so too, we reasoned, it is difficult to learn to construct a proof unless you have something you want to prove. We wanted to share with our students the joy, and the frustration, of mathematical discovery. We wanted to immerse them in exploration of mathematical phenomena, to have them discover things for themselves, to have them make conjectures and construct arguments in support of those conjectures. We wanted them to discover things that they understood might well be false and which, therefore, needed proof. We wanted them to encounter a broad range of mathematical phenomena early in their mathematical careers and to take with them a solid base of experience on which to build future understanding.

Our course, which we called the Laboratory in Mathematical Experimentation and which students called “the Lab”, succeeded beyond any of our expectations. After just one year, we found students much more likely to read mathematics actively, more likely to dive in and “mess around” with a hard problem, more likely to ask questions and look for patterns, and more likely to formulate an argument clearly. Students who have taken the Lab do better in the real analysis and abstract algebra courses required for the mathematics major than those who have not. Moreover, both students and faculty enjoy the course. We find it easy to teach: reading papers and grading is onerous, but preparation is trivial and office hour time is modest. Students enjoy setting their own agendas and inevitably become caught up in their investigations. The atmosphere of collegiality and shared inquiry sharpens students’ interest in mathematics and helps them think of themselves as mathematicians. We now require the course of every mathematics major.

This book grows out of the Lab course and consists of a collection of sixteen laboratory investigations in mathematics accessible to beginning college students. Each investigation invites students to observe and to look for patterns and encourages them to establish language to describe, conjecture, and analyze the phenomena under study. Each investigation leads to mathematics which students will encounter in later courses and seeks to supply the student with a repertoire of concrete examples to nourish their intuition. Most will also result in the student discovering some things she believes to be true and wants to prove, but cannot. In a typical offering of the Lab, students do six or seven of the investigations.

This book could be used to offer a course like ours. It could also be used to supplement other courses or as a source for students’ independent projects. The sixteen investigations are almost all independent of each other, and most do not require calculus (the exceptions to both statements are noted in the Introduction). All but two of the investigations require a computer (or programmable calculator). The accompanying instructor’s manual says more about each of the investigations. It also gives much more detail about the Lab course.

This work is truly a collective effort. Every mathematician and statistician in our department has had a hand in shaping it and the course from which it grew. When it comes to the Lab, we also consider J. William Bruce of the University of Liverpool to be an honorary department member, since while on sabbatical at Mount Holyoke he taught the Lab, made important contributions to every chapter he used, and contributed several additional projects.

It is a pleasure to acknowledge the support of the National Science Foundation, first for laboratory computers, then for curriculum development and writing, and finally for dissemination through their Undergraduate Faculty Enhancement program. We are also grateful for NSF’s insistence that we form an Advisory Board for this project. Advice, suggestions and encouragement from Thomas Cecil (College of the Holy Cross), Gregory Fredricks (Lewis and Clark College), Gregory Hill (University of North Texas), and Jacob Sturm (Rutgers University, Newark) have been invaluable, and we thank them. We also gratefully acknowledge support from the Sloan Foundation and Hewlett Packard.

We also thank several other colleagues. Mizan Kahn, now at Eastern Connecticut University, taught and commented on an early version of the Lab. More recently Patrick Fitzpatrick, a sabbatical visitor from University College Cork, taught the Lab and contributed many ideas, including the “warm-up” exercise on diagonals of rectangles. Our students have also helped us in many ways, especially the junior and senior majors in the fall of 1988 who helped design and test the projects used in the very first offering of the Lab: Tessa Campbell, Julie Derynda, Barbara Hswe, Kristine Kusek, Kathleen Malone, and Ke Wu. We received many valuable suggestions from the participants in our NSFUFE workshop in June 1996, as we were preparing the final version of this text. We thank Mysore Jagadish and Pedro Suarez (Barry University), Olusola Akinyele (Bowie State University), Terrence Bisson and Donald Girod (Canisius College), Barbara Reynolds (Cardinal Stritch College), Alan Levine and Ben Shanfelder (student) (Franklin and Marshall College), John Kellett (Gettysburg College), Lynnell Matthews (Howard Community College), Kathy Kraft and Robert Woodle (Jamestown College), Douglas Burkholder and Mary Flagg (McPherson College), Arup Mukherjee and Ethel Wheland (Pennsylvania State University), Steve Cohen and John Currano (Roosevelt University), Donald Miller (Saint Mary’s College), Patricia Army and Barbara Becker (Saint Xavier University), Craig Bailey and Charles Hanna (U.S. Naval Academy), Bruce Lundberg (University of Southern Colorado), Jiu Ding (University of Southern Mississippi), Michael Evans (Washington and Lee University), Karl David and Arnold Shilepsky (Wells College), and Mark Janeba and Frank Zizza (Willamette University). Finally we thank our colleagues in other departments at Mount Holyoke and our spouses for their support and encouragement.

The mathematicians and statisticians at Mount Holyoke College
South Hadley, 1996