Thomas F. Banchoff is a geometer and has been a professor at Brown University since 1967.
His research areas include the geometry and topology of smooth and polyhedral surfaces in three- and four-dimensional space, as well as development and dissemination of Internet-based courseware for communication and visualization in undergraduate mathematics. He has a long-standing project on the history of the fourth dimension, including a biography of Edwin Abbott Abbott , "The Man Who Wrote 'Flatland'".
The following list contains a description of my current pedagogical research from my successful proposal for the NSF Director's Award for Distinguished Teaching Scholar, followed by a summary of my research contributions in the past 38 years, and a list of projects that tie together my research areas and expository writing and undergraduate teaching.
Summary of the Project for the DTS Director's Award for Distinguished Teaching Scholar
Interactive Internet-Based Mathematics Teaching and Learning is the latest version of an ongoing project that is now ready for active and broad dissemination. The term "interactive" plays multiple roles in this project: 1) Teachers and students interact with geometric phenomena by means of powerful, flexible, locally-developed, accessible and easy-to-use Java applets, both for classroom presentations and for student explorations; 2) In a paperless setting, students do their homework and examinations online using html shortcuts requiring almost no learning time, and they receive timely comments from instructors, often leading to further interchanges; and 3) After appropriate delays, students can read the responses of their classmates, as well as instructor comments, and engage directly in discussion or group work over the web.
The communication software that makes this interaction possible, as well as the demonstration software for illustrations and animations, has all been developed by teams of students, almost all undergraduates, in a series of NSF sponsored projects in the Division of Undergraduate Education and the Research on Learning and Education program of the Education and Human Resources division. Six years ago the software for this approach required sessions in a workstation laboratory using proprietary communications programs requiring sophisticated word processing. As a result of the most recent project, the communications software has been rewritten in a much more stable and powerful form, and in a form that can be transferred easily to other teaching situations. A small number of html shortcuts make it possible for students to begin entering text and simple mathematical expressions almost immediately. Tutorials written by student assistants make it easy for students without sophisticated computer background to upload scanned drawings and to include images and links in their responses. Recent improvements enable students to include interactive illustrations from Geometer's Sketchpad.
The demonstration software includes several features not available on many other general purpose proprietary packages. Teachers and students can easily modify model programs to change functions and parameters, and to include other functions, either in the same or in a different window, responding to the same parameter changes and rotations in three-space in linked windows. Once a user finds a particularly appropriate example, it is possible to save the entire state and enter it as a button in a homework or exam response, so the viewer can enter the program in precisely the configuration that was chosen and proceed to investigate the phenomena further. Students who become very familiar with the software immediately become potential resources and collaborators with the teacher, producing increasingly elaborate and sophisticated demonstrations for use in classrooms and in presentations to researchers and general audiences.
Course Materials Development generally proceeds in three steps over a three-year period, with the first summer devoted to drafting a set of laboratory demonstrations and accompanying text for use in the next academic year in the PI's classes, with a detailed evaluation of results. The second summer refines those materials and features them in a workshop at the MathFest or similar conference to prepare teachers who will use them in different ways at their institutions. The third summer uses the evaluations of the various experiences to create an electronic document that can be used by teachers without workshop preparation. A summary evaluation assesses the effectiveness of the materials in enhancing the quality of teaching and learning.
Schedule of Course Development
Differential Geometry is in the second stage as of the summer of 2004. Multivariable Calculus will start its first stage in 2004, and Linear Algebra will begin in 2005. In Summer 2006, the topic will be a combination of Euclidean and non-Euclidean Geometry and some Combinatorial Topology. In each of these courses, the PI has experience using computer-generated materials and earlier versions of the communication software. Student assistants are chosen from among those who show special talent for mathematics and computer science in course using this approach.
Workshops and Conferences
The workshops for familiarizing potential users of the software and electronic materials will be held in conjunction with the summer MathFests of the MAA, with additional meetings at the January Joint Mathematics Meetings. In 2004, the PI will make a presentation to the NExT (New Experiences in Teaching) fellows at the Providence MathFest, and announce dissemination plans in the special session on web use designed by WebSIGMAA, the MAA Special Interest Group on that topic. We will mount a library exhibit at Brown University on "The Man Who Wrote 'Flatland'", and invite a discussion on the use of the Abbott/"Flatland" database in liberal arts mathematics courses. At the 2005 Joint Meetings in Atlanta, there will be an informational session for representatives of schools in the area to discuss a possible semester-long project in the spring of 2006, with teachers from various institutions carrying out projects, starting with a conference and meeting monthly to compare experiences.
The DTS award will act as seed money for a larger proposal, possibly in the Adaptation and Implementation program of the NSF. The PI would request a one-semester leave of absence and seek a visiting position at the University of Georgia and/or Spelman College. Other participating schools might include Georgia Tech, Emory, Morehouse, Agnes Scott, Georgia State, and the Paideia School. Possible locations for subsequent experiments of the same sort are the San Francisco Bay Area, or the Cleveland-Pittsburgh area. Similar workshops will be held in MathFests and Joint Meetings, to parallel the development of course materials described above.
Liberal Arts Mathematics and Exposition of Mathematics will concentrate on developing electronic versions of several texts and other materials already in use by teachers in different parts of the country. The PI's Scientific American Library volume "Beyond the Third Dimension" has sold over 30,000 copies in paper format, and it has been used as a text in liberal arts math classes at a number of institutions, including Slippery Rock University, Stritch University, Union College, Dartmouth College, and Alma College. The electronic version to be developed in 2004 and 2005 will include a full set of interactive illustrations, as well as pedagogical materials for each of the nine chapters, including exercises, projects, and research topics. A parallel project is an electronic version of "The Fourth Dimension Simply Explained," written by Brown professor Henry Parker Manning in 1911 and due to be reprinted by Dover Publications in 2005 with a new introduction by the PI. A third project, to be completed in 2006, is a reworking and updating of the interactive poster "Math Spans All Dimensions" by the PI and Davide Cervone, which first appeared for Math Awareness Month 2000 when the PI was chair of the Joint Policy Board for Mathematics and president of the MAA.
Adaptability: The electronic materials and communication and visualization software can be used in many different ways in different kinds of courses. At the minimal level, teachers can suggest the use of electronic materials as optional supplements, and at the other end of the spectrum, they can use them as a text following a detailed syllabus. Minimally, teachers can rely on "canned" standardized demos on videotape or CD. At the next level, they can modify demos to suit particular purposes, and even design new demonstrations from the beginning, perhaps with the help of a student assistant recruited from an earlier class. At a higher level, students can interact with demonstrations on their own, modifying parameters and entering new objects. A special feature of the visualization software developed by the PI and assistants is the ability to "save a state," inserting a link into a homework or examination so that the instructor, and everyone else, can not only see the illustration, but also enter into the program where the first person left off and continue to investigate the phenomena at hand.
Standard software can allow students to participate in discussion groups and communicate by email with professors, and this minimal interaction can be enhanced by a file server in a mathematics department. Slightly more sophisticated servers can make it possible for students to hand in work online and receive commentary. It's also possible to use the software developed by the PI and his students to run a "paperless class," with all work handed in online, including examinations, and where, after an appropriate time delay, class members can see the work of all other students and the comments. In a workshop, the entire range of possibilities will be presented, and teachers can choose to try out some features or others in their classes.
Scalability: The full operation of interactive Internet-based teaching and learning requires the chance for the instructor to comment on individual student responses in a timely manner, and the opportunity for students to view responses from classmates. For a large class, this can become unwieldy, unless there is a way to break the group into smaller working sections, each monitored by a single instructor or teaching assistant. For classes using only the text and demonstration software with little student interaction, scalability is not such a significant factor. Workshops will explore ways of using the electronic materials and courseware in ways that are suitable for large or multi-section courses, a research challenge in itself.
Assessment and Evaluation
In the PI's current NSF-sponsored project in the ROLE program, there is special emphasis on developing effective means of assessing the success of the project in enhancing teaching and learning. This becomes even more important as we enter the dissemination phase of the project, as potential users want to know results of prior experience when they consider using a version of this approach suited to their needs and the needs of their students and institutions. We have data for more than a dozen courses that have used this approach to a substantial extent, mostly at Brown, but also in courses the PI has taught at Yale University, University of Notre Dame, and UCLA. To make better use of these data, and to obtain more sophisticated information, we have recently redesigned our pre- and post-questionnaires with the help of a professional market researcher who is a specialist in the creation of questionnaires and in the analysis of qualitative data. The information we have already received is helping in the reconfiguration of our approach, keeping in mind the different circumstances of our possible collaborators. The questionnaire used this past year (Spring 2003 and Fall 2003) has nine questions, each with several parts, covering 1) online assignments, 2) solution keys and hints, 3) online student interaction, 4) online communication with instructors, 5) timing of assignments, 6) examinations, 7) group work, 8) textbook, and 9) computer demonstrations. The results are displayed in spreadsheets showing the responses of students (columns) to given sub-questions (rows). For any question of particular interest, a content analysis is conducted by coding qualitative responses into categories suggested by the responses. A summary table on each question presents a count as opposed to percentages (for samples of less than 50 respondents), supplemented by verbatim comments that illustrate or emphasize specific points. This method raises our data collection and analyses to a professional level, representing a substantial improvement over earlier assessment efforts and allowing us to use quotations and anecdotes in context.
Sample Content Analysis: An illustration of the value of this assessment method is the analysis of responses to an important question about our approach, namely the "comfort" level students report, given the centrality of the element of viewing the work of their classmates, and instructor comments. The sample here is taken for Math 35, the honors multivariable calculus course for entering freshmen with the equivalent of a 5 in the BC Advanced Placement Calculus Exam. Completed course evaluations were recorded from 27 out of 37 students (73%). Learning from questionnaires collected in last spring's courses, we split the question about comfort level into two parts, one for homework and the other for examinations. We used content analysis us to look for patterns in students' responses and the results have been used to generate additional ways to improve the interactive process for students.
Several students reported changes in their attitudes over time. Most interesting is the student who said: "In the beginning I really did not like the system of reading other student homeworks. It made me feel really nervous and exposed and it made me want to leave problems blank rather than to put in an incorrect answer. As the semester progressed I realized what a useful tool it could be and I started reading other people's hw responses more and more and felt more comfortable with mine being read."
Another student addressed motivation: "It didn't really bother me that other people could look at my work, either at the beginning or the end of the semester. Sometimes I felt bad about the quality of the work that I handed in and I might have preferred that others not look at it, but it didn't concern me enough to make me want to change the system and the motivation to do a better job probably didn't hurt either."
As the numbers indicate, some students felt differently about exams: "I think I am more self-conscious about my exams. At first I didn't mind so much, but now I am beginning to think that there should be some element of privacy with exams, or a way to choose to have your exam available or not". A counter-intuitive response was: "I have always been self-conscious about having work of mine open to criticism, so I was slightly uncomfortable (about the homework being available)" but "On exams, I was able to put more time into refining my answer, so I didn't mind having people see that."
As a result of this analysis, we are modifying the tensor to allow individual students a choice: to make their exams and the instructors' comments unavailable for viewing by their classmates. Responses to the analysis of other questions have also led to modifications and refinements; we expect this to continue during the dissemination phase of our project.
Summary of Research and Associated Projects
Synopsis of Research Contributions
In terms of general areas, my research career deals with several major topics. In each case there is a set of papers in a particular decade, followed up by generalizations and/or extended expository treatments at later times. A unifying factor has been my career-long involvement with interactive computer graphics for investigating phenomena in differential and polyhedral geometry and for presenting illustrations, examples, and results in my undergraduate courses and in colloquia, conferences, and talks to broad audiences. In the following list, I collect papers by category, roughly in chronological order within each. There are slightly more than two items per year in the bibliograph, so the date of the item labeled n appeared roughly in the year 1965 + n/2.
1. Tight and Taut Mappings: My thesis , written under the direction of Prof. Shiing-Shen Chern, introduced the term "tight" into the subject of minimal total absolute curvature, and produced the first examples of tight embeddings of polyhedral surfaces in dimensions six and higher. Also from my thesis are the first examples in  of globally non-rigid tight polyhedral tori. The two-piece property was first developed in  and the spherical version in  was the first paper in what has become the study of "taut" mappings. Higher-codimension examples of tight polyhedral embeddings of spheres first appeared in . The first examples of tight polyhedral Klein bottles were constructed in , which also proved uniqueness for the six-vertex embedding of the real projective plane into 6-space, the counterpart of the uniqueness theorem for smooth surfaces of my mentor Nicolaas Kuiper and my colleague William Pohl. Tightness for surfaces with boundary was the thesis subject of Lucio Rodriguez, my first Ph.D. student, and intermediate tautness was first introduced in the thesis of another of my doctoral students, Eugene Curtin. A third Ph.D.student, Leslie Coghlan, classified tight general position mappings of the real projective plane. Wolfgang Kuehnel and I published two extensive investigations of his remarkable nine-vertex tight triangulation of the complex projective plane in  and . The story of the first decade in the theories of tight and taut immersions appears in . For a treatment of the entire subject over the past 30 years, see the survey article , written with Wolfgang Kuehnel, in the MSRI volume on "Tight and Taut Submanifolds" edited by Chern and Thomas Cecil, my former student and long-time colleague in the Clavius Research Group. That volume was dedicated to the memory of Kuiper and it included a posthumous article which Cecil and I helped to prepare for publication ,.
2. Critical Points and Curvature for Polyhedral Manifolds: My post-doctoral research program as a Benjamin Peirce Instructor at Harvard and a research associate with Kuiper at the Universiteit van Amsterdam introduced combinatorial methods in global differential geometry in  and , reprinted in the collection . The general theory of curvature for mappings into subspaces of higher dimension appeared two decades later in my article . In my paper  with Floris Takens, I presented the first results on critical point theory for polyhedral surfaces with no natural Gauss mapping, and with behavior different from a smooth embedding or immersion.
3. Geometry of Characteristic Classes: In 1974, right after my first sabbatical,  and  presented the first published proofs of the basic result that the number of triple points of a generic immersion of a surface into three-space is congruent modulo 2 to the Euler characteristic. This result has been the starting point for many investigations by other mathematicians, and it led to my work on the Stiefel-Whitney classes for polyhedral manifolds in , the Whitney duality theorem  with Clint McCrory, and combinatorial formulas for normal Euler classes of polyhedra , also with McCrory. The Whitney Duality Theorem for polyhedral manifolds was treated in the thesis of my Ph.D. student Ockle Johnson. Three years ago, he and I together published a major paper on normal Euler classes for polyhedral surfaces in four-space . A related set of theorems is contained in my article with Frank Farris on Euler numbers and complex points for smooth surfaces in four-space . A generalization of the triple point theorem for stable mappings with singularities is a critical component of my work  with McCrory and Terence Gaffney on tritangent planes of Space Curves.
4. Global Geometry of Curves and Polygons: This category includes articles in an ongoing research program. Total Central Curvature of Curves  was first presented to the student group at the University of Amsterdam. For the joint paper  with William Pohl, I contributed an explicit elementary argument in the case of curves in the plane. Polyhedral catastrophe theory  is the first part of a paper that has yet to be given, most appropriately as an interactive electronic document. The Fabricius-Bjerre theorem in the polygonal case in the plane  leads to generalizations to double tangency theorems for pairs of surfaces in 4-space . Self-Linking for Polygons  includes global results for torsion of smooth space curves as well, carried over in  to theorems of Milnor and Jacobi. My article  with Edwin Beckenbach's was his last joint paper, treating phenomena for curves related to optics. Three papers with Peter Giblin , ,  initiated a study of finite models for symmetry sets, in particular piecewise circular curves (with another paper in process). My joint paper  with James White established properties of space curves under conformal transformations and introduced the notion of an osculating tube. These ideas appear again in the solution of a problem posed by Kuiper in my most recent research publication .
5. Geometry of Smooth and Polyhedral Surfaces: Two early papers investigated submanifolds of the bicylinder boundary, one on minimal surfaces  and the other on foliations of knot-complements . My monograph  with Clint McCrory and Terence Gaffney examined "Cusps of Gauss Mappings" in a way that led recently to an electronic version, illustrated with full-color animations. This monograph included an expanded version of the paper  with Rene Thom, correcting and expanding an example he had considered. My only joint paper with Kuiper  was a study of geometrical class and degree involving the construction of key examples in the analytic, smooth, and polyhedral categories.
6. Computer Graphics and Differential Geometry: (This topic necessarily overlaps heavily with the descriptions of my contributions to teaching.) My research on computer graphics and the geometry of submanifolds began in 1968, shortly after my arrival at Brown, in collaboration with Charles Strauss, using the computer graphics technology developed in his Ph.D. thesis with Andries van Dam. Various aspects of this collaboration are described in , , and , culminating in my 45-minute invited address  at the International Congress of Mathematicians in Helsinki in 1978. Starting in 1980, collaboration continued in cooperation with undergraduate students in computer science , , , , , , , , , , and . A major collaboration with two colleagues in Applied Mathematics and a CS graduate student produced a significant investigation of linear Hamiltonian systems . A related project used graphics to analyze the geometry of the Hopf mapping from the three-sphere to the two-sphere, to analyze regular polytopes in 4-space , and to modify an argument of Ulrich Pinkall to give an elementary construction of tori in 3-space representing any conformal type . Beginning in 1990, I have collaborated with my former graduate student Davide Cervone on a series of projects , , , , , and , described more fully in several parts of this proposal.
7. Miscellaneous Topics: My joint paper with Lou Kaufmann on "Immersions and mod 2 Quadratic Forms"  was awarded a Lester Ford prize from the MAA in 1978. The central argument of my article with Michael Rosen, "Periodic Points of Anosov Diffeomorphisms" , has been included as appendix in the Dynamical Systems text of Zbigniew Nitecki. The results of a two-year project on data visualization supported by the Office of Naval Research are contained in .
Synopsis of Teaching Contributions
From the beginning of my teaching career, I have experienced a great deal of satisfaction from the responses of my students to my undergraduate courses. In several articles over the years I have described my teaching methods and the courses I have designed for non-majors as well as majors, The first was an eight-lecture series in a summer institute on Applications of Elementary Calculus  in 1971, later reprinted by the MAA, and the most recent was my presentation in response to the Deborah and Franklin Tepper Haimo national award for distinguished college and university teaching . Most of my writings about contributions to the scholarship of teaching and learning are concerned with the use of computer graphics in teaching courses at all levels. In 1974 and 1978 I published papers ,  with my computer science colleague Charles Strauss about the use of computer graphics in research and teaching, and in 1978, I was asked to present our work in a invited 45-minute session at the International Congress of Mathematicians in Helsinki  in the section on history and pedagogy, the first time that computer graphics in mathematics teaching and research was presented before an international audience. Beginning in 1980, nearly all of my collaborations in computer graphics were with a series of talented undergraduate assistants, most of whom have gone on to distinguished careers in industry and in teaching. A major paper  in 1981 with David Salesin, a senior, and Steve Feiner, a graduate student, presented a new animation language, with applications to surfaces in four-space. There followed a series of papers co-authored with student assistants on student-generated software for workstation laboratories in differential geometry and in multivariable calculus ,, , , , and . I also co-authored a linear algebra text with John Wermer that included topics from the grometry of four-space , now in its second edition . In recent years, since the advent of interactive graphics on the Internet, most of my articles on pedagogy and expository mathematics have been collaborations with Davide Cervone, starting with my sabbatical stay at the Geometry Center at the University of Minnesota in 1974. Examples include the totally electronic journal "Communications in Visual Mathematics", two virtual art gallery projects , ,, and "Math Awareness Month April 2000"[to appear]. Cervone is also primarily responsible for the illustrations in my Scientific American Library volume "Beyond the Third Dimension", the text for my liberal arts mathematics course , now in its second edition , , .
One set of contributions that do not show up on the publication list are the films and videotapes that we have produced over the years. "The Hypercube: Projections and Slicing" with Charles Strauss (1978) is a classic, still used widely in secondary schools and colleges throughout the country. The other black and white films from the Helsinki Congress talk (repeated at a second showing by demand and shown twice more at the ICM in Warsaw in 1985) have all been remade in one form or another. Recent films shown widely at the major graphics conferences include "The Hypersphere: Foliation and Projections" with David Laidlaw, Huseyin Kocak, and David Margolis  and "Fronts & Centers" in collaboration with student assistant Nick Thompson and Composer Gerald Shapiro.
For the past five years, we have made most of our contributions to classroom teaching in the form of Internet-based courses in multivariable calculus, linear algebra, differential geometry, and the liberal education course "The Mathematical Way of Thinking: The Fourth Dimension". Starting with programs developed by Dan Margalit, now a graduate student in mathematics at the University of Chicago, we have continued to develop software for communication between students and instructors and among students. Evidence of the peer regard for this project is the fact that we have just received word from the NSF program officer in the ROLE program (Research on Learning and Education) that our proposal "received high praise from reviewers". I am in contact with the cognizant program officer negotiating the final budget arrangements. (I am aware that the final decision is made by the agency and I expect to hear final word soon.) Here is the short description for that proposal:
Project Summary for "Interactive Internet-Based Mathematics Courses with Geometric Visualization Software": Over the past five years, the Principal Investigator and student assistants have developed a series of Internet-based courses in a related set of undergraduate mathematics courses utilizing locally developed software for geometric visualization and for communication between students and instructors and among students. These new models for teaching and learning change the timing and the level of interactivity for instructors and students, and increase the ability of students to handle complex geometric phenomena. These innovations will be further developed, tested, and evaluated during the period of the proposal. The aim is to produce effective software that can be used in a wide variety of institutions with students at different levels, for mathematics and for its applications to physical science, computer science,and engineering.
Proposed Projects for the DTS Award
In the Synopsis of Research Contributions, there are seven different topics representing areas where I have written research articles. Associated with each of these areas, I can identify challenging problems that I have brought up in my undergraduate classes over the past 35 years, modifying the presentation depending on the level of the class. These problems exemplify the kinds interaction between research in mathematics and teaching of undergraduates that has become characteristic of my writings on education and my dozens of presentations each year. I propose to produce hypertexted interactive articles on at least eight of them over the four years of the award. These are thought of as separate projects at the present time, although it will be quite good if these presentations can be accumulated into a single volume at the end of this period. As is usual and appropriate, all of these projects will involve collaboration with undergraduate students as well as with other colleagues who share a commitment to the interaction of research and undergraduate teaching. I will describe eight of these projects below to illustrate the range of problems and the possibilities for presenting them in hypertext form with substantial Internet-based interaction. As elaborated below in the budget justification, the reason for listing eight is that in our experience, it takes a half year to bring any of these projects to completion!
1. Tight and Taut Mappings and the Grid Square Two-Piece Property: I don't know very many mathematicians who can describe their thesis problems to a general audience, but I can. After a few months of studying the global geometry of smooth surfaces satisfying a condition called "minimal total absolute curvature, I discovered that this condition is equivalent to a very elementary notion, the Two-Piece Property. An object in space has the TPP if every cut by a long straight knife separates it into at most two pieces. An egg or an orange has the TPP but a curved banana or a fork does not. A half cantaloupe has the TPP but a quarter cantaloupe does not. A doughnut or bagel has the TPP, and so does a half ba
Results from Prior NSF Support:
The most recent NSF Grant was:
NSF Grants: #REC-9814593 (9/98 - 8/01, $111,760) and # REC-0107035 (8/01 - 8/04, $553,678). "Interactive Internet-Based Mathematics Courses with Geometric Visualization Software" is scheduled to end in August 2004, and the funding for the PI and other senior personnel ended at the end of the summer of 2003. During the three years of this grant, the PI and his collaborators have made substantial contributions in all of the areas listed in the proposal, to the extent that we now have stable and robust versions of our communication and visualization software so that it is ready for a major dissemination effort.
There have been grants from the NSF stretching back to 1962, and a four-year period of funding from the ONR in the early 1980s.