Every course should strive to:
- Present key ideas and concepts from a variety of perspectives;
- Employ a broad range of examples and applications to illustrate and motivate the material;
- Promote awareness of connections to other subjects (both in and out of the mathematical sciences), and strengthen each student ability to apply the course material to these subjects;
- Introduce contemporary topics from the mathematical sciences and their applications, and enhance student perceptions of the vitality and importance of mathematics in the modern world.
Key Ideas and Concepts from Varied Perspectives
Project Intermath is an interdisciplinary mathematics project that is creating curricula at the United StatesMilitaryAcademy, CarrollCollegeGeorgiaCollege & StateUniversity, HarveyMuddCollege, MacalesterCollege, University of Redlands, and the Texas Southern Consortium. By working with professors from science, engineering, mathematics and computer science departments, the project aims to foster the creation of interdisciplinary courses that demonstrate the interdependence of mathematics and science. For example, at the United StatesMilitaryAcademy , first semester students study the concept of change from both a discrete and a continuous point of view. At the end of the semester students must model and solve particular problems by using a discrete dynamical system and by using a differential equation. Students then compare and discuss the appropriateness and the results of the two approaches. Within the four-course core program, students at the United States Military Academy also examine mathematical topics from the perspectives of linear versus nonlinear and stochastic versus deterministic. At Carroll College, a 4-class core consisting of a total of 18 credit hours covers many of the topics seen in the first two years of a traditional curriculum, including differential and integral calculus, multivariable calculus, differential equations, and linear algebra. The core also includes topics not usually seen early, if at all: discrete dynamical systems, partial differential equations, probability, and statistics. Concepts are threaded together in and between classes to help students develop a deeper understanding of how different branches of mathematics are intertwined. The website contains complete texts for over 40 modeling problems developed at the United StatesMilitaryAcademy site.
Although a number of the textbooks produced during the calculus reform movement are no longer in print, both mainstream and reform texts now consider the concepts of calculus from a variety of perspectives: not only the symbolic, but also the graphical, numerical, and verbal. The Calculus Consortium at Harvard Newsletters discuss issues involved in teaching calculus. Many calculus texts now come with software to enhance student understanding from a variety of perspectives.
A good source of ideas on how to teach linear algebra from various perspectives is Resources for Teaching Linear Algebra, edited by David Carlson, Charles R. Johnson, David Lay, Duane Porter, Ann Watkins, and William Watkins, MAA Notes vol. 42.
The concept of function can be regarded from many different perspectives and is important in all undergraduate mathematics courses. The editors of The Concept of Function: Aspects of Epistemology and Pedagogy (Harel & Dubinsky, 1992) contributed to the body of research on learning the function concept in order to assist in instructional approaches. Key Aspects of Knowing and Learning the Concept of Function by Marilyn Carlson and Michael Oehrtman is a recent online article that provides a broad view of the subject.
Victor Donnay, BrynMawrCollege, developed a PowerPoint presentation describing how computer visualization can be used to give an intuitive understanding of complex ideas in modern mathematics.
Promote Awareness of Connections between Mathematics and Other Subjects
Dan Maki (Indiana University Bloomington) and Bart Ng (Indiana University-Purdue University Indianapolis) co-direct the NSF-funded project Mathematics Throughout the Curriculum. The website includes links to a prototype course Analytical Problem Solving and a set of Home Pages for Developing Courses, which contain additional information about courses that relate mathematics to the life sciences, business and economics, the humanities and social sciences, and the physical sciences and engineering. A newsletter provides additional information about the project.
The MAA’s Journal of Online Mathematics and its Applications (JOMA) contains peer-reviewed articles, class-tested, web-based learning materials, and self-contained, dynamic, single-purpose learning tools. Many of these illustrate a range of examples and applications and connections between mathematics and other subjects. Some recent articles are Special Relativity and Conic Sections, Designing Attribute Acceptance Sampling Plans, and Art and Design in Mathematics.
DukeUniversity’s Connected Curriculum Project collects and develops interactive learning materials for mathematics and its applications, with applications to biology, chemistry, economics, engineering, environmental sciences, epidemiology, and physics. Each application is keyed to the level of mathematics used.
MAA Online’s Innovative Teaching Exchange contains several articles illustrating applications of mathematics in a range of courses. ’Stuck in Traffic in Chicago: A World Wide Web Projectâ? by M. Carroll and Elyn Rykken, has students apply Riemann sums from calculus to construct time estimates for radio traffic reports. In ’Volumes and History: A Calculus Project Involving Reading an Original Source,â? Elyn Rykken and Jody Sorensen use historical applications to enliven a calculus course. In ’The Trial of the Semester: A New Way to Introduce Newton's Law of Cooling,â? Joel Foisy describes how he has his students act out a murder mystery scenario.
The MAA’s Digital Classroom Resources provides a select collection of learning materials that are available without charge through the site. These materials have been classroom tested and peer reviewed. Many items in the library include editorial reviews and links to a moderated discussion group focused on the materials.
The MAA Bookstore has a number of books that give a range of mathematical applications, including Applications of Calculus, edited by Philip Straffin (MAA Notes # 29), Cryptology, by Albrecht Beutelspacher, Elementary Cryptanalysis by Abraham Sinkov, Environmental Math in the Classroom, edited by Bernard Fusaro and Patricia Kenschaft, Geometry at Work, edited by Catherine Gorini, Linear Algebra Gems, edited by David Carlson, Charles R. Johnson, David C. Lay, A. Duane Porter, and Problems for Student Investigation, edited by Michael Jackson and John Ramsay.
The entry for the Consortium for Mathematics and Its Applications (COMAP) in the bibliography contains additional information about incorporating real-world applications into mathematics courses.
The MAA’s online journal Convergence is a new online magazine that provides resources to help teach mathematics using its history. Reinhard Laubenbacher, David Pengelley, Jerry Lodder, and others at New MexicoStateUniversity have developed a large collection of instructional materials to teach mathematics using original historical sources. Other books that link mathematical topics with their history include William Dunham’s Journey Through Genius: The Great Theorems of Mathematics and The Calculus Gallery, Simon Singh’s Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem and The Code Book: The Science of Secrecy from Ancient Egypt to Quantum Cryptography, Rudy Rucker’s Infinity and the Mind: The Science and Philosophy of the Infinite, and Marcia Ascher's Ethnomathematics. A Multicultural View of Mathematical Ideas. Judith Grabiner of PitzerCollege developed two general education courses with an emphasis on history: Mathematics, Philosophy, and the ’Real World,’ and Mathematics in Many Cultures.
In a note to Project NExT participants, Doris Schattschneider, Moravian College, gave the following list of websites for courses linking mathematics, art, and design:
* Survey course on mathematics in art and architecture by Paul Calter at Dartmouth College
* Course on Mathematics in Art by Helmer Askalan
* Course in Mathematics and Art by Marc Franz
* Jill Britton's website on Symmetry and Tessellations, with annotated links to many other sites on these and related topics
* Totally Tessellated (can be accessed from Britton’s site)
* A mini-site on Escher's work and related math
* An exhibit by artists whose work has been influenced by M.C. Escher featured at the Escher Centennial Congress in Rome in 1998
* A website about harmony and proportion by John Boyd-Brent, M.A, Royal College of Art
* This website, related to a ’technical’ paper by D. Schattschneider and N. Dolbilin, has Java applets that allow users to manipulate flexible tilings.
Some additional websites with information about courses on mathematics and art: Pattern (Pippa Drew and Dorothy Wallace, Dartmouth College), Math and Art (Janice Sklensky, Wheaton College), Mathematics, Logic, and Symmetry and Modules for Spherical and Hyperbolic Geometry (Reza Sarhangi, formerly at Southwestern College, now at Towson University), Mathematics, Art, and Aesthetics (Annelisa Crannell, Franklin and Marshall College), and Math and the Art of M.C. Escher (Anneke Bart, St. Louis University). The Bridges Organization posts information about mathematical connections in art, music, and science, and the AMS hosts a website on Mathematical Imagery. A collection of modules developed by Marc Franz and Annelisa Crannell is available through a link on the Viewpoints website. A new version of Annelisa Crannell’s course on mathematics and art is available as of late summer 2007 through a link on her homepage.
Three websites with syllabi for courses on mathematics and music: Math and Music (Julia M. Wilson, SUNY Fredonia), Mathematics and Music (David Wright, Washington University), Mathematics and Music: The Cosmic Harmony (Bill Alves and Michael Orrison, Harvey Mudd College).
Most textbooks for general education mathematics courses include sections that connect mathematics with other fields. For instance, The Heart of Mathematics: An invitation to effective thinking by Edward B. Burger and Michael Starbird discusses the mathematics of bar codes, cryptography, geometry and art, fractals and chaos, and the likelihood of coincidences. A syllabus for an interesting course that makes some use of this text is from Sarah Greenwald, Appalachian State University. Another text, Using and Understanding Mathematics by Jeffrey O. Bennett and William L. Briggs, University of Colorado at Boulder, contains sections on financial management, modeling a variety of real-world situations, mathematics and art, mathematics and music, mathematics and politics, and mathematics and business. See also the listings in this section under ’Introduce Contemporary Topics.â?
Introduce Contemporary Topics
Robert Devaney, BostonUniversity, is a leader in promoting instruction in the contemporary topic of dynamical systems. In addition to his books and articles, talks, and professional development institutes, he has been director of the National Science Foundation's Dynamical Systems and Technology Project since 1989. The goal of this project is to show students and teachers how ideas from modern mathematics such as chaos, fractals, and dynamics, together with modern technology, can be used effectively in the high school and college curriculum.
The University of Maryland University College offers Mathematics ’ Contemporary Topics and Applications as both an in-class and distance-learning first-year course. The course is a survey of contemporary topics in mathematics, centering on applications and projects. Topics include measurements, rates of growth, basic statistics, the mathematics of political power, the geometry of the solar system, and computer arithmetic. The goals state that after completing this course a student should be able to cite elements of good statistical design, undertake elementary statistical analysis, and recognize and explain the shortcomings of unsound methods of statistical analysis; mathematically analyze situations involving the weighting of power in various voting structures and implement apportionment of power strategies; and use the Pythagorean theorem and properties of similar triangles to calculate sizes of and distance between objects, including astronomical objects.
StetsonUniversity offers a wide variety of courses that meet the general mathematics requirement, including many that discuss contemporary topics such as chaos and fractals, game theory, and cryptology.
Both of the popular texts For All Practical Purposes, produced by the Consortium for Mathematics and Its Applications, and Excursions in Modern Mathematics by Peter Tannenbaum and Robert Arnold aim to convey insight about topics in contemporary mathematics and its applications to undergraduate students who have limited mathematical backgrounds. Topics in these books include the mathematics of voting, fair division, and apportionment, applications of graph theory to management science, fractal geometry, and statistics.
At Mount Holyoke College George Cobb teaches a course on the Markov Chain Monte Carlo method (MCMC), ’a very general and powerful method for computer simulation of situations that are too complicated to handle using more conventional mathematical methods. MCMC has become a very active area of research at the interface of computer science and statistics, and has had a powerful impact on the practice of data analysis. As a method for computer simulation, MCMC has very broad applicability. As a branch of mathematics, MCMC offers a number of compelling surprises ’ structures that on a concrete level seem quite different, but, viewed at the right level of abstraction, turn out to be different versions of the same idea.â?
In the article ’Geometric Photo Manipulationâ? Tom Farmer shows how calculus and linear algebra can be used to manipulate photographs, a contemporary application with which many students have experience, thanks to currently available software.
G.H. Hardy once proudly asserted that number theory would never be applied. Yet today number theory has a range of important applications. Among these are cryptography (see, for example, lecture notes from two cryptography courses by Ed Schaefer at Santa Clara University, the RSA website, and recent textbooks in number theory and discrete mathematics), and error detection using check digits and error-correcting codes (see, for example, Numbers and symmetry: An Introduction to Algebra by Bernard L. Johnston and Fred Richman and Contemporary Abstract Algebra by Joseph Gallian).
Enhance Perception of Vitality and Importance of Mathematics
The World Wide Web provides a wealth of examples of the use and applicability of mathematics, but searching for appropriate illustrations can be time consuming. There are several sites that focus on providing good examples for instructors: Plus, an Internet magazine that aims to introduce readers to the beauty and the practical applications of mathematics; Mathematical Moments, an AMS program that offers a series of pdf files and podcasts to promote appreciation and understanding of the role mathematics plays in science, nature, technology, and human culture; and the Math Forum, a center that provides resources, materials, activities, person-to-person interactions, and educational products and services to enrich and support the teaching and learning of mathematics.
Chance News is a monthly, on-line newsletter that provides abstracts of articles from current newspapers, the media, and journals, and suggests discussion questions for class use. It also includes links to related resources at other web sites. Since 1992, Chance News has been maintained by J. Laurie Snell of DartmouthCollege. The examples are current and can be used for student motivation, for class discussion, and as exercises in an introductory statistics course or probability course. The website contains all issues of Chance News as well as information on signing up for the newsletter by e-mail.
Additional information and resources on communicating the breadth and interconnections of the mathematical sciences are in Part 2, Section C.3.