David M. Bressoud August, 2006
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C.3: Provide a broad view of the mathematical sciences
All majors should have significant experience working with ideas
representing the breadth of the mathematical sciences. In particular, students
should see a number of contrasting but complementary points of view:
Majors should understand that mathematics is an engaging field, rich in beauty, with powerful applications to other subjects, and contemporary open questions.
The mathematics major, especially through the first two years, has traditionally placed very heavy emphasis on the continuous phenomena of analysis, on the algebra of mathematical representations, on exact solutions, and on the theoretical framework of mathematics. I have no argument about their centrality and importance, but too often students emerge without even an awareness of discrete phenomena, geometric reasoning, stochastic processes and approximate solutions, or the connections to problems arising outside of mathematics. In previous columns, I addressed the need for statistics early in the mathematics major and the potential of discrete mathematics as a vehicle for building interest and enthusiasm and for teaching proofs and the role of careful definitions. In this column, I want to concentrate on the geometric, the visual treatment of mathematics that is given too little attention in the standard curriculum.
Several people have developed courses in transformational geometry that are appropriate for students in the first two years: Geometry by David. A. Brannan, Matthew F. Esplen, and Jeremy J. Gray at the Open University [1] and Continuous Symmetry by Roger Howe and William Barker [2]. For a broad overview of what is possible in an undergraduate course in geometry, I recommend Joseph Malkevitch’s “Geometry in Utopia” [3], a massive bibliography organized into 68 categories. It was last updated five years ago, but is still very useful.
Additional links to geometrical resources can be found in the CUPM Illustrative Resources, including David Royster’s Hyperbolic Geometry Links page, Michael Reid's Polyomino page, Torsten Sillke’s Tiling and Packing results, John C. Polking’s The Geometry of the Sphere page, The Geometry Center’s Tessellation Resources, and Geometric Dissections on the Web.
Important as they are, courses in geometry run against the obstacle that the curriculum of the first two years is already overcrowded. No one wants to relinquish several variable calculus, differential equations, linear algebra, or the department’s bridge course to make way for geometry. But this need not be a choice of one or the other. One of the great advantages of transformational geometry is that it motivates and draws upon a great deal of linear algebra, while also offering an alternate bridge course, one that leads into modern algebra. In fact, a linear algebra course that is not heavily geometric is a linear algebra that fails to communicate the essence of this subject.
Geometric representations also have an important role to play in calculus and differential equations. It is unfortunate that analytic geometry has been squeezed out of both the high school and college curricula, but calculus and differential equations give ample opportunities to sneak it back in. Even reading analytical information such as the sign of a derivative or an approximate value of a definite integral from the graphical representation of a function is a skill that too few students learn while taking calculus.
The Illustrative Resources has many other links, to the Bridge Project at Oregon State that developed geometric labs for use in several variable calculus, and to various other projects that link algebra and geometry, number theory and geometry, and complex variables and geometry. The resources are there. We owe to our students to draw upon them.
[1] David. A. Brannan, Matthew F. Esplen, Geometry,
Cambridge University Press, 1999. Review at www.maa.org/reviews/geometry.html
[2] Roger Howe and William Barker, Continuous Symmetry, preprint. Information available at academic.bowdoin.edu/faculty/B/barker/dissemination/Continuous_Symmetry.pdf
[3] Joseph Malkevitch, Geometry in Utopia, web page, www.york.cuny.edu/%7Emalk/utopia.html
Do you know of programs, projects, or ideas that should be included in the CUPM Illustrative Resources?
Submit resources at www.maa.org/cupm/cupm_ir_submit.cfm.
We would appreciate more examples that document experiences with the use of
technology as well as examples of interdisciplinary cooperation.
David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul, Minnesota, he was one of the writers for the Curriculum Guide, and he currently serves as Chair of the CUPM. He wrote this column with help from his colleagues in CUPM, but it does not reflect an official position of the committee. You can reach him at bressoud@macalester.edu. |