ConnectionsAuthor(s): James CaseySource: The Mathematics Teacher, Vol. 94, No. 8, CONNECTIONS (November 2001), p. 628Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/20870831 .

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Reader Reflections

Connections This focus issue on "Connections" is an important contribution to

understanding and teaching mathematics. It draws welcome attention to the energizing re

lationships that develop between mathematics and the other sub jects, as well as emphasizes the connections that exist within mathematics itself. Teachers can make good use of examples of both types of connections.

Examples in which advances in physics have led to important developments in mathematics, and vice versa, abound. Also, engineers constantly employ results from mathematics, for example, in

dynamical systems, structural mechanics, cardiovascular fluid mechanics, signal processing, computational modeling, and cryptography; conversely, problems arising in practical areas often give rise to new mathematical questions.

The importance of the inter connections that exist among different areas within mathematics can hardly be overestimated. I learned the mathematics that I know in separate pieces, both in school and at the university. Only later, and very gradually, did I

appreciate the essential unity and

organic interconnectedness of mathematics. Partly for historical reasons, mathematics is delivered to students in the standard dis joint pieces. Only when we study analytic geometry do we first

clearly see that geometry can be connected with algebra. Then we learn that matrix algebra can be used to describe geometrical transformations, including rotations and stretchings. As

undergraduates, we learn that calculus can be used in describing such important geometrical concepts as curvature.

At the most advanced level, research mathematicians are dis covering deep connections among

such apparently disparate fields as number theory, group theory, complex analysis, modern geometry, and topology. Perhaps mathematics is trying to tell us

something when, for a proof of Fermat's last theorem, which several great mathematicians believed to be an isolated problem, she transports us to a distant region full of exotic concepts. (See Fermat's Enigma, by Simon Singh [New York: Doubleday & Co., 1997].) Such surprising inter connections hint at a labyrinthine network of roads, tunnels, bridges, and canals that traverse the entire

landscape of mathematics. A

problem that seems insurmount able when viewed in its native setting can sometimes be solved

by transporting it over this net work to a more hospitable clime.

Simple intuitions in one region can lead to surprising discoveries in a connected region.

In David Hilberth address in 1900 to the International Congress of Mathematicians in Paris (translated in the Bulletin of the

American Mathematical Society, vol. 8,1902, pp. 437-79), he turned to the question of the unity of mathematics. Faced with the tremendous growth that was

taking place in all aspects of the field, Hilbert asked, somewhat rhetorically, whether mathematics was doomed to the same fate as those other sciences that had

split into separate branches. The

practitioners of the branches hardly understood one another, and the branches were becoming ever more loosely connected. Hilbert did not believe that mathematics would face that fate: he stated that to him mathematics was "an indivisible whole, an

organism whose vitality is con ditioned upon the connection of its parts." According to Hilbert, as mathematics grows, harmony, uniformity, and unexpected relationships develop; and these

characteristics tend to bind it

tightly into an organic whole. In "The Architecture of Mathe

matics," published in 1950 in The American Mathematical Monthly, the group of mathematicians known as Nicholas Bourbaki again asked, but with increased urgency, whether the exuberant growth of mathematics would inevitably lead to splintering or whether, alter natively, a cohesive organism could be maintained. Since no mathe matician can be expected to keep up with all new developments in mathematics, preserving unity is indeed a daunting problem. Bourbaki proposed that mathe matical knowledge be organized around a number of "great struc tures." These abstract structures are arrived at by noticing com mon patterns in various parts of mathematics. Thus, the process of adding real numbers shares a common logical pattern with the process of composing displacements in space. Each of these two processes is associative: a neutral, or identity, element exists; and an inverse element exists. This structure is one of the most per vasive in all mathematics, both pure and applied, and is called the group structure. Another common structure is the order structure, typified by "at most

equal to" for numbers, and "is a subset of" for sets. Simpler struc tures can be combined with one another to form complex structures, which in turn can be combined in a manner resembling the way that a complex organism, such as the human body, forms from its molecules, cells, and organs. We may use Bourbaki's abstract structures to better understand the anatomy and physiology of

mathematics.

How can we use these ideas in teaching? Almost everyone would agree that teaching students about sets or groups is pointless before they have had some

experience with concrete illustra tions of these concepts. Instead, we can teach them to search for common patterns in preselected portions of mathematics?and physics, too?especially patterns that appear to be unrelated at first glance. In other words, our

objectives should be to sensitize

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students to the existence and pervasiveness of abstract structures and to encourage them to discover these fundamental forms for themselves, so that they can begin to explore the internal harmonies and organic unity of mathematics.

James Casey jcasey@newton. me. berkeley

.edu

University of California at Berkeley

Berkeley, CA 94720

Connecting with geometry My vision of teaching mathematics is to meaningfully connect mathe

matics with other disciplines and with the world around us, as well as to connect the different topics in mathematics with one another.

My goal is to help my students make these connections and to show them that mathematics is connected with their lives and with the world around them. I want my students to be continu

ously intrigued by, and drawn into, the connections between geometry and science, art, design, communication, language, history, architecture, and engineering. I want them to see the relationship between circles and the gears of a

bicycle, the connections between

graphic design and geometric construction, and the inter connectedness of the medians of a

triangle and the center of gravity of a cantilevered balcony. My goal in teaching is to connect my stu dents with the structure, relationships, and principles of mathematics and the integral relatedness of this structure to our lives, as well as to make their

experience of these connections an active encounter.

I have created Web sites for two distance-learning classes that I teach over the Internet in

E-School, a grant-funded project through the Hawaii Department of Education. One E-School class, Connecting Geometry, can be seen at www.kl2.hi.us./~csanders. The

purpose of Connecting Geometry is to help students "discover the

beauty and wonder of geometry, and the many connections with the world: in science, art, navigation and architecture, through the Internet and through collaboration with other geometers."

The other E-School class, (Continued on page 716)

628 MATHEMATICS TEACHER

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