Devise criteria and procedures for fair addition of a congressional district to a state in a way that will minimize disruption of current districts while creating new districts that are relatively compact (non- gerrymandered) and of nearly equal size.

Employing Computers

            It has been clear for many years that technology has changed priorities for mathematics. Much of traditional mathematics (from long division to integration by parts) was created not to enhance understanding but to provide a means of calculating results. This mathematics is now embedded in silicon, so training people to implement these methods with facility and accuracy is no longer as important as it once was. At the same time, technology has increased significantly the importance of certain parts of mathematics (e.g., statistics, number theory, discrete mathematics) that are widely used in information-based industries.

            Many mathematics teachers have embraced technology, not so much because it has changed mathematics but because it is a powerful pedagogical tool. Mathematics is the science of patterns (Devlin, 1994; Steen, 1988), and patterns are most easily explored using computers and calculators. Technology enables students to study patterns as they never could before, and in so doing, it offers mathematics what laboratories offer science: a source of evidence, ideas, and conjectures.

            The capabilities of computers and graphing calculators to create visual displays of data have also fundamentally changed what it means to understand mathematics. In earlier times, mathematicians struggled to create formal symbolic systems to represent with rigor and precision informal visual images and hand-drawn sketches. However, today's computer graphics are so sophisticated that a great deal of mathematics can be carried out entirely in a graphical mode. In many ways, the medium of computers has become the message of mathematical practice.

            Finally, and perhaps most significantly, computers and calculators increase dramatically the number of users of mathematics--many of whom are not well-educated in mathematics. Previously, only those who learned mathematics used it. Today, many people use mathematical tools for routine work with spreadsheets, calculators, and financial systems--tools that are built on mathematics they have never studied. For example, technicians who diagnose and repair electronic equipment employ a full range of elements of functional mathematics--from number systems to logical inferences, from statistical tests to graphical interpretations. Broad competence in the practice of technology-related mathematics can boost graduates up many different career ladders.

            This poses a unique challenge for mathematics education: to provide large numbers of citizens with the ability to use mathematics-based tools intelligently without requiring that they prepare for mathematics-based careers. Although mathematicians take for granted that learning without understanding is ephemeral, many others argue that where technology is concerned, it is more important for students to learn how to use hardware and software effectively than to understand all the underlying mathematics. But even those who only use the products of mathematics recognize the value of understanding the underlying principles at a time when things go wrong or unexpected results appear. In a functional curriculum where, for example, algebra emerges from work with spreadsheets, the traditional distinction between understanding and competence becomes less sharp.