TABLE OF CONTENTS
Part II. 1
THE NATURE OF MATHEMATICS
Mathematics by nature is both a pure, theoretical adventure of the mind and a practically applied science. This dichotomy allows the theoretical mathematician to "do mathematics for mathematics sake" and the applied mathematician to "use mathematics as a tool" to solve real problems.
Mathematics has developed into an immense system comprising, according to Mathematical Reviews (1992), more than 60 categories of mathematical activity. Mathematical ideas have an unusually long life. The Babylonian solution for quadratic equations is as useful today as it was 4,000 years ago. Like other sciences, mathematics reflects the laws of the material world around us and serves as a powerful instructional tool for understanding nature. However, mathematics is also characterized by its independence from the material world. The abstract nature of theoretical mathematics gave birth even in antiquity to the fundamental dichotomy of mathematics as an object of study and also as a tool for application.
Mathematical ideas are both enduring and expanding. New mathematical ideas are built on other, older mathematical ideas or propositions. An analogy can be made to "continuous improvement," where current practices (in this case, ideas) can be improved upon, given new effort and time. Usually improvement does not occur without effort, and it typically does not occur quickly. Often problems are solved, and new areas of mathematics created, by looking at old problems in new ways.
In 1993, Andrew Wiles, an American mathematician, proposed a proof of Fermat's Last Theorem, first posed in 1637. This problem is simple to state. It says that there are no positive whole numbers that solve the equation:
where n
is greater than 2. Where n = 2, solutions are easy to find. For example:
In the case where n = 2, this algebra problem becomes the familiar Pythagorean Theorem of Geometry.
Wiles began to consider this problem when he was ten. It absorbed his youth and focused him on a career in mathematics. Building on the work of others and considering the problem from the point of view of a seemingly disparate field of mathematics related to elliptical curves, Wiles posed a proof of this famous theorem.
When Wiles offered a solution to Fermat's Last Theorem in 1993, news flashed on e-mail messages (computer mail) across the world. Almost instantaneously, mathematicians around the globe knew of the proposed proof and began questioning and exploring Wiles's work. The spread of Wiles's work via technology enabled rapid global conversations about its validity.
All students can learn and use challenging content. Most of the mathematics that students encounter in school seems as theoretical at first glance as Fermat's Last Theorem appears to be. Students ask "How will I ever use this?" Their education in mathematics should both give them a glimpse into the theoretical nature of mathematics, and an opportunity to relate the mathematics that they are studying to their everyday lives.
By its nature, there are interrelations and connections among the areas of mathematics such as algebra and geometry. A common example is showing an inequality on the number line. Relating 40 < x < 60 to a number line is an easy way to see the relationship. Extending the number line to include negative numbers, and relating "below zero" to temperature is an intuitive introduction and natural extension for elementary and middle grades students.
Many such hands-on manipulative models help students extend their understanding of operations and mathematical properties. Using mathematical structure and relating past knowledge and experience to new knowledge and experience encourages students to construct mathematical ideas and develop mathematical power.
On the applied side, most of the mathematics that students encounter does have practical uses, in fields such as telecommunications (satellites, fiber optics), transportation, and engineering. Making connections between mathematics and the students' real world includes incorporating examples of projects across subject areas.
Students deserve a response to their questions "What's this stuff good for?" and "Who cares?" To answer such questions, those who influence teaching and learning science and mathematics should remember this advice:
If you want to make a course interesting, then you should study something of interest.
Earth Algebra, M. Sims
This advice applies to all grade levels and to learning beyond the traditional years of school as well. The "something of interest" may come from students' real-world experiences or from other disciplines, often science.
A central goal for all levels of mathematics education is the development of mathematical power for all students. Students should acquire an effective awareness of both the spirit and the uses of mathematics and a growing understanding of the breadth of the mathematical sciences and their deep interconnecting principles. In particular, mathematical power includes the ability to explore, conjecture, and reason logically; to solve non-routine problems; to connect concepts within mathematics and between mathematics and real-world situations; and to read, write, listen, and speak mathematically. Acquiring mathematical power also requires the development of personal self-confidence and a disposition to pursue, and use quantitative and spatial information in solving problems and making decisions. Students' flexibility, perseverance, interest, curiosity, and creativity also affect the acquisition of mathematical power.