Abstract
The fundamental characteristic of mathematics is its abstraction. Its idealizations have been found to give surprisingly accurate descriptions of phenomena in the real world. Thus mathematics is endowed with incredible power to serve a broad range of diverse areas. The same mathematical tools, differently interpreted, can be applied to seemingly unrelated subjects to solve problems in a great variety of disciplines. Any educational program must address these important features of mathematics in a balanced way. One of the serious problems with the NCTM standards is that the essence of the subject is lost. Problems are viewed as relevant and interesting to students only if they are related to familiar daily experiences. Unless they are tied in with theory, however, isolated problems may well fragment the discipline. The power of mathematics vanishes when nothing occurs to show that the concrete is an illustration of the abstract. We have a magnificent opportunity to enlighten our students, leading them to the realization that mathematics is an intrinsically beautiful and exciting discipline in its own right. Its structure gives us substantial flexibility for teaching. We may use either the applications to develop the abstraction or, conversely, start with the abstraction as a lead to application. By turning at various times in either direction, we can provide great mathematical enrichment for our students.
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