mathematical idea by using plates of cookies. Whether the symbols represent the concrete objects or vice versa depends upon where the child starts. Both symbols and objects, however, represent a mathematical idea that is independent of the particular representation used.

The remainder of this section considers one particular representation system for numbers, the decimal place-value system, which is a significant human achievement. It should be emphasized, however, that representation systems arise out of human activity, and much mathematical insight can be gained by considering the genesis and development of the representation systems of the Egyptians, the Babylonians, the Mayans, or other cultures. Our intent here is more modest: to describe issues of mathematical representation by focusing on the representation system that is the major focus of school mathematics. It should also be emphasized that a representation system discussed previously, the number line, also deserves significant attention. In fact, the main unifying and synthesizing point of the previous section was that the number systems of school mathematics, which remain often fragmented and disjointed in the perceptions conveyed by school curricula, are in fact all subsystems of a single system, which has a geometric model that is the foundation of later analysis and geometry.

Grouping and Place Value

To use numbers effectively, to speak about them, or to manipulate them requires that they have names. Modern societies use decimal place-value notation in daily life and commerce. With just 10 symbols—0, 1, 2,…, 9— any number, no matter how big or small in magnitude, can be represented. For example, there are roughly 300,000,000 people in the United States. Or the diameter of the nucleus of an atom of gold is roughly 0.00000000034 centimeters. The decimal system is versatile and simple, although not necessarily obvious or easily learned. The decimal place-value system is one of the most significant intellectual constructs of humankind, and it has played a decisive role in the development of mathematics and science.

Over the centuries, various notational systems have been invented for naming numbers. To represent numbers symbolically, the ancient Hindus developed a numeration system that is based on the principles of grouping19 and place value, and that forms the basis for our numeration system today. In this system, objects are grouped by tens, then by tens of tens (hundreds), and so on. Hence, this numeration system is a base-10 or decimal system. These are nontrivial ideas that took humankind many centuries to invent and refine.

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