Decimal Notation is the product of thousands of years of cultural evolution. Given humankind's innate sense of number (unity, pair, triad, etc.), scribes developed many different systems for representing numerical concepts. These systems were ref ined over the eons, culminating in our present Hindu-Arabic notation. This notation combines the decimal base of the Egyptians with the positional notation of the Babylonians and the concept of "an empty space" (or zero) as formalized in India. While th e world sustains many languages and alphabets, a single "grammar of size and order" is virtually universal. An appreciation of the remarkable properties of this universal number system is essential for those who convey arithmetic concepts to children.
Long Division has been maligned as an anachronism that represents "the bad old days" of algorithm-based instruction. There is, however, much to be learned from an understanding of this algorithm. Just as multiplication corresponds to repeated add ition, so does division correspond to repeated subtraction. The long division algorithm provides a highly efficient way of performing such repeated subtractions. In secondary schools long division is needed to establish the connection between fractions and repeating decimals. In algebra, it arises as a tool for finding solutions of certain algebraic equations. Even in the presence of calculators, long division is an important part of arithmetic instruction.
Fractions and Decimals provide us with useful choices and alternatives in the representation of non-whole numbers. Decimal notation tends to simplify addition and subtraction, while use of fractions tends to simplify multiplication and division. Yet it is important to be able to perform all four operations in both systems and to understand the interrelationships that arise. Some of the most important properties of "real numbers" are based on the interplay between fractions and decimals. The add ition of fractions leads to the concepts of Least Common Multiple (LCM) and Greatest Common Divisor (GCD). In this way, the existence of these alternative forms of notation have greatly enriched arithmetic.
Polyhedra are 3-dimensional geometric figures that are bounded by planes. In their rich mathematical and aesthetic variety, they can be used to provide an introduction to geometric concepts while also developing visualization skills in 3 dimension s. Many aspects of the mathematical order and patterns they embody can be conveyed at the elementary school level. Polyhedra can also be to develop the vocabulary that will be required in future efforts to formalize geometric concepts.
Area has its roots in practical problems involving land measurement and taxation. Mathematically speaking, the area of a plane geometric figure is a nonnegative number with two properties: (I) Congruent figures have equal areas, and (II) If a figu re is partitioned into two pieces, the area of the figure is the sum of the areas of the two pieces. These two simple rules enable us to calculate the areas of a host of geometric figures. These rules are also closely related to probability theory and l ead to interesting connections between geometry and probability.
Probability is a numerical measure of the likelihood that a particular event will occur. Mathematical rules for adding and multiplying such measures provide us with an ability to deal with many interesting situations involving uncertainty. A star ting point for "the mathematics of uncertainty" is the definition of probability in terms of events with equally likely outcomes. Given such a definition, we are able to attach meaning to the operations of + and x and to provide our students with a sound basis for approaching problems arising in the secondary school probability curriculum.
Combinatorics is a collection of techniques that extend counting. By interpreting "M x N" as a technique for counting the number of objects in M sets of N objects, multiplication becomes the first combinatorial tool. Based on such an interpretati on of multiplication, combinatorics goes on to develop other counting rules such as permutations and combinations. These counting rules are central to an under-standing of probability. They also arise in algebra, geometry, and many other areas of mathem atics.
Irrational Numbers correspond to familiar concepts such as "the square root of two" and "pi." While the decimal representations of rational numbers either terminate or repeat after a certain point, the decimal representations of irrational numbers are infinite and nonrepeating. Other interesting properties of irrational numbers are related to their geometric representations as given by the early Greeks. Here irrational numbers were related to the concept of "incommensurability" and correspond to rectangles that cannot be tiled by squares.
Matrices are rectangular arrays of numbers that enjoy many uses. They can be introduced in terms of their applications in diverse areas such as demography, economics, and probability. However, they can also be regarded as a variant of our number system, one in which familiar properties such as associativity and commutativity have to be revisited with care.
Functions can be conceptualized in several ways. Geometrically we can identify functions with their graphs. Closely related are the rules and tables of values that are used to construct such graphs. These concepts are all contained in the mathem atical definition of functions as sets of ordered pairs. An understanding of these alternative ways of conceptualizing functions can help students deal with inverse functions, the composition of functions and their iteration, all of which are important i n the pre-calculus curriculum.
Proof is essential to the coherence of mathematics. Without it we cannot be sure that what we are doing is true and logically consistent. Through the study of geometry students can (like the early Greeks) come to understand how a sound mathematic al structure is erected, one building block at a time. They can come to appreciate the role of axioms as the foundation upon which this structure rests and the role of proof as the mortar that holds the structure together. There is a legitimate debate o n how much emphasis should be placed on proofs in teaching both algebra and geometry. One way of developing perspectives on this question is to build some mathematical structures using both formal (axiomatic) and intuitive methods. Such exercises can he lp students appreciate the difference between a proof and a convincing, but nonrigorous, heuristic argument.
Fractals are mathematical objects of fractional (non-integer) dimension. They often correspond to geometric figures that exhibit self-similarity - i.e., similarity to some proper subset of themselves. Determining the dimension of a fractal can de velop geometric skills and reinforce the concept of logarithm. The connection between fractals and chaos tends to be a subtle one, but there are situations in which it can be illustrated at the high school level. While one can find suggestions of self-s imilarity in nature, fractals are mathematical objects that do not exist in the natural world.