• Generality. Does the representation apply to broad classes of objects? Finger representations are not general. The number line is quite general, allowing the representation of counting numbers, integers, rationals, and reals. If digits on both sides of the decimal point are included, the decimal place-value representation of numbers is completely general in the sense that any number may be so represented.

  • Clarity. Is the representation unambiguous and easy to use? Representations should be clear and unambiguous, but that is often established by convention—how the representation is commonly used. (See Box 3–8.)

Box 3–8 Clarity of Representations

For simplicity of use, representations should be as clear and unambiguous as possible. Much of that clarity is not inherent in the representation, however, but is established through convention. For example, the expression 3+4×5 is ambiguous on its face because there is no explicit indication of whether to perform the multiplication or the addition first.* One might be tempted to proceed simply from left to right. The conventional order of operations, however, dictates that multiplication and division precede addition and subtraction, so 3+4×5 is evaluated as 23=3+(4×5) and not 35=(3+4)×5. In the middle grades and high school, as algebraic symbolism is introduced, the letter x and the multiplication symbol×can be confused, especially in written (rather than typeset) work. This ambiguity is solved in part by omitting multiplication signs, using parentheses or juxtaposition instead. Thus, xy means x times y, and 5(3) means 5 times 3.

But that practice creates another ambiguity. In the notation for mixed numbers, means It does not mean Furthermore, juxtaposing symbols to indicate multiplication creates confusion in high school mathematics with the introduction of function notation, where f(4) looks like multiplication but instead means the output of the function f when the input value is 4. The ambiguities of such standard notations can interfere with learning if they are not acknowledged, explained, developed, and understood.


Try a few different calculators. Scientific calculators typically perform the multiplication first, but simpler “four-function” calculators usually perform the addition first.

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