Integer Number Lines in U.S. School Mathematics - Integer Number Lines

When number lines contained and described only the integers, without emphasis on relativity in the illustration, the text was classified as using “integer number lines” in this paper. Integer number lines differ from relative number lines in that they illustrate zero and integers as numbers, not just relative numbers. These integer number lines appeared in arithmetic and algebra texts toward the end of the nineteenth century and early in the twentieth century.           

George Wentworth

Although the period of the “New Math” (from about 1957 to about 1975) is often regarded as the time when number lines first began to be widely taught (Bossé, 1996; McIntosh et al., 1992), it should be noted that the following example comes from a textbook authored by George “Bull” Wentworth in the 1890s. Wentworth is an example of an author who did not include negative integers or number lines in arithmetic books (see, e.g., Wentworth, 1893), but did include negative integers and number lines in his algebra books (see, e.g., Wentworth, 1898). Wentworth (1898) began a chapter on positive and negative numbers in his New School Algebra with a description of “The Natural Series of Numbers,” in which he described the natural numbers on a number line. This was a number line with numbers 0 through 11 and no negative numbers.

Wentworth stated that one cannot subtract 5 from 2 using this series of numbers, but expanded the natural numbers to the set of positive and negative integers (Wentworth, 1898—see Figure 3).

Figure 3. An integer number line in George Wentworth's New School Algebra (1898, p. 40).

Wentworth (1898) referred to this number line as illustrating two series of numbers, positive and negative numbers, which formed “the algebraic series of numbers.” Wentworth described the order of the numbers as they “moved” right as “increasing” or “ascending” in order, and as they moved left as “decreasing.” He also introduced the absolute value, or magnitude, of a number without making reference to the absolute value as the distance from zero on the number line. Instead, he wrote:

Every algebraic number, as +4 or –4, consists of a sign + or – and the absolute value of the number. The sign shows whether the number belongs to the positive or negative series of numbers; the absolute value shows the place the number has in the positive or negative series. (p. 35)

Referring to the algebraic series, Wentworth also described the “double meanings” of the signs + and –. He maintained that the signs + and – refer not only to the operations of addition and subtraction, but also to the series to which the number belongs. He wrote about the “opposition” or the opposite nature of the positive and negative integers. Although he provided an illustration and description of the integer number line, including order and magnitude, in his number line illustrations, he remained strictly within the integers and did not extend his use of number lines to fractions. For this reason, Wentworth offered an example of what I call an “integer number line.”

For the teaching and learning of operations about negative integers, Wentworth used the number line to give meaning to addition and subtraction of integers. With an illustration of the integer number line provided, he stated that the sum +2 + (–3) may be thought of as counting along from +2 three units in the negative direction; that is, to the left. “It is, therefore, –1” (p. 36). Of the texts explored in this paper, this is the first text that used the number line to explain operations about the integers. However, Wentworth’s explanations of multiplication and division with integers was based on rules that made no reference to number lines. Although there was a focus on procedures for operations, the distinction between the natural numbers and the integers was a notable contribution by Wentworth (1898) to school mathematics. Also noteworthy was Wentworth’s use of an integer number line.