Integer Number Lines in U.S. School Mathematics - Algebraic or Contextual Emphasis

In the nineteenth century, the positive and negative integers were commonly presented in texts in contextual practical situations (e.g., debts, eastings/westings) or in the context of algebra. These types of texts were considered to have an algebraic or contextual emphasis. Such texts included negative integers in an introduction or discussion, but they never included any illustration of a number line, or even written descriptions of properties of positive and negative integers (e.g., order, direction). Rather, these texts had an algebraic or contextual emphasis.

Emphases in Algebra Texts

When introducing or discussing pure-algebraic topics, such as solving equations, authors often focused on the need for negative integers in algebra (see, e.g., Perkins, 1848). Negative integers were presented as numbers which were needed to solve some equations that were previously thought to be impossible to solve (e.g., 2 + x = 1). Warren Colburn (1831), for example, introduced negative integers in sections concerned with writing and solving algebra equations. Although his introduction to negative integers emphasized a purely algebraic approach, later in the same book he drew attention to the usefulness of negative integers in the context of gains and losses in monetary transactions. Despite the use of context, Colburn’s focus remained on rules, or procedures, for operations with signed numbers, and no number line illustrations were provided or properties of number lines (i.e., ordinality, directionality, relativity, density) mentioned.

Charles Davies (1858) presented the negative integers solely in relation to their usefulness in algebra. In Davies’s texts, negatives were presented solely in the context of writing, simplifying, or solving algebraic equations. Davies, in contrast to Colburn, did not utilize contexts like monetary situations.

Benjamin Greenleaf’s The National Arithmetic, first published in 1836, and other books which Greenleaf authored (or was named as author) and which continued to be published even into the 1880s (see, e.g., Greenleaf, 1862, 1866, 1877, 1880) did not include explicit descriptions of negative integers. However, Greenleaf’s algebraic texts included negative integers as solutions to equations—but there was no suggestion that a number line could be useful. Similarly, Joseph Ray, a best-selling North American mathematics textbook author between 1835 and 1900, included the treatment of negative integers in his algebraic texts, rather than in his arithmetic texts (see,.e.g., Ray, 1837, 1952). Ray discussed negative integers within the contexts of latitude and temperature, and offered rules for operations on positive and negative integers in his algebra texts, but there was no discussion about characteristics or properties of the negative integers or number lines (Ray, 1852, 1866). Ray (1866) provided the following contextual explanation of negative integers:

Thus, if a merchant’s gains are positive, his losses are negative; if latitude north of the equator is +, that to the south is –; if distance to the right of a certain line is +, that to the left is –; if elevation above a certain point is +, that below is –; if time after a certain hour is +, time before that hour is –; if motion in one direction is +, motion in an opposite direction is –; and so on. (p. 29)

Although Ray did mention the “opposite” nature of negative numbers, his Although “opposites” is an important aspect to the development of the number line, the explanation is an example of a purely contextual description in an algebraic text.

Emphases in Arithmetic Texts

In the early 1850s, Horace Mann, a prominent educator in North America, co-authored an arithmetic text that included a contextual consideration of negative integers (Mann & Chase, 1851). In Mann and Chase’s (1851) text, the negative integers were only described in the context of a thermometer. The only question related to negative integers posed by Mann and Chase (1851) was: “If the temperature below 0° on the Russian scale is marked +, and the temperature above 0° is marked –, at what points will the number be the same, on the Russian and Centigrade scales?” (p. 371). Contexts, such as temperature, invite consideration of the historical evolution of the real-number line because they refer to a scale, which can be related to the number line.