# Integer Number Lines in U.S. School Mathematics - Contextual Emphasis with Number Line Properties

Author(s):
Nicole M. Wessman-Enzinger (George Fox University)

Some descriptions in the texts, although they made no reference to number lines or number scales, included contextual descriptions that included descriptions of at least one (or more) of the number line properties (i.e., order, direction, relativity, density). These types of texts were recognized as having a contextual emphasis with number line properties. Although such descriptions may have alluded to properties like relativity, these texts rarely recognized negative integers as actual numbers.

### Direction

Robinson's New Elementary Algebra (1875, 1876) is a text with contextual emphasis with number line properties. Robinson’s texts included some conceptual underpinnings not aligned with a number line entirely, but involving ideas of direction. For example, Robinson (1875), in the following passage, made use of the concept of “direction”:

We cannot, numerically, take a greater quantity from a lesser, nor any quantity from zero, for no quantity can be less than nothing. Hence in the last two examples, the answer, –5a, is not 5a less than nothing, but 5a applied in the opposite direction to +5a. To subtract a quantity algebraically, is to change the direction to which it is reckoned or applied. (pp. 34–35)

In this excerpt, Robinson referred to the directional aspect of negative numbers. Directional aspects were intuitively important for the historical development of the number line, especially in relation to different directions on a number line and to moving along a number line. Although Robinson (1875, 1876) referenced the directionality of the integers, he made no reference to a number line, a number scale, or even a thermometer scale.

Robinson (1875) also offered an example of the conceptual struggles that even authors experienced with understanding the nature of negative integers. He put forward a problem, solved it, got a negative answer, and interpreted the “negative” answer in the following way:

The following examples will illustrate negative results: 1. A man worked for a person 10 days, having his wife with him 8 days and his son 6 days, and he received 10 dollars and 30 cents as compensation for all three; at another time he worked 12 days, his wife 10 days, and son 4 days, and he received 13 dollars and 20 cents; and at another time he worked 15 days, his wife 10 days, and his son 12 days, at the same rates as before, and he received 13 dollars and 85 cents. What were the daily wages of each? Ans. Husband, 75 cents; wife, 50 cents; son, –20 cents. The sign minus signifies the opposite to the sign plus. Hence the son, instead of receiving wages, was at an expense of 20 cents a day, and the language of the problem is thus shown to be incorrect. (p. 161)

In this excerpt, Robinson seemed to assume that the negative result was “incorrect” and in subsequent sections he suggested how the problem could be posed so that such an “error” could be avoided. Robinson (1875) posed another similar problem:

What number is that whose fourth part exceeds its third part by 12? Ans.  –144. But there is no abstract number –144, and we cannot interpret this as debt. It points out the error or impossibility of [the problem], and by returning to the problem … we perceive that a fourth part of [a] number cannot exceed its third part; it must be, its third part exceeds its fourth part by 12, and the enunciation should be thus: What number is that whose third part exceeds its fourth part by 12. Ans. 144. Thus do equations rectify subordinate errors, and point out special conditions. (p. 162)

From these excerpts it can be assumed that Robinson did not recognize negative integers as actual numbers, even though he described the directionality of them. Robinson was not alone in holding such views. For example, Sherwin (1842) made the following revealing comment about negative integers:

It may happen, in consequence of some absurdity or inconsistency in the conditions of a problem, that we obtain, for a result or answer to the question, a quantity affected with the sign . Such a result is called a negative solution …. Negative results not only indicate some absurdity or inconsistency in the conditions of a question, but also teach us how to modify the question, so as to free it from all inconsistency (p. 119).

The recognition of negative integers as real numbers was of central importance in the development of the number line. Robinson, Colburn, Greenleaf and Ray were not the only mathematics textbook authors who experienced difficulty in making the necessary conceptual leap (Bishop, Lamb, Philipp, Schappelle, & Whitacre, 2011; Bishop, Lamb, Philipp, Whitacre, Schappelle, & Lewis, 2014). Although these authors did not recognize negative integers as actual numbers, they nevertheless seemed to be aware of their directional aspects.

### Order and Relativity

Loomis (1857) emphasized contextual aspects of negative numbers. Specifically, Loomis utilized context to describe order and relativity of the integers. Loomis began his introduction to the negative integers with the following:

The term subtraction, it will be perceived, is used in a more general sense in algebra than in arithmetic. In arithmetic, where all quantities are regarded as positive, a number is always diminished by subtraction. But in algebra, the difference between two quantities may be numerically greater than either. Thus, the difference between +a and –b is a + b. The distinction between positive and negative quantities may be illustrated by the scale of a thermometer. The degrees above zero are considered positive, and those below zero negative. From five degrees above zero to five degrees below zero, the numbers stand thus: +5, +4, +3, +2, +1, 0, –1, –2, –3, –4, –5. The difference between five degrees above zero and five degrees below zero is ten degrees, which is numerically the sum of the two quantities. (p. 17)

In this excerpt, the description included the order of the negative integers through the context of the thermometer. However, the ordering of the positive and negative integers is not discussed outside of the context of the thermometer. In addition to this description, Loomis also commented on relativity:

It has already been remarked, in Art. 5, that algebra differs from arithmetic in the use of negative quantities, and it is important that the beginner should obtain clear ideas of their nature. In many cases, the terms positive and negative are merely relative. They indicate some sort of opposition between two classes of quantities, such that if one class should be added, the other ought to be subtracted. Thus, if a ship sails alternately northward and southward, and the motion in one direction is called positive, the motion in the opposite direction should be considered negative. (pp. 18-19)

In this description the relativity of the negative numbers is described. Using context, Loomis supported relativity when he wrote, “one direction is positive, and the opposite direction should be considered negative” (p. 19). That means that either north or south directions could be regarded positive as long as the other direction is regarded as negative. Both ordinal and relative properties of numbers are mentioned, within a contextual emphasis.

Nicole M. Wessman-Enzinger (George Fox University), "Integer Number Lines in U.S. School Mathematics - Contextual Emphasis with Number Line Properties," Convergence (February 2018)