Lag-Time Perspective

The history of mathematics abounds with historical investigations about mathematicians and expositions about their mathematics (e.g., Schubring, 2005). Although mathematicians and their mathematics represent an essential component to understanding the history of mathematics, from an educational perspective it is also important to take into account the history of mathematics education and the history of applied mathematics (e.g., Wessman-Enzinger, 2014; Hertel, 2016). Clements and Ellerton (2013, 2015) explicitly distinguished between research mathematics, service mathematics, and school mathematics; they reflected on the implications of how the differences between these categories may have affected the history of school mathematics. They posited that although each of the histories of these areas of mathematics (i.e., research, service, and school) need to be researched, and although unique stories associated with those histories need to be told, there will, nevertheless, be important intersections.

A key component of Clements and Ellerton’s (2013, 2015) theory is the concept of “lag-time.” Lag-time points to the fact that although mathematicians may have developed new forms of mathematics, it has often been the case that such innovations have taken many years—often decades and even centuries—before they have entered the realm of school mathematics. In terms of the number line, we know the inauguration of the number line is often credited to mathematicians of the 1600s (e.g., Wallis, 1685; Núñez, 2017), but we need to find where it emerged within school mathematics in the United States of America, and elsewhere. A lag-time perspective applied to this study provides insight to look beyond the 1600s for use of the number line in school mathematics.

This paper will explore the evolution of the number line in the United States by looking through the lens of school mathematics. The investigation assumes a lag-time theoretical perspective. Although mathematicians may have developed and even used the number line with negative integers in the seventeenth and eighteenth centuries (Thomaidis, 1993), this does not mean that the number line with negative integers entered school mathematics at the same time.

A Grounded Theory Lens & Textual Analysis

Because the number line may or may not have played a key role in mathematics education during the nineteenth century, texts and curriculum documents from that century were examined for emerging themes using a grounded theory approach (Corbin & Strauss, 2008). The analysis presented in this article distinguishes between arithmetic texts and algebra texts aimed at making sense of the number line in school mathematics during this time period. The 30 arithmetic and algebra texts selected for examination in this study consisted of well-known texts of this time period; see Appendix A for a complete list (Ellerton & Clements, 2017).

The textual analysis conducted with these arithmetic and algebra texts included a qualitative analysis using grounded theory (Corbin & Strauss, 2008). For each text, I first examined the sections specifically about integers or integer operations; then I examined the texts page-by-page for use of negative integers or number lines. Initially, I analyzed the texts for use of number line illustrations or no use of number line illustrations. After the first pass through the texts, I noted a diminutive use of number line illustrations. For this reason, I took annotations and notes for each text specifically about integer use and number line use. After reflection on these notes alongside the research literature, I developed a definition of number line properties, which is described next. Then, because of the uncommon use of number line illustrations, I analyzed the texts again using a set of categories that incorporated the definitions of these number line properties. I then examined the categories further, alongside the original texts, notes, and annotations, and modified the categories. The framework that is presented within this paper emerged from this analysis.

Definition of Number Line

A number line in this paper is conceived to be a representation that illustrates the one-to-one correspondence of the real numbers to the points on the number line. The real number line consists of inherent necessary attributes for the teaching and learning of number and also the use of ordinality, directionality, relativity, and density. The ordinality of the number line refers to the attribute of the number line that numbers or points are placed in an increasing order on the number line (Bofferding, 2014). The number line also illustrates directional attributes (e.g., east/west, right/left, up/down). It extends indefinitely in two directions, conventionally with a positive direction to the right and a negative direction to the left. The relativity of the number line refers to the attribute that the points on the number line are relative to other points (Gallardo, 2002). For example, the number, +1, is to the right of zero and its position on the number line therefore depends upon where the zero is placed. The density of the number line refers to the property that there is an infinite set of real numbers represented on the number line, and an infinite set of real numbers between any two real numbers (Merenluoto & Lehtinen, 2004). These attributes or properties of a number line are intuitive to the mathematician, but not obvious observations for a neophyte learner. It is important that those conducting both historical and psychological investigations relating to the number line are conscious of these properties. In the qualitative analysis I conducted, I considered these properties to be necessary components of this historical investigation of the number line in school mathematics.

As described on the preceding page, I studied the sections containing negative integers in 30 popular arithmetic and algebra texts used in U.S. schools during the 19th century (Ellerton & Clements, 2017; Karpinski, 1940 [see Note]). Although handwritten cyphering books, prepared by U.S. school students in the eighteenth and nineteenth centuries, provide an important historical perspective on implemented curricula (see, e.g., Hertel, 2016), none of the 450 U.S. cyphering books in the Ellerton-Clements collection included an illustration of a number line showing negative integers or discussion about number lines (Clements, personal communication, January, 2017). It became obvious, therefore, that I would need to consult popular U.S. school arithmetic and algebra textbooks from the nineteenth century (see Appendix A for a list of texts studied) for evidence relating to educators’ thinking about the possible uses of the number line in schools.

Various categories of uses of the number line with positive and negative integers emerged: algebraic or contextual emphasis, contextual emphasis with number line properties, number line descriptions, relative number lines, and integer number lines (see Table 1).

Table 1. Categories of Number Line Use in U.S. Arithmetic and Algebra Texts

	Category of Number Line Use		Description
	Algebraic or Contextual Emphasis		Positive and negative integers introduced in texts in contextual practical situations (e.g., debts, eastings/westings) or in the context of algebra exclusively. These texts did not include illustrations of a number line or written descriptions of properties of a number line.
	Contextual Emphasis with Number Line Properties		Some descriptions in the texts, which, although they made no reference to number lines or number scales, included contextual descriptions that highlighted at least one number line property (i.e., order, direction, relativity, density).
	Number Line Descriptions		These texts included descriptions for positive and negative integers that mirrored properties of a number line, even though actual illustrations of number lines were not present. Such descriptions contrasted with the previously-mentioned categories in the sense that the text contained context-free number line descriptions and theoretical accounts of number line properties.
	Relative Number Lines		Descriptions of negative integers in the texts included illustrations of number lines that were notably different from the aforementioned categories. However, not only did the accompanying descriptions vary, but also the illustrations of the number lines themselves included attributes of relativity. For example, some illustrated number lines did not include zero. If the number line did not include zero and focused on the relativity of the integers, then the text was classified as using a “relative number line.”
	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 study. 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.

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Note: The texts examined were from the Ellerton-Clements collection (see, e.g., Clements & Ellerton, 2015).

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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.

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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.

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In some texts, part of the descriptions for positive and negative integers in algebra textbooks mirrored properties of a number line, even though actual illustrations of number lines were not present.  These texts were considered as “supporting” number line descriptions. Such descriptions contrasted with the previously-mentioned accounts in the sense that the text that contained context-free number line descriptions supported theoretical accounts of number line properties.

Descriptions of number lines, or of properties of number lines, in arithmetic and algebra texts in the 1800s varied in their complexity, as is evident in the following examples from texts written by Jeremiah Day and John Farrar.

Jeremiah Day

Jeremiah Day (1773-1867) was Head of Mathematics at Yale College before he became President of that institution. In 1814 he became the first North American author to write a textbook entirely devoted to algebra and aimed at North American students (Ellerton & Clements, 2017; Karpinski, 1940). Day's An introduction to algebra, being the first part of a course of mathematics, adapted to the method of instruction in American colleges (1814) provided an extensive number line description. Although he began his treatment of the positive and negative integers by referring to contexts, he eventually transitioned to more theoretical and context-free discussion. The following excerpt is Day’s contextualized account:

If, for instance, the profits of trade are the subject of calculus, the gain is considered positive; the loss will be negative; because the latter must be subtracted from the former, to determine clear profit. If the sums of a book account, are brought into an algebraic process, the debt and credit are distinguished by opposite signs. If a man on a journey, is by any accident necessitated to return several miles, this backward motion is considered to be negative, because that, in determining his real progress, it must be subtracted from the distance which he has travelled in the opposite direction. If the ascent of a body from the earth be called positive, its descent will be negative. These are only different examples of the same general principle. In each of the instances, one of the quantities is to be subtracted from the other. When a boat, in attempting to ascend a river, is occasionally driven back by the current; if the progress up the stream, to any particular point, is considered positive, every succeeding instance of forward motion will be positive, while the backward motion will be negative. (pp. 21-23)

Within his discussion of various contexts for negative integers, Day discussed the “relative” nature of the negative integers. He began with: “The terms positive and negative, as used in the mathematics, are merely relative” (p. 22), and concluded his explanations of positive and negative integers with a number line description drawn from Euler and Newton:

When a quantity continually decreasing is reduced to nothing, it is sometimes said to become afterwards less than nothing. But this is an exceptionable manner of speaking.* [References were made to Newton & Euler in a footnote.] No quantity can be really less than nothing. It may be diminished, till it vanishes, and gives place to an opposite quantity. The latitude of a ship crossing the equator is first made less than nothing, and afterwards contrary to what it was before. The north and south latitudes may therefore be properly distinguished, by the signs + and –; all the positive degrees being on one side of 0, and all the negative on the other; thus, +6, +5, +4, +3, +2, +1, 0, –1, –2, –3, –4, –5, &c. The numbers belonging to any other series of opposite quantities may be arranged in a similar manner. So that 0 may be conceived to be a kind of dividing point between positive and negative numbers. On a thermometer, the degrees above 0 may be considered positive, and those below 0, negative. (p. 24)

The above passage provides an example of a text used commonly in school mathematics that listed numbers as if they were on a number line, in order, described 0 as a dividing point, described two sides of the number line as positive and negative, and referred to the context of a thermometer (see, e.g., Figure 1). Although an extensive description, it is not a completely developed explanation. There is no evidence, for example, that Day recognized that negative integers could be subsumed into rational numbers or real numbers.

The above excerpt from Day can be interpreted in terms of the lag-time theoretical perspective. As stated previously, Jeremiah Day was a major early nineteenth century American educator and the first North American scholar to publish a text specifically devoted to algebra. His book would appear in many editions (Karpinski, 1940). It is interesting to note that his text, which was essentially written as a text for students, and not for mathematicians, offered a number line description which lagged in both time and development from texts offered by mathematicians such as Euler, Newton, and Wallis.

Figure 1. A passage from Jeremiah Day's Introduction to Algebra (1814, p. 24) included an ordering of integers from +6 to –5. Although Day’s Algebra was widely utilized, this 1814 edition has been rarely examined (Swetz, 2014).

John Farrar

John Farrar (1779-1853) was Hollis Professor of Mathematics at Harvard University between 1807 and 1836 (Ellerton & Clements, 2017; Karpinski, 1940). In his book, An introduction to the elements of algebra, designed for the use of those acquainted only with the first principles of arithmetic. Selected from the algebra of Euler (1821), Farrar provided an extensive “number line description” which attempted to differentiate between natural numbers and integers, as well as touching on the density of the number line. Farrar (1821) first introduced negative integers using the contexts of debts and temperature. After descriptions in these contexts, he then stated:

In the same manner therefore as positive numbers are incontestably greater than nothing, negative numbers are less than nothing. Now we obtain positive numbers by adding 1 to 0, that is to say, to nothing; and by continuing always to increase thus from unity. This is the origin of the series of numbers called natural numbers; the following are the leading terms of this series: 0, +1, +2, +3, +4, +5, +6, +7, +8, +9, +10. But if instead of continuing this series by successive additions, we continued it in the opposite direction, by perceptually subtracting unity, we should have the series of negative numbers: 0, –1, –2, –3, –4, –5, –6, –7, –8, –9, –10, and so on to infinity. (p. 5)

In this excerpt, Farrar (1821) not only offered a definition of the natural numbers, but also appealed to the perceptual or visual nature of ordering the negative integers. A “less than nothing” description of negative was provided with a footnote that expounded on this with contextual explanations. Farrar introduced the following “definition” of the integers:

All of these numbers, whether positive or negative, have the known appellation of whole numbers, or integers, which consequently are either greater or less than nothing. We call them integers to distinguish them from fractions, from several other kinds of numbers of which we shall hereafter speak (p. 6).

After defining natural numbers and integers and referring to the perceptual nature of ordering the integers, Farrar presented a description that describes the density of the real numbers or number line. Farrar stated:

All these numbers, whether positive or negative, have the appellation of whole numbers, or integers, which consequently are either great[er] or less than nothing. We call them integers, to distinguish them from fractions, and from several other kinds of numbers of which we shall hereafter speak. For instance, 50 being greater by an entire unit than 49, it is easy to comprehend that there may be between 49 and 50 an infinity of intermediate numbers, all greater than 49, and yet all less than 50. We need only imagine two lines, one 50 feet the other 49 feet long, and it is evident that there may be drawn an infinite number of lines all longer than 49 feet, and yet shorter than 50. (p. 6)

When Farrar developed the operations with the integers, he did not refer to a number line. The treatment and description of a number line remained solely the description of the identity of negative numbers themselves.

Descriptions of negative integers in early arithmetic and algebra texts that included illustrations of number lines were notably different from the aforementioned categories. However, not only did the accompanying descriptions vary, but also the illustrations of the number lines themselves included attributes of relativity. For example, some illustrations of number lines did not include zero. If the number line did not include zero and focused on the relativity aspects, then the text was classified as using a “relative number line.”

In their A School Algebra, Fletcher Durrell and Edward Robbins (1897) provided an example of a “relative number line” (see Figure 2). After presenting positive and negative integers in contexts (e.g., debts, temperature), Durrell and Robbins provided a “First Law for + and – taken together” with a “relative number line.” The number line provided in their text highlighted various points indicated by letters (K, B, O, A, and F). The point O was considered to be the initial position with positive and negative integers distinguished by walking to the right and left of this point. Rules were then presented for operating (e.g., “The sign – applied twice to a given positive quantity gives a + result.” (p. 22)), and these rules were then described in the context of this relative number line.

Figure 2. A relative number line in Fletcher Durrell and Edward Robbins' A School Algebra (1897, p. 20).

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.

As the nineteenth century progressed, then, the concept of negative integers was gradually developed and refined in U.S. school mathematics textbooks, usually with algebraic or contextual emphases, and occasionally there was direct reference to what might be called a “number line.” The real number line, as it is presently conceived in the Common Core State Standards, had not emerged and instead we had merely an “integer number line” for which density properties were not considered. How the negative integers and the use of the number line are integrated into school mathematics is still an ever-changing, ever-present issue (e.g., Herbst, 1997).

Significance

A noteworthy finding of the investigation described in this paper is the willingness of nineteenth century authors to give number line descriptions, without actually showing a physical number line. Often these descriptions were associated with contextual descriptions. Another significant finding was the use of what I have called the “relative number line,” which is not found in modern curricular documents or textbooks (e.g., CCSSO & NGA, 2010). Yet, the integers can be thought of as being related to each other (Gallardo, 2002) and their insertion on a number line, particular when paired with contexts, is relative (Durrell & Robbins, 1897).

From Past to Current Trends

Using a historical lens to interpret the past can help educators understand the future. Authors of school mathematics textbooks that appeared during the New Math era in the 1960s advocated for use of number lines in elementary and middle school classrooms (see, e.g., Fass & Newman, 1975; Wheeler, 1967). In fact, Rosenthal (1965), in Understanding the New Mathematics, maintained that children are capable of conceiving of negative integers and of extending the number line themselves. Rosenthal wrote:

On seeing the number line with counting numbers on it, an imaginative elementary-school child recently started talking about ‘left numbers’ and how they behaved (p. 66).

By 1989, NCTM was suggesting that negative integers be introduced in Grades 6–8. Then, in 2000, NCTM suggested informal, contextual discussion and introductions of negative integers in Grades 3–5 (see, e.g., Featherstone, 2000). However, current standards (CCSSO & NGA, 2010) suggest that the teaching and learning of negative integers begin in Grade 6 without operations, and in Grade 7 operations on integers should be introduced.

The Common Core State Standards include recommendations for number line use with positive fractions. For example, CCSSO and NGA (2010) suggest that students work with number lines in grade three, and should learn to “understand a fraction as a number on the number line” (p. 24) and to “represent fractions on a number line diagram” (p. 24). Thus, elementary students will be exposed to whole numbers, fractions, and number lines prior to instruction on negative integers. Arguably, then, the present standards support the idea that the number line be not fully expanded in both directions, from zero, until the middle grades. Accordingly, the Common Core State Standards suggest a context-only introduction of negative integers followed by number line use in two different standards for Grade 6, a learning sequence which mimics historical development. The actual standards in the Common Core State Standards document in relationship to integers for Grade 6 are listed below:

• Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
• Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. (CCSSO & NGA, 2010, p. 43)

Texts in the New Math era and the NCTM (2000) standards made innovative and influential steps by advocating the teaching and learning of negative integers and the number line in elementary school. Yet, in the Common-Core era, the negative integers have been pushed back to the middle grades despite the fact that number lines are to be introduced in the elementary school.

From high school algebra to elementary school, and now from elementary school to middle school, the teaching and learning of negative integers, with the inclusion of the number line, have a back-and-forth history on where these topics belong in school mathematics. I believe that negative integers should be introduced to young learners at around the same time as they are introduced to number lines. Such a change will not come easily, however—considering the findings of this historical investigation, which has revealed that the integer number line took several hundred years to find a place in school mathematics, the need to link negative integers and number lines in elementary and middle schools is not something which could be expected to happen overnight.

Historical Implications: Lag-time and the Story of the Number Line

Mathematics educators and mathematics education historians often credit the use of number lines as a pedagogical tool in schools to Max Beberman and Bruce Meserve (1956). Others more immersed in mathematics history point to Wallis (1685) and the inauguration of the number line—although this overlooks the difference between mathematics and mathematics education. However, the evidence presented in this article supports the idea that the use of the number line evolved over time in the United States of America, as there was significant lag-time and variation in use of the number line. The use of number lines did not suddenly emerge during the New Math era after centuries of mathematicians’ use. Clements and Ellerton’s (2013, 2015) theory of lag-time provided justification to look deeply at nineteenth-century U.S. school arithmetic and algebra texts for number line use; and this examination has revealed that there was significant lag from the time that mathematicians first utilized number lines to when their use became commonplace in schools. In fact, the use of integer number lines for the teaching and learning of negative integers in schools slowly became a reality over a period of nearly 200 years.

Intended and Implemented Curricula

Qualitative accounts of only arithmetic and algebra texts are shared throughout this paper because no cyphering-book evidence proved to illustrate use of negative integers or number lines. This subtlety (i.e., 450 cyphering books with no number lines or descriptions of number lines), presented at the start of the section of this article titled "Evolution of the Use of Number Lines in U.S. School Mathematics," points to the distinction between intended and implemented curricula. If one takes the cyphering tradition to represent implemented curricula and the texts to represent intended curricula, then we cannot ever be confident that number lines made a presence in school mathematics in the nineteenth century. Although some of the categories illustrated that these U.S. school mathematics texts began to formalize the definition of the number line, without evidence present in cyphering books, we cannot say that we have direct historical evidence of the presence of number line implementation in school mathematics, but rather only intended curricula. Thus, those who theorize historically about mathematics curricula and school mathematics should consider tightly distinguishing between intended and implemented curricula.

Future Research

This slow development of the use of number lines within school mathematics, in comparison to mathematicians’ use, has implications for historical research. A subsequent project to extend the conversation on the teaching and learning of integers and number lines to other sets of numbers – first the set of rational numbers, and then the set of irrational numbers and the real number line – will be important. This paper has drawn attention to the slow, but persistent, evolution of attempts by educators to extend and link the number line concept to the teaching and learning of the negative integers. Studying the evolution of human developments and conceptions of mathematical objects is important, but the difficulty of designing, implementing, and interpreting research into pedagogical uses and developments of mathematical objects should not be underestimated. There is a definite need for more historical studies that have a school mathematics focus.

Pedagogical Implications: Complexity of the Number Line

For contemporary teachers of mathematics at all levels, the idea that number line properties emerged from these nineteenth century texts without an abundance of illustrations provides opportunity for reflection. First, it points to the complex and nuanced nature of the number line as a pedagogical tool. Yes, research points to how children make use of it and research points to the wealth of affordances of using it. But what may we, as modern mathematics teachers, be missing or taking for granted about this nuanced number line? The intention and detail given to relativity in both descriptions and the relative number line illustrations point to some of the complexity that we ask children and students to assume as they immediately engage with a modern illustration of an integer number line. Integers are, after all, relative numbers on a continuous scale with properties such as order and density between integers. The idea of relativity is often lost in standards, modern curricula, and even in our culture. With positive integers traditionally on the right side of the number line and negative integers traditionally on the left of the number line (or positive integers at the top of a vertical scale), we lose—as a modern society and culture—the essence of the relative nature of both the integers and the number line (e.g., relativity of the number line is not mentioned in the Common Core State Standards). Thus, one pedagogical implication for mathematics teachers at all levels is to consider having students generate or create properties of the number line as they engage with negative integers or the number line for the first time. Or, provide students with illustrations of number lines that counter traditional notions of typical numbers (e.g., negative integers on the right side). Students may even spend time examining the historical texts themselves that have illustrations of relative number lines. For example, the relative number line from Durrell and Robbins (1897) in Figure 2 could be compared to the integer number line from Wentworth (1898) in Figure 3.

Lag-time and Modern Mathematics Curricula

The lag-time perspective presented at the beginning of this article in relation to integer number line use points to the difficulty of introducing new mathematics to school mathematics. Not only did it take research mathematicians many centuries to agree that it made sense to use negative numbers (not to mention the difficulties of zero), but there are centuries between Wallis’s 1685 introduction of the number line and the physical illustrations by Durrell and Robbins in 1897 and Wentworth in 1898. Although there is more to the number line than just an illustration, as evidenced by descriptions of number line properties in the other categories, this lag-time and complex use of number lines offers two implications. First, as teachers, we need to understand that the historical story of the number line is a long one. This type of understanding may change how we choose to engage with children, prospective teachers, or students as we ask them to use a number line as a model. Second, thinking about lag time of the vetted number line begs the question: What other mathematical topics are “lagging” in our modern curricula? If the number line, a prevalent and crucial mathematical tool, has had such a complex evolution of use in school mathematics, we have to consider other tools in school mathematics that may also be as useful, but have not made it to our curricula yet. 

When considering what models and content are most appropriate in schools it is interesting to consider the implications lag-time phenomena may have on current pedagogy and content in school mathematics. Although I am not suggesting the need for a “new” New Math, it does seem interesting to consider the role of the number line in school mathematics in 1713 versus the role of the number line in 2018. The 300-year period has seen qualitatively different forms and uses of the number line evolving in school mathematics, and this points to the possibility of related modern mathematical developments making their way into school mathematics. To reflect on this, given the current trend toward mandated curricula, is important. What mathematics has been developed? And, what is its place in modern school mathematics?

John Wallis (1685), a British mathematician, may have been the first to offer a physical illustration of a number line (Núñez, 2017), although I would maintain his was an illustration of a relative number line. This article draws attention to the lag-time between the work of mathematicians like Wallis and the use of the number line with negative integers in U.S. school mathematics. The descriptions of how number lines have been used in different ways highlight a trend from absence to descriptions, to formality, to illustrations. In mathematics education, teachers and their students now talk about the number line in a wide range of contexts—anthropology, cognitive science, pedagogy, and history, to name just four. This article has, I hope, added a missing piece to this conversation, by examining the number line through the lens of school mathematics. Understanding and discussing the gaps between forms of pure and applied mathematics, on the one hand, and school mathematics, on the other, is an important development toward more robust, scholarly discussions—not only with respect to negative integers and number lines, but also to mathematics and mathematics education, in general.

Acknowledgments

The author would like to thank Dr. Nerida Ellerton and Dr. Ken Clements for access to their comprehensive collections and also for the personal communications about the texts. The author would also like to thank Dr. Janet Beery and the anonymous reviewers for the helpful feedback that made the revision process delightful. 

About the Author

Nicole M. Wessman-Enzinger, PhD, is an Assistant Professor in the School of Education at George Fox University. She teaches content and pedagogy mathematics courses for future teachers. Her main research interests are situated in children and prospective teachers' cognition about negative numbers. She views the history of mathematics education as a tool with which to make sense of students' thinking and contemporary issues.

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Table 2. Arithmetic and Algebra Texts Examined for Number Line Use with Integers

Year		Author	Book Title
1809		Simpson	A treatise of algebra: Wherein the principles are demonstrated, and applied in many useful and interesting inquiries, and in the resolution of a great variety of problems of different kinds. To which is added, the geometrical construction of a great number of linear and plane problems, with the method of resolving the same numerically
1814		Day	An introduction to algebra, being the first part of a course of mathematics, adapted to the method of instruction in American colleges
1821		Farrar	An introduction to the elements of algebra, designed for the use of those acquainted only with the first principles of arithmetic. Selected from the algebra of Euler (2nd ed.)
1831		Colburn	An introduction to algebra upon the inductive method of instruction
1832		Young	An elementary treatise on algebra, theoretical and practical; with attempts to simplify some of the more difficult parts of the science, particularly the demonstration of the binomial theory in its most general form; the solution of fractions of the highest orders; the summation of infinite series, &c. intended for the use of students
1833		Day	An introduction to algebra, being the first part of a course of mathematics, adapted to the method of instruction in the American colleges (11th ed.)
1837		Ray	Eclectic series-newly improved. Ray’s arithmetic: Part third. Being the author’s eclectic arithmetic, on the inductive and analytic methods of instruction; designed for common schools and academies
1839		Bailey	First lessons in algebra, being an easy introduction to that science designed for the use of academies and common schools
1842		Sherwin	An elementary treatise on algebra, for the use of students in high schools and colleges
1843		Day & Thomson	Elements of algebra, being an abridgment of Day’s algebra adapted to the capacities of the youth, and the method of instruction, in schools and academies
1848		Perkins	A treatise on algebra, embracing, besides the elementary principles, all the higher parts usually taught in colleges; Containing moreover, the new method of cubic and higher equations as well as the development and application of the more recently discovered Theorem of Sturm
1848		Thomson	Day and Thomson’s series. Elements of algebra, being an abridgment of day’s algebra, adapted to the capacities of the young, and the method of instruction, in schools and academies
1851		Mann & Chase	Arithmetic, practically applied, for advanced pupils, and for private reference, designed as a sequel to any of the ordinary text-books on the subject
1852		Ray	Eclectic education series. Ray’s new higher algebra. Elements of algebra, for colleges, schools, and private students
1853		Thomson	A key to the abridgment of Day’s algebra; containing many explanations, the answers to all the questions, together with a statement and solution of the more difficult problems
1856		Leach & Swan	An elementary intellectual arithmetic, containing numerous original contractions in multiplication
1857		Loomis	Treatise on algebra (12th ed.)
1858		Davies	Elements of algebra: On the basis of M. Bourdon: Embracing Sturm’s and Horner’s theorems, and practical examples
1862		Greenleaf	Key to Greenleaf’s new elementary algebra
1866		Greenleaf	The national arithmetic, on the inductive system, combing the analytic and synthetic methods; forming a complete course of higher arithmetic
1866		Ray	Eclectic educational series. New elementary algebra. Primary elements of algebra, for common schools and academies
1867		Greenleaf	Introduction to the national arithmetic, on the inductive system, combining the analytic and synthetic methods; in which the principles of the science are fully explained and illustrated
1875, 1876		Robinson	New elementary algebra: Containing the rudiments of the science for schools and academies
1877		Greenleaf	Manual of intellectual arithmetic: An independent treatise upon the basis of mental arithmetic
1879		Olney	Olney’s two-book series. A practical arithmetic for intermediate, grammar, and common schools
1880		Greenleaf	New practical arithmetic; in which the science and its applications are simplified by induction and analysis
1892		Milne	Standard arithmetic: Embracing a complete course for schools and academies
1893		Wentworth	A practical arithmetic
1897		Durell & Robbins	A school algebra
1898		Wentworth	New school algebra