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

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.