Descriptions of the Integer Number Line in United States School Mathematics in the 19th Century

For contemporary mathematics educators, the number line often serves as a key pedagogical tool for representing real numbers (e.g., Saxe, Diakow, & Gearhart, 2013). Mathematicians and mathematics educators alike support the use of number lines with young children (e.g., Council of Chief State School Officers & National Governors Association [hereafter “CCSSO-NGA”], 2010; Wu, 2011). Even curricula, such as the widely used text Everyday Mathematics (University of Chicago School Mathematics Project, 2001), went so far as to recommend that number lines be hung on classroom walls in all grades, starting in Kindergarten, as a way to facilitate the learning of negative integers (Smiddy, 2008). Number lines are a foundational tool in mathematics, especially for the teaching and learning of negative numbers (e.g., Herbst, 1997; Martínez, 2006; Peled & Carraher, 2008).

As research on student thinking about the teaching and learning of negative integers grows (e.g., Bofferding, 2014; Featherstone, 2000), there has been increased scholarly interest in the use of historical epistemologies to provide suitable theoretical frameworks for investigating student thinking on negative integers (see, e.g., Bishop, Lamb, Philipp, Schappelle, & Whitacre, 2011; Bishop, Lamb, Philipp, Whitacre, Schappelle, & Lewis, 2014). There can be no doubt that the concept of a number line is foundational to this scholarly conversation (e.g., Bofferding, 2014; Saxe et al., 2013). Consequently, it is imperative for mathematics educators to investigate and reflect on various historical perspectives with respect to the use of the number line in school mathematics (Hefendehl-Hebeker, 1991; Hitchcock, 1997).

Although historical developments of a concept may not parallel psychological developments (Sfard, 1991, 1995), wider perspectives on and a better understanding of the past offers educators perspectives on and an understanding of the present, and therefore may help them make decisions wisely for the future. As Sfard (2008) pointed out, “one becomes … bewildered when one notices the strange similarity between children’s misconceptions and the early historical versions of the concepts” (p. 17). In this sense, studying the number line from a historical perspective in school mathematics may provide insight into school mathematics currently.

It is interesting to note that very few models other than the number line have been used to investigate the teaching and learning of negative integers (Peled & Carraher, 2008). Yet, we know very little about the emergence of the number line historically, especially in relation to the development of school mathematics. This article provides a historical perspective on the number line as a mathematical object, which can provide insight into the teaching and learning of negative integers.

Although some mathematicians had conceived of the number line in the seventeenth and eighteenth centuries (e.g., Wallis, 1685 [see Note]), most mathematicians and educators during the period 1600–1900 did not refer to number lines when attempting to make sense of operations on negative integers (Heeffer, 2011; Núñez, 2017). Rather, mathematicians and educators made sense of negative integers by using contexts, such as debts, or they would incorporate geometrical approaches within explanations. Heeffer (2011) presented historical evidence that illustrated how mathematicians struggled to link number lines with operations (e.g., division) with negative numbers.

Purpose

Although extensive research has been conducted in the realm of the history of negative numbers (e.g., Henley, 1999; Schubring, 2005), in recent years research honoring student thinking with respect to the negative numbers has gained momentum (see, e.g., Bofferding, 2014; Bishop et al., 2011; Bishop et al., 2014; Bishop, Lamb, Philipp, Whitacre, & Schapelle, 2016). Yet, specific attention to the historical development of the number line within the teaching and learning of integers has been lacking. Indeed, Heeffer (2011), who has conducted historical research incorporating both number lines and negative numbers, affirmed in a footnote, “A systematic study of the number line in mathematics education needs to be undertaken” (p. 864).

This article begins to address this gap in the history of mathematics education and negative numbers by presenting research that explores the use of the number line in United States (U.S.) arithmetic and algebra texts during the nineteenth century. Specifically, the study reported here addressed the question: In what different ways were number lines presented in nineteenth century texts which were aimed at supporting the teaching and learning of negative integers?

Note: John Wallis (1616-1703) is often credited with the invention of the number line (see, e.g., Heefer, 2011; Núñez, 2017). Núñez (2017) provides a copy of Wallis’s number line as it appears on page 265 of his Treatise of Algebra (1685). The author of this article considers Wallis’s number line to be a relative number line, which will be discussed later in the article.