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Integer Number Lines in U.S. School Mathematics - Evolution of Their Use

Nicole M. Wessman-Enzinger (George Fox University)

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.

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

Nicole M. Wessman-Enzinger (George Fox University), "Integer Number Lines in U.S. School Mathematics - Evolution of Their Use," Convergence (February 2018)