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Integer Number Lines in U.S. School Mathematics - Discussion

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

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?

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