The Number Line: Unifying the Evolving Definition of Number in K–12 Mathematics Brigitte Lahme, Cam McLeman, Michael Nakamaye, and Kristin Umland Introduction content. But for a comprehensive K–12 curriculum, def- Beginning in the spring of 2016, a team of more than thirty initions and notation must evolve alongside student un- mathematics teachers, mathematics educators, and mathe- derstanding, and diagrams and other representations must maticians at Illustrative Mathematics began to write a change over time to accommodate and facilitate the expan- grades 6–8 mathematics curriculum that was released as sion of these ideas. In this article, we discuss some of the an open education resource in the summer of 2017. Since mathematical and pedagogical nuances of making K–12 then the team has expanded, and Illustrative Mathematics curriculum-wide decisions about the selection and adop- released a high school curriculum in the summer of 2019 tion of a coherent and rigorous set of definitions, notation, and will release a grades K–5 mathematics curriculum in and graphical conventions that both hold up to mathemat- the summer of 2021. The four authors of this article are ical scrutiny and also serve the needs of the students to mathematicians who have worked together since the in- whom they are introduced. ception of Illustrative Mathematics in 2011 on these and In particular, we will focus our attention on curriculum other K–12 mathematics projects. design issues surrounding the development of the real Over the time we have worked together, we have en- numbers from kindergarten to grade 12, keeping in mind joyed the surprising variety of interesting mathematical is- the work that students might do in later years. Because our sues that arise when writing about K–12 mathematics. For work is rooted in the Common Core State Standards for example, choosing a unique best definition for a given Mathematics [1], we constrain our discussion to the devel- mathematical concept may be a sensible undertaking for opment of a K–12 mathematics curriculum designed with a textbook covering a single semester or a year’s worth of these standards in mind, and all subsequent discussion of grade-level work refers to the sequential development of Brigitte Lahme is a professor of mathematics at Sonoma State University. Her the mathematics in these standards. email address is [email protected]. Cam McLeman is an associate professor of mathematics at the University of Numbers and the Number Line Michigan–Flint. His email address is [email protected]. Developing the notion of a number requires attention to Michael Nakamaye is Mathematician in Residence with Illustrative Mathemat- both the relevant mathematics and how students learn that ics. His email address is [email protected]. mathematics. Students need some understanding of the Kristin Umland is the Vice President for Content Development at Illustrative Mathematics. Her email address is kristin real number system by the time they complete high school, @illustrativemathematics.org. and yet the real numbers are notoriously more subtle to Communicated by Notices Associate Editor William McCallum. define and work with than is typically evident from their For permission to reprint this article, please contact: treatment in instructional materials. None of the classi- [email protected]. cal constructions or definitions from a real analysis course (via Cauchy sequences, Dedekind cuts, or as the unique DOI: https://doi.org/10.1090/noti1959 OCTOBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1465 ordered field satisfying certain axioms) are particularly use- the whole numbers by placing the number 2 on the op- ful for most K–12 students (though neither are they far posite side of 1 from 0 so that 1 is equidistant from both, from our minds, e.g., infinite decimal expansions as and so on. Later, students will reason through the loca- Cauchy sequences of finite approximations). Of course, tion of the positive rational numbers on the number line even a set-theoretic mathematical definition of the count- by partitioning the segment from 0 to 1 into equal-length 1 ing numbers is out of the question, and yet the need re- pieces to identify unit fractions (fractions of the form 푏 , mains: students learn about and make use of the counting for 푏 a positive integer). Then, by adding a whole num- numbers starting in kindergarten and expand and refine 1 푎 ber, 푎, of copies of 푏 they can place any fraction 푏 , just their notion of what a number is many times in the twelve as the location of the whole number 푎 was deduced from years of schooling that follow. Students need working def- the location of 1. Negative rational numbers are located initions of the various types of numbers they encounter by finding the point that is the same distance frombut that allow them to reason deductively within the frame- on the opposite side of 0 as its positive counterpart. The work of their current stage of learning. Because the concept real numbers are located via an exploration of decimal ex- of a number expands from year to year, students’ working pansions (with some hand-waving around limits). Thus, definitions must evolve accordingly over time. as students’ understanding of number evolves from whole H. S. Wu advocates that we define a real number asa numbers to rational numbers to real numbers, we enlarge point on a number line [3]. However, even if this is the def- the subset of points on the number line that correspond to inition adopted in later grades, students in kindergarten numbers, eventually filling out that initial Euclidean line and grade 1 need something more concrete to reason from. as a visual representation of the set of all real numbers. We agree that number line diagrams should be the primary A number line diagram is a drawing that represents an in- representation of numbers once students are sophisticated terval of the number line. The question of how to draw enough to learn about and interpret these types of repre- number line diagrams is tied intimately with pedagogical sentations. By tying any understanding of a number to the decisions about the use of the number line itself, and so number line, students can expand their understanding of we are left with a slew of questions. Among others, of prin- the real number system in an intuitive but not mathemati- cipal interest are the following: cally misleading way. While we must attend to long-term goals for how we want students to think about mathemati- • When should number lines first be introduced? cal concepts, we must also respect the intellectual work ap- • What conventions for drawing number line dia- propriate to each grade level to ensure that students have grams should be followed at each grade level? a strong foundation on which to build a more abstract un- • Should operations be represented on a number derstanding of number in later grades. Curriculum writers line diagram? If so, how? must pay careful attention to the evolution of the work- We address these and other questions below. ing definitions that students will use at various stages of learning. We assert that a good curriculum will help stu- Working Definitions and Representations for dents build their understanding of the counting and ratio- Number Systems nal numbers as telling “how many/how much” and then We begin by describing the development of the real num- help them connect this understanding to the representa- ber system across grade levels. Students study the whole tion of a number as a point on the number line. This numbers in kindergarten through grade 2, the positive ra- stance raises a number of important mathematical ques- tional numbers in grades 3–6, the rational numbers in tions related to the use of number line diagrams in cur- grades 6 and 7, and the real numbers in grade 8 and be- riculum materials. yond (although they are introduced to 휋 in grade 7, they For our work the number line begins as a line in the usual do not explore the more general notion of nonrational Euclidean sense, which is used as a geometric model for numbers until grade 8). Table 1 shows the way in which the real numbers. Two distinct points are chosen and iden- the number system expands across K–8. Beyond the count- tified with the numbers 0 and 1 (which establishes amet- ing numbers, students first study a new set of numbers as ric on the line). As new sets of numbers are introduced, objects in their own right before they study the four arith- a process for locating them on the number line is also in- metic operations (+ − × ÷) in the context of each system. troduced, so that the difference between the two numbers To support the understanding expected of students in is reflected in the displacement on the line between the later grades, early elementary students need working defi- two corresponding points. The mechanism for locating nitions for the whole numbers, addition, and subtraction further points is grade-level dependent: in early grades stu- that are developmentally appropriate, robust enough to dents locate points on the number line that correspond to support mathematical reasoning, and engineered to admit 1466 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 9 Grade K 1 2 3 4 5 6 7 8 accessible at this age (see [2]). Discrete representations are + + + + − the primary way that students in grades K and 1 represent ℤ ≥0 − − − × ÷ numbers and operations, and a number path can provide a + − + − + − ℚ≥0 numbers bridge to continuous representations like number line dia- × × ÷ × ÷ grams. Figure 2 can be viewed as an intermediary between + − ℚ numbers × ÷ Figures 1 and 3.