Number Lines to Support Mathematical Understanding

Abstract

The number line has been proposed as a central construct used by students to solve a range of mathematics problems. Given the capacity of number lines to represent all real numbers and to be used in a variety of contexts, there have been calls to increase the use of number lines in mathematics instruction. However, due to the recency of these recommendations coupled with the myriad ways in which number lines can be used, little hands-on guidance has been provided how best to leverage number lines in mathematics instruction. This special series details five areas in which teachers can use number lines to support acquisition of critical mathematics concepts and skills.

Proficiency in mathematics impacts students’ postsecondary success, employment opportunities, and overall well-being (Carroll et al., 2017; Cogan et al., 2019). After all, mathematics is used in nearly every aspect of our lives. Telling time, determining travel distances, and taking appropriate dosages of medications are a few examples of how mathematics is prevalent in daily activities. Yet, most students in U.S. schools are not achieving mathematics proficiency in elementary, middle, or high school as measured by the National Assessment of Educational Progress (NAEP; U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, 2022). For students with learning disabilities (LD) in mathematics, proficiency with core mathematics concepts and skills is even more difficult to attain. On the 2022 NAEP report card, 16% of students with disabilities performed at or above levels of proficiency in mathematics, whereas 39% of students without disabilities performed at or above proficiency in mathematics. These grim outcomes provide a strong rationale for using effective teaching practices in mathematics for all students, and especially for students who have the largest gaps between current achievement and grade-level mathematics proficiency.

Within the past two decades, there has been a surge in mathematics intervention research for students in grades K through 12 who struggle with academic content (Jitendra et al., 2021). Effective instructional practices have emerged from intervention research for students with or at-risk of LD in mathematics. To that end, the Institute for Education Sciences (IES) has published two practice guides focused on identifying high-leverage practices for teaching core mathematics concepts and skills to struggling learners (Fuchs et al., 2021; Gersten et al., 2009). Among the six practices recommended in the most recently published guide was the use of the number line to facilitate learning of grade-level concepts and skills. This practice had a strong evidentiary basis, with 14 high-quality intervention studies showing positive effects on student mathematics outcomes when number lines were used (Fuchs et al., 2021).

One hypothesis for the strong outcomes associated with the number line is that students who are proficient in mathematics can access a mental number line when solving mathematics problems, which has been constructed through a strong understanding of the magnitude of numbers. Prior research has indicated that students with LD in mathematics have deficits in number magnitude understanding (Gersten et al., 2005). Thus, students with LD in mathematics may benefit from using a physical number line when problem solving because it provides a scaffold when a mental number line has not sufficiently developed. For instance, the number line provides a visual representation of the relative magnitude of numbers from zero, as well as from other numbers. This visual may be particularly useful when teaching the magnitude of fractions, for example, so students can begin to understand the size of fractions in relation to nearby whole numbers (Fuchs et al., 2013). Representing different sets of numbers (e.g., whole numbers, fractions, decimals, and irrational numbers) on the same number line can encourage students to conceptualize all numbers as belonging to a unified number system (Siegler & Lortie-Forgues, 2014).

Because of the number line’s meaningful effect on developing students’ numerical magnitude understanding, its use has strong potential to impact student mathematical learning across a range of mathematics domains. Therefore, this special series in Intervention in School and Clinic is devoted to showcasing the versatility of the number line for special educators who work with students with LD in mathematics. Specifically, the special series provides special educators with explicit recommendations for how to incorporate number lines when teaching fundamental concepts and skills across grade levels and mathematics domains, including teaching whole numbers and operations, fractions, and measurement.

Sutherland et al. begin the series with an article focused on the early elementary grades (i.e., kindergarten to second grade). The authors present strategies for how special educators can use number line representations to support students’ development of early number sense, focusing initially on numerical magnitude and transitioning to early concepts of addition and subtraction. Sutherland and colleagues conclude the article with suggestions for how to promote understanding with multi-digit addition and subtraction computation problems.

Rodrigues et al. offer the second article, putting forward suggestions for how special educators can use number lines to support fraction magnitude understanding among students with LD in mathematics in the elementary and intermediate grades (i.e., third to sixth grades). The authors center the article on the six validated recommendations identified in the recent IES practice guide (Fuchs et al., 2021). Using the recommendations as a backdrop, Rodrigues and colleagues present practice activities for building students’ understanding of fraction magnitudes, fraction equivalence, and the comparison of fractions.

The third article is by Gersib et al., who describe using number lines in the context of early measurement instruction for students in the upper elementary grades. Although research in measurement intervention is limited, there is an abundance of evidence for using the number line to support whole number and rational number skill development. Therefore, the authors demonstrate how the number line can support mathematics development across a host of measurement topics, including distance, time, intervals, liquid volume, mass, and money.

Next, Rojo et al. present a teaching sequence for introducing decimal notation. Research has demonstrated that teaching rational number magnitude facilitates learning of more complex skills, such as fraction to decimal translation (Resnick et al., 2016). The teaching sequence places an emphasis on using the number line to teach base-10 fraction magnitude prior to teaching decimal place value. The authors recommend that teachers provide opportunities for students to place different number sets (i.e., fractions and decimals) on the same number line, which lays the foundation for conceptualizing a unified number system.

The final article, by Lesner et al., illustrates how the number line can be used as a visual representation when teaching fraction arithmetic. Students who lack conceptual understanding of fractions (i.e., fraction sense) will struggle with fraction arithmetic, a common issue for students with LD in mathematics (Jordan et al., 2017). Using the number line as an instructional tool can provide a scaffold for understanding basic fraction concepts while students are learning rigorous fraction arithmetic (Schumacher et al., 2018). The article demonstrates how the number line can scaffold understanding of addition, subtraction, multiplication, and division of fractions.

Overall, and as demonstrated by these articles, the number line is a powerful tool for mathematics teaching and learning. By leveraging the number line across settings and grade levels, schools and teachers can provide a coherent framework for students with LD in mathematics to work with number systems on an array of problems. While the examples listed in this series are not exhaustive, the hope is that the recommendations provide teachers with useful examples for incorporating the number line in various settings.