Number Sense on the Number Line

Abstract

A strong foundation in early number concepts is critical for students’ future success in mathematics. Research suggests that visual representations, like a number line, support students’ development of number sense by helping them create a mental representation of the order and magnitude of numbers. In addition, explicitly sequencing instruction to transition from concrete to visual to abstract representations of mathematics concepts supports students’ conceptual understanding. This column describes and illustrates how teachers can use number lines and features of explicit and systematic instruction to support students’ early development of number sense.

Keywords

lesson planning number line number sense visual representation explicit instruction

Young children understand the meaning of numbers from a very early age. Without knowing it, they develop a firm grasp of concepts such as cardinality (“I have three candies.”) and the ordinal meaning of numbers (“I came in first place. You came in second.”). Quickly, they begin to understand concepts such as quantity (“I have a lot of candies.”) and can begin to make quantity comparisons (“No fair! You have more candies than me.”). These concepts form the basis of children’s development of number sense, or their understanding of what numbers mean and their ability to flexibly use numbers to make comparisons and perform operations (Berch, 2005; Gersten & Chard, 1999).

Some young children may have difficulty grasping these early number concepts. To prevent future difficulties, these children need instructional support to help them make connections between their concrete experiences and abstract mathematics concepts (Geary, 1993). Children who continue to have persistent difficulties may be at risk for mathematics disabilities. Research suggests that incorporating visual representations and using critical features of explicit and systematic instruction support the opportunities that young children who are at risk for mathematics difficulties, including those who have been diagnosed with learning disabilities (LD) or mathematics difficulties (MD), have to access and learn from core mathematics instruction (Bryant et al., 2008).

This column describes and illustrates how teachers can use visual representations and features of explicit and systematic instruction to incorporate number lines into their instruction to support students’ early development of number sense. Specifically examined are (a) the value of using visual representations, focusing on the number line, to illustrate mathematics concepts, (b) an explicit and systematic instructional sequence for using the number line as a tool to model and develop children’s number sense, (c) a sample of student work, and (d) a sample lesson plan illustrating the use of a number line for teaching children how to make quantity comparisons. The information presented in this article is derived from research on instructional practices designed to support students with or at risk for LD or MD (Fuchs & Fuchs, 2001; Fuchs et al., 2008; Morgan, Farkas, & Maczuga, 2015; Powell & Fuchs, 2012).

Developing Number Sense

Even before children enter kindergarten, many have developed a strong informal sense of numbers through play, interactions with older children or adults, and direct interactions with their environment. Often, their understanding of natural and whole numbers grows from their desire to find patterns and resolve problems as they work with and manipulate concrete objects through activities that investigate counting, comparing, sorting, joining, and sharing (Baroody, 2000; Clements & Sarama, 2014). These experiences lay the foundation for developing a formal understanding of mathematics in school.

In early school experiences, mathematics instruction is intended to help children begin connecting these informal representations with formal mathematics (Frye et al., 2013). Children start to associate the quantity and verbal representation of numbers with the symbolic representation expressed as a numeral and word (see Figure 1), link the names of geometric shapes with the formal attributes that generalize across representations of the shape, and sort objects based on specific criteria. By the time these children reach kindergarten, many have already mastered basic number identification, shape recognition, and counting of single-digit numbers (Engel, Claessens, & Finch, 2013).

Figure 1.

Children start to associate quantity and verbal representation of numbers with the symbolic representation expressed as a numeral and word.

With many children coming to kindergarten already having made meaningful connections between their informal and formal mathematical knowledge, kindergarten teachers may need to carefully consider the content they prioritize during instruction. In a recent study with more than 11,500 children from across the country, Engel et al. (2013) found that 95% of the students had already mastered fundamental concepts such as basic counting and shape recognition by fall of kindergarten, and about 65% had mastered patterns and measurement. However, many teachers spent upwards of 13 days per month teaching these concepts that students had already mastered, whereas very few days per month were spent teaching more advanced mathematics concepts, such as understanding place value and developing proficiency with addition and subtraction. Only the small percentage of students who had not made these connections between informal and formal mathematics before entering kindergarten benefited from instruction focused on these foundational concepts. For the majority of students, however, focusing instruction on these foundational concepts had a negative effect on their achievement at the end of the year (Engel et al., 2013).

Because children’s early mathematics skills are important predictors of future achievement in mathematics and other subjects (Claessens & Engel, 2013), these results should be considered a call for action for designing mathematics instruction that is responsive to the needs of all students in early elementary classrooms. Recent research points to the value of using visual representations of mathematics concepts for supporting the development of students’ mathematics understanding. Similar findings have been found for students in general education (National Council of Teachers of Mathematics, 2014), those who are at risk, and children with LD or MD (Gersten et al., 2009). As such, a solution to the challenge of designing mathematics instruction that is responsive to all students’ needs could be as simple as changing the instructional tools that teachers use to teach early mathematics concepts. An approach is described for using a number line during early elementary mathematics instruction that can be responsive to individual students’ learning needs while simultaneously enhancing students’ conceptual understanding of numbers.

Why Are Number Lines So Powerful?

The Common Core State Standards for Mathematical Practices suggests that students should be provided with opportunities to use appropriate tools strategically as they learn about mathematics (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). These tools include a wide range of materials such as models, visual representations, calculators, and software programs. A particularly powerful tool that supports students’ understanding of numerous mathematics concepts is the number line (Gersten et al., 2009). A number line is a visual representation that illustrates the order and magnitude of numbers. The number line is both correlational and causally related to arithmetic learning (Booth & Siegler, 2008) and can be used to teach a variety of concepts. For example, students can learn the ordinal relationship of numbers by learning that each number has a specific location on the number line and that subsequent numbers are sequenced in a specific pattern. Similarly, as students learn that numbers represent a distance or quantity from zero, they learn about cardinality. And, as students learn that each equal-size interval on a number line represents a specific unit, they can extend their counting skills from (a) counting a number of objects in a set to (b) counting units of length (Van de Walle, Karp, & Bay-Williams, 2012). Learning these concepts and skills by working with number lines may help students develop a mental representation of the order and magnitude of numbers that can then be used to make comparisons, understand place value, and model mathematical operations (Diezmann & Lowrie, 2006). As Engel et al. (2013) noted, purposefully targeting these concepts during early mathematics instruction may support students’ conceptual understanding of more advanced concepts as early as kindergarten.

Students with LD or MD may enter elementary school without proficiency in foundational mathematics concepts. For these students, visual representations like the number line can help them translate their informal mathematics knowledge to formal symbolic representations. Research points to the positive outcomes for students with or at risk for LD or MD when instruction is carefully sequenced to transition from concrete to visual to abstract representations (Gersten et al., 2009; Witzel, Mercer, & Miller, 2003). Moreover, students with LD in mathematics are less accurate than their typical peers in their number line representation and, consequently, may benefit from explicit and systematic instruction on how to interpret and use the number line (Geary, Hoard, Nugent, & Byrd-Craven, 2008). As such, incorporating the number line during early elementary mathematics instruction may help learners with and without disabilities develop proficiency with early mathematics concepts.

Using the Number Line in Instruction

With some advanced preparation, number lines can be easily incorporated into early elementary mathematics instruction. Explicit and systematic instruction provides a framework for designing instructional routines that use number lines. A series of general steps follows that describes how to incorporate the number line into early elementary mathematics instruction (see Figure 2). Next, the recommended steps are used during the planning of a lesson on basic mathematical operations. Finally, an illustrative example including a sample of student work and a sample lesson plan for magnitude comparisons is presented.

Figure 2.

General steps for incorporating the number line into mathematics instruction.

The first step identified in Figure 2 is to consider the objective of the mathematics lesson and whether a number line would serve as a useful model. Many topics, including counting sequences, quantity comparisons, and operations, can be effectively modeled on a number line (Van de Walle et al., 2012). As with any instructional model, the number line is a tool for teaching the mathematical content; the number line itself is not the content. Therefore, the first step is to make sure that the number line can be used as an effective tool to build students’ conceptual understanding, given the focus of instruction.

It is important to recognize that some students may have difficulty understanding how to interpret a number line. These students may struggle to shift their thinking from counting the number of objects in a collection to counting units of length (Van de Walle et al., 2012). Because a number line measures distances from zero (like a ruler), a focus on the unit length may help young learners and students with LD or MD focus on the spaces and not the hash marks or numerals. A number path is another tool that can be used to teach children about unit length and may help bridge their understanding to number lines. Playing board games that use number paths (e.g., Chutes and Ladders) can also help introduce the concept of length. Siegler and Ramani (2009), for example, found that playing board games with number paths helped students from low socioeconomic backgrounds develop a better concept of numerical magnitude, which subsequently helped them to estimate more accurately on a number line. Similarly, Cuisenaire rods of the same color can be connected to build a number path that illustrates the number of units within a specific length (Dougherty, 2008). These types of representations of number paths may also help students shift from the concrete representation when counting units of length to the visual representation of a number line (Van de Walle et al., 2012).

The second step for using a number line during instruction is to prepare the number line models that the teacher and students will use. Because equally spaced intervals are an important attribute of a number line, it is difficult to hand draw an accurate number line, for both the teacher and the student. Instead, post a number line on the wall and have practice number lines ready for the students. In addition, include arrows on both ends of the number line so that students see that the number line is infinite.

Third, organize instruction to include explanations and ample practice using the number line. Using the features of explicit and systematic instruction as a guide, instruction should begin with a clear demonstration that includes a verbal description of the teacher’s thinking and reasoning. Next, the teacher can engage students in guided practice. Because some students may not be familiar with the number line representation, the teacher needs to build in multiple opportunities for students to practice using the number line (Dougherty, Flores, Louis, & Sophian, 2010). Students with LD or MD may benefit from scaffolded practice opportunities that include both student verbalizations and corrective feedback (Gersten et al., 2009). Once students demonstrate proficiency with guided practice, students can engage in more independent practice.

Fourth, and finally, compose high-level questions that challenge all students to think and talk about the content (Thornton, Langrall, & Jones, 1997). Kazemi and Hintz (2014), as well as the National Council of Teachers of Mathematics (2014), suggested that students’ mathematical reflections may be increased when asked questions that require them to think deeply about mathematics, such as, “Why do you think that?” and “How do you know?” These open-ended questions provide students with the opportunity to explain their thinking and reasoning about the mathematics concepts being taught. Such opportunities to verbalize are essential for supporting students’ understanding when other modes of communication (i.e., writing) may not be instructionally appropriate (Doabler et al., 2014).

Incorporating the Number Line

The instructional sequence presented in Figure 2 could be extended to model mathematics content using the number line, such as basic operations. Therefore, Figure 3 presents a series of steps to consider when incorporating the number line into instruction focused on basic operations.

Figure 3.

Steps for incorporating the number line into basic operations mathematics instruction.

Consider, for example, the objective of modeling addition. Because addition is about joining numbers, a number line could provide a powerful visual representation by modeling what the joining of two (or more) lengths looks like. Once the number lines are prepared and students know how to use the number line, the teacher can begin to model the process of addition. Here, the teacher draws two distances with different colored markers on the number line (see Figure 4). Next, the teacher discusses the length of each segment by posing questions such as, “What is the length of the gray line? How do you know?” Once students understand the length of both segments, a teacher can facilitate a discussion by asking students questions such as, “What is the total length of the gray and black lines?” and “Why do you think that?” Teachers can probe students’ thinking further by asking questions that illustrate the relationship between addition and subtraction (e.g., “If I took away the gray line, how much would be remaining?”) and number properties (e.g., “If we rearranged the gray and black lines, what is the total length? Is it the same or different? Why does this happen?”).

Figure 4.

What is the total length of the gray and black lines?

Modeling Addition: A First-Grade Example

In a representative vignette (see Figure 5), a discussion is presented between a first-grade teacher and her student. This vignette starts with practice and extends into talk with the teacher representing the numbers 6 and 12 on the number line followed by guided questioning to elicit student thinking. The student generates a response but the teacher explicitly prompts the student to model the mathematical concept on the number line. Subsequently, the student explains how she arrived at the answer using the representation she created on the number line and describes addition in relation to units of length. As the teacher continues to ask questions, the student elaborates on her understanding of the relationship between addition and subtraction and the commutative property of addition.

Figure 5.

This vignette provides an example of a teacher working with a first-grade student using the number line to show 6 + 12 = 18.

Making Comparisons: Sample Lesson

Applying these ideas for using the number line as a tool to support mathematics instruction together with features of explicit and systematic instruction, the lesson (see Figure 6) illustrates how a number line can support students in making quantity comparisons. It is important to note that evidence-based instructional design features, such as those used in this lesson, improve the ability of students with or at risk for LD or MD to learn from core instruction (Bryant et al., 2008; Doabler, Nelson-Walker, Fien, & Baker, 2012). Features of explicit and systematic instruction in this lesson (Archer & Hughes, 2011) include (a) the big idea, or broad conceptual mathematics topic addressed in the lesson, (b) explicit reference to the content standards addressed by the lesson, (c) a clear statement of the lesson objective so that students understand the goal of the learning activity, (d) scripted language that the teacher can use to help introduce the mathematical concept using the number line, and (e) sample questions to facilitate more in-depth mathematical discussions with students. In this lesson, these features of explicit and systematic instruction, as well as the number line, are used to provide students with a visual representation of the order and magnitude of numbers while providing a framework that teachers can use to elicit and respond to students’ thinking about mathematics. Moreover, by using the number line to teach this concept, teachers support students’ conceptual understanding of magnitude and extending that understanding to make comparisons.

Figure 6.

This is a sample lesson that illustrates how teachers may use the number line as a tool to compare two numbers.

Conclusion

This column illustrated how the number line can be used as an instructional model to enhance and support the foundational conceptual understanding of early mathematics concepts for all students, including students with LD or MD. For students with LD or MD or who are at risk, the number line may provide an important link between concrete representations of early mathematics concepts and more abstract mathematical notation. Some students may need explicit instruction on how to interpret the number line or may need to work with a number path prior to using the number line. In addition, some students may benefit from instructional adaptations such as constraining the range of the number line (e.g., 0–5), using graph or grid paper to visualize the equal intervals on the number line, or using a raised number line for tactile stimulation (e.g., outline the number line with glue and allow to dry). These adaptations should support students’ access to the content and should coincide with allowable accommodations.

Systematically modeling mathematics concepts on a number line and teaching students how to represent relationships on a number line may support students’ understanding of early mathematics concepts. For example, number lines can support students’ understanding of counting and magnitude, part–whole relationships, and mathematical operations such as addition, subtraction, and multiplication by making these concepts concrete (Diezmann & Lowrie, 2006). It is important that these early mathematical or number sense concepts form the basis of early mathematics performance, which is strongly predictive of later mathematics achievement (Claessens & Engel, 2013). Moreover, systematically incorporating precise visual representations like the number line into mathematics instruction as early as kindergarten and continuing across the grades will not only support the development of students’ understanding of increasingly complex mathematics concepts (e.g., proficiency with operations and algebraic thinking) but also equip students with a mathematical tool that can support their thinking over time.

Footnotes

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

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