# The Number Line: An Overview

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2 The Big Ideas Learning to Think Mathematically with the Number Line As noted above, the number line stands in contrast to other manipulatives and mathematical models used within the number realm. Some of the reasons for developing the number line as a foundational tool are illustrated below as key ideas for this textbook. Key Concept #1: The linear character of the number line. The number line is well suited to support informal thinking strategies of students because of its inherent linearity. In contrast to blocks or counters with a set-representation orientation, young children naturally recognize marks on a number line as visual representations of the mental images that most people have when they learn to count and develop understanding of number relationships. It is important to note the difference between an open number line (shown below) and a ruler with its predetermined markings and scale. An open number line: The open line allows students to partition, or subdivide, the space as they see fit, and as they may need, given the problem context at hand. In other words, the number line above could be a starting point for any variety of number representations, two of which are shown below: the distance from zero to 1, or the distance from zero to 1. Once a second point on a number line has been identified, the number line moves from being an open number line, to a closed number line. In addition, the open number line allows for flexibility in extending counting strategies from counting by ones, for example, to counting by tens or hundreds all on the same sized open number lines. Closed number lines: 1 1 Key Concept #2: Promoting creative solution strategies and intuitive reasoning. A prevalent view in math education reforms is that students should be given freedom to develop their own solution strategies. But to be clear, this perspective does not mean that it is simply a matter of allowing students to solve a problem however they choose. Rather, the models being promoted by the teacher should themselves refine and push the student toward more elegant, sophisticated, and reliable strategies and procedures. This process of formalizing mathematics by having students recognize, discuss, and 6

3 internalize their thinking is a key principle in math education reforms, and is one that can be viewed clearly through the use of a model like the number line a tool that can be used both to model mathematical contexts, but also to represent methods, thinking progressions, and solution strategies as well. As opposed to blocks or number tables that are typically cut off or grouped at ten, the open number line suggests continuity and linearity -- a representation of the number system that is ongoing, natural, and intuitive to students. Because of this transparency and intuitive match with existing cognitive structures, the number line is well suited to model subtraction problems, for example, that otherwise would require regrouping strategies common to block and algorithmic procedures. Key Concept #3: Cognitive engagement. Finally, research studies have shown that students using the empty number line tend to be more cognitively active than when they are using other models, such as blocks, which tend to rely on visualization of stationary groups of objects. The number line, in contrast, allows students to engage more consistently in the problem as they jump along the number line in ways that resonate with their intuitions. While they are jumping on the number line, they are able to better keep track of the steps they are taking, leading to a decrease in the memory load otherwise necessary to solve the problem. For example, imagine a student who is trying to solve the problem: Under the traditional addition algorithm, or with base-1 manipulatives as well, the regrouping strategy, or the carry the one algorithm is significantly different cognitively than thinking of this addition problem as a series of jumps. Specifically, the student might represent the problem as: 1) Start at 39; 2) Jump 1; 3) Jump 1 more; 4) jump three. These steps can easily be mirrored on the number line as the student simply traces her thinking with a pencil = Like any mathematical tool, the more teachers are aware of both the benefits and constraints of the model, the more likely they are to use it effectively with students. Throughout this book, the previous big ideas though theoretical in nature are drawn upon repeatedly as students view and subsequently manipulate various open number lines that are used to represent numerous mathematical contexts and operations. Teaching Ideas A large portion of this book is devoted to helping students develop a rich sense of numbers and their relationships to one another. The number line is centrally related to this task. As noted above, perhaps the most important teaching point to convey regarding the number line is the notion that, unlike a ruler, it is open and flexible. Given this starting point, students will quickly recognize that they need to create their own 7

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