Numbers and the Base-Ten System 

What are numbers and where do they come from? What humans consider to be numbers has evolved over the course of history, and the way children learn about numbers parallels this development. When did humans first become aware of numbers? The  answer is uncertain, but it is at least many tens of thousands of years ago. Some scholars  believe that numbers date back to the beginning of human existence, citing as a basis for  their views the primitive understanding of numbers observed in some animals.  In this chapter, we discuss elementary ideas about numbers, which reveal surprising intricacies that we are scarcely aware of as adults. We will study the base-ten system—a  remarkably powerful and efficient system for writing numbers and a major achievement  in human history. Not only does the base-ten system allow us to express arbitrarily large  numbers and arbitrarily small numbers—as well as everything in between—but also it  enables us to quickly compare numbers and assess the ballpark size of a number. The  base-ten system is familiar to adults, but its slick compactness hides its inner workings.  We will examine with care those inner workings of the base-ten system that children must  grasp to make sense of numbers.  In this chapter, we focus on the following topics and practices within the Common Core  State Standards for Mathematics (CCSSM). 

Standards for Mathematical Content in the CCSSM  

In the domain of Counting and Cardinality (Kindergarten) young children learn to say and  write small counting numbers and to count collections of things. In the domain of Numbers and Operations in Base Ten (Kindergarten through Grade 5), students learn to use  the powerful base-ten system. This system starts with the idea of making groups of ten  and gradually extends this idea to the greater and to the smaller place values of decimals.  

Standards for Mathematical Practice in the CCSSM  

Opportunities to engage in all eight of the Standards for Mathematical Practice described  in the CCSSM occur throughout the study of counting and the base-ten system.

The following standards are especially appropriate for emphasis while studying counting and the  base-ten system:  

• 2 Reason abstractly and quantitatively. Students engage in this practice when  they make sense of number words and symbols by viewing numbers as representing quantities and when they use numbers to describe quantities.  

• 5 Use appropriate tools strategically. The base-ten system represents numbers  in a very compact, abstract way. By reasoning with appropriate tools, such as  drawings of tens and ones or number lines that show decimals, students learn to  make sense of the powerful base-ten system. 

 • 7 Look for and make use of structure. The base-ten system has a uniform structure, which creates symmetry and patterns. Students engage in this practice when  they seek to understand how increasingly greater base-ten units can always be created and how the structure of the base-ten system allows us to compare numbers  and find numbers in between numbers. 

Chapter Summary