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## Adding It Up: Helping Children Learn Mathematics (2001) Center for Education (CFE)

### Citation Manager

. "11 Conclusions and Recommendations." Adding It Up: Helping Children Learn Mathematics. Washington, DC: The National Academies Press, 2001.

 Page 418

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Adding + It Up: Helping Children Learn Mathematics

#### Using the Number Line

Students often view the study of whole numbers, decimal fractions, common fractions, and integers as disconnected topics. One tool that we believe may be useful in developing numerical understanding and in making connections across number systems is the number line, a geometric representation of numbers that gives each number a unique point on the line and an oriented distance from the origin, depicting its magnitude and direction. Although it may be difficult to learn, the number line gives a unified geometric representation of integers and rational numbers within the real number system, later to be encountered in geometry, algebra, and calculus. The geometric models of operations afforded by the number line apply uniformly to all real numbers, thus presenting one unified number system. The number line may become particularly useful as students are learning about integers and rational numbers, for it may help students develop a sense of the magnitudes and relationships of those numbers in a way that is less clear in other representations:

• Because it can serve as a tool for simultaneously representing whole numbers, integers, and rational numbers, teachers and researchers should explore effective uses of the number line representation when students learn about operations with numbers, relations among number systems, and more formal symbolic representations of numbers.

#### Expanding the Number Domain

Students currently encounter the expansion of the number domain by starting with whole numbers, gradually incorporating fractions, and only much later expanding the domain to include negative integers and irrational numbers. That sequence has a long history, but there are arguments for an alternative. For example, expanding the whole numbers to take in the negative integers in the early grades would allow students to do more with addition and subtraction before venturing into the rational number system, which requires multiplication and division. Systematic study of this alternative is needed:

• Teachers, curriculum developers, and researchers should explore the possibility of introducing integers before rational numbers. Ways to engage younger children in meaningful uses of negative integers should be devel oped and tested.

 Page 418
 Front Matter (R1-R20) Executive Summary (1-14) 1 Looking at Mathematics and Learning (15-30) 2 The State of School Mathematics in the United States (31-70) 3 Number: What Is There to Know? (71-114) 4 The Strands of Mathematical Proficiency (115-156) 5 The Mathematical Knowledge Children Bring to School (157-180) 6 Developing Proficiency with Whole Numbers (181-230) 7 Developing Proficiency with Other Numbers (231-254) 8 Developing Mathematical Proficiency Beyond Number (255-312) 9 Teaching for Mathematical Proficiency (313-368) 10 Developing Proficiency in Teaching Mathematics (369-406) 11 Conclusions and Recommendations (407-432) Biographical Sketches (433-440) Index (441-454)