Questions? Call 800-624-6242

| Items in cart [0]

PAPERBACK
price:\$34.95

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

### Citation Manager

. "3 Number: What Is There to Know?." Adding It Up: Helping Children Learn Mathematics. Washington, DC: The National Academies Press, 2001.

 Page 82

The following HTML text is provided to enhance online readability. Many aspects of typography translate only awkwardly to HTML. Please use the page image as the authoritative form to ensure accuracy.

Adding + It Up: Helping Children Learn Mathematics

By means of somewhat lengthy reasoning, you can find out how to do arithmetic with integers. But are the regularities observed about the whole number system (the rules in Box 3–1) still valid? Going through the cases again will show that they are. So not only has the number system been extended from the whole numbers to all integers, but the arithmetic in the larger system looks very similar to arithmetic in the original one in the sense that these laws are still valid.

Moreover, there are some new notable regularities that describe how the new numbers are related to the original ones. These are summarized in Boxes 3–2 and 3–3.

The extension of whole numbers to integers is an example of the axiomatic method in mathematics: basing a mathematical system on a short list of key properties.

Something much more dramatic is also true. One can show that, if the goal is to extend addition and multiplication from the whole numbers to the integers in such a way that the laws of arithmetic of Boxes 3–1 and 3–2 remain true, then there is only one way to do it. And the rules in Box 3–3 describe how it has to work. Recipes laboriously constructed by means of some sort of concrete interpretation of negative numbers are all completely dictated by this short list of rules of arithmetic. This uniqueness is a striking exhibition of the power of these rules—that they capture in a few general statements a large chunk of people’s intuition about arithmetic. The extension of whole numbers to integers is an example of the axiomatic method in mathematics: basing a mathematical system on a short list of key properties. Its most famous success is the Elements of Euclid for plane geometry. Since Euclid’s time, axiomatic schemes have been constructed to cover most areas of mathematics.

Another rather striking thing has happened during this extension from whole numbers to (all) integers. The reason for making the extension was to

 Box 3–2 Additional Properties of Addition Additive identity. Adding zero to any number gives that number. For example, 3+0=3 and 0+3=3. In general, m+0=m, and 0+m=m. Additive inverse. Every number has an additive inverse, also called an opposite. The opposite is the unique number that, when added to that number, gives zero. For example, the opposite of 3 is –3 because 3+–3=0; the opposite of –4 is 4 because –4+4=0. In general, –s is the unique solution m for s+m=0.
 Page 82
 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)