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## 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.

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 Page 77

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Adding + It Up: Helping Children Learn Mathematics
 Box 3–1 Properties of the Arithmetic Operations Commutativity of addition. The order of the two numbers does not affect their sum: 3+5=8=5+3. In general, m+n=n+m. Associativity of addition. When adding three (or more) numbers, it does not matter whether the first pair or the last pair is added first: (3+5)+4=8+4=12=3+ 9=3+(5+4). In general, (m+n)+p=m+(n+p). Commutativity of multiplication. The order of the two numbers does not affect their product: 5×8 produces the same answer as 8×5. In general, m×n=m×n. Associativity of multiplication. When multiplying three or more numbers, it does not matter whether the first pair or the last pair is multiplied first: 3×(5×4) is the same as (3×5)×4. In general, (m×n)×p=m×(n×p).
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 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)