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

### Citation Manager

. "6 Developing Proficiency with Whole Numbers." Adding It Up: Helping Children Learn Mathematics. Washington, DC: The National Academies Press, 2001.

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Adding + It Up: Helping Children Learn Mathematics
 Box 6–10 Model for Multidigit Subtraction: 1444–568=?

The goal is to fix the top number so that every top digit is larger than the corresponding bottom digit. The second major step is then to subtract in every column. This subtraction can also be done in any direction. Methods B and C clarify that the top number is a single number that must be rewritten in a form equivalent in value but ready for subtraction in every column. This rewriting can reduce the otherwise frequent “top from bottom” error.73

The drawing in Box 6–10 shows how students can make a quantity drawing to show both aspects of multidigit subtracting. Making such drawings initially can help students develop their own procedures or help them make sense of an algorithm presented by someone else. Again, such drawings should, when used, be linked to a numerical method and not just used to calculate an answer.

##### Multiplication Algorithms

There is much less research on children’s understanding of multidigit multiplication (and division) than of addition and subtraction. Sample conceptual teaching lessons have been published for multiplication, and some alternative methods of instruction have been explored.74 A preliminary learning progression of multidigit procedures that fosters children’s invention of

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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)