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

### Citation Manager

. "8 Developing Mathematical Proficiency Beyond Number." Adding It Up: Helping Children Learn Mathematics. Washington, DC: The National Academies Press, 2001.

 Page 273

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Adding + It Up: Helping Children Learn Mathematics

Box 8–6 Two Methods for Solving Equations

 Cover-Up Method Work Backward (Undoing) Method 2x+9=5x 2x+5=11 Cover up 9. Undo adding 5; subtract 5 from 11; Since 2x+(cover up)=5x, 2x=6 9=3x. Undo multiplying by 2; divide 6 by 2; Cover up x. x=3. Since 3 times (cover up) equals 9, x=3.

For equations involving multiple operations, such as 3x+4–2x=8, they erroneously generalized their method and simply undid each operation as they came to it. For example, they would take 8, divide it by 2, add 4, and then subtract 3. (They had to ignore the last operation of multiplication because they had run out of operands.) A preference for the undoing method of equation solving seemed to work against the students when they were later taught the procedure of performing the same operation on both sides of an equation. The students who preferred the undoing method were, in general, unable to make sense of “performing the same operation on both sides.” The instruction seemed to have its greatest impact on those students who had an initial preference for the informal method of substitution and who viewed the equation as a balance between left and right sides. This observation suggests that learning to operate on the structure of a linear equation by performing the same operation on both sides may be easier for students who already view equations as entities with symmetric balance and not as statements about a calculation on the left side and the answer on the right.

Despite the considerable body of research on creating meaning for the transformational activities of algebra, few researchers have been able to shed light on the long-term acquisition and retention of transformational fluency. In one study, students were able to produce a meaningful justification for

 Page 273
 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)