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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.

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Adding + It Up: Helping Children Learn Mathematics

tities are unknown, (b) functions describing geometric patterns or numerical sequences, and (c) expressions of the rules governing numerical relationships (see Box 8–1 for an example of each).

Proficiency with representational activities involves conceptual understanding of the mathematical concepts, operations, and relations expressed in the verbal information, and it involves strategic competence to formulate and represent that information with algebraic equations and expressions. Hence, facility with generating expressions and equations combines two of the strands of mathematics proficiency.

The second kind of algebraic activities—the transformational or rule-based activities—includes, for instance, collecting like terms, factoring, expanding, substituting, solving equations, and simplifying expressions. These activities are largely concerned with changing the form of an expression or equation to an equivalent one using the rules for manipulating algebraic symbols. For example, in solving the equation 4(x+3)=2x+19, you can replace the expression 4(x+3) by the equivalent expression 4x+12. Subsequently, by subtracting 2x and then 12 from both sides, the equation 4x+12=2x+19 can be replaced by the equivalent equation 2x=7; finally, dividing both sides by

 Box 8–1 Representational Activities of Algebra There are 3 piles of stones; the first has 5 less than the third, and the second has 15 more than the third. There are 31 altogether. Find the number in each pile. Say to yourself what you see in the picture sequence. T hen state a rule for extending the sequence of pictures indefinitely. The sum of two consecutive numbers is always an odd number. Can you show why, using algebra? SOURCES: Bell, 1995, p. 61; Lee and Wheeler, 1987, p. 160; Mason, 1996, p. 84. Used by permission of Elsevier Science and of Kluwer Academic Publishers.
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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)