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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

in the problem, they might be asked to use a table of values in which the first column points to a position in the sequence and the second column gives the corresponding number of dots.67

 Sequential position (x) of the triangular number Number of dots (y) in the triangular number 1 1 2 3 3 6 4 10 · · · · · ·

Two kinds of rules describe this table. One, the recursive rule, is based on an analysis of the growth occurring in the right-hand column. For the nth triangle, add n dots to the number of dots in the previous triangle. But this right-hand regularity, which is not too difficult to detect, is easier to say in words than to symbolize algebraically. The other kind of rule, the closed form, requires analyzing both columns together to try to determine a relationship between a member of the left-hand column and the corresponding member in the right-hand column. Algebra students have more difficulty deriving the latter rule, y=x(x+1)/2, than the former.68

The use of computer technology can enable students to engage in activities like those above without having to generate or transform algebraic equations on their own.69 But students have to learn how to use the equations produced by the technology to make predictions, even if they do not actually generate them by hand.

Through an emphasis on generalization, justification, and prediction, students can learn to use and appreciate algebraic expressions as general statements. More research is needed on how students develop such awareness. At the same time, more attention needs to be paid to including activities in the curriculum on identifying structure and justifying. Their absence is an obstacle to developing the “symbol sense”70 that constitutes the power of algebra.

### Algebra for All

Because of advancements in the use of technology and its prevalence today, a greater understanding of the fundamentals of algebra and algebraic reasoning is viewed as necessary for all members of society, including those

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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)