Instructional programs from
prekindergarten through grade 12 should enable all students to
 create and use representations to organize, record, and communicate
mathematical ideas;
 select, apply, and translate among mathematical representations to solve
problems;
 use representations to model and interpret physical, social, and
mathematical phenomena.

Representations are necessary to students'
understanding of mathematical concepts and relationships. Representations allow
students to communicate mathematical approaches, arguments, and understanding to
themselves and to others. They allow students to recognize connections among
related concepts and apply mathematics to realistic problems.
To become deeply knowledgeable about fractions,
for example, students need a variety of representations that support their
understanding. They need to understand various interpretations of fractions,
such as ratio, indicated division, or fraction of a number. They need to
understand other common representations for fractions, such as points on a
number line.
Some forms of representationsuch as diagrams,
graphical displays, and symbolic expressionshave long been part of school
mathematics. Unfortunately, these representations and others have often been
taught and learned as if they were ends in themselves. This approach limits the
power and utility of representations as tools for learning and doing
mathematics.
It is important to encourage students to
represent their mathematical ideas in ways that make sense to them, even if
those representations are not conventional. At the same time, students should
learn conventional forms of representation in ways that facilitate their
learning of mathematics and their communication with others about mathematical
ideas. The integration of technology into mathematics instruction further
increases the need for students to be comfortable with new mathematical
representations.