Teachers should analyze students' representations and carefully listen to their discussions to gain insights into the development of mathematical thinking and to enable them to provide support as students connect their languages to the conventional language of mathematics. The goals of the Communication Standard are closely linked with those of this Standard, with each set contributing to and supporting the other. These representations are methods for communicating as well as powerful tools for thinking. The process of linking different representations, including technological ones, deepens students' understanding of mathematics because of the connections they make between ideas and the ways the ideas can be expressed. Teachers can gain insight into students' thinking and their grasp of mathematical concepts by examining, questioning, and interpreting their representations.
Young students represent their mathematical ideas and procedures in many ways.
They use physical objects such as their own fingers, natural language, drawings, diagrams, physical gestures, and symbols. Through interactions with these representations, other students, and the teacher, students develop their own mental images of mathematical ideas. Representations make mathematical ideas more concrete and available for reflection. Students can represent ideas with objects that can be moved and rearranged. Such concrete representations lay the foundation for the later use of symbols.