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. Students' representations are often insightful and many times resemble more-conventional representations. The tap helps the student connect the zero with the empty tens column. The following account of a lesson, drawn from a classroom experience, illustrates that what children do and say as they find answers and represent their thinking gives teachers information about their levels of understanding. Two cats, then three frogs, four turtles, and five fish joined the rooster for a total of fifteen animals.