What We Know

Representations are tools to help students understand and think about mathematics. Representations include the conventional symbol systems of mathematics (including base 10 numeration, coordinate plane, number lines, algebraic conventions) and materials or pictures used to ground abstract ideas in concrete representations (such as manipulatives or computer simulations). Tools - language, materials and symbols - should be thought of as "amplifiers of human capabilities" (Bruner, 1966, p. 81).

Research has demonstrated the effectiveness of using manipulatives in mathematics classrooms for students of all ages. The use of manipulatives produces higher mathematical achievement among students (Sowell, 1989; Suydam & Higgins, 1977). In addition, the use of tools can help students make mathematical connections. Using familiar tools when solving unfamiliar problems can help students connect previously known mathematical concepts to new ones (Hiebert et al., 1997).

The use of tools can influence students' thoughts (Vygotsky, 1978). However, tools cannot take students to higher levels of mathematical thinking. Students' abilities to abstract, or make mental representations, are closely connected to their abilities to represent; they are able to represent only as well as they are able to abstract. As a result, educators should focus on raising students' abilities to abstract mathematically in order for them to more successfully understand and use conventional mathematical symbols (Kato, Kamii, Ozaki, & Nagahiro, 2002). Hiebert and Carpenter (1992) reiterate this idea: "To think about mathematical ideas we need to represent them internally, in a way that allows the mind to operate on them" (p. 66).

Hiebert and his colleagues (1997) provide two reminders for educators using representational tools with their students:

• The meaning of tools is a constructed meaning; it is not inherent to the tools themselves. As a result, the meaning may not be immediately obvious to students. Allow time for students to create this meaning; do not just provide it for students in a demonstration.
• At the same time that students are making sense of what a given tool means, they are also working to understand how the tools can help them grasp the given mathematical concept. Teachers should allow students time and encouragement for both types of exploration. Yackel (2000) cautions, however, that educators should be sure to keep the focus of the acitivity on the use of symbols for developing and communicating meaning.