tive becomes just one more thing to learn rather than a process leading to a larger mathematical learning goal.57
When used well, manipulatives can enhance student understanding. They can, for example, enable teachers and students to have a conversation that is grounded in a common referential medium, and they can provide material on which students can act productively provided they reflect on their actions in relation to the mathematics being taught.58 The base-10 blocks that Kurt is using in Ms. Kaye’s class provide both student and teacher with a way to discuss the problem that would have been more difficult without the blocks. Research on four successful projects aimed at teaching multidigit number concepts and operations through a problem-solving approach found that, although different in approach, the projects treated the use of conceptual supports, whether manipulatives or diagrams, in similar ways.59 Each project provided sustained opportunities for students to construct connections between the conceptual support, the written symbols, and the number words and to use the object-word-symbol triad in solving multidigit addition and subtraction problems. Manipulatives also help students correct their own errors.60 The evidence indicates, in short, that manipulatives can provide valuable support for student learning when teachers interact over time with the students to help them build links between the object, the symbol, and the mathematical idea both represent.
Although calculators are used more frequently than manipulatives in grades 4 and 8, the use of calculators is more controversial in mathematics lessons in grades pre-K-8 than are manipulatives, particularly in the elementary grades. Although mathematics educators have advocated the appropriate use of calculators since the 1970s, persistent concerns have been expressed that an extensive use of calculators in mathematics instruction interferes with students’ mastery of basic skills and the understanding they need for more advanced mathematics.61
A large number of empirical studies of calculator use, including long-term studies,62 have generally shown that the use of calculators does not threaten the development of basic skills and that it can enhance conceptual understanding, strategic competence, and disposition toward mathematics. A meta-analysis of 79 research studies on the effects of calculator was conducted in 1986 and extended in 1992 with nine additional studies.63 This analysis found that with the exception of the fourth grade, students at all