Research-Informed Answers for Mathematics Education Leaders The National Council of Supervisors of Mathematics Improving Student Achievement Series No. 11/Spring 2013 Improving Student Achievement in Mathematics by Using Manipulatives with Classroom Instruction Manipulative materials help students make sense of abstract ideas, provide students ways to test and verify ideas, are useful tools for solving problems, and make mathematics learning more engaging and interesting by lifting mathematics off textbook and workbook pages. Burns, 2007 The NCSM Improving Student Achievement Series is a set of position papers designed to provide research-based practices for school and district mathematics education leaders. Our Position It is the position of the National Council of Supervisors of Mathematics (NCSM) that in order to develop every student’s mathematical proficiency, leaders and teachers must systematically integrate the use of concrete and virtual manipulatives into classroom instruction at all grade levels. This position can be accomplished when leaders and teachers: • Understand that manipulatives are not toys but are powerful learning tools which build conceptual understanding of mathematics; • Use research to guide instructional use of manipulatives; • Provide sustained professional learning opportunities in the use of manipulatives; and • Recognize that learners—both adults and students— progress through varying levels of proficiency as they use manipulatives before they can realize their full impact. The Common Core State Standards emphasize that concrete models are an essential tool for learning mathematics across all grade levels, K–12. This assertion is articulated most clearly in the Standard for Mathematical Practice 5, “Use Appropriate Tools Strategically,” where students choose from concrete models (including manipulatives) and technology. Beyond this, the standards regularly suggest using models in initial steps of learning mathematics before students move to other representations. Therefore, students should have a variety of manipulatives and tools available to them at all times. Manipulatives used in classroom instruction are physical objects handled by individual students and small groups. Virtual manipulatives are important tools for teacher modeling and demonstration and, additionally, provide students access to manipulatives both inside and outside of the school day via computers. However, virtual manipulatives do not replace the power of physical objects in the hands of learners. Research That Supports Our Position John van de Walle and his colleagues (2012) define a mathematical tool as, “any object, picture, or drawing that represents a concept or onto which the relationship for that concept can be imposed. Manipulatives are physical objects that students and teachers can use to illustrate and discover mathematical concepts, whether made specifically for mathematics (e.g., connecting cubes) or for other purposes (e.g., buttons)” (p. 24). Moyer et al. (2000) define virtual manipulatives as “digital objects that resemble physical objects and can be manipulated with a mouse of a computer” (p. 372). For example, virtual versions of Cuisenaire Rods and Tangrams are readily available online for instructional purposes. In the opening quote, Marilyn Burns (2007) provides four reasons manipulative materials are fundamental to mathematics instruction. These ideas appear repeatedly

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