A mathematical manipulative is an object which is designed so that a student can learn some mathematical concept by manipulating it. The use of manipulatives provides a way for children to learn concepts in developmentally appropriate, hands-on ways. Mathematical manipulatives are used in the first step of teaching mathematical concepts, that of concrete representation. The second and third step are representational and abstract, respectively. Mathematical manipulatives can be purchased or constructed by the teacher. Examples of commercial manipulatives include Tangrams; Cuisenaire rods; Numicon patterns; Diene's blocks; interlocking cubes; base ten blocks; pattern blocks; colored chips; links; fraction strips, circles, blocks, or stacks; and geoboards. Examples of teacher-made manipulatives used in teaching place value are beans and bean sticks or bundles of ten Popsicle sticks and single Popsicle sticks.
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