Manipulatives are concrete objects that are commonly used in teaching mathematics. They include attribute blocks, geometric shapes of different colors and sizes that may be used in classification or patterning tasks; plastic counting cubes for solving simple addition and subtraction equations; base ten blocks for representing and performing operations on multidigit numbers, such as 321 + 104; and fraction pieces, which can be used to represent different fractional concepts and relationships. Most authorities would agree that manipulatives play a helpful role in teaching math, especially in the teaching of concepts. The National Council of Teachers of Mathematics (NCTM), an organization that has been highly influential in math reform efforts in the United States, strongly advocates the use of manipulatives. However, because manipulatives typically are used as part of a much broader program of math instruction, it has sometimes been unclear in research studies how much benefit children derive specifically from manipulatives as opposed to other features of instruction.
Many different kinds of manipulatives are commercially available, and it is also possible to make them using common objects, such as craft sticks, beans, or buttons. In using manipulatives to teach basic operations involving whole numbers, it is important to use objects that are uniform and that accurately represent base-ten relationships (e.g., a “ten” should be ten times as big as a “one,” rather than using only color to show tens vs. ones). A mat for organizing manipulatives and for children to work on is also essential. When children begin learning two-digit and three-digit numbers, the mat is organized from right to left in columns of ones, tens, and hundreds, to reflect the way that numerals are written.
Not all authorities are equally enthusiastic about the use of concrete manipulatives, but there is widespread agreement about the importance of developing conceptual understanding in math. For example, an understanding of basic number concepts, such as being able accurately to count objects, should precede learning written numerals; an understanding of the meaning of multiplication should precede memorizing multiplication tables. Focused assessments should distinguish whether children are struggling with concepts or with other math skills, such as automatic recall of facts. Conceptual understanding can be developed through the use of visual or pictorial representations as well as through concrete manipulatives. Computer-based “virtual” manipulatives are also increasingly available.
There can be some pitfalls to manipulatives, especially for struggling students. Manipulatives are potentially confusing if their presentation is haphazard, disorganized, or lacking appropriate guidance and instruction from the teacher. They can result in considerable time spent off-task or on activities that are not directly relevant to the needs of certain children. For instance, youngsters who do not understand the basic concept of multiplication could well benefit from grouping objects to represent simple equations, such as 3 x 6. However, children who have a good conceptual understanding of multiplication, but who do not know the computational algorithm–the written series of steps–for solving multidigit problems (e.g., 22 x 48), are not likely to benefit from continued use of manipulatives for multiplication. Rather, they need explicit teaching of the algorithm and opportunities for practicing it with carefully chosen written examples. Some children find concrete manipulatives a source of distraction and may do much better with visual or pictorial representations. Monitoring the performance of groups of children may also be easier when visual or pictorial representations are used.
Struggling students, including those with learning disabilities, are consistently found to benefit from instruction that is explicit and systematic. Important math concepts and skills should be directly and clearly taught; the sequence of instruction should emphasize learning of prerequisite skills prior to higher-level skills; instruction should take into account research on mathematical development, for example, that problems and numbers involving 0 are typically more confusing to children than those not involving 0 (e.g., writing 308 is more difficult than writing 328); and sufficient opportunities for practice are needed for children to develop automaticity with new skills. If properly used and appropriately integrated with this type of instruction, manipulatives can be very helpful in concept development, as part of a broader math program for youngsters with learning disabilities.