Classification skills can be learned before a student is able to count. Given a bag of buttons, a student can sort them according to color, size, or shape. Geometric shapes are easily learned if a student is able to feel and count the edges of cardboard and foam shapes. Later the student can make his own shapes by gluing toothpicks onto paper. With manipulative activities like these, it won't be long before the kindergarten student will be identifying pentagons and octagons.
Many students have a difficult time understanding fractions.
It is important for students in the early grades to cut and color parts of a whole when naming or adding fractions. It is just as important for the older student to manipulate fractions when multiplying and dividing them. This activity will help most students understand why the product of two fractions is smaller than one or both of the factors. Manipulatives are important when you teach for the understanding of math concepts. They are the concrete objects you provide in order to transfer understanding to the abstract level.