Starting in the primary grades, objects that enhance counting skills include beads, beans and blocks. It is important that the objects be uniform in shape and size so students learn that they have equal value. Students can learn how addition increases value while subtraction decreases. Fact families can be modeled using manipulatives, such as adding two counters plus three counters to get a total of five. Subtraction can be viewed as the inverse, or undoing of addition. Base ten blocks model digits ranging from single digits to thousands. Students have concrete models of how ones are grouped to make tens. Understanding place value helps students develop complex addition and subtraction problems using carrying and regrouping strategies.
Pattern blocks consist of polygons that are proportional in size including triangles, squares, parallelograms, trapezoids and hexagons. The blocks help students explore, compare and classify geometric properties. The National Council for Teachers of Mathematics recommend using geometric modeling to develop spatial reasoning skills. Fractional skills are also enhanced through the use of the pattern blocks. The smaller blocks are proportional to the larger blocks by halves, thirds, fourths, sixths and twelfths. Skills such as comparing, ordering and finding fractional equivalences are enhanced through visual understanding.
By middle school, many mathematical concepts and skills have been developed, but the use of manipulatives can still add to the development of critical thinking skills. Square tiles can be used to model area and perimeter problems and to understand the relationship between the two. Rods of different lengths and Geoboards provide concrete representations of congruence, similarity and the Pythagorean relationship. By holding three dimensional shapes such as prisms and pyramids, students can feel the faces, edges and vertices instead of memorizing how many there are.
Through the use of interactive white boards and computers, the use of manipulatives has expanded into the virtual world. Students are exposed to a range of manipulatives that are unlimited. Students can now watch as larger grids of base ten blocks are broken apart virtually to model subtraction with regrouping. Shapes are transformed to model flips, slides and turns. Fractional models can be shaded using a virtual paint brush. Motivation levels are elevated as learning becomes a fun and interactive activity.