In mathematics education, a manipulative is an object which is designed so that a learner can perceive some mathematical concept by manipulating it, hence its name. The use of manipulatives provides a way for children to learn concepts through developmentally appropriate hands-on experience. The use of manipulatives in mathematics classrooms throughout the world grew considerably in popularity throughout the second half of the 20th century. Mathematical manipulatives are frequently used in the first step of teaching mathematical concepts, that of concrete representation. The second and third steps are representational and abstract, respectively. Mathematical manipulatives can be purchased or constructed by the teacher. Examples of common manipulatives include

number line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...

Cuisenaire rods Cuisenaire rods are mathematics learning aids for students that provide an interactive, hands-on way to explore mathematics and learn mathematical concepts, such as the four basic arithmetical operations, working with fractions and finding diviso ...

; fraction strips, blocks, or stacks;

base ten blocks Base ten blocks, also known as multibase arithmetic blocks (MAB) or Dienes blocks (after their creator, mathematician and educationalist Zoltán Pál Dienes), are a mathematical manipulative used by students to learn basic mathematical concepts in ...

(also known as Dienes or multibase blocks); interlocking linking cubes (such as Unifix);

construction set A construction set is a set of standardized pieces that allow for the construction of a variety of different models. The pieces avoid the lead-time of manufacturing custom pieces, and of requiring special training or design time to const ...

s (such as

Polydron A construction set is a set of standardized pieces that allow for the construction of a variety of different models. The pieces avoid the lead-time of manufacturing custom pieces, and of requiring special training or design time to const ...

and

Zome The term ''zome'' is used in several related senses. A zome in the original sense is a building using unusual geometries (different from the standard house or other building which is essentially one or a series of rectangular boxes). The word "zome ...

tool); colored tiles or

tangram The tangram () is a dissection puzzle consisting of seven flat polygons, called ''tans'', which are put together to form shapes. The objective is to replicate a pattern (given only an outline) generally found in a puzzle book using all seven pie ...

pattern block Pattern Blocks are a set of mathematical manipulatives developed in the 1960s. The six shapes are both a play resource and a tool for learning in mathematics, which serve to develop spatial reasoning skills that are fundamental to the learning of m ...

s; colored counting chips;

numicon In mathematics education, a manipulative is an object which is designed so that a learner can perceive some mathematical concept by manipulating it, hence its name. The use of manipulatives provides a way for children to learn concepts throug ...

tiles; chainable links;

abaci The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool which has been used since ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the Hi ...

such as "rekenreks", and

geoboard A geoboard is a mathematical manipulative used to explore basic concepts in plane geometry such as perimeter, area and the characteristics of triangles and other polygons. It consists of a physical board with a certain number of nails half driv ...

s. Improvised teacher-made manipulatives used in teaching place value include

bean A bean is the seed of several plants in the family Fabaceae, which are used as vegetables for human or animal food. They can be cooked in many different ways, including boiling, frying, and baking, and are used in many traditional dishes thr ...

s and bean sticks, or single popsicle sticks and bundles of ten popsicle sticks. Virtual manipulatives for mathematics are computer models of these objects. Notable collections of virtual manipulatives include The National Library of Virtual Manipulatives and the

Ubersketch The Ubersketch is a moniker for a collection of sketches created in The Geometer's Sketchpad by thPRISM-NEO projectwhich mimic ( virtual) manipulatives, such as the ones found aThe National Library of Virtual Manipulatives. The editor of the col ...

. Multiple experiences with manipulatives provide children with the conceptual foundation to understand mathematics at a conceptual level and are recommended by the

NCTM Founded in 1920, The National Council of Teachers of Mathematics (NCTM) is a professional organization for schoolteachers of mathematics in the United States. One of its goals is to improve the standards of mathematics in education. NCTM holds an ...

. Some of the manipulatives are now used in other subjects in addition to mathematics. For example, Cuisenaire rods are now used in language arts and grammar, and pattern blocks are used in fine arts.

In teaching and learning

Mathematical manipulatives play a key role in young children's mathematics understanding and development. These concrete objects facilitate children's understanding of important math concepts, then later help them link these ideas to representations and abstract ideas. For example, there are manipulatives specifically designed to help students learn fractions, geometry and algebra. Here we will look at pattern blocks, interlocking cubes, and tiles and the various concepts taught through using them. This is by no means an exhaustive list (there are so many possibilities!), rather, these descriptions will provide just a few ideas for how these manipulatives can be used.

Base ten blocks

Base Ten Blocks are a great way for students to learn about place value in a spatial way. The units represent ones, rods represent tens, flats represent hundreds, and the cube represents thousands. Their relationship in size makes them a valuable part of the exploration in number concepts. Students are able to physically represent place value in the operations of addition, subtraction, multiplication, and division.

Pattern blocks

Pattern blocks Pattern Blocks are a set of mathematical manipulatives developed in the 1960s. The six shapes are both a play resource and a tool for learning in mathematics, which serve to develop spatial reasoning skills that are fundamental to the learning of ...

consist of various wooden shapes (green triangles, red trapezoids, yellow hexagons, orange squares, tan (long) rhombi, and blue (wide) rhombi) that are sized in such a way that students will be able to see relationships among shapes. For example, three green triangles make a red trapezoid; two red trapezoids make up a yellow hexagon; a blue rhombus is made up of two green triangles; three blue rhombi make a yellow hexagon, etc. Playing with the shapes in these ways help children develop a spatial understanding of how shapes are composed and decomposed, an essential understanding in early geometry. Pattern blocks are also used by teachers as a means for students to identify, extend, and create

pattern A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric shapes and typically repeated l ...

s. A teacher may ask students to identify the following pattern (by either color or shape): hexagon, triangle, triangle, hexagon, triangle, triangle, hexagon. Students can then discuss “what comes next” and continue the pattern by physically moving pattern blocks to extend it. It is important for young children to create patterns using concrete materials like the pattern blocks. Pattern blocks can also serve to provide students with an understanding of fractions; because pattern blocks are sized to fit to each other (for instance, six triangles make up a hexagon), they provide a concrete experiences with halves, thirds, and sixths. Adults tend to use pattern blocks to create geometric works of art such as mosaics. There are over 100 different pictures that can be made from pattern blocks. These include cars, trains, boats, rockets, flowers, animals, insects, birds, people, household objects, etc. The advantage of pattern block art is that it can be changed around, added, or turned into something else. All six of the shapes (green triangles, blue (thick) rhombi, red trapezoids, yellow hexagons, orange squares, and tan (thin) rhombi) are applied to make mosaics.

Linking cubes

Like pattern blocks, interlocking cubes can also be used for teaching patterns. Students may use the cubes to make long trains of patterns. Like the pattern blocks, the interlocking cubes provide a concrete experience for students to identify, extend, and create patterns. The difference is that a student can also physically decompose a pattern by the unit. For example, if a student made a pattern train that followed this sequence: the child could then be asked to identify the unit that is repeating (red, blue, blue, blue) and take apart the pattern by each unit. Also, one can learn addition, subtraction, multiplication and division,

guesstimation ''Guesstimate'' is an informal English portmanteau of ''guess'' and ''estimate'', first used by American statisticians in 1934 or 1935.Tiles are one inch-by-one inch colored squares (red, green, yellow, blue). Tiles can be used much the same way as interlocking cubes. The difference is that tiles cannot be locked together. They remain as separate pieces, which in many teaching scenarios, may be more ideal. These three types of mathematical manipulatives can be used to teach the same concepts. It is critical that students learn math concepts using a variety of tools. For example, as students learn to make patterns, they should be able to create patterns using all three of these tools. Seeing the same concept represented in multiple ways as well as using a variety of concrete models will expand students’ understandings.

Number lines

To teach integer addition and subtraction, a

is often used. A typical positive/negative number line spans from −20 to 20. For a problem such as “−15 + 17”, students are told to “find −15 and count 17 spaces to the right”.

References

Sources

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External links

NCTM's Official Website

* ttp://www.didax.com/virtual-manipulatives-for-math Didax Virtual Manipulatives* Manipulatives for negative numbers at Wikiversity {{Authority control Educational toys Early childhood education materials