The Purpose of Manipulatives for Learning Maths

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“Manipulatives can be important tools in helping students to think and reason in more meaningful ways. By giving students concrete ways to compare and operate on quantities, such manipulatives as pattern blocks, tiles, and cubes can contribute to the development of well-grounded, interconnected understandings of mathematical ideas.”

Stein and Bovalino (2001)

The use of manipulatives has often been recommended in the teaching of subjects like Mathematics because they enhance and deepen your child’s understanding. It has been said that a picture is worth a thousand words. In the teaching of abstract concepts, manipulatives provide your child that picture. Manipulatives allow your child to move from concrete experiences to abstract reasoning.

For example, children learning about fractions may find it difficult to grasp that 1/2 is larger than 1/8 because, in their understanding of numerical values, 2 is smaller than 8. By observing a physical representation of 1/2 compared to 1/8, they can visualise why 1/2 is larger than 1/8.

The Benefits of Using Maths Manipulatives

Studies demonstrate that children using manipulatives make gains in the following areas (Heddens; Picciotto, 1998; Sebesta and Martin, 2004):

• verbalizing mathematical thinking
• discussing mathematical ideas and concepts
• relating real-world situations to mathematical symbolism
• working collaboratively
• thinking divergently to find a variety of ways to solve problems
• expressing problems and solutions using a variety of mathematical symbols
• making presentations
• taking ownership of their learning experiences
• gaining confidence in their abilities to find solutions to mathematical problems using methods that they come up with themselves without relying on directions from the teacher

Children using manipulatives in specific mathematical subjects are also more likely to achieve success than those who do not:

• Some children need to use manipulatives to learn to count – Clements, 1999.
• Using manipulatives increases students’ understanding of place value – Phillips, 1989.
• Students learning computational skills tend to master and retain these skills more fully when manipulatives are used as part of their instruction – Carroll and Porter, 1997.
• Using manipulatives has been shown to help students reduce errors and increase their scores on tests that require them to solve problems – Carroll and Porter, 1997; Clements, 1999; Krach, 1998.
• Students who have appropriate manipulatives to help them learn fractions outperform students who rely only on textbooks when tested on these concepts – Jordan, Miller, and Mercer, 1998; Sebesta and Martin, 2004.
• Students who have appropriate manipulatives to help them learn fractions also have significantly improved achievement when tested on ratios when compared to students who do not have exposure to these manipulatives – Jordan, Miller, and Mercer, 1998.
• Research indicates that students who used manipulatives in their mathematics classes have higher algebraic abilities than those who did not use manipulatives – Chappell and Strutchens, 2001.

Source: Research on the Benefits of Manipulatives

The use of manipulatives deepens the understanding of concepts and relationships, makes skills practice meaningful, and leads to retention and application of the information in new problem-solving situations.

Resources: Math Manipulatives from ETA Hand2Mind

Manipulatives in Action

Here’s an example of manipulatives being used to help children understand the mathematical concept:

3 Stages of Learning

Jerome Bruner identified three stages of cognitive representation:

1. Enactive, which is the representation of knowledge through actions.
2. Iconic, which is the visual summarization of images.
3. Symbolic representation, which is the use of words and other symbols to describe experiences.

Children need to go through each stage before they can move on to the next. In Math, this is referred to as the concrete-representational-abstract sequence of instruction:

• Each math concept is first modeled with concrete materials.
• Children are given opportunities to practice new skills using concrete materials.
• When your child has mastered the concept using concrete materials, the math concept is then modeled at the representational level. Concrete materials are replaced with images that represent the concrete objects previously used.
• Children practice the math concept using the representational drawing solutions.
• When your child has mastered the math concept using representational drawing solutions, the math concept is finally modeled at the abstract level.
• Children practice and master the concept at the abstract level before moving to a new math concept.

The use of manipulatives supports the learning of math in the concrete stage. Some examples of mathematical manipulatives that help in this stage include:

• Base 10 blocks
• Link cubes
• Math counters
• Place value trains

When the concept has been mastered at the concrete stage, your child is ready to move on to the representational stage. This stage involves the use of image representations, such as:

• Number lines
• Hundred squares
• Arrays

When the concept has been mastered at the representational stage, your child is ready to move on to the abstract stage.

This final stage involves the use of abstract representations with numbers and symbols, for example:

• Column method
• Grid method
• Chunking