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.

Image Source: Year 6 Math – Fractions, Decimals, Percentage

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:

Image Source: Teacher

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:

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:

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:

Published by Shen-Li

SHEN-LI LEE is the author of “Brainchild: Secrets to Unlocking Your Child’s Potential”. She is also the founder of (a website on parenting, education, child development) and (a website on Right Brain Education, cognitive development, and maximising potentials). In her spare time, she blogs on Forty, Fit & Fed, and Back to Basics.

7 thoughts on “The Purpose of Manipulatives for Learning Maths

  1. Hi… my child is 1 yr old…currently we r doin glenn method digitally.. wen shall i introduce abstract numbers and counters (age appropriate)..i never tested her nt sure about her grasping in dot cards…kindly guide me

    1. Hi Rahila,

      Once you have completed the first round of the Doman program, you can start introducing abstract numbers and other math facts. Counters and other physical manipulatives can be shown alongside your flashcard program. Please be mindful of small parts, especially if your child has a habit of putting things into her mouth. If you have a large abacus – the ones with ten beads to a row and 100 beads in all, you can start with that.

      Please remember that Doman specifically instructs not to test your child to see what she has learned. You can play games where you can present your child is a math problem, e.g. 5 dots plus 6 dots and then show her two cards (11 dots and some other number of dots) and ask her to pick the answer. It must be done in a fun and engaging manner. Children dislike being tested and can be put off easily. The distinction between testing and playing math games is quite a fine line. You must maintain a cheerful disposition and respond positively even if your child chooses the wrong answer. Simply tell her which was the correct card and move on. I hope that helps.

      1. Thank you very much dr… She is one year old… Hw to show abstract number… Whether along with dot cards… And kindly guide me when to introduce phonics… Alphabet sounds..


      2. Hi Rahila,

        The phonics materials are available on the supplementary activities and materials –

        You can show quantities by sliding the beads along the abacus. If you’re showing say five, slide 5 beads across and tell your child this is five. If you’re showing say 20, slide ten from the first row and ten from the second row across and say this is 20. This is how you can show your child what quantities are. It’s similar to the dot cards but with physical objects.

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