Math 7-9

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Math 7-9 > Strategies > Multiple Representations: Manipulatives

Multiple Representations: Manipulatives


Providing physical and virtual representations of numbers and math concepts helps activate mental processes. Learners understand more deeply through hands-on experimentation and exploration with manipulatives. Research has found that manipulatives are effective when students are in the concrete operational stage -- where seeing an actual object supports their ability to think about it. As students become able to think about an object without it being there (formal operational stage, which usually begins at 11 years or older), manipulatives start to become less effective but can still offer some cognitive support.

Use It in the Classroom

Watch how this middle school student uses manipulatives to practice equations and problem solving. By using dice and playing pieces, she demonstrates her thinking by talking through her problem solving process, further supporting moving these concepts and skills into Long-term Memory.

  • While it is important for teachers to choose the best manipulative for a specific learning objective, such as fraction strips for Proportional Reasoning or dice for Statistical Reasoning, they should also let students explore in an open-ended way, developing their Mathematical Flexibility as they generate multiple solutions to problems. Teachers can also use manipulatives to launch rich discussions and to clarify students' misconceptions of content.
  • Design It into Your Product

    Videos are chosen as examples of strategies in action. These choices are not endorsements of the products or evidence of use of research to develop the feature.

    Learn how products such as DragonBox Numbers allow learners to playfully explore digital manipulatives to strengthen their number sense. Through various games using the manipulatives, learners can build Geometric Reasoning and enhance Algebraic Thinking.

  • Research has shown that virtual manipulatives are as effective for students as physical ones. Through interacting with virtual manipulatives by resizing, sliding, flipping, color coding, and even turning them into 3D objects, students practice their Geometric Reasoning.
  • Factors Supported by this Strategy