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Mathematical structures: the power of representations.



What Are Representations?

Representations are powerful tools used in mathematics education to expose the underlying structure of mathematical concepts. They help learners visualize and comprehend abstract ideas by providing tangible, visual, or symbolic ways to engage with mathematical content. These representations can take various forms, such as objects, pictures, diagrams, or manipulatives.


Historical Roots

The use of representations is not a recent phenomenon. Think back to your childhood – perhaps you remember using counters or other physical objects to understand mathematical ideas. Throughout history, mathematicians and educators have employed representations to convey complex concepts. Today, primary classrooms commonly feature tools like tens frames, Dienes blocks, Cuisenaire® rods, and bar models.





Dispelling Myths

Let’s address some common misconceptions about representations:

  1. Myth: Manipulatives and pictures are only for younger children and low prior-attainers.

  • Reality: Representations benefit learners of all ages. Even research mathematicians use them to explain their thinking. Teaching for mastery emphasizes their use across primary and secondary levels to deepen understanding.

  1. Myth: Representations help children perform calculations.

  • Reality: While objects can assist in calculations, the true purpose of representations lies in revealing mathematical structure. For instance, instead of merely counting groups of counters, we use them to demonstrate the structure (e.g., three groups of five counters) and build efficient calculation methods.

  1. Myth: Children should always choose their own representations.

  • Reality: Teachers often select the most appropriate representations to illuminate specific structures. Giving students free rein with manipulatives may hinder their understanding.

  1. Myth: Each concept demands a specific representation (e.g., fractions as pizzas).

  • Reality: Teaching for mastery employs multiple representations to reveal different facets of a concept. By using diverse tools, we deepen students’ grasp of mathematical ideas.


Practical Strategies

Here are practical strategies for incorporating representations in primary mathematics:

  1. Start Early: Introduce representations from the early years. Use concrete objects, drawings, and diagrams to make abstract concepts accessible.

  2. Choose Wisely: Select representations that align with the specific mathematical structure you want to highlight. Consider the context and purpose.

  3. Model Thinking: Demonstrate how to use representations effectively. Show students how to transition from relying on them to mentally visualizing mathematical relationships.

  4. Explore Variety: Experiment with different tools – from visual models to digital representations. Encourage students to explore and discuss their insights.


Remember, representations are not mere teaching aids; they are windows into mathematical beauty. By embracing them, we empower students to unlock the hidden structures that underpin mathematics.

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