In mathematics, mastery learning is a method of instruction that focuses on helping students achieve a high level of understanding and proficiency in a particular concept or skill. One way to achieve mastery learning is by using representations and structures.

**What are Representations and Structures?**

Representations are ways of showing or representing mathematical concepts. They can be concrete, such as manipulatives, or abstract, such as symbols. Structures are the relationships between mathematical concepts. They can be seen in the way that numbers and shapes are related to each other, or in the way that equations are structured.

**Why are Representations and Structures Important?**

Representations and structures are important because they help students develop a deep understanding of mathematical concepts. When students see a mathematical concept represented in multiple ways, they are able to see the connections between the different representations. This helps them to understand the concept more deeply.

Structures are also important because they help students to see the patterns in mathematics. When students see the patterns in mathematics, they are able to generalize their knowledge to new situations. This helps them to become more proficient in mathematics.

**Different Types of Representations and Structures**

There are many different types of representations and structures that can be used in the classroom. Some common types of representations include:

Concrete representations: These are representations that can be seen and touched. They can be used to help students visualize mathematical concepts. Some examples of concrete representations include manipulatives, such as blocks and counters, and real-world objects, such as money and clocks.

Pictorial representations: These are representations that are drawn or painted. They can be used to help students visualize mathematical concepts. Some examples of pictorial representations include diagrams, graphs, and charts.

Symbolic representations: These are representations that use symbols. They can be used to represent mathematical concepts in a concise way. Some examples of symbolic representations include numbers, letters, and equations.

There are also many different types of structures that can be used in the classroom. Some common types of structures include:

*Number sense*: This is the ability to understand and work with numbers. It is important for students to have a strong number sense in order to be successful in mathematics.*Spatial reasoning*: This is the ability to think about and visualize shapes and objects in space. It is important for students to have strong spatial reasoning skills in order to be successful in geometry and other spatial concepts.*Algebraic thinking*: This is the ability to think about and solve problems using equations and formulas. It is important for students to have strong algebraic thinking skills in order to be successful in algebra and other more advanced math courses.

**Using Representations and Structures in the Classroom**

There are many ways to use representations and structures in the classroom. Here are a few examples:

When teaching a new concept, start by using concrete representations to help students visualise the concept. Once students have a good understanding of the concept, you can move on to using pictorial and symbolic representations.

When students are working on problems, encourage them to use multiple representations to solve the problem. This will help them to see the problem from different perspectives and to develop a deeper understanding of the solution.

Create a variety of representations and structures that students can use to learn mathematics. This will help to ensure that all students have access to the information they need to be successful.

**Conclusion**

Representations and structures are important tools that can be used to help students develop a deep understanding of mathematical concepts. By using a variety of representations and structures in the classroom, teachers can help all students succeed in mathematics.

**Free Downloadable Staff Meeting on Representations and Structures**

I have created a free downloadable staff meeting on representations and structures. This meeting includes information on the importance of representations and structures, as well as examples of how to use them in the classroom. You can download the meeting here:

I hope this blog post and staff meeting have been helpful. If you have any questions, please feel free to contact me.

Very useful to remind staff what CPA looks like 👍