Variation theory suggests that learners learn best when they are presented with a variety of examples of a concept or skill, and when they are encouraged to compare and contrast these examples. It allows them to understand what makes circle a circle and not a sphere or why a triangle is scalene rather than isosceles.

There are two main types of variation: **conceptual variation** and **procedural variation**.

Conceptual variation involves varying the essential features of a concept, while keeping the non-essential features constant. For example, we might look at the variation of different elephants and draw generalisations that they are large, trunked mammals (but we should be careful here as so is a mammoth) and so we really focus in on what makes an elephant an elephant and not a mammoth. For maths, you could teach the concept of a triangle by showing students different triangles of different sizes and shapes. The essential feature of a triangle is that it has three sides, so all of the triangles you show students would have three sides. The non-essential features of a triangle are its size and shape, so you could vary these features without changing the essential feature.

Procedural variation involves varying the way in which a skill is performed, while keeping the goal of the skill constant. For example, you could teach the skill of multiplication by showing students different ways to multiply numbers together. You may vary one of the factors and notice the impact on the product. The goal of multiplication is to find the product of two or more numbers, so all of the methods you show students would need to achieve this goal. The non-essential features of multiplication are the methods used to multiply the numbers together, so you could vary these methods without changing the goal of the skill (this was covered in the blog on mathematical fluency).

Variation theory has been shown to be an effective way to teach mathematics. Studies have shown that students who are taught using variation theory are better able to understand and apply mathematical concepts. Variation theory can also help students to develop their mathematical reasoning skills.

Here are some tips for using variation theory in your primary school mathematics lessons:

Start by identifying the key concepts or skills that you want your students to learn.

Create a variety of examples of these concepts or skills.

Encourage your students to compare and contrast the examples.

Ask questions that help your students to identify the essential and non-essential features of the concepts or skills.

Provide opportunities for your students to practice the concepts or skills in a variety of ways.

Variation theory is a powerful tool that can help you to teach mathematics effectively. By using variation theory, you can help your students to develop a deep understanding of mathematical concepts and skills.

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