Unlocking fluency: building automaticity through varied practice

Fluency in mathematics

Fluency in mathematics goes beyond memorising facts and formulas; it is about developing a deep understanding of concepts and building confidence to apply them in varied contexts. The Mental Arithmetic series helps pupils achieve this by using principles of Variation Theory and Intelligent Practice—two powerful approaches that make practice both meaningful and effective.

In this post, we explore these principles and explain how they help pupils build automaticity, paving the way for long-term success in maths.

What is Variation Theory?

Variation Theory focuses on how changes to a problem’s structure or context can deepen understanding. Instead of presenting pupils with repetitive drills, it introduces subtle variations in problems, encouraging them to identify patterns, connections and underlying principles. For example, a series of addition problems might vary the placement of numbers, use different number types (e.g. whole numbers, decimals) or present the same concept in worded and symbolic forms.

Using this approach in the Mental Arithmetic series ensures that pupils are not just practising but actively engaging with maths. By varying aspects of a question, pupils are challenged to think critically, while the rich blend of question types in each book improves vocabulary, reasoning skills and understanding of numbers.

What is Intelligent Practice?

Intelligent Practice moves beyond quantity to focus on the quality of questions. Rather than solving a large number of identical problems, pupils work through a carefully selected set of questions designed to deepen their thinking and apply their knowledge in diverse ways.

The Mental Arithmetic series embodies this principle by combining arithmetic, number vocabulary and problem-solving questions in a logical, progressive format. This approach keeps practice purposeful, ensuring that every question reinforces learning and highlights areas for improvement.

How these principles build automaticity

In maths, automaticity can be defined as the ability to perform basic processes quickly and accurately—a critical skill for tackling more complex mathematical problems. By repeatedly engaging with a wide range of topics and varied question types, pupils develop automaticity as their mathematical understanding and skills grow.

In the Mental Arithmetic series, the breadth of topics covered encourages pupils to apply their full knowledge and skills in answering each question, demonstrating what they know as well as what they may have forgotten. Question types change frequently to improve pupils' vocabulary and reasoning skills, and topics are revisited to compound knowledge and understanding. Only once pupils have completed enough intelligent and varied practice will they begin to feel confident in maths.

Why it matters for fluency and confidence

When pupils achieve fluency and automaticity, they free up cognitive resources for higher-order thinking. This means they can focus on solving multi-step problems or tackling unfamiliar challenges without getting stuck on basic calculations. With over 1,500 questions per book, the Mental Arithmetic series provides the volume of questions needed to help pupils develop these skills.

 For more information about the Mental Arithmetic series and how it can support your pupils, visit: www.schofieldandsims.co.uk/ma.

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