• Fluency demands more of students than memorisation of a single procedure or collection of facts. It encompasses a mixture of efficiency, accuracy and flexibility.
• Quick and efficient recall of facts and procedures is important in order for students to keep track of sub-problems, think strategically and solve problems.
• Fluency also demands the flexibility to move between different contexts and representations of mathematics, to recognise relationships and make connections, and to make appropriate choices from a whole toolkit of methods, strategies and approaches.

• ### Purpose of Fluency Practice

To enable students to gain confidence and competence in the carrying out of a method so they can attend to more complex ideas.

Craig Barton,  Reflect, Expect, Check, Explain

• ### What is fluency?

“We consider someone to be fluent in a technique, skill, idea, concept or facts at the point at which they no longer need to give attention.”

Mark McCourt, Teaching for Mastery

• ### Developing Number Fluency – What, Why and How

Russell (2000) spells this out in more detail and suggests that fluency consists of three elements:

Efficiency – this implies that children do not get bogged down in too many steps or lose track of the logic of the strategy. An efficient strategy is one that the student can carry out easily, keeping track of sub-problems and making use of intermediate results to solve the problem.

Accuracy depends on several aspects of the problem-solving process, among them careful recording, knowledge of number facts and other important number relationships, and double-checking results.

Flexibility requires the knowledge of more than one approach to solving a particular kind of problem, such as two-digit multiplication. Students need to be flexible in order to choose an appropriate strategy for the numbers involved, and also be able to use one method to solve a problem and another method to check the results.

So fluency demands more of students than memorising a single procedure – they need to understand why they are doing what they are doing and know when it is appropriate to use different methods.

• ### Developing mathematical fluency: comparing exercises and rich tasks

Achieving fluency in important mathematical procedures is fundamental to students’ mathematical development. The usual way to develop procedural fluency is to practise repetitive exercises, but is this the only effective way? This paper reports three quasi-experimental studies carried out in a total of 11 secondary schools involving altogether 528 students aged 12–15.

### Increasingly Difficult Questions

Increasingly Difficult Questions website

These increasing difficult questions are a great source of fluency practice and they also interweave other concepts for retrieval practice.