Mental mathematics: An introduction

Help your pupils improve mathematical thinking skills in number, algebra, geometry and statistics.

What is mental mathematics?

Almost all of mathematics could be described as ‘mental’ in the sense that engaging in a mathematical task involves thinking. Thus every mathematical problem a pupil tackles must involve several stages of mental mathematics.

Pupils actively involved in mental mathematics might be engaged in any combination of:

• interpreting
• visualising
• analysing
• synthesising
• explaining
• hypothesising
• inferring
• deducing
• judging
• justifying
• making decisions.

These ideas are prevalent throughout mathematics and underpin mathematical processes and applications.

If the definition is so wide ranging, how have we produced a few brief resources with this title? The answer is that we have been very selective! The ‘mental mathematics’ supported through the teaching approaches described in these resources is aimed at a subset of mental mathematics in its broadest sense.

We have chosen a few key areas likely to influence pupils’ progress beyond level 5. These selections have been informed by recent annual standards reports from the Qualifications and Curriculum Development Agency (QCDA) and the experience of teachers and consultants. The initial ideas have also been supported by classroom trials.

Helping pupils to improve the way they process mathematics mentally

The activities in these resources are designed to increase opportunities for pupils to:

• work collaboratively
• engage in mathematical talk
• learn from one another.

Individual pupils will be at different stages but all pupils develop some strategies for processing mathematical ideas in their heads. Often pupils develop and enhance their understanding after they have tried to express their thoughts aloud. It is as if they hear and recognise inconsistencies when they have to verbalise their ideas.

Equally, new connections can be made in a pupil’s ‘mental map’ when, at a crucial thinking point, they hear a different slant on an idea. A more discursive way of working often allows pupils to express a deeper and richer level of understanding of underlying concepts that may otherwise not be available to them.

In this way pupils may:

• reach a greater facility level with pre-learned skills, for example, becoming able to solve simple linear equations mentally
• achieve a leap in understanding that helps to complete ‘the big picture’, for example, seeing how the elements of a function describing the position-to-term relationship in a sequence are generated from elements in the context of the sequence itself.

Is mental mathematics just about the starter to the lesson?

The activities are intended to support the main part of the lesson.

Developing mental processes is not simply about keeping some skills sharp and automating processes through practice. The activities described in these resources support the main part of the lesson.

Developing a mental map of a mathematical concept helps pupils to begin to see connections and use them to help solve problems. Developing the ability to think clearly in this way takes time.

Once in place, some aspects of mental mathematics can be incorporated into the beginning of lessons as a stimulating precursor to developing that topic further.

Is mental mathematics just about performance in mental tests?

Improving mental mathematics will impact on pupils' progress.

Using these materials will help pupils to perform more successfully in tests, but the aim is more ambitious than that. Developing more effective mental strategies for processing mathematical ideas will impact on pupils’ progress in mathematics and their confidence to apply their skills to solve problems.

Secondary teachers recognise the importance of pupils’ mathematical thinking and application, but few have a range of strategies to support their development. The expectations described, and the activities suggested in the accompanying mental mathematics resources, aim to create a level of challenge that will take pupils further in their thinking and understanding.

These materials should provide the chance for pupils to interact in such a way that they learn from each other’s thinking, successes and misconceptions, and thereby become increasingly confident and independent learners.

Applying mathematics in real-life contexts

Improving mental mathematics will improve pupils’ confidence to apply what they know.

Most commonly, pupils will use mental mathematics in solving problems as they occur in their lives, in other areas of their studies and as they prepare for the world of work. To support pupils in doing this, teachers will frequently need to set both large and small mathematical problems in real, purposeful and relevant contexts.

Pupils will need to solve increasingly complex and unfamiliar problems using mathematics, apply more demanding mathematical procedures during their analysis and do so with increasing independence.

These materials support teachers in planning a structured and progressive approach to do this. If learning is planned with mental mathematics as a significant element, pupils will develop increasing confidence in applying mathematics.

Can mental mathematics involve paper and pencil?

Progress may not appear as written output. Gather evidence during group work by taking notes as you listen in on group discussions. Feed these notes into the plenary and use them in future planning.

Mathematical thinking involves drawing on our understanding of a particular concept, making connections with related concepts and previous problems, and selecting a strategy accordingly. Some of these decisions and the subsequent steps in achieving solutions are committed to paper and some are not. When solving problems, some of the recording becomes part of the final solution and some will be disposable jottings.

Many of the activities involve some recording to stimulate thinking and talking. Where possible, such recording should be made on large sheets of paper or whiteboards. This enables pupils, whether working as a whole class or in pairs or small groups, to share ideas. Such sharing allows them to see how other pupils are interpreting and understanding some of the big mathematical ideas. Other resources such as diagrams, graphs, cards, graphing calculators and ICT software are used in the activities. Many of these are reusable and, once developed in the main part of a lesson, can be used more briefly as a starter on other occasions.

The materials

Each attainment target in mathematics is addressed through its own resource, divided into separate topic areas. In each topic, there is a progression that illustrates expectations for mental processes, broadly from level 5 to level 8.

Mathematical ideas and pupils’ learning are not simple to describe, nor do they develop in a linear fashion. These are not rigid hierarchies and the degree of demand will be influenced by the context in which they occur and, particularly for the number topics, by the specific numbers involved. The aim is that you will adjust the pitch of the activities that are described within the resources.