Developing mental strategies

Teachers cannot see what is going on inside a learner’s head, but they can hear it and see the results. Learners can describe, show and explain their thinking using counting objects, shapes, patterns, measuring tools or something else. To find out what learners are thinking, you may need to listen in when learners talk to or show their work or ideas to a partner or a group, ask learners questions or simply ask learners to explain. See below for some suggested questions. As Stage 2 progresses, learners will be increasingly able to explain their thinking. Encourage the use of spoken and written words, diagrams, symbols and other written work, often alongside practical resources. Teachers need to introduce and model the relevant vocabulary so that learners have the words they need to talk about their strategies. Class teachers should value learner’s intuitive methods as well as introducing methods and approaches for learners to practise. Learners then have the opportunity to choose which strategy to use and develop their own. Any method that works is a correct method. It is important that you provide regular opportunities for learners to explain and discuss their methods, so they share ideas with one another and acquire a range of mental strategies. Not all learners will carry out a mental calculation in the same way, but some methods are more efficient and reliable than others. If you allow time for learners to discuss, explain and compare different methods you can support them to choose and use efficient methods. Learners will see the need for methods that can be applied generally, and this eventually leads towards using standard written methods. This helps to raise learners’ awareness of other strategies and develop their confidence and fluency. This could be through mini plenaries within the session or during a closing plenary. Mental strategies are not restricted to calculations. Visualising patterns, shapes and quantities is a key skill which supports learners to develop their mental imagery and strategies across all areas of mathematics. As learners manipulate objects, they are internalising their actions and making sense of the mathematics. There are many patterns in our counting sequence. Learners need to recognise and internalise these so that they can use them to develop their understanding of how our number system works and how it all fits together. Learners will then use this information to predict the next number when counting in ones, twos, fives and tens. Counting in multiples of 2, 5 and 10 will support learners as they develop their understanding of multiplication and division. As learners begin to learn about place value in numbers, they will develop a sense of the size of numbers so that they can compare and order them. Learners need to experience quantities of objects and measures to support estimating. Learners will have already experienced estimating measurements using non-standard measures such as their own foot or hand span. With some experience of standard units of measure such as centimetres, grams and litres, learners will begin to estimate measurements. The same is true of numbers. As learners extend their counting range to 100, they also need to experience collections of 10, 20, 50 and 100 objects before they can estimate amounts. With some experience of a number line, they will begin to estimate the position of a number when only a few key positions are marked. In Stage 2, learners begin to calculate with numbers up to a total of 100. Learners add and subtract 2-digit numbers, without regrouping ones or tens. This gives learners lots of opportunities to practise and begin to recall number facts for all numbers up to 9, since this is the maximum number

of ones and of tens that learners can total to or subtract from without regrouping. These number facts underpin later development as learners move on to regrouping ones and tens with numbers up to 3 digits in Stage 3. The restriction on regrouping of ones or tens limits learners when estimating the result of a calculation. When adding or subtracting ones to or from a 2-digit number, learners will be working within the same ten, so there is little to estimate. When adding or subtracting tens to or from a 2-digit number, the process of estimating by looking only at the tens is the same as the process of adding or subtracting tens when there is no regrouping. Learners can make some general statements about number answers being less than or greater than 50, for example, but they are likely to have completed the calculation mentally and really be talking about their solution. It is only when adding or subtracting two 2-digit numbers that learners can truly begin to estimate the result of their calculations, to alert themselves to any errors. Estimating using rounding to the nearest 10 is introduced once learners have some experience of adding two 2-digit numbers or subtracting a 2-digit number from a 2-digit number. At first, learners will count on or back in tens, but they will move on to calculating mentally. This key mental strategy for estimating will be developed further in Stage 3. Use starters to practise basic skills such as counting, patterns and the features of shapes. Focus on an area that is relevant for the session. For example, if you are about to introduce learners to a new 2D shape such as a pentagon, hexagon or octagon, revise the features of known 2D shapes in the starter. This helps learners to recall and use the appropriate vocabulary before applying this to the new shape. If learners struggle with an aspect of the starter, identify the misconception and clarify that before moving on to the main session.

Key starters

Count forward and back in twos, fives or tens from any start number. Use different voices, for example: deep like a giant, squeaky like a mouse, quiet as a whisper, loud as a countdown, slow as a creature waking up, fast as a bee flitting from flower to flower. There is no need to count from 0 to 100. Build this up gradually by starting from where learners are secure and extend to less familiar numbers. Focus on the difficult areas, such as crossing to the previous ten when counting back. Sometimes use a pendulum for support. This is particularly useful when counting in ones and twos. Practise and consolidate the rapid and accurate recall of addition and subtraction number facts to 9.

Continue to practise and recall complements to 10 and 20. Revise and share mental strategies for tackling a number problem, perhaps discussing what learners did in a previous session. Look at an array and describe it with a number sentence. Ask, for example: Can you think of a repeated addition/multiplication/division number sentence for this array? Can you find all the number sentences for this array? Revisit a concept to revise what it means and how to use it, for example, fractions. What is a fraction? How do you find one? Can you do it in a different way? What is half? What is a quarter? How can you find a half or a quarter of a shape or number? Can you do it another way? Extend a pattern at either end or identify errors. Ask, for example: What comes next in my pattern? Why doesn’t this come next? Is there a mistake? If so, where is it and how could you correct it?

Explore shapes, for example, put a 2D shape in a bag and expose just a small part of it. Ask, for example: What do you see? What shape could it be? What shape can’t it be? Why not? Which 3D shapes have a face this shape?

Key plenaries

Share different mental strategies used in the session. Ask learners to explain why they used that method or approach and if they would use the same approach again. Discuss other approaches that could have been used. Discuss estimates. Ask learners how they decide upon their estimate, sharing ideas. Complete a puzzle based on the session. Ask learners how they found their solution. Use ‘If I know this, what else do I know?’ to revise the main session and link to other areas of mathematics.

Ask open questions such as, ‘If the answer is 53 (or pyramid or something else), what could the question be?’

Make a conjecture and ask whether it is always, sometimes or never true. Ask learners to explain how they know. • Reinforce the correct use of mathematical vocabulary.

Asking learners to explain their mental strategies can happen at any time during a session. Useful questions include:

How do you know how many without counting?

How did you find your answer?

Can you do it a different way?

Can you show me what you mean?

How do you know? • Why can’t the answer be …?

By the end of Stage 2, learners should be able to:

Count to and back in ones, twos, fives and tens (0 - 100).

Use place value to help develop a sense of the size of a number.

Regroup a number in different ways.

Use patterns of similar calculations.

Recall number facts to 9 and number bonds for 10 and 20.

Know the relationship between addition and subtraction.

Know the multiplication tables for 1, 2, 5 and 10.

Estimate quantities and measures.

Extend a pattern, notice and correct mistakes in patterns.

Recognise, describe and use some features of 2D and 3D shapes. • Link 2D and 3D shapes to everyday objects. This is not an exhaustive list, but these are key areas which will be built upon in Stage 3.