City of Edinburgh Council Mental Agility Guidelines

City of Edinburgh Council Mental Agility Guidelines

Introduction

The ability to calculate in your head is an important part of mathematics. It is also an essential part of coping with society’s demands and managing everyday events.

City of Edinburgh Council’s Mental Agility Guidelines have been adapted from a National Strategies Primary (Department of Education England and Wales) publication entitled “Teaching Learners to Calculate Mentally (2010)”.

The documents will be in two parts covering:

Part 1

Principles of Teaching Mental Agility

Part 2

Lines of Progression

(this part is in electronic format to allow teachers to print and use the relevant pages for their groups).

Principles of Teaching Mental Agility

This section identifies the principles that underpin teaching: for example, encouraging children to share their mental methods, to choose efficient strategies, and to use informal jotting (where necessary) to keep track of the information they need when calculating. It also looks at the role of questioning.

What is Mental Agility?

For many people, mental agility is about doing arithmetic; it involves rapid recall of number facts – knowing your number bonds to 20 and the multiplication tables to 10 x 10.

Rapid recall of number facts is one aspect of mental agility but there are others. This involves presenting children with calculations in which they have to work out the answer using known facts and not just recall it from a bank of number facts that are committed to memory. Children should understand and be able to use the relationship between the four operations and be able to construct equivalent calculations that help them to carry out such calculations.

Learning key facts “by heart” enables children to concentrate on the related calculation which helps them to develop calculation strategies. Using and applying strategies to work out answers helps children to acquire and so remember more facts. Many children who are not able to recall key facts often treat each calculation as a new one and have to return to first principles to work out the answer again. Once they have a secure knowledge of some key facts they can use these to help them access more problems e.g. using known facts such as number bonds and times tables.

The following teaching principles apply for teaching mental agility:

• Every day is a mental mathematics day, with regular time given to teaching mental agility strategies.
• Provide practice time with frequent opportunities for children to use one or more facts that they already know to work out more facts.
• Introduce practical approaches, with concrete materials and images, and jottings which children can use to carry out calculations as they secure mental strategies.
• Engage children in discussion when they explain their methods and strategies to you and their peers, and build children’s confidence by discussing and learning from their mistakes.

Revisiting mental work at different times in the daily mathematics lesson, or even devoting a whole lesson to it from time to time, helps children to generate confidence in themselves. Look out too for opportunity to introduce short periods of mental calculation in other lessons our outside lessons when queuing for P.E etc.

Regular short practice keeps the mind fresh. Mental calculation is one of those aspects of learning where – if you don’t use it you will end up losing it!

How do I help children to develop a range of mental strategies?

Learners are likely to be at different stages in terms of the number facts that they have committed to memory and the strategies available to them for figuring out other facts.

When teachers are focussing on a specific mental strategy they should think carefully about the questions they will present to children to try and encourage them to use the desired strategy. By doing this, and by discussing all strategies used by the children it should be possible to “arrive at” the most efficient strategy. It is important to talk about what strategy is being used, either by its correct terminology if children can understand this or by using another name (sometimes the name of the child who suggested it) so it can be referred to in the future.

What practical equipment can support mental agility?

Children will not be able to visualise and “see” how something works if they have not had any practical experiences to draw on, or been shown any models and images that support the approaches being taught.

It is important to provide suitable resources for learners to manipulate and explore before they are able to move away from these and solve an abstract problem.

Resources to support mental agility can include:

• Counters
• Counting sticks
• Number lines (numeral tracks and numeral rolls)
• Arithmetic racks (rekenreks)
• Hundred squares (both complete and empty)
• Bundling sticks (and base ten materials)
• Ten frames and five frames
• Arrow cards and digit cards
• Interactive whiteboard

Screening of resources helps children with the transition from concrete to abstract.

The importance of finger and dot patterns

Can mental calculations involve pencil and paper?

Mental calculation is not about the use of algorithms. Children however might want to use pencil and paper to support their mental calculations:

• by writing the problem down so they can visualise it (if the question was given orally)
• by showing their method so they can explain it to others

Using an Empty Number Line (Addition and Subtraction Strategies)

The empty number line is a powerful tool for developing children’s calculation strategies and developing their understanding.

Here are two examples of the calculation 61 + 29 = 90 represented on the empty number line:

+20 +9

61 81 90

This is an example of a jump strategy, where we start at 61 and jump on 20 then 9 to arrive at our answer.

+30

-1

61 90 91

This is an example of an over-jump strategy, where we start at 61 and jump on by 30 to get to 91 and subtract one as we wanted to add on 29 only.

Addition and Subtraction Strategies

Children who are confident with addition and subtraction use a variety of strategies to solve problems. Our aim is that our children have a selection of non count-by-one strategies, and are able to choose and use the most efficient one for the problem they are facing.

Seven Mental Strategies for addition and subtraction (with a suitable example and an empty number line demonstration where appropriate):

SplitSplit tens and ones, add/subtract them separately, and then recombine.

e.g. 86 - 21 = 6580 – 20 = 60 and 6 – 1 = 5, so answer is 65.

JumpBegin from one number, jump tens then jump ones (or ones then tens).

+10+10

e.g. 43 + 20 = 63

43 53 63

Over-jumpBegin from one number, overshoot the jump, and then compensate.

e.g. 46 + 19 = 65 +20 -1

46 65 66

Split-jumpSplit tens and ones, add/subtract tens, add first ones and then jump second ones.

e.g. 37 + 22 = 5930 + 20 = 50, then 50 + 7 + 2 = 59 (jump on from 50)

Jump to the decupleBegin from one number, jump to the nearest decade, jump tens, then jump remaining ones.

+2 +20 +3

e.g. 38 + 25 = 63

38 40 6063

CompensationChange one or both numbers, add/subtract, then compensate.

e.g. 18 + 19 = 37 Use 20 + 20 = 40, then subtract 2 (for 18) and 1 (for 19); 40 – 2 – 1 = 37

TransformationChange both numbers while preserving the result, then add/subtract.

e.g. 49 + 17 = 6649 + 17 is the same as 50 + 16 = 66

Using Arrays (Multiplication and Division Strategies)

Arrays are formed by arranging a set of objects into rows and columns. They are useful when exploring multiplication and division.

Exploring 4 x 3Exploring 12 x 3 = (10 x 3) + (2 x 3)

Children can begin by building their own arrays using dot cards (e.g. 2-dot cards) or different coloured counters (e.g. 2 of each colour) and can explore the following questions:

• How many dots are there on each card (or how many counters are there of each colour)?
• How many cards are there (or how many colours are there)?
• How many dots (or counters)are there altogether?

The teacher can then present arrays and ask the following questions:

• How many rows are there? (What can you say about each row?)
• How many columns are there? (What can you say about each column?)
• How many dots are there altogether?

Depending on whether the child can count in the relevant multiplies or not, they may need to count in ones or may be able to count in multiple steps to work out how many dots there are altogether.

Teachers can then move to screen parts of the array.

Arrays are also ideal for teaching the commutativity of multiplication. Simply by rotating the array we can show that the answer to the multiplication is the same (i.e. there are the same number of dots) no matter which way round the sum is.

How do I Collect Responses from Learners?

AifL approaches, which may include the following should be in evidence in all classrooms:

• Asking for no hands up (children can show they are ready with thumbs up if the teacher wishes) and choosing someone randomly (using names on lollipop sticks) or specifically (by targeting a specific child).
• Giving children digit cards or number fans so they can all display their answer.
• Asking children to write their answer (and strategy?) on a small wipe board.

Remember that the strategy is as important as the answer so the teacher should ask more than one child for their answer and strategy. It is also important to discuss and investigate any wrong answers so all the children can learn from these and improve all their strategies for the future.

How does Higher Order Thinking fit with mental agility?

Mental calculation is more than just recalling number facts. In itself, this is an important skill that aids learners to concentrate on their calculations, the problems and the methods involved. However, if they are to develop their ability in mental maths, they need to apply thinking skills to the problems presented to them and deepen their understanding.

Remembering

• What is 3 add 7?
• What is 6 x 9?
• How many days are there in a week?
• What fraction is equivalent to 0.25?

Understanding

• Which is bigger, ¾ or 0.8?
• Is 14 an even number?
• What is the better approximation for ½ of 593 - 250 or 300?

Applying

• Tell me two numbers that have a difference of 12.
• If 3 x 8 is 24, what is 6 x 0.8?
• What is 20% of £3?
• What is the remainder when 31 is divided by 4?

Analysing

• The number 6 is 1 + 2 + 3, the number 13 is 6 + 7. Which numbers to 20 are the sum of consecutive numbers?
• On a 1 to 9 key pad, does each row, column and diagonal sum to a number that is a multiple of 3?
• The seven coins in my purse total 23p. What could they be?

Evaluating

• If 6 x 7 = 42 is 60 x 0.7 = 42?
• Are these all equivalent calculations: 34 – 19, 24 – 9, 45 – 30, 33 – 20, 30 – 15?

Creating

• How might we count a pile of sticks?
• How could we find 20% of a quantity?
• How could we test a number to see if it is divisible by 6?

What types of questions might I ask?

In the table below are examples of closed questions and open questions. Closed questions generally have just one correct answer, while open questions usually offer alternatives and may have a number of different answers, each of which is correct for a different reason. Each type of question has their use and purpose.

The use of questions is generally to:

• Prompt thinking and get children started.
• Probe understanding and establish the confidence and security of the children’s knowledge, skills and understanding.
• Promote thinking to set a new challenge, problem or line of enquiry that the children can follow.

The importance of the type of questions learners are presented with, when developing their mental agility skills, cannot be overestimated.

Closed questions help establish specific areas of knowledge, skills and understanding; they often focus on children providing explanations as to how and why something works and can be applied when identifying and developing approaches and strategies for a particular purpose.

Open questions help to generate a variety of alternative solutions and approaches that offer children a chance to respond in different ways; they often focus on children providing explanations and reasons for their choices and decisions, and a comparison of which of the alternative answers are correct or why strategies are more efficient.

Appendix – Further Reading

• “Teaching Learners to Calculate Mentally” – National Strategies Primary (Department of Education England and Wales) 2010

• “Teaching Number – Advancing Children’s Skills and Strategies” – Robert Wright, Jim Martland, Ann Stafford, Garry Stanger

With thanks to the working group:

Sophie Allen, Pirniehall Primary SchoolAndrew Blaikie, Carrick Knowe Primary School

Helen Donaldson, Cramond Primary SchoolLouise Stevenson, DO Early Number

Tracy Urquhart, Liberton High SchoolKatherine Watson, Sciennes Primary School

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