• Thompson, I. (1999) Mental Calculation Strategies for Addition and Subtraction, Mathematics in School, Volume 28 Issue No. 5

• Thompson, I. (2007) Deconstructing calculation methods. Part 2: Subtraction, MTO2, ATM

A thought provoking article by Ian Thompson, who deconstructs the primary national strategy’s approach to written subtraction.

(This article is only available to full ATM members on the site.)

• Sugarman, I. (2007) The same difference, MT202, ATM

Ian Sugarman highlights a mental strategy for subtraction overlooked by the primary strategy, that offers a more child- friendly alternative to decomposition.

(This article is only available to full ATM members on the site.)

• Houssart, J. (2000) The role of number resources in the daily mathematics lesson, Centre for Mathematics Education, Open University

This article is based on interviews with Key Stage Two teachers about how they use number resources. It concentrates on one resource advocated by the Numeracy Strategy, the number line and one resource which now has a lower profile, base ten blocks. Particular attention is paid to the use of resources in teaching calculation. Findings show that most teachers make use of the number line, often to teach calculation. Base ten blocks are used for teaching calculation by some teachers, though approximately half make no use of them. However, closer analysis suggests that teachers have different levels of understanding both about how number lines might aid calculation and about the place of standard written methods of calculation.

• Beishuizen, M. (1997 ) Two types of mental arithmetic and the empty numberline, Dept of Education, Leiden University, The Netherlands

The empty numberline was introduced as a new model in Realistic Mathematics Education (RME), after discussions in the Netherlands how best to improve the basic skills up to 100. This research explores the value of the number line and looks at alternative models.

Pupils should be taught to:

Activity A - Interactive Base Ten Blocks

This software shows a clear, interactive version of the use of base ten blocks for addition of two 3-digit numbers. It clearly demonstrates regrouping, and has a useful focus sheet for pupils to guide their work.

Activity B – Twenty divided into Six

This task from Nrich requires children to arrange a pack of 20 cards numbered 1-20 into 6 unequal piles of the same total. A good activity for consolidating mental addition skills and encouraging children to discuss their approaches and ideas. Could it be extended to larger numbers?

Activity C - Images of addition and subtraction

This activity comes from the NCETM Secondary Magazine, and looks at the range of models and images used for addition and subtraction. Challenge the children to sort the cards and think about all the skills they are using.

Activity D – Models and Images

‘Slidey- box’ cards, number trios and the function blocks Interactive Teaching Programme (ITP) can all be used to support children in understanding the concept of inverse operations for addition and subtraction.

Activity E - Reach 100

This activity from Nrich requires children to use their knowledge of addition and subtraction to add several two digit numbers to reach a target total.

Activity F – Estimating differences

In pairs, children take turns to circle two numbers from a grid such as this:

They agree a time period in which to estimate the difference between the two numbers and check the range in which the estimate falls on the chart below. Play continues until all of the numbers are used, and the player with the most points wins.

• Solve addition and subtraction two-step problems in contexts, deciding which operations and methods to use and why

Activity G – Bar Models

This resource provides a useful introduction to the bar model approach for calculating, with lots of ideas for problems to use in the classroom. Additional resources can be found at the ‘Thinking Blocks’ website

Examples of what children should be able to do, in relation to each (boxed) Programme of Study statement

add and subtract numbers with up to 4 digits using the formal written methods of columnar addition and subtraction where appropriate

estimate and use inverse operations to check answers to a calculation

solve addition and subtraction two-step problems in contexts, deciding which operations and methods to use and why

Children should be able to carry out practical tasks such as that represented here in an Australian classroom.

Children were asked to individually run the class market stall. They were told they could use mental strategies or the whiteboard provided to assist them in their calculations. The customer (their teacher) would come to purchase some items. Each child was asked to solve a transaction problem involving a single item (calculating change – subtraction) and then a transaction involving two items (adding together values and then calculating change or two subsequent subtractions). They were also asked to explain their thinking and asked how to give the change in a different way (representing money values in various ways).

Children should be able to solve problems such as:

• I have read 134 of the 512 pages of my book. How many more pages must I read to reach the middle?
• There are 8 shelves of books. 6 of the shelves hold 25 books each. 2 of the shelves have 35 books each. How many books altogether are on the shelves?
• I think of a number, subtract 17, and divide by 6. The answer is 20. What was my number?
• You start to read a book on Thursday. On Friday you read 10 more pages than on Thursday. You reach page 60. How many pages did you read on Thursday?

• NCETM - Developing column subtraction in key stage 2

• Market Stall, Australian Curriculum, Assessment and Reporting Authority, Work sample 12.