Maths to share - CPD for your school
Following on from Maths to share - addition in the last issue of the Primary Magazine, this month we explore the operation of subtraction. Teachers often report that children experience a great deal more difficulty with subtraction than they do with addition. Feedback from analysis of national, end-of-key-stage tests also supports this.
So why does it cause so many problems? Are pupils not taught the key skills of subtraction in a structured, progressive way? Are they not given sufficient time to practise and consolidate those skills? Do we not provide opportunities for them to apply these skills in different contexts? Is teachers’ subject knowledge of the teaching of subtraction sufficient for the task? Ask colleagues to consider where they think the problems lie and discuss as a group.
Are we directly teaching mental calculation skills? Ask colleagues to share the mental calculation strategies for subtraction that they teach. How much time do they allow for the children to practice these? Ask colleagues teaching in Years 5 and 6 whether the compact written methods take over from mental methods when they are formally taught.
Ask them to consider the strategies they might use in answering the following questions. Some suggestions of how they might be solved are shown here. Before sharing them with staff, ask them to write down the methods they would use and also those they would expect the children to use:
9.5 - 8.6
The skill of choosing a method appropriate not only to the operation, but also to the numbers involved is one which adults often don’t realise they are using. Ask colleagues to consider what is involved in the direct teaching of these mental calculation strategies. Suggestions might include:
£12.75 - £7.49
- ‘count up’ from 8.6 to 9.0, then from 9.0 to 9.5
- calculate 95 – 86, then adjust using knowledge of place value
- use image of a number line
Find the difference between 840 and 421
- round £7.49 to £7.50. Subtract this from £12.75, then add 1p back on
- use mental image of a number line to know whether to add or subtract the 1p
50 013 - 6 078
- recognition that 420 is half of 840. 421 is one more than half, so the difference is one less than half, i.e. 419
4 400.32 - 20.08
- might be tempted to use a more formal vertical method, based on decomposition. errors likely due to the number of ‘zeros’
- counting on might be more efficient e.g. adding 922 to get to 7 000 and then 43 000 and 13
- deal with whole numbers and decimals separately, i.e. 4 400 – 20 and 0.32 – 0.08
- adopting a structured approach so that skills are developed systematically
- modelling a strategy using an image, a model, or a ‘real life’ scenario
- using an error or less efficient strategy as a starting point for demonstrating a better strategy
- encouraging children to compare strategies and improve them
- showing how to use a known fact in developing a strategy
- providing the children with a ‘prompt’ to help them recall and then use an appropriate fact.
It is important that pupils are encouraged to build up a bank of these skills to draw upon when most appropriate. Progression through their teaching does not mean that previous skills are replaced, merely added to.
It is sometimes the very early calculation skills that are the most difficult to teach. A progression through early skills for subtraction might look like this:
Many schools now have a ‘Calculation Policy’ or ‘Route Through Calculation’, outlining their approach to the teaching of calculation in school. Often these take account of expanded and more compact written methods, but not necessarily the progression in these early skills. If possible, take the time as a whole staff to review the school’s policy and ensure that everyone is fully supportive of it.
Something to Share in Issue 4 of the Primary Magazine explored the importance of using effective models and images in mathematics to support pupils’ learning. Ask colleagues to list the different models and images they use in the classroom and consider the reasons for using them when teaching subtraction. They might suggest:
- helps the teacher in demonstrating or modelling a calculation strategy or provides a physical representation of a mathematical concept or operation
- enables children to do a calculation or use a strategy which they could not do without assistance
- can keep all pupils involved and engaged
- can help children to visualise what is happening.
If not already familiar, show colleagues the NCETM online Self-evaluation Tool for Mathematics Content Knowledge. It supports teachers in checking their understanding of the mathematics they are teaching and explores ideas on how to develop their practice further. Following this session, ask them to use the tool to evaluate their level of confidence in teaching subtraction-related concepts. They will need to explore the topics Counting and Understanding Number, Knowing and Using Number Facts and Calculating. Encourage everyone to feed back at a future meeting and share any areas for concern. It is likely that similar areas for development will emerge – and what better way to learn than together!
Anghileri, J.: 2000, Teaching Number Sense. London: Continuum (pp46 – 66)