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Primary Magazine - Issue 95: Achieving Fluency in Addition and Subtraction Facts


Created on 13 February 2017 by ncetm_administrator
Updated on 20 June 2017 by ncetm_administrator

 

Primary Magazine Issue 95
 

Achieving Fluency in Addition and Subtraction Facts

by Clare Christie, maths lead for a federation of two Bristol schools, and Teaching for Mastery Lead for Boolean Maths Hub

Why we focus on fluency in addition and subtraction facts

  • A defined set of addition and subtraction facts builds the basis of all additive calculation, just as times tables are the building blocks for all multiplicative calculation. For example:
  • If children are not fluent in these facts, then when they are solving more complex problems, the working memory is taken up by calculating basic facts, and children have less working memory to focus on solving the actual problem (see Is It True That Some People Just Can’t Do Math? by the cognitive scientist Daniel Willingham). So fluency in basic facts allows children to tackle more complex maths more effectively.
  • The importance of fluency is recognised in the national curriculum, and since 2016, children’s fluency is more heavily tested by SATs.
  • Children need to be taught strategies to solve these facts. Conferencing I have done over the last few years show that most children don’t magically become fluent in these facts even in KS2, particularly for those facts which bridge 10. If they aren’t explicitly taught to solve e.g. 6 + 7 by thinking ‘double 6 and one more’ or to solve 12 – 8 by using 'find the difference' strategies, then many children will get stuck on inefficient counting based approaches.
  • Counting on approaches are not only less efficient, they are associated with lower attainment in maths as well. Research by Tall and Gray (1994) found what our own extensive conferencing has shown: higher attainers tend to use known facts or derived fact strategies, and lower attainers are much more likely to use counting based approaches to solve addition and subtraction facts.

Which facts have we focused on?

The full set of addition facts is:

(You can download the grid used by the author of this article in her schools here.)

And here are the corresponding subtraction facts:

Note that in subtraction facts not all subtractions within 20 are root facts, e.g. 17 - 5 is not considered a root fact (7 - 5 is the root fact for this).

The majority of these facts are learned in Years 1 and 2.

In Reception, children become fluent in working with totals to 5 (though not presented as number sentences), e.g. “Show me 5 on your hands. Now show me 5 in a different way.”

Year 3 continue to focus on securing fluency in facts which bridge 10, and subtraction facts which bridge 10 in particular. Although this is a Year 2 objective, my feeling after many hours teaching and reflecting on factual fluency is that aiming for real fluency in subtraction facts such as 14 – 9 and 13 – 5 (where fluency is an answer in three seconds) for each and every child in Y2 is unrealistic. We feel that unless we are honest about that and accept the need to secure these facts in Year 3, we risk having children who never become secure in this.

Does fluency just mean memorisation?

Not necessarily – when you conference adults on how they solve addition and subtraction facts, almost all adults rely on very quick use of strategies to solve some of them. Reflect carefully on the set of addition and subtraction facts shown: which have you memorised and which are you very quickly deriving? We’ve taken fluency to mean ‘getting an answer pretty quickly and with limited demands on working memory’, aiming for an average of three seconds or less per fact. My work conferencing fluent children in KS2 who were working at this speed showed:

  • Most facts which didn’t bridge 10 were memorised – the children reported ‘just knowing’ that 4 + 5 = 9 or 2 + 6 = 8 for example.
  • For facts which bridge 10 the picture is more complex, and many of the facts which bridge 10 were quickly derived using strategies (but still in less than three seconds!).
    • Double 6, 7. 8 and 9 were always memorised in fluent children
    • Many fluent children also reported ‘just knowing’ that 9 + 3 = 12 and 8 + 4 = 12 and related this to their times table/skip counting knowledge.
    • Fluent children in the year groups conferenced (up to Year 4) generally reported using strategies for many of the other facts. 8 + 9 is an example of a fact that actually very few people (either adults or children) have memorised. Of the many hundreds of teachers I have asked, only about five per cent report ‘just knowing’ that 8 + 9 = 17. Most fluent people solve this through very quickly applying a strategy: bridging through ten, near doubles or compensating (adding 10 and subtracting 1).

As a reference point, this grid is a good example of the approaches taken by a fluent, high attaining Year 4 child to each of the addition facts: he doesn’t use a counting approach for any of the facts, but he has certainly not memorised them all either (K= Known fact; S= Strategy):

Why not try this on some of your children? Find out how they solve each of the 121 facts. They are all written out on this PDF – just print and cut them out. I use these sorting circles with the children, and have found they very quickly get the idea once I have given them an example of a known fact (the vast majority of KS1 and KS2 children will just know that 5 + 5 = 10 for example), a strategy fact (e.g. calculating 6 + 5 by relating it to 5 + 5) and a counting based approach. If children say they would use counting, don’t bother getting them to solve the fact – you will be there all day. If they say strategy, I find it interesting & helpful to ask what strategy they have used: my notes on the child whose grid is shown above for example showed that he added numbers with a difference of 2 by relating to doubling the number in between (e.g. 6 + 8 = 7 +7; 5 + 7 = 6 + 6). It takes about 15 minutes to conference one child, and is time very well spent.

How do children become fluent?

As mentioned above, children need to be TAUGHT strategies to derive the facts! An interesting piece of research (Thornton, 1978) showed that teaching strategies is more effective in securing fluency in addition and subtraction facts than taking a rote memorisation approach. That is to say, even if your aim is memorisation, the most effective way to get there is through the teaching of strategies. There is a huge amount to unpick in this and you need to consider how children are going to become fluent in each and every fact. For example, we want children to just know that 4 + 2 = 6 and 9 – 2 = 7 etc so we need to teach children that when we add 2 or subtract 2 we are moving to the next/previous even number (if starting on an even) or odd number (if starting on an odd). Without being taught this, many children will count. e.g. for 9 – 2, “nine, eight, seven”. Being able to do this without counting requires being able to count fluently in odd numbers (as well as in the more commonly practised even numbers), something we realised we were not previously teaching our children to do.

We mapped out a teaching progression so we could identify when every individual fact was being taught, and discussed and agreed teaching approaches for each of these fact groups. In Year 1 we teach strategies for facts within 10 (steps 1 – 7) and in Year 2 we teach the bridging ten facts (steps 8 – 11).

  1. Adding 1 (e.g. 7 + 1 and 1 + 7)
  2. Doubles and near double of numbers to 5 (e.g. 3 + 3, 4 + 5, 5 + 4)
  3. Adding 2 (e.g. 4 + 2 and 2 + 4)
  4. Number bonds to 10 (e.g. 8 + 2 and 2 + 8)
  5. Adding 0 to a number (e.g. 3 + 0 and 0 + 3)
  6. Adding 10 to a number (e.g. 5 + 10 and 10 + 5)
  7. The ones without a family 5 + 3, 3 + 5, 6 + 3, 3 + 6 (these pairs of facts are the only ones which don’t fit in any of the other families, though the last two can be related to counting in 3s)
  8. Doubles of numbers to 10 (e.g. 7 + 7)
  9. Near doubles (e.g. 5 + 6 and 6 + 5)
  10. Bridging (e.g. 8 + 4 and 4 + 8)
  11. Compensating.

The last three of these strategies can often be used interchangeably, e.g. for 8 + 9, some people will use near doubles (e.g. 8 + 8 + 1), some will use bridging (e.g. 8 + 2 + 7) and some will use compensating (8 + 10 – 1).

NB: before the children are ready to learn bridging as a strategy, they need to be able to partition all single digit numbers. Adding 8 + 5, for example, by bridging through ten requires children to partition 5 into 2 and 3. We do an enormous amount on partitioning single digit numbers all through Year 1.

There is even more pedagogy involved in supporting children to become fluent in subtraction facts then there is for addition facts, beyond the scope of this article.

Once children have been taught the strategies, they need to move on to PRACTICE of the facts, Remember, for many facts the ultimate aim of the practice is memorisation, while for others the aim of the practice is increasing speed and fluency in the applied strategy. We have a software package which children use to practice which encourages them to aim for an average of three seconds or less per fact. This means that in a two-minute practice session the children should be recalling at least 40 facts: the more you practice the quicker they get, and the quicker they get, the less time it takes out of the lesson.

Generally, for practice we focus on:

  • Practising the set of facts being learnt (or just learnt) in isolation for a few days
  • Mixing these up with all previously learnt facts.

We use a mixture of a software package, practice sheets in class and flash cards/smartboards to give the children practice, in fact very similar approaches to those we take in phonics sessions. In fact, we’ve found thinking about the structured and systematic approach we take to the teaching of phonics in general is a good analogy for thinking about structured and systematic teaching and learning of these strategies and facts.

Although working on securing fluency in addition and subtraction facts might sound basic and dry, we have found children really enjoy both the discussion and reasoning that the learning of strategies involves, and the confidence they get from having these building blocks in place. In one memorable conferencing session with a middle attaining Year 4 child, who had been part of a class revisiting the teaching of addition facts for a month, I asked how she would work out 6 + 9:

“I’d probably add one to make 10, then add the other 5 to make 15,” she said before continuing, “but of course the other way you could think about it is that 6 is two groups of 3 and 9 is 3 groups of 3, so if you add them together you get 5 groups of 3 which is 15.”

This was a child who just four weeks before had been reliant on counting based strategies and had solved 6 + 9 by putting “nine in my head and counting on.” It was really exciting to see she had gained not just greater confidence in basic arithmetic, but had come to see patterns and connections with a much much wider reach than this.

References:

Gray, E.M. and Tall, D.O. (1994): Duality, Ambiguity and Flexibility: A Proceptual View of Simple Arithmetic.

Thornton, C.A. (1978): Thinking Strategies in Basic Fact Instruction.

Willingham, D.: Is It True That Some People Just Can’t Do Math?.

 

 
 
 

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Comments

 


14 July 2017 13:23
I'm coming back to this again as I think it's so important. Just wanted to share that I did a lesson with Y1s where we sorted calculations into 4 different strategies +/1, +/10, doubling or 'making 10'. It really showed me where their gaps and understanding were and I hope it helped them to see that there are lots of useful ways we can use facts flexibly to calculate. I've got mixed Y1/2 next week and I'm revisiting near doubles, +/9 (compensation) and partitioning to bridge 10. The last one seems to be the trickiest as, like you say, they need to be really secure at partitioning 1 digit numbers before they can do this. We have some confident children who seem to do this naturally without much teaching input but others who can't see it yet. We use Numbergym too but currently don't have the hardware to use it in class - only at home and as usual, some do and some don't. Will keep trying though!
By Misseast
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20 June 2017 11:41
We have now included a link to the fact grid in the content above.
19 June 2017 16:25
Very interesting article. Please could anyone tell me where I might find the 'fact' grid for Y1/Y2 at the top of the page. It would be very useful to use both in class and as homework.

Thanks Alex
By redkites2017
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01 May 2017 20:13
Hi there Laurie, lovely to hear from you. I hope all is well. Sorry it's taken me a while to get back to you. Do feel free to take and use any of this in your PD. I'm happy to hear you found it interesting! I agree that the basic conferencing and sorting is so powerful and totally fascinating. I also agree with you that counting is essential in terms of understanding the number system. The way I think about it in my own practice now is that counting is counting and calculating is calculating, and using the former for the latter isn't that helpful but absolutely, both need developing. Now I have my own four year old case study at home it's been great to see all of this outside the classroom setting too. The curse of being the child of a maths teacher! Thanks for the Boaler link which I hadn't specifically seen. Lots that she says chimes with my own experience but I've been reading her stuff on finger counting with interest and wonder if I would draw slightly different conclusions from the same research. One day I'd love to do more formal research on all of this! Take care and thanks for the note here! Very best wishes to you.
By clarech
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30 March 2017 10:13
Clare, thank you for your ever insightful thoughts about the teaching and learning of arithmetic. There's a PhD thesis in there somewhere isn't there? I'd like to incorporate some of this into some CPD. I think the sorting activity would be a really interesting one from a Maths Lead monitoring point of view.

By using this tool with a sample of pupils across the school it might provide some useful data about whether pupils are reliant on counting strategies rather than related facts. This will help leads to consider how teachers and the school's teaching approaches need to be developed for fluency in number facts.

I have only skim read but also of interest is this clip from Jo Boaler about number sense. Her research showed that low attainers ofter did a different type of maths because they only had counting strategies. https://www.youcubed.org/what-is-number-sense/

Counting is an essential tool early on in the development of number but it is not the only tool. It helps pupils with their ordinal placement of additive properties but this needs to develop, as you say into part-whole relations, and unitising that are such important elements for fluency.

I find the practice examples in the Shanghai Maths Project books by Collins, really useful for developing these bigger ideas of additive reasoning.

Have a great Easter Clare. x
13 March 2017 20:10
Thanks Clare, if only it were that easy, eh?! Had a quick look and there is lots of support material on the website anyhow, I'll keep an eye out for further forthcoming articles. Love the look of Number Gym!
By stquinnl
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12 March 2017 23:43
Hi Clare,

You mentioned,
"There is even more pedagogy involved in supporting children to become fluent in subtraction facts then there is for addition facts, beyond the scope of this article."

I would love to see your work on the strategies behind securing the subtraction facts. Until now I have mainly focused on the Fact Family Triangles and the Family of Facts related to each addition fact.
11 March 2017 14:35
And thanks for the link Connor
By clarech
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11 March 2017 14:31
stquinnl, meant to say...I don't have a 'comprehensive world view' (!) on the teaching of multplication/division facts in the same way as I do for addition/subtraction facts and I don't know the literature nearly as well - however I think there may be an article coming up in a future magazine.
By clarech
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10 March 2017 05:49
Interesting article Clare. I like the counting in odd numbers idea for helping to secure adding 2 to the numbers 3,5,7 and 9.

Check out Ben Harding's - Big Maths - approach that has simplified your list of 100 facts down to 36.
09 March 2017 16:26
Thanks for clarifying. I totally agree that if children are secure with these this will really help them in KS2.
By suziej
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09 March 2017 15:48
Hi there Suzie. Seeing as historically we had had quite a lot of children coming up to Year 3 who were still doing bridging additions by counting (e.g. 6 + 7) we thought we would start by aiming for this to be secured by the end of Year 2 as a first step! Again, I am not saying it is impossible to secure these in Year 1 but we just decided to focus on it in Year 2. As content can be moved within key stages this is still in keeping with the requirements of the NC. Once you break down the mini steps involved in bridging or near doubles there is quite a lot involved, particularly around decomposition of single digit numbers so this is a big focus in Year 1. As well as doing addition and subtraction facts within 10 and decomposition of single digit nos, Year 1 will then apply this to PV (e.g. adding multiples of 10, like 30 + 40), as well as doing load on understanding place value in 2 digit numbers (5 + 20, 25 - 20, 25 - 5 etc). They also do other extrapolating of addition facts (I know 3 + 4 = 7 so 13 +4 = 17, 23 + 4 = 27 etc etc) and similar with subtraction. The aim is for all of that to take place without counting in 1s. We've worked really hard on Y1 - 2 progression, and feel if children are coming up with that secured it is a really good start in Y2. They tackle addition facts through ten right at the start of Y2. Hope that makes some sense.
By clarech
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09 March 2017 12:51
I love this. I totally agree with the focus on mental fluency.

Can I just clarify the key for the addition facts? You have put 9+7 as a fact you learn in year 2 but in the National Curriculum they are learning facts up to 20 in Year 1 so I am just interested how you have separated the Year 1/2 facts. I look forward to hearing from you. Many thanks.
By suziej
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09 March 2017 08:41
A great article...it will help me explain to colleagues why it is worthwhile spending time teaching strategies for 'easy' calculations and not just rushing through to using bigger numbers in ks1🙂 Do you have any material to share, or can you suggest some reading material, on fluency in multiplication and division? Many thanks!
By stquinnl
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09 March 2017 08:34
Hi there Ilena. Glad you find it helpful. We use Number Gym, and in particular Bond Builder and have found it fantastic. Sections 4 and 6 of Bond Builder focus on addition and subtraction within 10, and 13 and 15 focus on the addition and subtraction facts which bridge 10 (e.g. 5 + 7, 12 - 8). I am sure there are others out there, but we liked this because you could see the pedagogy shining through (Bond Builder starts with dot matching for example) and it was good value. The children have logins they can use at home and it allows you to download spreadsheets with average time for each fact set for each child. Ideally I would love a programme that breaks fact groups down even further than this but I haven't found it yet!
By clarech
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08 March 2017 22:47
This is a great article for cpd. You mentioned software packages in the article. Are there any in particular ones that are helpful?
By ilenahoward
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08 March 2017 20:36
As the lead of Number Basics at my school I found this really useful and am going to try conferencing our gifted and talented pupils and those children who are middle achieving to identify the differences in their strategies and known facts.
08 March 2017 14:35
This looks really fascinating and useful thanks.
By Misseast
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