A common priority on a maths subject leader action plan will read something along the lines of… ‘To develop pupils’ fluency and reasoning skills’. Delving into this sort of priority with subject leads and teachers is always a fascinating one. Let’s take the fluency aspect. (I am in NO way suggesting that fluency is something entirely separate from reasoning – in fact far from it – but let’s make arithmetic fluency the focus for now)
For some, there will be a lot of fact recall going on in classrooms – early morning activity sheets where children come in and have to complete ‘number bonds to 10’ or perhaps a list of times tables facts. For others, there will be things developing across the school such as maths meetings or fluency sessions, where key concepts are being rehearsed and revisited on an almost daily basis with the teacher modelling and carefully crafting questions to pick up misconceptions and developing fluency that way.
While we are on the subject of early morning activity sheets – there is a note of caution here in that practice doesn’t always make perfect but practice DOES make permanent. Take a child who is coming in every morning and inserting a 4 as the missing number:
7 + ? = 10.
If this is never picked up by anyone but merely filed in a folder of ‘early morning fluency work’, said child may be limited in their strategy choice to solve something like 7 + 84 (from the KS1 2019 Paper 1), where the number bonds to 10 could play a key role in using an efficient regrouping strategy: 7 + 3 + 81 = 91.
I would like to illustrate how simple models could be used to rehearse and deepen understanding of key facts to develop fluency for children in KS1, to get to the point of being able to quickly solve the calculation above.
‘Recall all number bonds to and within 10 and use these to reason with and calculate bonds to and within 20, recognising other associated additive relationships’ is one of the KS1 TAFs for children working at the expected standard.
So let’s look at how this skill could be rehearsed. With the whole class together and the teacher leading the session, take a look at this model – the humble (yet powerful) tens frame.
How many counters are there altogether?
How do you know?
Do the blank spaces help?
Replies the children may give:
Child 1: I think it is 6 because there are 4 gaps. 4 fewer than 10 is 6.
Child 2: I can see a group of 2, a group of 3 and 1 more counter. 2 add 3 add 1 is 6.
The children are able to see ‘the ten’. There are 10 spaces on the tens frame to be filled. They are able to see how many are already filled with counters and therefore the number of gaps gives them that ‘how many more?’ piece of information. Use of the speaking frame and the vocabulary provided enables the children to precisely explain what they can see or what they notice, allowing them to answer questions and think more deeply.
Why not bring in some part whole relationship modelling here?
The part whole cherry model for example:
The 2-part cherry model illustrates the thinking of Child 1 with the 10 squares being the whole, the 6 being the counters and the 4 being the gaps.
The 3-part model then represents the thinking of Child 2.
A note on cherry models – from work with schools on diagnostic test analysis in recent weeks, children seem to be very thrown by the part whole cherry models being rotated (a bit like when a square is rotated and the children no longer identify it as a square). So do throw in a ‘sideways’ cherry every now and again. The parts still combine to make that whole, no matter which way up the model. I’m not sure that in my classroom, I would have allowed a diagonal cherry model to be on the working wall but maybe… in the interests of flexible thinking… I would allow it now.
While on the topic of children understanding maths models in a flexible way, how often do you present the children with the equals sign in different places in a calculation? This, again, challenges their understanding of ‘part’ and ‘whole’. So for example, if we only present calculations in the form of 6 + o = 10, how would they fare if presented with 10 = 6 + o?
Anyway, back to the slide above… a further adaptation to this could be to give the children a part whole cherry model and ask them what the tens frame could look like. Are there multiple layouts?
Throughout, the children responding to the questions are using reasoning to identify the number of counters. For example, one child is looking at the groups they can see and applying this part whole understanding to identify the total number. The other was using their understanding of the gaps in the model to identify ‘how many fewer’. They are being provided with the language required to reason precisely and the teacher is able to teach this using the visual model alongside.
Now, let’s adapt the model slightly and look at how it could be used to develop fluency in mental calculation strategies.
Presented on an interactive board, the counters can be moved to model how to calculate 7 + 6, using understanding of those number bonds to 10.
So for this example, the speaking frame would be used to explain that:
“I can regroup 6 into 3 and 3.
I can add 3 to 7 to “think 10”.
Then add 3 to 10 to total 13.”
A possible model to enable all children to access this could be:
‘My turn’ – teacher models
‘Our turn’ – all say in unison
‘Your turn’ – children say independently of the teacher
As this is repeated across several sessions and the children become familiar with the language and the process, the ‘my turn’ part may become omitted from the sequence.
This would be modelled along with the explanation and the model would end up like so:
How many more to 10?
Which number shall we regroup?
Now we have made 10, what else do we need to add?
Why would it not be useful to regroup the 6 into 4 and 2?
The children are now applying their knowledge of the bonds to 10 from above to calculation. Later use of this regrouping strategy without the visual model will rely on the children’s recall of two facts: 7 + 3 = 10 and 6 = 3 + 3. Use of the double sided counter image here helps the children to see ‘where the 6 is’ once it has been regrouped to make calculating easier.
It wouldn’t necessarily need to be the 6 that is regrouped here. The 7 could be regrouped into 4 and 3, leading into a conversation with the children around the commutativity of addition.
The calculation (and therefore number of counters) could be changed several times across several sessions before adapting as suggested below.
The addition of the familiar part whole model could form the link, moving from the children being able to ‘see it’ and ‘say it’ to being able ‘write it’ as a calculation.
The skills of reasoning can be taught throughout the development of this strategy using sentence starters such as:
I have noticed that… there are 3 gaps in the first ten… so we need to move 3 counters from the 6.
I already know that… 7 and 3 more is ten… so I need to regroup the 6 into 3 and 3.
My strategy is different because… I would regroup the 7 into 4 and 3 to add the 4 to the 6 to make 10.
If the development of fluency and reasoning is of interest, hopefully this has provided some food for thought in terms of a simple model to use.
- The language is being modelled with the visual model alongside.
- The structure of the maths is exposed.
- The children are provided with speaking frames to support with clearly articulating thought or strategy.
- Misconceptions can be quickly picked up and adaptations made to the fluency session or in fact the main maths lesson the following day if relevant.
- The act of having to retrieve key facts on a regular basis leads to stronger long term memories to be drawn upon when problem solving. The children are more likely to have the ‘facts at their fingertips’.
We have been delighted by the response to our new HfL maths fluency materials and training.
3 new dates have been added for: Developing effective fluency sessions in maths
A one day training course which includes a full suite of materials for the school, including 18 teacher PowerPoints, and guidance document and staff development resource (a PowerPoint which the school can adapt and use for INSET / staff meeting).
29th April - The Gateway, Aylesbury
Click here for further details.