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#### Effective Professional Development: Surviving and Thriving by Deliberate Design

Vina Sharma discusses the recommendations of the EEFs Effective Professional Development guidance report

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by Staffordshire Research School

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Teaching for mastery is now a well-embedded notion in Primary teaching, focusing on a secure development of skills across the domains of fluency, mathematical reasoning and problem-solving. The NCETM describes one of its central tenets as “focusing on mathematical relationships and making connections” and in practice, an important element of the shift towards mastery, was the renewed focus on exploring with manipulatives and a move away from any belief that, where once manipulatives were only to be used by children who needed additional support; now they were for all. The EEF research review for Maths in Early Years and KS1 supports the view that “manipulatives can be powerful tools for supporting young children to engage with mathematical ideas” and although there are many maths materials available, I have come to wonder whether we are missing one of the most useful tools in our armoury: fingers.

In my discussion with many primary teachers, across different phases, there seems to be an idea that using fingers is something to be avoided. Something for babies, to be left behind once children reach a level of maturity, where they can use ‘the proper stuff’. But we only have to look to our colleagues in the East, to witness the truly amazing things that can be achieved with fingers, which could help to place them firmly back in the game. Fingers are cheap; children use them effortlessly from their earliest experiences with Maths and they travel well: all children have fingers. And now, with a renewed focus on subitising as part of the 2021 EYFS framework, fingers could be a material of substantial use. Additionally, because fingers are linked physically to the body, they can greatly support the development of spatial reasoning skills, which underpin mathematical fluency and success. Clements and Sarama state that “spatial deficits affect children’s ability to comprehend numeric quantities and magnitudes.” (2009) If we know and understand that “spatial visualisation has proven to be particularly important for mathematics learning and achievement” (Paying Attention to Spatial Reasoning), then surely it makes sense for such approaches to become part of the fabric of our approach, forming “a missing link across that curriculum” (National Research Council, 2006).

In my own school, we wanted to develop our approach to see if using fingers as a representational tool, would continue to support children’s fluency and spatial reasoning skills and so began with an evaluation of how practitioners perceived finger-use, our observations of children and any current finger-use, and an exploration of how we might develop this idea further. Like many settings, we knew that children were already using fingers in their earliest numerical experiences, as they developed security of cardinal and ordinal aspects of number. Nursery and Reception children were regularly working with initial representations of quantity through finger-songs, which allowed them to rehearse the ordinal aspect of the number sequence, but also deepen this learning by exploring representations of the numbers within them (cardinality). We added games such as ‘grow and show’, where we gave a number and asked children to grow it, by counting 1 finger to each number until they reached the goal and then quickly show it, by subitising with their fingers. This was an effective way to assess children’s number knowledge, visual representation skills and dexterity. And this idea could be easily extended include other manipulatives, but also developed into pictorial representations.

However we also wanted to explore how we might travel beyond the seemingly simple finger representations of numbers within 10, in order to support the development of our older children in KS1. Here we began to think about how numbers can be represented in different ways – and it is this element, which can help to inform the way we teach children to use their fingers in number-work. In KS1, subitising remains a key theme and fingers can continue to be used to support composition and comparison. At this stage, the emphasis moves to consider the importance of the “five and bit” structure; anchoring subitisation within a base of 5 because children subitise well within this range. It just so happens that this is intrinsically linked to the five fingers that children have on each hand: allowing children to represent a base of 5, to which other fingers can be added.

Fingers can also support the exploration of missing number problems, such as “I have 3: how many more to make 5?” because first children realise they are constructing patterns, which involve all of the fingers on one hand. Once secure this can easily be extended to problem-solve with the ‘5 and bit’ structure above. However, there came a time when we needed to question how far this method would track, as children began to explore larger numbers, with more complex construction and of course, when adding and subtracting within 100. We only have ten fingers. But this does not mean that we can only add and subtract within this range, we simply needed to rethink the way we were asking children to represent numbers (or components of numbers) and restructure our initial teaching to support this. This is where substituting representations becomes important. And this is not a new concept. We do this all the time with 10, which is not a static number. 10 changes position in the decimal system, moving from right to left and it also changes representation: altering from 10 to become 1 (a representation of one 10). Therefore using this idea, we found we could easily offer substitutions to children in the same way, allowing them to continue to use their fingers. There is also a direct comparison here with money (often an abstract concept for younger children) – where quantities are represented with singular coins (i.e. 5 as one 5p coin and 10 as one 10p coin).

Imagine that as we count 5 using our right hand, instead of representing this quantity with 5 fingers raised (four fingers and a thumb); we did so using only the thumb. Raise the index finger as 1 and then each successive finger to the little finger and you will have 4. To show 5, we lower the remaining fingers and release the thumb. The thumb then becomes a representation of this group: 5. In order to be successful with this idea, children need to be able to visualise the quantity 5 in their minds and understand that it is being represented differently (i.e. as 1) – but this is what subitisation is all about. If children master it deeply enough, they not only recognise quantity quickly, they can visualise it within other representations (such as the monetary values given above).

If we were to continue counting to 10, we would return again to the index finger of the same hand (5 + 1 is 6). The numbers 6 to 9 could be counted in the same way, reinforcing the 5 and bit structure, but leaving the left hand free to be able to continue the count to 10. Representing the 10 (like the 5) is then key. We need to show children that 10 is not static: that it moves position and changes representation, by being shown as 1. One ten. And we have found successfully, that even children as young as Reception, are capable of understanding this concept. The 1 ten is represented using the first finger of the left hand. This is important because in terms of spatial reasoning and visual representation, it shows the two important elements of this transition:

1. That the ten changes form/representation from 10 units to 1 ten.

2. That the ten is not static and moves to new positions upon each new group (i.e. each group of 10: 10, 100, 1000 etc). This can help to prepare children for future work with the columns of the decimal system.

Then, continuing to use this approach proves easy for the continued exploration of numbers within 20 and of course all the way to 100. And because each new cycle repeats in a similar way, we encourage children to develop secure internal and visual/physical representations of the relationships between these numbers. The number 26 is counted in the same way as the number 76, we simply additionally hold up the thumb as a representation of 5 (in this case 50, because tens are counted on the left hand).

Even in its earliest stages, as we observed children working with this approach, we saw a growing confidence across the ages, in fluency and reasoning, where children began to more readily build and discuss relationships between numbers, with a greater awareness of pattern: 10 can be shown as ten 1s, one 10, two 5s and five 2s. And it is this flexibility of thinking and representing, which can really aid children to master number skills and develop speed and automaticity in their working, creating connections with both body and mind.

Even in its earliest stages, as we observed children working with this approach, we saw a growing confidence across the ages, in fluency and reasoning, where children began to more readily build and discuss relationships between numbers, with a greater awareness of pattern.

Fingers therefore have gained our renewed importance in our work with children. Supported by research, children should be encouraged to use their fingers “as [an] important manipulative” (EEF) in a range of mathematical investigation and certainly beyond their early and initial explorations. Although in its initial stages, returning to ‘fingernosis’ (NCETM), has continued to have a positive impact on our children, in all groups. I heartily encourage you all to reconnect with the use of fingers as a vitally important mathematical representation, ensuring that they are used to their fullest capacity and not simply relegated to something children should perhaps have grown out of.

*James Gray **is an Advanced Skills Teacher at St. Leonards’ Primary School, Staffordshire & Evidence Lead in Education for the Staffordshire Research School.*

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Vina Sharma discusses the recommendations of the EEFs Effective Professional Development guidance report

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