Rachel Rayner is a Mathematics Adviser at HfL and is one of the KS1 Number Sense and Fluency project leads in Hertfordshire and Cambridgeshire.  The project aims to support teachers to develop pupils’ retention of facts and how we can help them use learned facts flexibly.  The project has been hugely successful and findings have been presented at conferences and journals. In this blog, Rachel turns her attention to what the greater depth judgement actually means and what kinds of opportunities can be used to foster it.

I’ve spent a lot of time in schools recently considering with teachers whether they have pupils working at greater depth in Year 2 but also what that might look like in Year 1.  Part of this work has, understandably been with schools who are fully expecting to be moderated this year and would like their books to reflect evidence for pupils they suspect could achieve the greater depth tag.  Why so nervous?  Well the landscape for maths has shifted in terms of expectation, whereas before L3 might be judged by acceleration into new coverage, speed and accuracy which seem easier to tick off, now ‘Greater Depth’ seems a little hazier, perhaps just out of reach. Indeed the language of judgement gives us the weightiest indicator with the change from high attainer to working at greater depth.  In terms of scaled score versus Interim Teacher Assessment Framework (ITAF), there seems a difference in expectation too. This has left schools feeling uncertain about their own judgements.  I have plenty of sympathy for schools and the greatest respect for the teachers who are questioning their judgements and recognising the shift.

So what is greater depth?  To look at the ITAF itself is our first action where we can see the slight shifts in expectation for directly relate-able criteria e.g.

Working at expected standards

The pupil can subtract mentally a two-digit number from another two-digit number when there is no regrouping required (e.g. 74 − 33).

 Working at greater depth

The pupil can work out mental calculations where regrouping is required  (e.g. 52 − 27; 91 – 73).

This is helpfully specific.  But beyond the criteria, I found the exemplification materials very illuminating in terms of national expectations.  Three words jumped out at me when I read through them –

Reasoning

Deduction

Complexity

For a pupils to be judged at greater depth I’d say we are looking for pupils who are confidently able to independently deal with increases in all three areas.  Let’s deal with these in turn.


Deduction It’s not just Sherlock Holmes who needs to deduce but it’s useful to think of how he make deductions.  He collects the clues around him, looks for patterns and seeks relationships, then…and this bit is important…he usually makes a huge leap of faith based on those clues and patterns – not entirely elementary. I realise it is a skill that we talk about a lot in reading, but pupils need opportunities to demonstrate those leaps of faith in mathematics too, a page of practice is great for retention but we need other opportunities to conjecture and generalise based on clues and patterns too.  I like the simplicity of the examples in the exemplification materials here though I would have liked more of a focus on following own lines of enquiry – it is well structured and perhaps a bit leading?  Nevertheless bearing in mind the pupil’s age a leap of faith has happened here in the last question.

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Have a look at the work of this Year 1 pupil who during her reasoning demonstrates that she is able to deduce a simple generalisation from patterns she was working with – she made the leap independently where other pupils didn’t.  Now you might say, ‘but you can halve 7.’ – you can’t halve seven counters which is what she was working with and is yet to experience halving 7 apples with her teacher.  So we can only perhaps expect deductions with the patterns and clues that we have.

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Y1 pupil from St Gregory’s Catholic Primary School, Northampton.


Complexity In the ITAF this is evident through comparison of the statements here

 Working at expected standards

The pupil can recognise the inverse relationships between addition and subtraction and use this to check calculations and work out missing number problems (e.g. Δ − 14 = 28).

Working at greater depth

The pupil can solve more complex missing number problems (e.g. 14 + □ – 3 = 17; 14 + Δ = 15 + 27).

The complexity here is confidence in dealing with more numbers and with an increased appreciation of equality and operation.  Below the exemplification materials give us further clarification about what pupils’ work might look like here.

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I always enjoy seeing the pupils reasoning as here on the bar.  But the accompanying comment makes me wonder how procedurally this was taught and the bar seems to have been pre-drawn for the pupil. It seems this pupil worked from left to right without due concern of the numbers involved and personally I couldn’t be sure equality was understood due to the pre-drawn bar.  So whilst this is more complex calculation, the child has been guided to a procedural acknowledgement of equal expressions with a conceptual model, ‘The pupil had to find a total for one side of the equivalence before calculating the missing number.’  I’m not sure it displays depth of complexity in terms of number sense, I would be happy if this was the child’s choice of approach. However, perhaps this teacher has since dealt with a more multi-strategy approach and this will have only been part of the evidence for this child.

I’d also like to suggest some other classic opportunities for pupils to increase complexity with missing number and inverse operations could be provided for.

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Obviously all of these can be adapted and examples like these are available on the internet but I love to see how pupils grapple with these few examples. Great evidence here for meeting the requirement I believe.

We can also have a look at what this looked like in the actual assessments and provide similar examples working with the pupils to decide what strategies would be useful here. Not only that but it provides some lovely consolidation of partitioning numbers.

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Paper 2 KS1 Reasoning 2016


Reasoning I think we can see that deeper complexity and opportunities to deduce is going to result in greater opportunity to reason.  Working with the amazing teachers that I do, in the number sense fluency projects and beyond I know that this is paying real dividends in terms of pupils’ motivation and voices in mathematics.  Anyone who says to me that Y1 and 2 pupils are too young to develop their written and pictorial reasoning are missing the delightful responses that enrich pupil experience and understanding (see the Y1 example above).  Additionally the examples above show that reasoning isn’t always about words, non-explicit reasoning is important too.  Take the number pyramids earlier – plenty of reasoning available there.   The reasoning paper also provides this lovely example e.g.

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Paper 2 KS1 Reasoning 2016

To encourage reasoning with examples like this I might ask pupils – ‘What is a really stupid number it couldn’t be? Why?’ before asking what it must be.  In non-explicit reasoning the pupils will be asked to deduce from the clues they see and pull from the knowledge they have and that takes multiple experiences to secure.


And finally independence means that we need to allow them to edit their work in just the same way as writing, we can indicate there are errors but not exactly what and where.  I like this, they are able to take control and develop responsibility for checking.  My favourite technique for this is ‘Seek and Destroy’ where we draw a box around some correct and incorrect responses and indicate how many errors there are then pupils seek and destroy them.  It cuts down on all of that ticking you do and makes them think more deeply too – now that’s what I call a win win!


References

2016 Key stage 1 mathematics Paper 2 – reasoning © Crown copyright and Crown information 2016

2016 teacher assessment exemplification: key stage 1 mathematics – working at greater depth within the expected standard.© Crown copyright and Crown information 2016