*Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas [1]*

I have, for some time now, had an obsession with certain elements of the mastery approach to mathematics:

- Variation
- Multi-strategy discussions
- Bar modelling

I am not going to attempt to define what mastery is or even peek inside the can of worms concerning assessment and mastery. Instead, what I want to discuss is how these elements manifested themselves in the 2017 KS2 SATs papers.

I frequently hear from Year 6 teachers that they don’t have time to teach following the ‘mastery approach’ (whatever this may be). They don’t have a year to get children prepared for SATs and regardless of which side of the formal testing fence you sit on: they are a reality. Good results are vital for many of the schools I have the privilege of working in. However, having spent an unhealthy amount of time unpicking the KS1 and KS2 2017 Maths SATs papers, I want to highlight that there are a number of strands of the mastery approach which the test developers are exploiting – a blunt but effective way of drawing our attention to them. Performing well in SATs and developing learning so that it creates a deep, rich and interconnected view of the mathematics landscape are not mutually exclusive things: you can have both.

**Mix it up to deepen understanding**

*Variation in presentation; the same core idea being presented in a range of different ways.*

There are many examples of variation in presentation in the KS2 SATs papers. At the most fundamental level are the purposes of the different papers: Paper 1 is purely arithmetical (do they have the skills to calculate?); Papers 2 and 3 are the so called ‘reasoning papers’ (can they apply their maths skills?). But if we look a little deeper, the variation becomes more complex, probing the children’s understanding of the core concepts.

Let’s take division as an example. Approximately one quarter of the questions in Paper 1 are division. But even here they have managed to mix it up, starting off with a not so gentle one:

Dividing anything by one is a very abstract idea, one that many children and adults would struggle to contextualise into a real life scenario. This is exploring how deeply the children understand division, avoiding the misconception that division always makes the answer smaller. Simple for those who have explored division in multiple ways…

…yet deeply confusing for those who are just trying to click through steps – how can the answer be the same? Many might be inclined to think the answer is zero or one.

Switching attention for a moment to Papers 2 and 3, division comes in many different disguises:

Contrast Paper 2 Q19 (above) with Paper 3 Q2:

The question above uses traditional division language (‘share’, ‘equally’), but, despite using ‘share’, it is a grouping structure – how many groups of 16 can I take from £80?

Pausing for a moment, we should consider the difference in challenge level between the three calculations below from Paper 1 (a similar pattern appeared in the 2016 SATs Paper 1):

I believe that Q34 is the most straightforward – it involves applying an of understanding ‘x’ and ‘place value’ (15% x 100 = 15, so 15% x 1000 = 150). So why has this been placed further into the paper? It is considered more challenging because it uses the less frequently employed abstract symbolic, but the actual computation is much simpler. How many children have not explored multiplication in its different disguises?

SATs also remind us that numbers can be presented in different ways other than digits (also see the Roman Numeral problem Paper 2 Q15, but I could not bring myself to include it):

This question checks that children have a secure understanding of place value and the inclusion of the tricky number zero.

Paper 1 presents division calculations in both linear and division bracket forms. Children need to be aware that just because the division bracket has been used, does not mean they have to do a formal division calculation. This brings me nicely to the next key element: multi-strategy discussions.

**There are more ways than the formal way**

*Number sense/multi-strategy discussions.*

By Year 6, many children appear to throw their lot in with formal calculation methods. They click unthinkingly through the mechanical steps, not estimating or checking for appropriateness – just accepting that the magic formula knows best. Children should be engaged in multi-strategy discussions, not just identifying one way of solving a calculation but multiple ways **as only when we have a choice can we be efficient**.

Consider for a moment:

Now, please don’t get me wrong, I am not anti-formal methods; I just believe they have their place alongside a range of methods. There is no hierarchy.

This question could be completed using formal column subtraction. It could also be done by:

**Regrouping the subtrahend:**the amount being removed in a subtraction calculation.

9 subtract 3 and then subtract a further 0.45

**Regrouping the minuend:**the number from which an amount is being taken in a subtraction calculation. This is the less trodden pathway, but once you start to see the power of regrouping the minuend, you will love it! (Perhaps we need a more in-depth discussion another day.)

Take 3.45 from 4 (using our compliments to the whole), leaving 0.55.

Recombine this with 5, giving the answer of 5.55

**Equal difference:**add 0.55 to both the minuend and subtrahend – changing the calculation to a simpler but equivalent one.

9.55 – 4

I am sure that you have other ways of solving this calculation. And that’s the point – we all have different ways. If we don’t encourage children to playfully explore this world of numbers, we deny them the excitement that comes from being in control and manipulating numbers with understanding.

The third element takes aspects of both of the above.

**The confidence to have a go**

*Bar modelling.*

Papers 2 and 3 in the 2017 SATs were filled with beautiful bar modelling possibilities: questions where the language was ambiguous or non-traditional; where there appeared to be too little or too much information; and especially where we had the dreaded ‘*some’*. Here are two of my favourites:

This involves the children taking that leap of faith. *Amina has some seeds*. How many? Lots? A few? We just don’t know at this stage. So let’s hold on to what we don’t know – she has *some*. *For every 3 seeds Amina planted, only 2 grew*. Excellent, we have numbers. But more than that, we have a ratio:

*Altogether, 12 seeds grew. **How many seeds did Amina plant?*

All of this information can translate into a bar model very easily. The equality between the two statements now takes on a physical form, making spotting the operations required more obvious.

Now we have the option to take this into multi strategy discussions. How *could* we solve this calculation? Bar modelling doesn’t give the numerical answer: it just makes the relationship and operations more apparent. I would hope that in a classroom we would have a range of strategies suggested: short division, using known facts and place value patterns (27 ÷ 3 = 9), distributive law (90p ÷ 3 + 90p ÷ 3 + 90p ÷ 3) and so on…

We can engage our children in good mathematical debates over which strategy is preferable and why. We can mix up the language and presentation to ensure that the understanding is deep and linked. We can create models which not only help children understand what operations are required, but take their ability to reason to the next level and explore complex concepts from different directions. This will not only prepare the children to perform well in SATs, but encourage them to explore the maths landscape more widely for themselves. In this way, they will create their own maps and forge new pathways, rather than relying on Teacher SAT Nav.

**Charlie Harber**

*Charlie Harber is Deputy Lead Adviser for Primary Mathematics at Herts for Learning.* *Herts for Learning Ltd is the UK’s largest Schools’ Company, with 521 shareholder schools. We provide an ethical and informed approach to school support, within Hertfordshire and, due to an increasing recognition of our work, increasingly beyond our borders.*

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References:

Department for Education (2013) *The National Curriculum in England: Key Stages 1 and 2 framework document*

Department for Education (2017) *2017 National Curriculum Tests Key stage 2 Mathematics: Paper 1* *arithmetic*

Department for Education (2017) *2017 National Curriculum Tests Key stage 2 Mathematics: Paper 2 reasoning*

Department for Education (2017) *2017 National Curriculum Tests Key stage 2 Mathematics: Paper 3 reasoning*

[1] Mathematics Programmes of Study: Key Stage 1 and 2, National Curriculum, September 2013, page 3