**Louise Racher** is a Mathematics Teaching and Learning Adviser at HfL, in this she gives her own interpretation of what priorities teachers might have had leading up to KS2 SATs, and what the priorities might be for next year following the second year of “New Curriculum” SATs.

Year 6 teachers had their game face on for the second year of the newly revised end of KS2 SATs. Greater awareness of how the papers would be presented meant it was a slightly fairer fight and lessons learned from the previous year were taken on board and assimilated back into classrooms across the land.

### Let us begin with The Fight: Mental versus Formal written

In the red corner, we have mental strategies, the new up and coming calculation. Quick on its feet but for some a less popular fighter due to a change in direction being hard to swallow. In the blue corner, we have formal written, the sturdy and reliable strategy, favoured by many parents over the land. With one punch an answer can be knocked-out with little or no understanding, sometimes prone to error or used to winning a fight which could have been better won by using an alternative strategy.

One of the three aims of the curriculum is;

**Pupils (of all ages, not just primary children) will: ***become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately.*

This statement is open to interpretation as the words “practice” might imply practice through testing, perhaps timed. In addition, the word “recall” can be inferred as the need to learn facts by rote. When these two words are intertwined “practice of recall” then the message might be interpreted as teach facts and then use them repeatedly to allow pupils to become fluid (fast) at formal methods.

However, for some teachers it does not seem like it is a fair fight. They are trying to undo some of the pupils’ reliance on formal methods. As soon as the pupils are taught the formal method for the three operations, they feel they have reached the pinnacle. Finally, they have achieved the “best way” and for some it soon becomes the “only way”. From a perspective of someone who has only had a diet of formal methods throughout primary and secondary it can be really hard to make the shift. It works well, I get the right answer (mostly!) and it is usually fairly quick. Why would I need a range of mental strategies? If we are talking specifically about Year 6 pupils finding it difficult to make this shift (and the teachers finding it hard to shift them) you might consider reading Rachel Rayner’s blog SATs chat: Why did my children revert to written methods on Paper 1?

The challenge is for teachers to help those pupils make the shift. To allow pupils to have a full repertoire of strategies at their fingertips. Teachers are demanding that their pupils are more playful with number. If you are interested in looking at some worked examples then read more in Rachel Rayner’s blog: Are you and your children playful with number?

“To the person without number sense, arithmetic is a bewildering territory in which any deviation from the known path may rapidly lead to being totally lost.”Dowker (1992)

his quote sums up (no pun intended) many mathematical experiences. Traditionally teachers rarely asked pupils to look at the whole calculation, to evaluate and estimate to allow choice for the best strategy. We want a strategy to leap out at them, to feel natural and effortless. Much like learning to drive a car. At first you have to think so hard about each separate thing which needs to happen simultaneously, until it is so natural that you are barely aware of the different actions you are taking. Teachers frustration sometimes comes from trying to get the pupils to that point, already with an over reliance on strategies which are less efficient so early on in their school career.

Being playful with number can benefit all groups of pupils. If you are particularly interested in championing disadvantaged pupils you would definitely be interested to read the blog In Nicola Randall’s blog Tried and tested: Diminishing the difference at UKS2 where she discusses some of the barriers for pupils and how to overcome them. One aspect she discusses is the need to estimate. To be able to estimate you need to have a sense of the number and where it is placed within our number system. Whether it be a larger number or a number involving decimals it is important that pupil has a sense of whether that number is very close to another number which is easier to manipulate, this in turn means that rounding numbers becomes a natural skill.

If a pupil sees a calculation 1002 – 998 we want them to notice that both numbers are very close together. It would not be efficient, in this case to carry out column subtraction. yet that is what many pupils will do.

Take a look at these samples of calculation from a year 6 pupil.

In the case of this pupil, they had been encouraged a lot in class to show their working out and check it, even if they did it mentally. Perhaps as teachers, we sometimes further embed the use of formal strategies by asking pupils to do this. “How did you work that out?” we often ask pupils. But it might be too difficult to explain. Pupils might not know how and revert to using a formal method even though they have used a perfectly efficient mental strategy. Or perhaps, when being asked to “check”, they have used this strategy as a different way to check something they have already carried out mentally. This might undo their natural reflex to find an efficient method when they are then being asked to use an alternative, less efficient strategy to do so. As a teacher it is important for me to discuss the appropriateness of methods.

Strategy talk is an integral part of ensuring that pupils are equipped to talk about a range of mental strategies. Is this something you could build into your teaching to support pupils? Let’s consider the example of 1002 – 998. I would explain my strategy as equal difference. Imagine I am sliding those numbers on a number line, the calculation now becomes 1000 – 996 which is much easier to work out. If this was something we had explored and discussed in class then however the pupils choose to view it when asked “how did you work it out?” the response could be simply be, “equal-difference”. With a shared understanding no more would need to be said.

This blog will not focus on the different mental strategies and what we call them, and how we encourage pupils to use them as this has been covered in previous blogs. The main reflection I hope people might have after reading this is how we can encourage pupils to use mental methods, of course with the use of informal jottings being fully encompassed by this, but to reflect on how we ask and expect pupils to respond when we ask them to “check their answer” or to “how their working out”. Perhaps lifting this restraint for some pupils would help us see more clearly where they are able to use mental fluency and where they are not.

If we look at the balance of strategies from paper 1 in which they could have used these strategies either as a sole strategy or part of a combined strategy (hence the total being higher than marks available) it is clear which strategies would really support pupils in being successful at completing this test in the time constraints and with accuracy.

- Place value (12)
- Re-grouping (or partitioning) (8)
- Fact recall (of multiplication and division facts) (8)
- Doubling or Halving (8)
- Formal strategies (as explained in the Appendix of the National Curriculum) (7)
- Equal Sum or equal difference (2)
- Order/knowledge of Operations (2)
- Recognition of common multiples (2)

It is worth noting that even within the formal strategies there would be evidence that having flexibility in other approaches would enhance that. If we take the example from the last question on the paper 2242 ÷ 59.Many pupils would have listed some of the multiples of 59 to help them. If they had listed the first four: 59, 118, 177, 236 they would have seen for the final stage of the calculation that they needed to know 472 ÷ 59. Looking at that fourth multiple and doubling that to know that is 8 x 59 would support them in using those listed facts much more effectively. Therefore doubling could be used alongside the formal written procedure and would make this quicker and less prone to error for the pupils. It is also of note that many questions involved manipulating fractions. Therefore, the strategies to be applied depend on pupils having a good depth of knowledge of fractions.

This theme was carried through into the reasoning papers, with a heavy emphasis on place value throughout the paper, and emphasis on the popular strategies from paper 1 needing to be applied in papers 2 and 3. When comparing the reasoning papers from 2016 and 2017, place value was given a high emphasis in specific questions targeting this knowledge, with 11 marks available in 2016 and 8 marks in 2017. Ensuring that pupils have a strong basis of place value in a range of contexts is beneficial to all other number related skills and so is worth spending some time on.

To look at paper 2 and 3 is complex, as the pupils are being asked to apply a combination of skills to be awarded one mark. An example of this would be paper 2, question 7 where the pupils were asked to fill in missing information regarding years days and weeks, they needed to recall that information and then complete calculations: 60 ÷ 12, 72 ÷ 24, 84 ÷ 7. There again a combination of using facts they know and applying strategies to derive other facts was necessary. Also in paper 3, question 17, pupils needed to convert mixed numbers to fractions and then compare them.

When looking at the most suitable problem solving strategies across both papers it was noted that there were some which would have higher stakes than others: interpreting tables/graph, modelling out multi-step problems, working backwards were the top three themes which recurred.

Beginning with modelling out multi-step problems. In 2016, there was a total of 14 marks available for questions which really lent themselves to this strategy being used to work through the stages of the problem, compared to 17 marks in 2017. During consultancy in schools this year, it has by far been the most in demand training from schools.Where it is beginning to be used, teachers and pupils both appear to be reaping the benefits. I especially noted its use not only for the multi-step problems which were longer, but also for some seemingly shorter problems which it might be easy to omit if unsure how to tackle it.

An example would be question 14 on paper 3.The question indicated that 3 pineapples cost the same as 2 mangoes. One mango costs £1.35. How much does 1 one pineapple cost pupils were asked. Even by restating the information in a bar and doing no other working out I can visually see what I need to do.

Or take the example of the question involving ratio, which have featured in many past papers. In question 14 in Paper 2, pupils were told Amina planted some seeds. For every 3 seeds Amina planted only 2 seeds grew. Altogether, 12 seeds grew. Once again, by simply knowing the appropriate model to draw and re-drawing this information it is much easier to see the solution.

Of course pupils do need to be able to do the calculation once they have drawn this out, but note how the numbers the pupils are working with are simple.

The next area to mention is how **interpretation of data **is used to vary presentation whether it is in the form of a chart or table. In 2016, 8 marks were available across papers 2 and 3, in 2017 10 marks. Of course, again, the pupils do need to manipulate numbers or times to then answer the question, but extracting the correct information could be the barrier for some. As statistics does not have a heavy emphasis in the National Curriculum in the amount of statements it could be easy to lack coverage in this area. As teachers are often under pressure to cover a large amount of skills in one year, ensure statistics features either through other lessons, such as science or as part of sequences within problem solving lessons, or as a way to apply certain calculation strategies. This will help ease that burden. Exploring a range of different data representations is also beneficial so the pupils can be flexible about how they view data and can extract the information regardless of the presentation.

**Working backwards** as a strategy also features in 2016, with 6 marks for questions where this strategy was most efficient and in 2017 5 marks. Helping pupils to recognise when they are using this skill and ensuring they can identify its use in a range of problem solving activities will support this recognition. The last question on paper 3 (when the pupils are flagging) demanded this strategy when finding first of all the volume of a cube, then having to work that skill backwards to find a missing side of a shape with the same volume by different dimensions.

If pupils were confident in those three areas and applied them well for the relevant question in the 2017 paper they could have scored 32 marks, which is just under half the marks they need. If you add in the additional 8 marks for place value then that takes it to 40 marks. Then with a strong score on paper 1, pupils would be close to meeting the expected standard (depending on the threshold of course).

In conclusion, there were many parallels to draw from 2016 and 2017, showing that the tests were evenly matched in terms of the expectation of strategies the pupils could use to be successful. This would mean they would not only meet the expected standards but also have the necessary skills to feel confident Mathematicians as they begin secondary school, and then enter the big wide world where these skills will be useful to them every day.

### References

Department for Education (2013) *The National Curriculum in England: Key Stages 1 and 2 framework document.* Available at: https://www.gov.uk/government/publications/ national-curriculum-in-england-primary-curriculum (Accessed: 19 June 2017).

Department for Education (2017) Keystage 2 Tests: 2017 Test Materials. Available at: https://www.gov.uk/government/publications/key-stage-2-tests-2017-mathematics-test-materials (Accessed: 19 June 2017).

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