Louise Racher is a Mathematics Adviser at HfL
“By three methods we may learn wisdom: First, by reflection, which is noblest; Second, by imitation, which is easiest; and third by experience, which is the bitterest.” Confucius.
As many practitioners ponder over the “new” KS2 tests, this article picks out some of our “noble” reflections on what would make a pupil confident to tackle the KS2 test without fear and trepidation. Pupils who met Age Related Expectations in 2016 (just over half of year 6 pupils nationally) demonstrated that they had a flexibility which allowed them to manipulate not only the calculations to find solutions with ease within the constraints of the time limit – but also had a good grasp of problem solving strategies. This enabled them to access some complex multi-step problems using higher order thinking skills and demonstrate that they were able to reason with confidence. This is really what we aspire to give all our pupils – to give them the skills which, when faced with complex situations can make the desired connections, and to draw on the relevant skills to help them find solutions.
Of course, the test questions will be different next year. But thinking about some very broad themes, and considering what may have been a barrier for some pupils, might help this year’s cohort to acquire a richer diet of skills and strategies to help them become more successful mathematicians. The two very broad themes are: mental fluency and problem solving. You might conjecture why the all-important ‘reasoning’ doesn’t feature within that, but as Jane Jones (National Lead for mathematics, Ofsted) recently discussed at the Better Maths Conference in June 2016 reasoning is the overlap between fluency and problem solving.
Firstly, mental fluency. While the National Curriculum explicitly outlines the progression towards the formal written methods across KS2, resulting in compact methods for all four operations, there is far less explicit progression that supports teachers to secure mental strategies. It is this lack of detail which means it is sometimes difficult for teachers to build progressively year upon year. If you consider the calculations across papers 1, 2 and 3 and take into account the number range within those, it is clear to see that over half of the calculations across all three papers could be worked out mentally. When discussing some of the examples with teachers, there will always be a range of possible approaches. These will be guided by your own experiences as you make connections to learning in the classroom. Consider an example; ‘468 – 9’. This was a calculation on the arithmetic paper and there could be several approaches to this. You may subtract 10, then adjust your answer to compensate. Perhaps you might partition the 9 into 8 and 1 to make it easier to bridge back to 460 and then finish by knowing that 1 less is 459. Of course, you could count back 9 or indeed use column subtraction. Just within that one example, you can see the variety of explanations and the range of language and skills which would need to be applied to make those strategies feasible. This brings us to two possible teaching points. Firstly, the importance of having a shared understanding of the language you might use to explain the range of strategies which will support pupils in discussing them within class. And secondly, the essential tools the pupils will need to confidently apply a range of strategies (such as partitioning in multiple ways, rounding, adjusting) aligned to the operation involved (arithmetic laws). Of course, this is just looking at one example. You might consider the paths you would take to solve some of the calculations in paper 1; you might think about the language you use when explaining them to someone else, and what skills you needed to apply to support this. This will all need to be interwoven into your teaching sequence to build towards the ultimate goal of pupils discussing multiple strategies for calculations.
The second broad theme is that of problem solving. Recognizing the strategies that will help you to solve a more complex problem. There are many strategies and if we are going to try and really promote these, once again thinking about the progression across the school in teaching these and helping pupils identify “types” of problem would ensure they build on those simpler problems as they move through the primary phase. You might consider some strategies which are quite broad and applicable in many circumstances to get the pupils started:
- Finding a starting point
- Drawing a model
- Working backwards
- Making connections
If we just illustrate each of those with a snippet of an example from the KS2 SATs it might help practitioners to generalise and to source other problem solving opportunities to rehearse these skills so pupils are proficient, not only in the strategy, but in identifying different types of problems.
The question below exemplified the need for the pupils to use their “tool-kit”. In this case multiplication and division facts but also to prioritise the information to help them get started and enable them to solve the calculation. So finding a starting point is the initial strategy and then the pupils need to apply their skills to help them work out the calculations.
Below is an example in which pupils are being tested on their understanding of inverse laws, as well as calculating and the strategy needed here is working backwards. Notice as well how they need to extract the number sentences from the worded problem as well.
Making connections is a broad term, and could also be labelled “use what you know” or “what do you notice”. Those questions where the information might be less obvious. Pupils are having to be observant and really break down the steps to help them reach the destination. Consider the question below from Paper 2. Pupils need to use their knowledge of number magnitude, to help them find mid-points. The number range is not particularly high, but there is potential for pupils to be distracted by other information which is irrelevant initially. So they need to ignore the negative numbers and the mercury in the thermometer initially. They need to consider where 25 will fall on the left hand scale. Estimating its position between the 20 and 30 will help them judge the scale of the numbers on the right.
Finally, we come to drawing a model as a possible way to help interpret the problem. Are pupils able to re-interpret the information in a more manageable way to help them unpick the missing information more gradually? In a possible interpretation, the bar model has helped represent a way to find the cost of 3 pencils, by halving the £1.68, then enabling the missing information to be identified. And in this case a possible mental strategy to be applied to find the cost of one rubber. There are alternate pathways – this is just one route which became more apparent through drawing the model in this way. If teachers build this model into teaching then pupils might consider re-drawing the problem to help them identify the missing information.
With these examples in mind, teachers might consider using these broad problem solving strategies across the year and help pupils to identify the skills within these strategies, as well as being able to identify the kinds of problems which would be examples within these categories.
Teachers might exploit opportunities to take one simple problem and then use a staggered approach of estimating and build up towards that more complex problem so that pupils are starting to make the connections which will really help them apply the skills more readily to unfamiliar problems. But there can be no doubt that, threaded through the development of mental strategies and problem solving approaches, reasoning needs to play an integral part within the teaching sequence.
I am reminded of that old saying: success breeds success. It is through carefully structured approaches that sequentially develop pupils’ successful strategies further that we can help them become yet more confident mathematicians who can reason why a selection makes most sense.
Achieving Related Expectations in Year 6 training is being run on November 3rd at Hertfordshire Development Centre 9-4pm.
Jane Jones, Lead HMI Mathematics, Better Maths Conference June 2016
Department for Education (2013) The National Curriculum in England: Key Stages 1 and 2 framework document. [Online] Available from: https://www.gov.uk/government/publications/national-curriculum-in-england-primary-curriculum [Accessed 22 July 2014].
2016 key stage 2 mathematics Paper 2 and 3 – reasoning © Crown copyright and Crown information 2016