1. Reasoning and Problem Solving Maths Café

3. The SIX key areas to consider based on our 2018 analysis are:
Securing calculation method for the four operations
Ensuring pupils are confident with a formal written method for each operation, but also have a wealth of mental calculation strategies to draw on is still of vital importance. Pupils who are too heavily reliant on written methods and not able to apply mental calculation strategies to some questions will not have enough time to complete the papers. Remember the arithmetic paper has 36 questions to answer in 30 minutes! Similarly, pupils have a lot to get through in the reasoning papers each allocated 40 minutes.

4. 2. Embedding the understanding of fractions
Assessing pupils’ understanding of fractions continues to be very prevalent in the SATs papers. This year we saw 22% of all marks allocated to the Fractions, Decimals and Percentages domain. There was clear emphasis on mixed numbers (in fact, 7 questions included a mixed number!) and we cannot fail to mention one of the most challenging questions this year, the final question of Paper 2, Q23, which involved pupils partitioning 58 2/3 to complete the long multiplication (58 x 24), then finding a fraction 2/3 x 24 and finally re-combine!

5. 3. Developing fluency in number and variation in practice
The 2014 Programmes of Study for mathematics has three clear aims, one of which is that pupils are fluent in mathematics. This year SATs papers saw an increase in questions that assessed pupils’ fluency and ‘mastery’ of mathematics. For example, Paper 3, Q9 required pupils to have a firm understanding of the relationship between numbers:

6. 4. Practising ‘show your method’ multi-step word problems
Multi-step word problems that require pupils to show their calculation method continue to have a heavy emphasis. This year a third of total marks were awarded to ‘show your method’ questions. It is essential therefore that pupils have regular opportunities to break down these more complex word problems into mini-calculation steps to gain as many marks as possible, even if they don’t always get the correct definitive answer.

7. 5. Understanding mathematical terminology and knowing key mathematical facts
It comes as no surprise that for pupils to be successful at solving the vast range of questions, they need to have a secure understanding of mathematical vocabulary and key mathematical conversion facts.

8. 6. Exposing pupils to a variety of problem solving and reasoning questions
Although there was a heavy focus on ‘word problems’, we saw a variety of problem solving question types being assessed:
Missing numbers - Paper 3, Q4 (finding three missing digits to make a correct addition).
Finding all possibilities - Paper 3, Q2 (missing colour combinations for a new team shirt).
Logic Puzzles - Paper 2, Q 21 (calculate the value of individual shapes from a design).
Visualisation puzzles - Paper 3, Q17 (sum of the dots on opposite faces of a die).
Reasoning questions - Paper 3, Q6 (identify ‘true’ statement about a big cat pie chart); Paper 2, Q9 (explaining why Cricket World Cup was not held every four years).
It is therefore essential that problems solving and reasoning is an integral part of all pupils’ daily maths diet and they are explicitly taught how to solve the rich variety of questions that are presented to them.

9. What do we mean by 'problem-solving skills'?
There are four stages of the problem-solving process: Stage 1: Getting started Stage 2: Working on the problem Stage 3: Digging deeper Stage 4: Concluding
By explicitly drawing children's attention to these four stages, and by spending time on them in turn, we can help children become more confident problem solvers. There are different ways in which learners might get started on a task (stage 1), but it is once they have got going and are working on the problem (stage 2) that children will be making use of their problem-solving skills.

10. Here are some useful problem-solving skills:
Trial and improvement
Working systematically (and remember there will be more that one way of doing this: not just the one that is obvious to you!)
Pattern spotting
Working backwards
Reasoning logically
Visualising
Conjecturing

11. The first two in this list are perhaps particularly helpful
The first two in this list are perhaps particularly helpful.  As learners progress towards a solution, they may take the mathematics further (stage 3) and two more problem-solving skills become important:
Generalising
Proving
Having reached a solution, stage 4 of the process then involves children explaining their findings and reflecting on different methods used.

12. How can we help children to get better at these skills?
Trial and improvement
Trial and improvement is perhaps an undervalued skill. Children can be reluctant to use trial and improvement as they sometimes feel they are only using it because they do not know the 'right' way to solve the problem in hand. In reality, trial and improvement involves trying something out, which will always give more insight into the context and therefore gives the solver a better idea of what to try next. Trial and improvement is often the start of working systematically.

13. Working systematically
In the context of problem solving, working systematically could be thought of as working in a methodical and efficient way which could clearly show others that a pattern or system is being used. This is important, for example, when a task entails finding all possibilities, or when it is helpful to structure a method for solving a problem.

14. Pattern spotting
During the problem-solving process, being able to identify patterns can save time. However it is by then asking why the pattern occurs, and by trying to answer this question, that learners gain greater insight into mathematical structures and therefore deepen their conceptual understanding.

15. Working backwards
Starting from the end might sound counterintuitive, but it can be an efficient way of solving a problem.

16. Logical Reasoning
Reasoning is fundamental to knowing and doing mathematics. We wonder how you would define the term? Some would call it systematic thinking. Reasoning enables children to make use of all their other mathematical skills and so reasoning could be thought of as the 'glue' which helps mathematics makes sense.

17. Visualising
Picturing what is happening in your mind's eye, or imagining what is happening or what might happen, is a skill which is perhaps not talked about very much in the classroom. Specifically drawing attention to instances when it might be used will raise learners' awareness of this skill so that they might use choose to use it themselves.

18. Conjecturing, generalising and proving
Conjecturing, or asking "What if..?" questions, is an important problem-solving skill. Knowing what to ask means that you understand something about the structure of the problem, and being able to see similarities and differences means you are starting to generalise.