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Developing mastery for all 
through everyday problem solving

Sarah-Anne Fernandes, September 2016

Developing ‘mastery’ in mathematics is currently a hot topic of discussion and high on the agenda within current school practice. But what does ‘mastery’ really mean? And how can it be effectively implemented in daily mathematics teaching? 
The NCETM rightly points out that mastery is ‘nothing new’ and the main objective of mastery is to develop a deepened and more secure understanding of mathematics. 
One important way to support pupils in developing this understanding is to provide them with regular opportunities to reason and problem solve in daily mathematics. This embeds understanding before accelerating pupils to new learning content that is not within their year group standard.
For example, if you had a pupil who could confidently recall all the facts for the five times table, rather than accelerating them to learn all the other times tables it would be more beneficial for them to solve this type of problem which involves them exploring the pattern relationship:
Crack the times table code
and find the value of the symbol used...
C x B = FB
F x B = GK
D x B = FK
A x B = JB
G x B = JK
B x B = GB
H x B = AB
J x B = B
E x B = AK

Would your pupils spot the pattern
with the last symbol in each calculation?

Would your pupils know where
the best place is to start? And why?

Furthermore, a key part of mastery is that this concept of deepening understanding should not be viewed as an area for only ‘more able’ pupils to engage in. Teachers should make problem solving and reasoning accessible to all pupils and make it an integral part of daily teaching. 

From working with a range of schools, there are two common barriers to effective implementation of problem solving that I have come across: 
  1. Teachers viewing problem solving tasks as time consuming activities that take a long time to embed within daily practice 
  2. Pupils not having experience with solving a wide range of problem solving tasks and consequently they have not built up effective skills to tackle such problems
To combat this my advice is to develop a ‘key bank of problem and reasoning strategies’ and roll out these different types of problem strategies in a systematic way. This allows pupils to develop the prerequisite skills needed to solve the different type of problems over time. 
The new Everyday Problem Solving and Reasoning series for Year 3 to 6 has been developed with this in mind. It provides 11 key problem-solving and reasoning strategies that teachers can use on a daily basis in their mathematics lessons.  
  1. Finding all possibilities 
  2. Finding rules and describing patterns 
  3. Logic puzzles 
  4. Real –life word problems 
  5. Reasoning: True or false 
  6. Reasoning: Explain how you know 
  7. Reasoning: Would you rather? 
  8. Reasoning: Odd one out 
  9. Reasoning: Always, sometimes, never true 
  10. Reasoning: Convince me – What’s the same? What’s different? 
  11. Reasoning: If the answer is X, what is the question? 
The series promotes ‘mastering’ year group content through regular problem solving opportunities rather than just accelerating to new mathematical concepts and objectives. 
Most importantly the 11 key strategies that are used to develop problem solving and reasoning DO NOT have to be long-winded investigative problems that go on and on for several lessons. 
Rather they can be short, sharp reasoning tasks that can be dropped into lessons at any point to promote high quality mathematical talk and reasoning. For example: 
Is it always true, sometimes true or never true
that a pentagon has five equal sides?
This reasoning strategy of ‘Always, Sometimes, Never true’ is a powerful way to develop deepened understanding. However, it must not be taken for granted that pupils are clear on how to articulate and explain their answers. In a future blog, we will explore a range of different reasoning strategies that can be used as ‘bite-size’ problems to develop mastery but also, more importantly look at how we can effectively model to the pupils how to articulate effective explanations!