What is Mathematics Mastery?

Mastery approaches to teaching maths is a current hot topic within education. ‘Mastery’ means acquiring a deep, secure and transferable understanding of maths. Our Educational Product Development Coordinator and in-house teacher, Joanne Moore, has outlined the 3 main elements for developing ‘mastery’ (also the 3 aims in the National Curriculum for Maths):

  • Fluency: to develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately.
  • Reasoning:  to follow a line of enquiry, conjecture relationships and generalisations, and developing an argument, justification or proof using mathematical language. 
  • Problem solving: to apply mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.

‘Mastery’ is built gradually and, in order to acquire the skills to a deepened level, learning moves through three different stages (based on Bruner, 1960); concrete, pictorial and abstract. 

1. Concrete

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When first introduced to an idea or skill, hands-on teaching resources enable the process to be acted out through experience. Whatever the resources are, they can be moved, grouped and rearranged to illustrate the problem. This is the foundation for conceptual understanding. This is a necessary stage for young children who are still thinking in concrete terms and learning through discovery.

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Learning Resources Owls on a Branch Counting Activity Set


2. Pictoral

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As a child’s experience and confidence grows in mathematics, they may no longer need physical objects to actually move around in order to understand and practise their maths skills. Instead, they draw them.  Children will visualise the concept and then record this pictorially using models, this may involve drawing pictures, using circles, dots or tallies. This encourages logical thinking. 

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Learning Resources About Time – Telling the time & understanding elapsed time



3. Abstract

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Once children have enough context of the maths skill or concept they will then begin to perform them at a symbolic level, using only numbers, notation and mathematical symbols to represent and solve the problem mentally.

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The three steps provide children with a deeper understanding of mathematical concepts and ideas and lays the foundations for effective problem solving in the future. It is important to understand that a child who uses abstract representations in one area may need concrete objects in another. On a different occasion, a child may need to revisit a concrete representation before moving on to a pictorial or abstract one. Therefore, it is important that children are free to use a variety of representations to support their maths learning.

Joanne Moore, Educational Product Development Coordinator and in-house teacher.

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