How children learn maths

Early years settings and schools should invest in developing practitioners’ own understanding of mathematics, their understanding of how children typically learn, and how this relates to effective pedagogy.⁵ This is important for realising the potential of the other recommendations in this guidance report.

Effective mathematics teaching requires knowledge of mathematics pedagogy and learning as well as of mathematics itself. This includes knowledge of how children learn mathematical concepts, connections between mathematical concepts, likely difficulties children may have, and different approaches to solving problems or tasks. Professional development should therefore focus on the integration of three areas: mathematics itself, children’s mathematical development, and of effective mathematical pedagogy.⁶

‘Self-regulationHow children monitor their emotions and thoughts, and adapt their behaviour in different circumstances.’ refers to the ability to understand and manage one’s own emotions, behaviour and thoughts in different situations. To successfully complete a mathematical task, children must be able to self-regulate, and so the development of self-regulationHow children monitor their emotions and thoughts, and adapt their behaviour in different circumstances. is consistently linked to successful learning in early mathematics.¹⁵

Metacognition is the ability to reflect on your own thinking processes and is closely related to self-regulationHow children monitor their emotions and thoughts, and adapt their behaviour in different circumstances.. Practitioners should encourage children to explain their thinking processes and strategies when solving mathematical problems. Such monitoring of problem-solving processes enables children to gain insight into their own thinking, learn from their errors, and develop their problem-solving skills.¹⁶

Practitioners can support children to develop these skills by describing the child’s strategies and approaches linked to thinking and learning. For example: ‘I can see that you are thinking really carefully about where the corners are on the jigsaw pieces—that could help you to find the right place for it’, or, ‘I can see you’re finding it difficult to concentrate, we could find a quieter place to work so that we won’t be disturbed. Practitioners may also talk through their own problem-solving strategies out loud whilst solving a problem, to model this thinking to children. This can involve modelling getting stuck and reflecting on different strategies.

The EEF’s guidance report ‘Metacognition and Self-Regulated Learning’ provides more detail on these skills and approaches.¹⁷

A substantial amount of research on how children learn mathematical concepts has revealed the complexity of mathematical development. Developing a secure grasp of early mathematical ideas takes time. Even if a child appears to be engaging successfully in mathematical activities (for example, reciting the count sequence), they may not have a full grasp of the underlying concepts (for example, the meaning of numbers in the count sequence). Children may also appear to have grasped an idea in one context but then fail to show that knowledge in a different context.¹⁸

However, research has suggested possible paths that children may follow in developing an understanding of a mathematical topic. ‘Developmental progressions’ are descriptions of the typical path that children tend to follow in developing an understanding of a mathematical topic.

There are several developmental progressions available;^{19 20 21} these vary in their focus and the amount of information they present. The diagrams below provide simple examples of progressions in number development, operation development, and geometry and spatial thinking. The spiral highlights the progression of individual skills or concepts that develop over time.

The diagrams are a spiral to convey that whilst there is some ordering in which these skills may emerge, development does not take place in clearly defined linear steps. Children may develop several skills in parallel and individual children may move through skills in different orders. However, to reach full understanding, children will need to master each of these skills. Developmental progressions can, therefore, be seen as approximate paths of the development of thinking, but not a clear linear progression for all.

While each spiral is presented as a separate diagram, there is considerable overlap in development across these topics. In particular, understanding of operations builds upon children’s understanding of number. Across all three topic areas, children must come to understand ideas of composition and decomposition—putting together and taking apart—as this is fundamental to both number and arithmetic (for example, part-whole relations, adding and subtracting) and geometry (for example, shape composition).

It is important that practitioners are aware of typical development of mathematical skills and concepts to inform teaching. This knowledge will support the implementation of the other recommendations in this guidance report.

Such knowledge can support practitioners to:²²

• have a good understanding of what children need to learn to progress;
• make judgements about the range of experiences that children may benefit from to develop a full understanding of mathematical topics;
• determine the developmental pre-requisites for a particular skill;
• assess a child’s level of understanding; and
• intervene at the appropriate levels of challenge and build on what children already know.

Clements, Douglas H. and Sarama, Julie (2014). Learning and Teaching Early Math: The Learning Trajectories Approach (2nd edn) (Studies in Mathematical Thinking and Learning Series, London: Routledge).

Also see https://www.learningtrajectori....

Researchers Douglas Clements and Julie Sarama have developed detailed developmental progressions that also include activities and tasks for each step to support children to achieve the next level of thinking, which are coined ‘Learning Trajectories’.

The Early Math Collaborative, Erikson Institute (2014). Big Ideas of Early Mathematics; What Teachers of Young Children Need to Know, Pearson.

This book by the Erikson Institute explores key concepts in early mathematics learning and implications for teaching, including activity ideas.

Gilmore, C., Gobel, S. and Inglis, M. (2018). An Introduction to Mathematical Cognition. (London: Routledge).

This book explores the research that seeks to understand how people come to understand mathematical ideas.

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