The principles for appropriate pedagogy in early mathematics: Exploration, apprenticeship and sense-making

Catherine Gripton, University of Nottingham, UK

Helen J Williams, Freelance Education Consultant, UK

Early mathematics pedagogy is about exploration, apprenticeship and sense-making.

• Exploration: Children and adults enjoy exploring mathematical ideas mentally and physically.
• Apprenticeship: Children learn through being mathematical with adults, learning mathematical words, ideas and behaviours from them
• Sense-making: Children make sense of their world through identifying patterns, making connections and seeing mathematical structure in their everyday experiences.

As practitioners, we support children’s early mathematical learning through our provision, interactions, pedagogies and beliefs. In this article, we reflect upon what is special about early mathematics practice. Recognising that no one set of practices is appropriate for all children, we instead explore the underpinning principles of effective and appropriate mathematics practice with young children. These are based on the Early Childhood Mathematics Group’s (a UK-based group of Early Years mathematics enthusiasts and experts, who work together to promote early childhood mathematics) pedagogy statement (ECMG, 2022).

Mathematical learning does not happen in isolation or separate to other areas of learning. It is an essential part of the holistic development of the child, learned in reference to personal, social, emotional, linguistic and other aspects of development. Mathematics practice that is relevant to the individual child and their world means that we use culturally and socially responsive approaches that build upon children’s unique experiences and are appropriate given all that we know about child development. While responsive, appropriate and holistic, these approaches are not generic. Mathematics is special. There are distinctive ways of thinking and behaving, as well as particular knowledge and concepts that make mathematics, well, mathematics. These require developmentally appropriate pedagogies that develop mathematical thinking, knowledge, concepts and behaviours (Downton et al., 2020), rather than a watered-down version of pedagogy designed for much older learners (Gifford, 2015). A key principle of early mathematics pedagogy is that it is appropriate for the child.

All children are entitled to be mathematical and learn mathematics

This is probably the most important principle underpinning appropriate mathematics pedagogy. Children are innately mathematical beings, able to distinguish simple visual forms from birth and categorise what they see (Goswami, 2015) and to solve problems in contexts that they understand (Gopnick, 2015). No child is ‘not ready’ for mathematics. To neglect a child’s mathematical abilities is to deny them part of who they are. Sensitive and supportive interactions with adults with whom they have strong, secure relationships support all our children’s mathematical development.

Mathematical development involves learning mathematical behaviours (or ways of working mathematically) as well as mathematical concepts

Children learn from adults that mathematics is active and enjoyable. There are ways of thinking and acting that really help with mathematics. Cuoco et al. (1996) refer to these as mathematical ‘habits of mind’, which include being pattern-spotters, experimenters, describers, tinkerers, inventors, visualisers and conjecturers. Supporting children to develop the statutory characteristics of effective teaching and learning (DfEDepartment for Education - a ministerial department responsible for children’s services and education in England, 2021) in mathematics is to help them to be problem-solvers who use these ways of thinking. Practitioners model these behaviours every day. We think aloud and verbalise what we wonder, what we notice, the patterns that we see and the connections that we make. We model ways of moving our bodies and manipulating objects, as well as representing our thinking using jottings, marks or drawings. In doing so, we make mathematical thinking visible to the children.

Young children are differently experienced and not differently able at mathematics

This links with our first principle, that all children are entitled to learn mathematics. When children are young, a few months’ difference in age is a difference in a wealth of experience and development. In the early years of life, children have differing access to mathematical experiences and ideas, having heard more or less mathematical vocabulary and mathematical chat, for example. They have had their attention drawn to number, pattern and geometry to differing extents and in differing contexts. Providing a broad range of rich mathematical experiences supports all children but particularly those who are currently less experienced. It compels us to provide opportunities rather than address ‘gaps’, which is the developmentally appropriate pedagogy that is needed by the children most at risk of under-achievement in mathematics.

Children are entitled to the full breadth of mathematical learning

As we work to support children to flourish in mathematics, assessment goals (from curriculum plans, mathematics schemes or the early learning goals (DfE, 2021)) can lead us to focus on number, which can leave less time and attention for pattern and geometry. Due to the interconnected nature of mathematics and the importance of developing mathematical habits of mind, development in one area supports and enhances achievement in another. Time spent developing spatial reasoning (Gifford et al., 2022), for example, supports children to interpret number lines (Gunderson et al., 2012). Engagement with pattern-copying, continuing, making and fixing errors, supports children to see patterns in all areas of mathematics and develop important early algebraic thinking (Papic et al., 2015). It is an important principle that all children receive a broad diet of mathematics in the Early Years and that pedagogy supports this, including connection-making within and between mathematical concepts.

Early mathematics practice is best shaped by practitioner knowledge of typical developmental trajectories in mathematics (Siemon et al., 2019)

These are ‘descriptions of the typical path that children tend to follow in developing an understanding of a mathematical topic’ (EEF, 2020, p. 9). They support decisions about what to offer, suggest or provide for children, now and next. They are not a curriculum map or a tick-list assessment. Learning trajectories are knowledge of key developmental milestones about which the practitioner thinks when making decisions about pedagogy. They guide us to notice and scaffold the mathematics concepts that will provide the firmest foundation for future mathematical learning. The most well-researched and comprehensive of these is Clements and Sarama’s learning trajectories (2021).

Young children learn mathematics through play and need time to:

(1) follow and develop their own choices and ideas

(2) play and interact with adults

(3) participate in adult-led episodes.

Early mathematical understanding is achieved during both child-initiated and adult-initiated play and other meaningful contexts such as routines. All children should enjoy daily moments where they explicitly engage with mathematical concepts and language. Guided play (Skene et al., 2022) can be particularly powerful, with its careful balance between adult intentions (clear focus and pedagogic flexibility) while retaining the space and time for child agency (play and choice).

Practitioners provide opportunities for mathematical learning through the environment and continuous provision

They also ensure that mathematical learning occurs in these contexts, through observation and sensitive interaction. Planning in the Early Years is varied and takes place over differing timescales. In the longer term it begins with deliberate choices made for the continuous provision, rich in possibilities for mathematical learning. Planning can also be in the moment, where pedagogy is determined and enacted immediately to draw out, emphasise or extend the mathematics in the current context. These mathematical pedagogies can be quite subtle, such as a quizzical look or a movement of an object, or can be more obvious, such as asking ‘I wonder if/why…?’. Routines such as snack-time, tidying and registration are authentic opportunities for mathematical learning, as are games, songs and stories. The mathematics in these contexts are meaningful to the child and provide them with a reason to problem-solve (Gripton and Williams, 2021). Large-scale outdoor resources such as water play, physical play, construction/blocks, loose parts, role-play and number tracks support important early mathematical learning. Scales, timers/clocks, measuring tapes, calculators and other mathematical equipment provide opportunities for foundational concept development long before children are expected to use this equipment accurately.

Children have a repertoire of ways in which to communicate their mathematical thinking.

All learners, but particularly young children, have a diverse range of ways in which to communicate their mathematical thinking with others. Children represent their ideas by placing and moving objects, by creating mathematical graphics (marks, drawings and symbols) and by using words and gestures. As practitioners, we use mathematical vocabulary in context, modelling gestures where appropriate, to support children’s concept formation. We support children to include standard symbols in their representations when appropriate and in meaningful contexts, such keeping score in a game. Children should have agency and ownership over their mathematical communication, providing us with a ‘window into their thinking’ (Davenall et al., 2021, p. 15).

Principles for appropriate pedagogy in early mathematics

• All children are entitled to be mathematical and learn mathematics
• Mathematical development involves learning mathematical behaviours (or ways of working mathematically) as well as mathematical concepts
• Young children are differently experienced and not differently able at mathematics
• Children are entitled to pedagogy that supports the full breadth of mathematical learning
• Early mathematics practice should be shaped by practitioner knowledge of the typical developmental trajectories in mathematics
• Young children learn mathematics through play and need time to: (1) follow and develop their own choices and ideas, (2) play and interact with adults, and (3) participate in adult-led episodes
• Practitioners provide opportunities for mathematical learning through the environment and continuous provision
• Children need to have a repertoire of ways in which to communicate their mathematical thinking.

Conclusion

The eight research-informed principles discussed above underpin appropriate early mathematics pedagogy. They are sufficiently flexible to apply across the diverse range of settings within Early Years education. While change is inevitable, we feel sure that these principles are sufficiently well informed to stand the test of time. New research, guidance and frameworks help us to develop specific resources and approaches, but the principles for appropriate pedagogy can help to guide the ways in which these are implemented in practice to most effectively support young learners to be confident, determined, creative and joyful mathematicians.