We are delighted to read in the new English document 'Best start in Mathematics' (Ofsted, 2024) a statement on calculations in Reception classes, with children 4 - 5 years of age. This is the first time in recent years that any 'official' comment on these important aspects of mathematics ('written' mathematics and calculations) has been made. The document emphasises,

     'It is not appropriate to expect number operations to be written down as equations, or sums, in the

     early years. Formal recording is introduced in Year 1. However, children might choose to experiment

     with and talk about their own ways of recording number operations.'

     But without guidance, or any idea of what children's 'experiments with and talk about their own ways' with representing number operations, or knowing what they might look like, such a statement poses a conundrum for Reception teachers.

     This is where the concept, pedagogy and practice of Children's Mathematical Graphics can help, and with numerous children's examples and with guidance on children's developing understanding of calculations (and what came before) in our publications and on this website. Our taxonomy also provides a means for adults to understand children's graphics, that may also be used to support formative assessment.

     In 2002 Ofsted reported that a weakness of mathematics teaching is that teachers relied too heavily on published schemes and worksheets, suggesting that children rarely have opportunities to represent their own mathematical thinking – including calculations, in their own ways (Carruthers & Worthington, 2004; 2008). More recently, Baroody et al., (2019) found that in recent years there has been a significant increase in the use of worksheets at the expense of play.

     It is important to emphasise that our research is based on numerous examples of ‘written’ (and represented) mathematics from children who were encouraged to really think about what they did and to build their understanding in ways that were personally meaningful to them. Young children need to be able to represent and communicate their thinking through their own early graphical marks and signs, well before they are thinking about calculations.

     In a previous study of 273 teachers of children in the 3 - 8 age-range (in Carruthers & Worthington, 2006), we found that fewer than ten per cent gave children opportunities to represent their own mathematical thinking in their own ways. Yet we know that children have great difficulties with understanding and using standard algorithms (Gifford, 1997; Hughes, 1986).

     As Thompson argues, children 'should be working in an environment  that attaches value to idiosyncratic written methods, and should be praised for inventing their own notations' (2008, p. 158). Children's own calculations are integral to our research (Carruthers & Worthington, 2004; 2008), and show how their earlier marks and signs develop over time into children's own number operations and problem solving, as they gradually come to understand and use the formal abstract symbolic language of written mathematics including calculations, at a deep level.

Note: To see examples of children's early marks and signs, click on the first link below. 

References

Baroody, A. J., Clements, D. H., & Sarama, J. (2019). Teaching and learning mathematics in early childhood programs. In C. Brown, M. B. McMullen & N. File (Eds.), Handbook of Early Childhood Care and Education (1st ed., pp. 329-353). : Wiley Blackwell.

Carruthers, E., & Worthington, M. (2003).: 'Research uncovers children's creative mathematical thinking', in Primary Mathematics 7(3),  p. 21-25. Mathematical Association, Autumn, 2003.

Carruthers, E. and Worthington, M. (2004). Young Children Exploring Early Calculation,

Mathematics Teaching, 187 (June), 30 – 34.

Carruthers, E., & Worthington, M. (2006). Children’s mathematics: Making marks, making meaning. (2^nd ed.). Sage.

Carruthers, E., & Worthington, M. (2008). Children's mathematical graphics: young children calculating for meaning. In I. Thompson. (ed.). Teaching and learning early number, (2nd edition), chapter 10. pp. 127-148. OUP.

Gifford:, S. (1997). 'When should they start doing sums?' A critical consideration of the 'emergent mathematics' approach' in I. Thompson. (ed) Teaching and learning early number, (1st edition), chapter 7, p. 75-88. Open University Press.

Hughes, M. (1986). Children and number: difficulties in learning mathematics. Blackwell.

OFSTED (2002) The First Three Years of the National Numeracy Strategy. HMI.

Ofsted (2024) Best start in life part 3: the 4 specific areas of learning. Gov.UK

Payler, J., Wood, E., Georgeson, J., Davis, G., Jarvis, P., Rose, J., Gilbert, L., Hood, P., Mitchell, H., & Chesworth, L. (2017). BERA- TACTYC Early Childhood Research Review 2003-2017. BERA, London.

Thompson, I. (2008). What do young children's mathematical graphics tell us about the teaching of written calculations? In I. Thompson. (ed.). Teaching and learning early number, (2nd edition), chapter 11. pp. 149-159. Open University Press.