1. Number and Quantity
This is the first part of an article first published in IMPACT, journal of the Chartered College of Teachers, 2019.
Over the last 20 years there have been significant developments in our understandings of how 3-7 year olds learn mathematics. Many of the international studies published during this time have practical relevance for the mathematics we plan for in Nursery, Reception and Y1 settings. They also bring into question the current weight being attached to number and recall over other areas of mathematics in the 2021 changes to the Early Learning Goals – the set of measures by which children are assessed in their Foundation Stage Profile - at the end of the Early Years Foundation Stage (EYFS). In this article I will explore the ‘what’ – looking at significant elements of research in relation to two ‘big ideas’ in early years mathematics – the relationship between number and quantity, and spatial reasoning - illustrating these with practical examples of how young children learn. I will also consider the ‘how’ - the development of children’s reasoning and mathematical thinking in relation to these big ideas.
Image from Back, J., Gifford S. and Griffiths, R. (2016). 'Making Numbers: Using manipulatives to teach arithmetic'. Oxford: OUP
The ‘what’
One significant development in our understanding of the early learning of number is the recognition that establishing a robust connection between numbers and quantities is not simple and takes time. The second area that is important in making sense of how mathematical understanding develops is the role of early spatial awareness in predicting later mathematical achievement.
· Connecting numbers and quantities
“Quantities and numbers are not the same thing.”
Nunes and Bryant (2009, p4)
In everyday life many number symbols we see and use are distinguishing labels, for example: bus numbers, shoe sizes, road numbers. Typically it is quite rare to see numbers attached to quantities, for instance, in the case of boxes of eggs or multipacks of crisps. The number on a 64 bus does not mean that 63 buses have already passed by, a shoe numbered “5” does not mean that I have four other pairs of shoes.
Nunes and Bryant go on to say:
“Because the connections between quantities and numbers are many and varied, learning about these connections could take three to four years in primary school.”
Nunes and Bryant (2009, p4)
This interprets as these connections continuing to be forged between the ages of 5 and 9 years of age. A key early conceptual development for children in the EYFS, as well as those in the first years of statutory schooling, is to help them establish a clear connection between numbers – including written numerals - and quantities (Lyons Price Vaessen Blomert and Ansari 2014). We have known since the work of Hughes in the 1980s that the move from the more concrete to the abstract is not simple (Hughes 1986) and that it takes children a long time to acquire secure number sense (Munn 1996). Yet much of what we do mathematically, certainly from the end of Y1, becomes quickly abstract. Children bring their own varied, rich knowledge about numbers to school (Trundley 2008), but they might not link their counting or number knowledge to quantity, or quantity to numerals, even though they will have had experience of both: “I’m 3.” “He’s got more than me.” (Mix, Sandhofer, Moore and Russell 2012). It is crucial for practitioners to continue to provide young learners well into Y2, with plentiful and varied opportunities to organise, estimate, count, compare, share out and label different objects, in many different contexts and for different purposes. They should also support them in counting through the ‘tricky teens’ – the numbers from 11 to 19 - which in English are unhelpfully irregular. By seeing as well as hearing the counting numbers above 20 to 100, where it all begins to regularise, practitioners can also help children see the patterns in the number sequence. As long ago as 1978 Gelman and Galistel’s research pinpointed five principles underpinning a thorough understanding of counting, underlining the complexity of learning to count:
the ‘one-to-one principle’ –one counting word only is attached to each object
the ‘stable-order principle’ –counting words are repeatable and said in a stable order
the ‘cardinal principle’ – the last counting word names the quantity of the set
the ‘abstraction principle’ – any collection can be counted, tangible or otherwise
the ‘order-irrelevance principle’ - the quantity of the set remains constant whatever order items are counted.
(Gelman and Galistel 1978).
Research points to the following as helpful in supporting young children’s number sense:
· Counting real objects that can be moved, rather than pictures of objects or ‘objects’ on a screen. ‘Estimation Stations’ are gaining popularity, with a box for self-written estimates (Thompson 2008).
· Counting into a container to draw attention to the seven individual items becoming one group of seven, for instance by asking: “How many are in the bag now?” (Bills, Latham and Williams 2007, Trundley and Williams 2020).
· Drawing attention to the cardinal principle by saying before the count begins “It looks like we’ve got about seven here – let’s count to find out.” (Mix, Sandhofer, Moore and Russell 2012, Johnson and Ginet 2019 ).
· Counting out from a larger collection, by asking “Can you fetch me nine buttons from that jar-full?” This is the key assessment of the ‘big idea’ of counting (Young-Loveridge 1991, 2011, Sarnecka and Lee 2009).
Alongside this, and somewhat contrarily, it is not helpful for adults to ask children to count when they have already subitised (recognised a small amount without counting), for example: “I can see 4 there and I’m 4”. Clements and Sarama (Clements 1999, Clements and Sarama 2009) have been researching the importance of conceptual and perceptual subitising since the 1990s and yet work with young learners has yet to find its way into many early years’ settings. Perceptual subitising is the ability we all have (along with some other mammals) to recognise up to four items without having to count these. It is difficult to identify more than four without arranging these into a recognisable or iconic arrangement. Conceptual subitisingbuilds on this knowledge to recognise larger groups: “There’s six, look, three here and three here”. Dice and dot patterns are easily subitised and children should be encouraged to recognise these without ‘counting to check’. These understandings link into important later work combining and partitioning numbers (Gifford 2018, Early Intervention Foundation 2018).
Gifford (2015) identified various aspects of early years provision that are particularly helpful in as contributing positively to children’s early number sense. These include the following: routines – snack time, making number labels, ‘stock taking’; games – outdoor tracks, skittles, keeping score; number rhymes and books – ensure that at least some include symbols; simple problem solving – e.g. sharing gold coins amongst pirates; playfulness - correcting mistakes, testing ideas; and ‘sustained shared thinking’ (Siraj-Blatchford, Sylva, Muttock, Gilden and Bell 2002) with adults: (https://eycp.essex.gov.uk/media/1181/positiverelationships-understanding_sustained_shared_thinking.pdf ).
REFERENCES
Bills, L., Latham, P. and Williams, H. (2007) Encouraging all learners to think. Mathematics Teaching 181. 14-16. Derby: ATM
Clements, D. (1999) Subitizing. What is it? Why teach it? Accessed here: https://www.researchgate.net/publication/258933161_Subitizing_What_is_it_Why_teach_it
Clements, D., and Sarama, J. (2009) Learning Trajectories in Early Mathematics - Sequences of Acquisition and Teaching. Accessed here: https://www.researchgate.net/publication/242686699_Learning_Trajectories_in_Early_Mathematics_-_Sequences_of_Acquisition_and_Teaching
Early Intervention Foundation (2018) Key competencies in early cognitive development: Things, people, numbers and words. Accessed here: https://www.eif.org.uk/report/key-competencies-in-early-cognitive-development-things-people-numbers-and-words
Gelman, R. and Gallistel, C., (1978) The Child’s Understanding of Number. Harvard University Press: Massachusetts
Gifford, S. (2005) Teaching mathematics to 3 – 5s: developing learning in the Foundation Stage Maidenhead: Open University Press
Gifford, S., (2015) Early Years Mathematics: How to Create a Nation of Mathematics Lovers? Accessed here: https://nrich.maths.org/11441
Gifford, S., (2018) Subitising. Accessed here: https://nrich.maths.org/14004
Johnson, D. and Ginet, L. (2019) The role of adult and environmental input in children’s math learning. In J. McCray, J. Chen and J.E. Sorkin (Eds) Growing Mathematical Minds. New York: Routledge
Lyons, I.A., Price, G.R., Vaessen, A., Blomert, L. & Ansari, D. (2014) Numerical predictors of arithmetical success in grades 1-6. Developmental Science 17(5) 714-726 DOI
Mix, K.S., Sandhofer, C. M., Moore, J. A., and Russell, C. (2012) Acquisition of the cardinal word principle: The role of input. Early Childhood Research Quarterly, 27, 274-283
Nunes, T., and Bryant, P. (2009) Key Understandings in Mathematics Learning. Paper 2: Understanding whole numbers. London: Nuffield Foundation
Sarnecka, B. W., & Lee, M. D. (2009) Levels of number knowledge during early childhood. Journal of Experimental Child Psychology, 103(3), 325-337
Siraj-Blatchford, I., Sylva, K., Muttock, S., Gilden, R. and Bell, D. (2002) Researching Effective Pedagogy in the Early Years, (REPEY) Research Report 356. London: Department of Education and Skills
Thompson, I., (Ed.) (2008) Teaching and Learning Early Number. (2nd Edition). Maidenhead: OUP McGraw Education
Trundley, R., (2008) The value of two. Mathematics Teaching 211, 17-21. Derby: ATM
Trundley, R. and Williams, H. J. (2020) Learning from Variation. London: BERA. Accessed here: https://www.bera.ac.uk/publication/learning-from-variation
Johnson, D. and Ginet, L. (2019) The role of adult and environmental input in children’s math learning. In J. McCray, J. Chen and J.E. Sorkin (Eds)
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