Reception teachers everywhere will recognise the scenario: Child ‘A’ can easily work with sets of structured or unstructured objects, adding or removing one to say the number that is one more or one less than a given number. However, recognising that the next number in the counting sequence for any given number is ALWAYS one more – or the complementary idea that the previous number in the counting sequence for any given number is ALWAYS one less – involves a gigantic leap of faith that they are not yet quite ready to make.
To an adult, the connection between the two concepts seems obvious. Yet, for the little people we teach, this leap involves a complex shift between cardinal and ordinal thinking, which must be carefully planned.
Cardinality, as defined by Gelman and Gallistel (1986), is the understanding that the last number you say when counting a set indicates the quantity value of that set. If one more component is added to or taken from the set, the cardinal value increases or decreases accordingly. On the other hand, Ordinality refers to where a ‘thing’ (usually a number) is ordered in relation to something else.
Concerns have been raised – by Haylock and Cockburn (2017) and others – about the possibility of over-emphasis on the cardinality of number in EYFS at the expense of the ordinal aspect. If this is indeed the case, it follows that children may not yet have developed a conceptual understanding that numbers are not just labels for concrete sets of ‘things,’ but that they can also be labels for more abstract points or positions on a continuum. This may, in turn, lead to scenarios like the one described for Child A, where the identification of one more/one less does not easily translate to numbers on a continuum.
It is worth noting that there has been a subtle shift in the wording around one more/one less in the new Development Matters non-statutory guidance, which seems to suggest a renewed focus on the ordinality of number.
Previously, guidance suggested that children working at the expected level in Reception would be able to say ‘the number that is one more than a given number.’ This could be achieved by simply counting out a set of manipulatives, adding one more and counting the new total. Cut to 2020, and it is suggested that children working at the expected level now need to ‘understand the ‘one more than/one less than’ relationship between consecutive numbers.’ This seems to tip the scales towards the ordinality of the relationship, a level of understanding that could easily be bypassed if working solely with the cardinality of sets.
Of course, it is not news to most EYFS teachers that the teaching of numbers should extend beyond labelling sets of ‘things.’ I can almost hear a chorus of experienced teachers shouting, “Obviously. That’s why we use number lines!” But even the shift from using concrete sets of manipulatives to the more abstract number line involves a huge conceptual leap. Is there a more nuanced progression of models that could help to bridge the gap?
Haylock and Cockburn (2017) suggest that number lines arranged vertically so that the numbers get larger as you go up and smaller as you go down, better help children focus on the ordinality of the relationship between numbers. I would add that using a concrete, vertical number line made from numbered stacking blocks provides a useful stepping stone to support the transitional thinking between cardinal and ordinal thinking.
Concrete, vertical number lines:
- Retain the visual link between numbers and the quantity value, or cardinality of the set, which can be lost in pictorial number lines;
- Introduce stable number order on a continuum, helping children to visualise the ordinality of numbers in relation to each other, which can be difficult when working with non-numbered concrete sets;
- Provide a physical representation of the idea that as numbers go up, increase, or get bigger (i.e. as the tower gets one more block), both the numbers and the tower get bigger.
- Provide a physical representation of the complementary idea that as numbers go down, decrease, or get smaller (i.e. as the tower gets one less block), both the numbers and the tower get smaller.
- Allow children to be hands-on and active in their exploration of this key concept.
If we want our youngest children to develop a rich network of mathematical connections on which to build future learning, we must be mindful not to continually reinforce one aspect of number at the expense of another. However, if we are not careful in planning the transition between two key aspects of thinking, large conceptual leaps might leave some of our children behind. When it comes to progression from cardinal to ordinal thinking in relation to one more/one less, concrete vertical number lines may be one tool to help bridge the gap.
Marie Birchall is a practising primary school teacher and postgraduate student specialising in primary school mathematics. She can be found on Twitter @marie_birchall.
References and further reading:
EARLY EDUCATION, 2012. Development Matters in the Early Years Foundation Stage (EYFS). London: Early Education. Available from: https://foundationyears.org.uk/files/2012/03/Development-Matters-FINAL-PRINT-AMENDED.pdf [Accessed 17 February 2021].
DEPARTMENT FOR EDUCATION, 2020. Development Matters in the Early Years Foundation Stage (EYFS), London: Early Education. Available from: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/971620/Development_Matters.pdf [Accessed 10th April 2021]
GELMAN, R. and GALLISTEL, C.R., 1986. The Child’s Understanding of Number. 2nd ed. Cambridge, MA: Harvard University Press.
HAYLOCK, D. and COCKBURN, A., 2017. Understanding Mathematics for Young Children. 4th ed. London: SAGE