A mother holds a small stack of four saucers in front of her child. She asks the child how many saucers she would have left if she took away two and systematically goes through several other instructions, requiring her to add or remove saucers, physically mirroring the discussion with the actual objects. The child, who is three years old, is then invited to lead the game, asking similar questions of her mother. The child is completely engaged, showing high levels of enjoyment, curiosity and, indeed, aptitude; there is no sense of pressure in either the setting or the delivery.
This, I would suggest, is not a common scenario between mother and child, and teachers will be all too aware that the foundations of mathematical understanding are not necessarily being laid at home. This may not be because parents are uninterested, but perhaps simply that they are too ready to submit to the acceptable cliche of pride in being awful with numbers. Either way, children don’t often gain a good grasp of concepts associated with numbers at home.
The idea behind the simple exercise above is that it fosters an understanding of quantity without counting. The ability to see numbers in this way is called “subitising”, a concept explored among others by one of the most influential educational theorists of the 20th century, Jean Piaget (1896-1980), who suggested that children develop in distinct, incremental stages determined by their maturity.
It comes from the Latin “subitus ” (suddenly) and refers to the ability to instantly grasp the whole of two or three in its entirety rather than counting up or down. The child can immediately see “how many” – but can only do this having successfully navigated earlier stages of counting on and recognising patterns.
A child (or indeed, adult) needs, though, to be able to hold the idea of twoness, threeness or fourness distinctly and separately in their heads to be able to manipulate numbers effectively (simply counting would be a concept of “oneness”). This is one of the first steps in genuine, versatile numerical understanding and, indeed, shows that the child is ready to move towards calculation. Recognising such small groups is, in fact, all you need – many adults can only hold concepts of up to five in their minds, outside of familiar, recognisable patterns such as on dice or playing cards.
The difficulty of holding the “idea” of a number in the mind is where the early use of concrete resources comes into its own, as it helps to overcome the essentially abstract nature of maths – symbols representing things that are not present. Concrete resources, such as saucers (but it could be anything), allow the imagination to fix or make real the idea of an abstract number, thus enabling the child to be able to operate intuitively with numbers through an inherent, visual awareness of pattern.
The results of not gaining a concept of numbers early on are all too apparent to teachers, who sometimes see year 3 children (seven to eight-year-olds) counting laboriously on fingers and thumbs when they should be fluently manipulating groups of numbers. If a child cannot subitise, then, unfortunately, some variation of counting on or back is their only option.
An analogy to reading stories is perhaps pertinent, seeing as the foundations of reading are much more frequently laid at home and children are far less likely to hear adults talk openly about being “rubbish at reading” in the way that many do about a lack of mathematical skill. Reading a picture book or a story to a child is something of a ritual of bedtime. As a result, many children have a daily engagement with the world of narrative, supported visually with pictures, but also emotionally through the positive experiences of reading intimately with a parent. Story is thus invested with feelings of specialness, security and attention that hopefully underpin a child’s own, ongoing exploration of books and narrative.
This is in no way a plea to replace a bedtime story with counting blocks but rather a suggestion that since counting does not take place in the home as much as reading does, teachers can build number play into everyday routines with children to help shore up the foundations of maths. They can do this right from the early years. The emphasis should be on play and could be replicated in many settings: some blocks or balls on a carpet during break or playtime. Books could be used for a subitising game – or an imaginary story or song based around numbers.
As a game, it is meant to be enjoyable and is also not something that can be forced. An ability to subitise rests upon a solid understanding of number and pattern that will have developed from previous experiences of counting and from the visual recognition of groups of different sizes. The girl at the start of this article could only subitise because she was at the right stage of development for her. In this, it is akin to the inferential comprehension of texts – the ability to read between the lines, which is achieved through immersion and exposure rather than applying a learned formula.
However, once the child has arrived at a stage in which they can subitise, the world of operations and calculation will unfold for them – and getting to this stage certainly can be carefully supported and encouraged. It will then lead naturally on to other mathematical abilities, such as being able to build repeated patterns into a sequence.
In this way, and by impressing on parents the importance of giving a dedicated time and space to number (as well as to story), we can establish a strong and enduring base for future mathematical understanding. Story and number can then be developed in tandem, seen as jointly important, and ideally jointly developed as the building blocks of learning.
So if children ever ask if three really is the magic number (NB look up schoolhouse rock from the 1970s on YouTube), you should be able to genuinely answer yes (… together with two, four, five…).
Will Wareing is the deputy director of education, innovation and learning at the Girls’ Day School Trust (GDST).
