Calculating is a mathematical process that children learn as their understanding of numbers and counting develops. Through their play, daily experiences and adult support, children start to understand that numbers have a sequence and relate to specific amounts. From there, they can progress to understanding the notion of 'one more' and 'one less' - the basic concept of calculation within the EYFS - before progressing to the more complex concepts of addition and subtraction.
These concepts are grouped together within Problem Solving, Reasoning and Numeracy and expanded upon in the assessment points for the Early Years Foundation Stage Profile (see box).
EARLY LEARNING GOALS: CALCULATING
- In practical activities and discussion, begin to use the vocabulary involved in adding and subtracting.
- Use language such as 'more' or 'less' to compare two numbers.
- Find one more or one less than a number from one to ten.
- Begin to relate addition to combining two groups of objects and subtraction to 'taking away'.
THE VOCABULARY OF CALCULATING
To engage in and develop an understanding of calculation, children need to extend their mathematical vocabulary for quantity and comparison to include:
- numbers one to ten and beyond
- zero, none, nothing
- many, much, more, most, how much/how many?
- little, less, few, fewer. (Note that few and fewer refer to objects that can be counted - toys, apples, cars - while little and less refer to objects that cannot be counted - play, food, traffic).
- one more, one less, the same, different, too many, enough, nearly
- add, take away, subtract, makes, leaves, total, together, altogether.
SUPPORTING LEARNING
To calculate, children must first have grasped the sequence of numbers and progressed to the more difficult task of counting how many objects/people there are. Only then can they begin to understand 'one more/less' and the idea of combining (addition) or reducing (subtraction) two amounts.
In all these aspects of a young child's mathematical development, effective learning comes through real and meaningful experiences, both in their daily routines and in their play. Children will learn a great deal more about calculating from filling an egg box with six dough eggs than they will from merely placing six pieces of dough on a card with the number six written on it.
An obvious time for learning is snack and mealtimes, where children will daily hear conversations and questions involving amount and calculations: 'Would you like more grapes/another potato/less custard?' 'There are only two apples left in the bowl. I'll need to get another four from the kitchen.' 'Jack and Dylan aren't here today, so we'll only have to set the table for six.' 'How many extra places have been set at the table?'
As for their play, consider, for example, the calculations that may develop in the sand and block play areas or role play: 'How many more jugfuls of water do you think you'll need to fill the bucket?' 'Will all the water fit in the bowl?' 'How many more blocks will you need to complete the wall?' 'You've given me two pence, but the ice creams cost six pence altogether.'
Adult role
To support a child's developing ability to calculate, it is important that practitioners:
- start with their ongoing observations of a child. Only careful and regular observations will enable practitioners to assess a child's current understanding, motivations and interests and to plan meaningful and stimulating contexts in which they can explore and build upon their existing skills
- create an environment in which children can develop as active and independent learners - children learn by doing, repetition and experimentation
- recognise that knowledge is not a set commodity that is passed from practitioner to child. Adult and child must co-construct a child's developing knowledge. Remember that children will use a range of strategies to solve large and small number problems
- think aloud as you make your own calculations in the course of the day, model making calculations when supporting the children's play and use appropriate mathematical language
- scaffold children's learning by asking questions that will edge them towards solving a calculation and take seriously their efforts, whether right or wrong.
- encourage children to use their fingers to represent quantities - they can simply raise or bend their fingers to add more or less (see Case Study)
- support children to represent combined quantities on paper, either by drawing pictures or symbols or by writing numerals.
ENABLING ENVIRONMENTS
Children need opportunities within their play and daily routines to organise, order, sequence and classify objects and quantities. Such experiences will help consolidate their understanding of number, develop their reasoning skills and ability to classify and make comparisons and, in turn, open up opportunities for calculating. Settings should, therefore:
- make available a range of natural objects such as shells, stones, sticks and pine cones
- provide a variety of different-sized bowls, baskets and containers
- display number lines, to build on children's understanding of number symbols, sequence and 'one more/less'
- set up role-play scenarios such as a shoe shop or grocery stalls
- create interactive displays. Encourage the children to add or take home one or more toy or object, or provide incomplete sets for the children to complete.
- share number rhymes with the children (include ones that ascend in number as well as those that count down)
- provide books that encourage children to try to calculate - for example, Five Little Ducks by Ian Beck (Orchard Books), The Shopping Basket by John Burningham (Red Fox) and One Little Teddy Bear by Mark Burgess (Picture Lions)
- involve the children in making their own snacks and organising snack and mealtimes, such as setting the table and serving themselves
- ask children to tidy away, say, five items each at tidy-up time, then encourage them to calculate how many more items they need to find to return to the cupboard if they have only put away three
- make cooking or baking a regular feature of their provision, as these activities open up so many opportunities for counting and calculating - 'How many spoonfuls do we need?' 'How many have you put in?' 'How many more tomatoes do I need to add?' 'Have you added too many?' (see More Information)
- involve the children in making lists, including shopping lists
- play games with '+' and '-' (see Case Study).
IN PRACTICE
How children learn to calculate, and the role of the practitioner in that learning, can best be illustrated through case studies, such as the two below (EYFS scale points for calculating are included in italics).
CASE STUDY 1: PLAYING DICE GAMES
Emma is playing a calculating game that is linked to her favourite story of Snow White and the Seven Dwarves. The practitioner has six identical boxes - one empty, the others with different amounts of objects inside - and shows Emma the contents of each. The practitioner then closes the lids and shuffles the boxes around and asks Emma to throw a dice with + and - symbols on it.
Emma throws the dice and the addition symbol comes up; she then chooses a box containing three items. The practitioner asks her 'How many more teddies will you need so that all seven dwarves have a teddy to take to bed?' Using her fingers, she replies 'four'.
In this example, we can see that Emma is learning about mathematical symbols and gaining an understanding of addition.
CASE STUDY 2: PLAYING WITH BLOCKS
Richard and Emmanuel, both aged 48 months, are trying to construct an underground car park, using blocks. The boys line up five unit-blocks vertically, then add a ramp at one end and place two parallel lines of longer blocks on either side of the five unit-blocks. Next, they place two unit-blocks across the top of the parallel blocks, creating a roof on part of the building.
Richard then tries to fit a large truck in the car park, but is unsuccessful. So, the boys begin to discuss ways of making the truck fit. Emmanuel says that the car park will need to be much bigger. Richard replies they will need to add more lines of unit-blocks. Emmanuel agrees, adding that they will have to build a much higher wall.
Despite their best efforts, it is clear to the practitioner observing them that the boys are struggling to calculate the height of the truck and the number of additional lines required. She, therefore, suggests that they use the truck to help them measure the extra lines needed.
The boys turn their attention to the size and shape of the truck (1 Responds to the vocabulary involved in addition and subtraction). Richard places the truck alongside the long block and says, 'It's too long!' He then tries to add three lines to the unit-block. 'It's too short, we need two more lines, and then we have to take away the big blocks,' he continues (5 Relates subtraction to taking away). Emmanuel then adds the additional lines (2 Recognises differences in quantity when comparing sets of objects). Next, he places the truck on to the surface and moves it backwards and forwards. 'We need four more ramps,' Emmanuel tells Richard. He finds ramps to add to the end of each row. Emmanuel then counts the rows of unit-blocks. 'We have five rows, and five ramps,' he tells Richard (3 Finds one more or one less from a group of up to five objects).
In this observation, the practitioner has taken time to talk to the children about what they are doing. She is aware that both boys have a good understanding of counting and measuring, and intervenes appropriately. Her careful observations of their self-initiated play have left her able to scaffold their thinking about size and shape. From this example, we can see that practitioners must use their understanding of child development in conjunction with their knowledge of a child's interests and emerging capabilities to inform their planning and support children's learning.
CASE STUDY 3: PLAYING WITH GLOOP
Ann-Marie and Sabrina, both aged 42 months, are observed playing with cornflour gloop, a self-initiated activity. They repeatedly scoop the gloop out of the bowls with their hands, let it drip into two duck-shaped pastry cutters, and then, once filled, remove the cutters and watch the shapes disintegrate.
The girls discuss the problem and pursue their separate investigations, as they cannot agree on a solution. Sabrina asks for the same size bowl as the practitioner, explaining that she wants to start again.
Ann-Marie takes the box of cornflour over to the pastry cutters, scoops three small spoonfuls directly into a cutter, counting '1, 2, 3', then adds a few drops of water into the mould (the mixture is still very powdery).
Next, she adds a little more water, counts '4,5,6,7,8' more scoops of cornflour and stirs the mixture with her fingers before scooping the almost solid, but crumbly, material on to her hands (6 In practical activities and discussion, begins to use the vocabulary involved in adding and subtracting).
After asking the practitioner's approval, she pours some water on to one of her hands, manipulates the cornflour from hand to hand, strokes it and says, 'It feels like clay now, and it's ready.' (8 Uses developing mathematical ideas and methods to solve practical problems).
Sabrina, in the meantime, has started again from scratch and asks the practitioner's advice on how best to measure the amount of water that she will need. The practitioner suggests either counting the water by the spoonful or pouring a little at a time from a measuring jug. As further assistance, she demonstrates pouring the water from the jug and points clearly to the numbers on it, explaining that they indicate how much water is in the jug.
Initially, Sabrina measures four ounces of water and pours it into her small bowl. 'It's too much!' she says, and pours it out. Then, almost acting like a scientist, she measures various quantities of water, observing what the different amounts of water look like. On one occasion, she fills the bowl to overflowing.
Next she fills the jug with three ounces of water, then adds three heaped spoonfuls of cornflour, clearly associating equal numbers, rather than equal quantities, of ingredients in her quest to solve the problem.
Ann-Marie rejoins her and suggests that she adds more cornflour. Both girls then discuss how much. Suddenly, Sabrina announces, 'I've got a very good idea!' and adds two more spoonfuls. As a result, her mixture begins to thicken.
Sabrina looks at the practitioner with a puzzled expression on her face and asks, 'When it's mixed altogether, what happens to all the water? Why does it disappear?' The practitioner explains that cornflour absorbs the water when stirred.
She encourages Sabrina to add two spoonfuls of cornflour to some water and to stir it with a lollypop stick to find out what happens. Sabrina does this willingly, adding 'It's getting harder!' (7 Finds one more or one less than a number from 1 to 10).
The practitioners suggests that Sabrina adds another spoonful and asks, 'Can you feel the mixture getting thicker?' 'It's very hard at the bottom, and not the top,' Sabrina replies. After adding another spoonful, the mixture solidifies completely. Sabrina scoops it out of the bowl and into the pastry cutter. (9 Uses a range of strategies for addition and subtraction, including some mental recall of number bonds).
In this example, we see how an effective practitioner supports children's explorations and attempts to calculate by modelling (using the measuring jug), explaining (what happens when more/less water is added to cornflour) and making suggestions (about measuring the water and cornflour).
She helps highlights cause and effect, scaffolds their thinking by offering information that builds on their previous understanding and supports their individual ideas, so enabling them to co-construct and think logically for themselves.
EYFS PROFILE ASSESSMENT SCALES: CALCULATING
1. Responds to the vocabulary involved in addition and subtraction in rhymes and games.
2. Recognises differences in quantity when comparing sets of objects.
3. Finds one more or one less from a group of up to five objects.
4. Relates addition by combining two groups.
5. Relates subtraction to taking away.
6. In practical activities and discussion, begins to use the vocabulary involved in adding and subtracting.
7. Finds one more or one less than a number from 1 to 10.
8. Uses developing mathematical ideas and methods to solve practical problems.
9. Uses a range of strategies for addition and subtraction, including some mental recall of number bonds.
REFERENCES AND FURTHER READING
- Athey, C (2009) (second edition) Extending Thought in Young Children: A Parent-Teacher Partnership. Sage
- Bruce, T (2010) second edition) Early Childhood: A Guide for Students. Sage
- Eaton, C (2011) Enabling Environments. Hampshire County Council
- Ebbutt, S, 'All about ... problem-solving' (Nursery World, 9 July 2009)
- Ebbutt, S and Skinner, C, 'All about ... number' (Nursery World, 3 May 2001)
- Ebbutt, S and Skinner, C, 'Mathematics in the EYFS - Counting' (Nursery World, 28 June 2007), 'Calculating' (26 July 2007), 'Addition and subtraction' (29 August 2007)
- Ebbutt, S, Latham, P, Mosley, F and Skinner, S 'All about ... measure' (Nursery World, 7 February 2002)
- Hughes, M (1990) Children and Number: Difficulties in learning mathemathics. Basil Blackwell
- Louis, S (2009) Knowledge and Understanding of the World in the Early Years Foundation Stage. Routledge
- Pound, L (2008) Thinking and Learning about Mathematics in the Early Years. Nursery World/Routledge
- Tassoni, P (2008) Practical EYFS Handbook. Heinemann
Stella Louis is an author, early years consultant for the London Borough of Southwark and Froebelian-trained tutor working in the UK and South Africa in an initiative funded by the Froebel Council. She is studying for a Doctorate in Education at Roehampton University, London
