Brecknock Primary School

Early Years Calculation Policy

Progression in Calculations – Early Years

Counting

Gelman and Gallistel’s five principles govern and define counting:

1. The one-one principle This involves the assigning of one, and only one, distinct counting word to each of the items to be counted. To follow this principle, a child has to be able to partition and repartition the collection of objects to be counted into two categories: those that have been allocated a number name and those that have not. If an item is not assigned a number name or is assigned more than one number name, the resulting count will be incorrect. 2. The stable-order principle To be able to count also means knowing that the list of words used must be in a repeatable order. This principle calls for the use of a stable list that is at least as long as the number of items to be counted; if you only know the number names up to ‘six’, then you obviously are not able to count seven items. So, a child who counts 1, 2, 3 for one particular collection of three objects and 2, 1, 3 for a different collection cannot be said to have an understanding of the stable-order principle – although such a child would appear to have an understanding of the one-one principle. However, a child who repeatedly counts a three-item collection as 2, 1, 3 does appear to have grasped the stable-order principle – although, in this case, has not yet learned the conventional sequence of number names. 3. The cardinal principle This principle says that, on condition that the one-one and stable-order principles have been followed, the number name allocated to the final object in a collection represents the number of items in that collection. To be considered to have grasped this principle, a child needs to appreciate that the final number name is different from the earlier ones in that it not only ‘names’ the final object, signalling the end of the count, but also tells you how many objects have been counted: it indicates what we call the numerosity of the collection. If a child recounts a collection when asked how many objects there are, then they have not yet grasped this principle. Until recently, it was generally assumed that a child understood the cardinal principle if, after counting a collection and being asked how many objects there were, they immediately repeated the last number name spoken. However, in 2004 Bermejo et al. showed that when children were asked to count a collection of five objects starting the count with the word ‘three’ many gave the answer ‘seven’, i.e. the last number name they had said. These three principles are considered by Gelman and Gallistel to be the ‘how-to count’ principles as they specify the way in which children must execute a count. The remaining two are ‘what-to-count’ principles, as they define what can actually be counted. 4. The abstraction principle This states that the preceding principles can be applied to any collection of objects, whether tangible or not. Obviously, for young children learning to count it is easier if the objects are tangible and, where possible, moveable, in order to help them to distinguish the ‘already counted’ from the ‘yet to be counted’ group. To understand this principle, children need to appreciate that they can count non-physical things such as sounds, imaginary objects or even the counting words – as is the case when ‘counting on’. 5. The order-irrelevance principle This principle refers to the knowledge that the order in which items are counted is irrelevant. It does not really matter whether the counting procedure is carried out from left to right, from right to left or from somewhere else, so long as every item in the collection is counted once and only once.

Objective and

Strategies

Counting

Children learn to count through games, images and songs as part of a stimulating indoor and outdoor environment. As children progress and understand more about values and the fixed nature of counting objects (no matter how many times they count the original set, the answer will remain the same), they will begin to ‘subitise’ objects when counting. Children will begin to look at the arrangement of objects and just ‘know’ how many things are there – it has become instinctive to recognise four objects as four – e.g. do you need to count the spots on a dice or do you know how many?

Children will develop to match the spots from one domino to the next, recognising the number straight away without having to count the spots.

Songs and rhymes

Counting up and down as chn eat them: 1, 2, 3, 4 etc 10, 9, 8, 7 etc

Construct stepped walls using building blocks,

learning to count forward and back as they move

toys from one step to another.

Counting whilst playing sport: '1 skip, 2 skips, 3 skips' '1 catch, 2 catches, 3 catches' Counting turns: 'My turn now, you’ve had 4!'

Games and activities should show that

anything can be counted (not only objects)

such as steps, claps or jumps.

Counting stairs. Incorporate counting into daily activities, putting images alongside to assist. Extend tasks through games

Lego Maths- counting and ordering

Counting whilst getting enough

cups/plates/cutlery/food

Count, match and order

Practical games e.g. pretending to be a frog to jump from 1-20

Count the number of people in the register: use fingers to give a physical and visual element to the experience.

Organise races on foot and using wheeled

vehicles, for which they receive rosettes to

develop a clear understanding of ordinal

numbers (1st, 2nd, 3rd…)

Objective and

Strategies

Addition and

Subtraction

Children should begin to

relate addition to counting on

from an amount and

subtraction to taking away

and counting how many

objects are left.

Key Vocabulary:

add, more, plus, makes, total, altogether, score, double, one more, two more, ten more how many more to make…? how many more is … than …? take, take away, leave, subtract, minus, equals, number sentence, count back, one less, two less, ten less; how many are left / left over? how many have gone? how many fewer is … than …?

Concrete

Children begin by using concrete

apparatus during practical

activities and play in

mathematically rich

environments: children cast dice,

play matching card games, roll

marbles into numbered

compartments etc. Use of

mathematical vocabulary is

modelled. This is reinforced by

chanting, rhymes and singing

songs.

Through games

Using fingers Adapted games

How many do you have

altogether? What if I add/subtract

a red car?

Songs to reinforce addition and subtraction e.g. Five Little Men in a Flying Saucer, Ten Green Bottles, Five Currant Buns

Adding and subtracting legs from

a spider and other animals

Use physical items and images

to help counting how many

children are outside. What if 'x'

goes outside too? What is ‘x’

comes inside?

Adding/subtracting fruit for break Use the vocabulary ‘more’ and

‘less’ to compare

Adding/ subtracting real objects

at the 'shop'

Objective and

Strategies

Addition and

Subtraction

Children should begin to relate

addition to counting on from

an amount and subtraction to

taking away and counting how

many objects are left.

Key Vocabulary:

add, more, plus, makes, total, altogether, score, double, one more, two more, ten more how many more to make…? how many more is … than …? take, take away, leave, subtract, minus, equals, number sentence, count back, one less, two less, ten less; how many are left / left over? how many have gone? how many fewer is … than …?

Concrete / Pictorial

Children are then shown pictorial representations which link to real contexts that they have experienced. Children play a range of stimulating and engaging games on the IWB to

investigate patterns and number sequences.

Have an understanding of what “more” and “less” means using number tracks and then number lines

Solve simple problems using fingers and introduce Numicon when appropriate.

Children can record adding and subtracting using Numicon by printing or drawing around the pieces. They combine pieces to add find number bonds and start to add without

counting.

A good understanding of place value is considered to be of paramount importance. This is supported by a wide range of practical equipment including base-10 apparatus,

100 squares, bead strings, place-value cards and number lines. Because pupils also require good instant recall of number facts, such as number bonds to 10, and, later,

multiplication tables, every opportunity is taken to develop these.

Children begin to combine groups of objects or pictures and use concrete apparatus. Later they relate subtraction to taking away and counting how many objects are left.

Children make a record in pictures, words, Numicon shapes or symbols of addition and subtraction activities already carried out.

Children are encouraged to read number sentences aloud in different ways: e.g. “Three add two equals 5” “Four plus 3 makes 7” “Five subtract one leaves four” “Six take away 3

equals 3” 6.

Objective and

Strategies

Addition and

Subtraction

Children should begin to relate

addition to counting on from

an amount and subtraction to

taking away and counting how

many objects are left.

Key Vocabulary:

add, more, plus, makes, total, altogether, score, double, one more, two more, ten more how many more to make…? how many more is … than …? take, take away, leave, subtract, minus, equals, number sentence, count back, one less, two less, ten less; how many are left / left over? how many have gone? how many fewer is … than …?

Pictorial / Abstract

Number tracks can be introduced to count up on and to find one more: lines can be used alongside practical apparatus to solve addition and subtraction calculations and word

problems. Children “jump” along the number line to “count on" and “jump” back to “count down” the number line.

Teachers should introduce the number line alongside the

pictorial representations already used and model using

concrete apparatus

Objective and

Strategies

Multiplication

The link between addition and multiplication can be introduced through doubling and reinforced through repeated addition of the same number.

Key Vocabulary: lots of, groups of, times, repeated addition, double, combine, twos, fives, tens

Concrete / Pictorial

The link between addition and multiplication can be introduced through doubling and reinforced through repeated addition of the same number. Teachers should encourage children to compare relative sizes of objects to also build up multiplicative thinking. Multiplication is best introduced through physical activities and games.

. Children should begin to count in twos, fives and tens, both aloud and with objects, such as Numicon (these can be drawn around or printed as a way of recording) or other concrete apparatus.

Children are given multiplication problems set in a real life context. They should be presented with a range of concrete and pictorial representations that encourage them to make links. e.g. “How many fingers on two hands?” “How many sides on three triangles?” “How many legs on four ducks?” “How many wheels are there altogether?” “How much money do I have?” Use number tracks, number lines and number sticks, alongside other concrete apparatus.

Pictorial / Abstract: begin to link to groups using pictorial images to represent the objects

Objective and

Strategies

Division Key Vocabulary: halve, share, share equally, one each, two each, three each, group in pairs / threes / tens, equal groups of, in equal parts, left, left over

Concrete / Pictorial / Abstract

Division can be introduced by practically halving or sharing an equal amount into 2 groups.

Children begin with mostly pictorial representations linked to real life contexts: Children need to see and hear representations of division as both grouping and sharing

Children have a go at recording the calculation that has been carried out with objects, verbally and then written down: e.g. by drawing pictures in groups or by arranging concrete apparatus into groups.

12 shared equally by 3 is 4

Grouping Model Mum has 6 socks. She grouped them into pairs. How many pairs did she make?

Sharing Model I have 10 sweets. I want to share them with my friend. How many will we have each?

Depth: Discuss remainders. Can

you share this bar of chocolate

fairly between five people?

Why?