Below I've listed the principles of counting as devised by

*Gelman and Gallistel*and noted the 'steps to success' derived from the

*2012 Development matters*. I've provided exemplar photographs where possible.

*Please note: I've tried to fit in examples and steps to achieving the principle into each section. You may not agree necessarily, its not meant to be used as a guide, simply as food for thought on the way we teach number.*

__Gelman and Gallistel's principles:__

**One-One correspondence:**Counting 1 object at a time, touching with their fingers/foot/etc. Each object is counted only once and with early counting, consider moving the object out of the way once counted.

*16-26 months: Say some counting words randomly.*

*22-36 months: Selects a small number of objects from a group when asked, for example, ‘please give me one’, ‘please give me two’.*

*22-36 months: Recites some number names in sequence.*

*30-50 months: Recites numbers in order to 10.*

*30-50 months: Knows that numbers identify how many objects are in a set.*

*30-50 months: Realises not only objects, but anything can be counted,*

*including steps, claps or jumps.*

*40-60 months: Counts actions or objects which cannot be moved.*

*40-60 months: Counts out up to six objects from a larger group.*

*40-60 months: Counts an irregular arrangement of up to ten objects*

*40-60 months: Finds the total number of items in two groups by counting all of them.*

*40-60 months: Estimates how many objects they can see and checks by*

*counting them.*

Children see numbers in random orders but know their name. |

**Stable order principle:**Say number names/count in order (reliance on knowing the number names before knowing the numerals).

*16-26 months: Says some counting words randomly*

*22-36 months: Selects a small number of objects from a group when asked, for example, ‘please give me one’, ‘please give me two’.*

*22-36 months: Recites some number names in sequence.*

*30-50 months: Uses some number names and number language*

*spontaneously.*

*30-50 months: Uses some number names accurately in play.*

*30-50 months: Recites numbers in order to 10.*

*40-60 months: Counts objects to 10, and beginning to count beyond 10.*

*40-60 months: Selects the correct numeral to represent 1 to 5 and then 1 to 10 objects*

**Cardinality:**Understand the last number counted is the

__total__for the group and not just a number name.

*16-26 months: says some counting words randomly.*

*22-36 months: Selects a small number of objects from a group when asked, for example, ‘please give me one’, ‘please give me two’.*

*22-36 month: Knows that a group of things changes in quantity when*

*something is added or taken away.*

*30-50 months: Uses some number names and number language*

*spontaneously.*

*30-50 months: Compares two groups of objects, saying when they have the same number.*

*30-50 months: Knows that numbers identify how many objects are in a set.*

*30-50 months: Sometimes matches numeral and quantity correctly.*

*40-60 months: Counts up to three or four objects by saying one number*

*name for each item*

*40-60 months: Selects the correct numeral to represent 1 to 5, then 1 to 10*

*objects*

*40-60 months: Estimates how many objects they can see and checks by*

*counting them.*

*40-60 months: Finds the total number of items in two groups by counting all of them.*

Jumps in a tyre |

**Abstraction:**Understand that anything can be counted, tangible or not.

*Birth-11 months: Notices changes in number of objects/images or sounds in a group of up to 3.*

*8-20 months: Has some understanding that things exist, even when out of sight.*

*16-26 months: Knows that things exist, even when out of sight.*

*22-36 months: Selects a small number of objects from a group when asked.*

*30-50 months: Knows that numbers can identify how many objects are in a set.*

*30-50 months: Beginning to represent numbers using fingers, marks on paper or pictures.*

*30-50 months: Realises not only objects, but anything can be counted.*

*40-60 months: Counts objects or actions which cannot be moved.*

*40-60 months: Counts and irregular arrangement of up to 10 objects.*

**Order irrelevance:**Children can begin counting on any object and the total will always be the same.

*8-20 months: Has some understanding that things exist, even when out of sight.*

*16-26 months: Knows that things exist even when out of sight.*

*16-26 months: Beginning to organise and categorise objects.*

*22-36 months: Selects a small number of objects from a group when asked.*

*22-36 month: Knows that a group of things changes in quantity when*

*something is added or taken away.*

*30-50 months: Compares two groups of objects and says when they have the same number.*

*30-50 months: Separates a group of three or four objects in different ways, beginning to recognise that the total is still the same.*

*30-50 months: Knows that numbers identify how many objects are in a set.*

*30-50 months: Realises that not only objects, but anything can be counted.*

*40-60 months: Counts out up to six objects from a larger group.*

*40-60 months: Counts objects or actions which cannot be moved.*

*40-60 months: Count out six objects from a larger group.*

*40-60 months: Finds the total number of two groups by counting them all.*

__The following are not principles but are things we do everyday which should be considered. In addition I have added Subitizing which our lovely Maths consultant talks about a lot:__

**Knowing that the total does not change regardless of the layout or configuration of objects.**

This follows on from the order irrelevance principle however it is important that children understand that by simply moving the objects around they remain the same.

(E.g. 5 children are in a line, they line up to go to the kitchen. On the way back they are not in the same order - but how many children are there now?)

*16-26 Months: Beginning to organise and categorise objects, e.g. putting all the teddy bears together or teddies and cars in separate piles*

*16-26 months: Knows that things exist, even when out of sight.*

*22-36 months: Begins to make comparisons between quantities.*

*22-36 month: Knows that a group of things changes in quantity when*

*something is added or taken away.*

*30-50 months: Knows that numbers identify how many objects are in a set.*

*30-50 months: Compares two groups of objects, saying when they have the*

*same number.*

*30-50 months: Separates a group of three or four objects in different ways,*

*beginning to recognise that the total is still the same.*

Count forwards as you dribble a ball around a cone. |

**Knowing that numbers (and groups of object) get bigger and increase as you count forward/add and get smaller/decrease as you count backwards/subtract**

This links to concepts such as more/less and full/empty. It is not solely applicable to number however it is important for children to understand that as they add more to a group of objects that the side of the group increases and the number associated gets bigger (learning link the two concepts).

*22-36 months: Uses some language of quantities, such as ‘more’ and ‘a lot’.*

*22-36 month: Knows that a group of things changes in quantity when*

*something is added or taken away.*

*30-50 months: Uses some number names accurately in play.*

*30-50 months: Recites numbers in order to 10.*

*40-60 months: Counts objects to 10, and beginning to count beyond 10.*

*40-60 months: Uses the language of ‘more’ and ‘fewer’ to compare two sets*

*of objects.*

*40-60 months: Says the number that is one more than a given number.*

*40-60 months: Finds one more or one less from a group of up to five objects,*

*then ten objects.*

*40-60 months: In practical activities and discussion, beginning to use the*

*vocabulary involved in adding and subtracting.*

Literal: 1 is smaller (and goes into) 2 etc. |

**Knowing that when a number exceeds 9 we group into sets of 10. E.g. 9 becomes 10, then 11, 12 with the 1 (1_) remaining constant until we reach 20, then the process repeats itself.**

When children count and write numbers beyond 10 they often recognise a pattern, e.g.

__1__1,

__1__2,

__1__3. But then you get 210 (twenty-ten). They need to understand that beyond 9, the number becomes a ten.

*22-36 month: Knows that a group of things changes in quantity when*

*something is added or taken away.*

*30-50 months: Recites numbers in order to 10.*

*40-60 months: Counts objects to 10, and beginning to count beyond 10.*

*40-60 months: Counts an irregular arrangement of up to ten objects*

*40-60 months: Says the number that is one more than a given number.*

*40-60 months: Finds one more or one less from a group of up to five objects,*

*then ten objects.*

**Subitizing:**Knowing the total number of objects in a set by sight, e.g. dots on a dice - we know the number without having to count. (Children can subitize to about 6/7 maximum). (CLICK here for a vLog on subitizing in action)

According to Haylock1 (2013) there are two kinds of Subitizing. The first is perceptual where children are able to recognise the total of a group of objects without any strategies or pattern. The second is conceptual where children have learned a pattern of objects in a particularly way such as on dice or dominoes.

*22-36 months: Begins to make comparisons between quantities.*

*40-60 months: Counts out up to six objects from a larger group.*

*40-60 months: Estimates how many objects they can see and checks by*

*counting them.*