Thursday, 30 October 2014

The principles of counting

The principles of counting are a fundamental part of mathematics.  They appear within the EYFS but are distributed throughout each stage.

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. 11,12,13.  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.



Children learn the layout of dots on items such as dice or ladybirds.

* 1 Understanding Mathematics for Young Children: A Guide for Teachers of Children 3-8 By Derek Haylock and Anne Cockburn