Place Value, Counting & the Number System Flashcards
Who made the five counting principles?
What are the 5 counting principles?
Gelman and Gallistel’s five counting principles
1) The one-one principle
2) The stable-order principle
3) The cardinal principle
4) The abstraction principle
5) The order-irrelevance principle
What does the The one-one principle mean?
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 re-partition 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.
What is 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.
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.
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’.
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.
The one-one principle in an educational setting?
This principle refers to the need of matching one counting word to each item in the set to be counted.
To understand the one-one principle, children will need to:
Recite the counting words in order
Coordinate the touch and oral count so that they happen at the same time. Pointing to/touching items and counting is important in the process of counting as it ensures that each item is included
Keep track of items that have been counted and those that have not been. Children find it easier to move items as they count to keep track and, therefore, find counting objects easier than pictures (Potter and Levy, 1968).
What are The stable-order principle to an education setting?
Children need to also learn to say the counting words in order. Usually young children’s counting ‘string’ will consist of the first few words learnt correctly, a group of correct words with some omissions followed by words chosen randomly (Fuson et al. 1982). Learning to count in the English language is complicated as it involves rote learning of words that do not have a recognisable pattern until the number fourteen.
Initially children may just be chanting words memorised through rhymes and stories with it not having much meaning. Increasingly, the order of words takes meaning and children will begin to realise that the order of counting words is always the same and must always be said in this order: the stable-order principle.
What are the cardinal principle in a primary setting?
Children often learn counting as a process without understanding that the purpose is to find out the total number in the set. In other words, not realising that the final number in the count is not just identifying and labelling the last item counted but that this final number is a representation of the total number of items. The cardinal principle usually develops after the one-one principle and the stable-order principle. It is, therefore, important for adults to make the purpose of counting clear emphasising the final count is representing the total amount. Suriyakham (2007) recommends the use of gesture at the end of the count to develop this understanding, for example, a circular gesture which includes the whole set and emphasis is put on the final count word.
What area The abstraction principle in an education setting?
This principle refers to counting when children are moving on from counting objects which they can see and touch to counting through hearing and imagining items as they say the words.
Most young children’s counting experience is limited to using simple counting objects with most five-year- olds counting money in ‘ones’ irrespective of its value (Carraher and Schliemann, 1990). This limited experience can affect the development of place-value concepts at a later stage so it is important from the early years to teach pupils to use the correct number names for money, for example, this is two pence.
What area area The order-irrelevance principle in a setting?
Understanding that the total number in a set of objects will be the same when objects are counted in another order is a complex concept for young children to understand. Children may need to understand the cardinal principle more fully in order to develop the order-irrelevance principle. Interestingly, if a puppet is used to change the order of objects, children are more likely to realise the total amount will be the same compared to when an adult changes the order.
What principles are how to count
What are the count principles?
How to count principles:
the one-one principle
the stable-order principle
the cardinal principle
What to count principles:
the abstract principle
the order-irrelevance principle
Counting in 100s - 1) What are the misconceptions?
1) Often when children are counting from 0 to 1,000 and they get to 900, they say ‘ten hundred’ next. Explain that although they have 10 hundreds, we say this as one thousand. Ask:
Continue the count 700,800,900, …What comes after 900?
How can assessing mastery be added to counting to 100
Children can count in 100s from 0 to 1,000 and back again. They should understand what 100 is and the different ways of representing it. They will write the numbers in both numerals and words.
How can an understanding of counting to 100 be developed?
Children who are struggling to count in 100s first need to understand what a 100 is. Give them a jar of 100 dice or other objects or get them to count out 100 cubes. Once they have done this give them multiple packs of 100. Use a base 10 hundred block for each pack of 100. The connection between the hundred block and the 100 items should be made clear.
