Here are the ideas I did not have time to share about how some people view the stage of mathematical understanding based on a paper by:

A Model of Early Number Development
Kaye Treacy
Department of Education and Training, WA
<Kaye.Treacy2@eddept.wa.edu.au>
Sue Willis
Monash University
<Sue.Willis@Education.monash.edu.au>

• The research overviewed in this paper suggests that children will not develop an understanding of number as a representation of quantity through counting alone. This development is a complicated process that involves the interaction of a number of different quantitative aspects of a child’s daily life. From the research discussed above, a model (see Figure 1) is proposed of how these different components interact to contribute to a child’s developing understanding of numbers. The placement of the different components within the model indicates the interaction of the various components through time, not at particular times. Each is discussed briefly below.
• Protoquantitive comparison: Children, from about two years of age, become able to associate relational words with their innate ability to compare two amounts. They are able to say which is bigger or which amount has more or which has the most. Resnick calls this protoquantitive knowledge, though other researchers do not use this term, instead talking about relational knowledge.
• Children’s ability to subitize small amounts seems to develop out of their early ability to compare quantities. Children learn to associate a particular number word with a particular quantity. Starkey and Cooper (1995) found that by age two most children in their study could subitize one, two and three items, at three and a half years of age children subitize up to four items and by five they subitize up to five items. Sophian, Wood and Vong (1995), however, suggested that three and four year olds could subitize up to six items.
• At about two years of age, children begin to learn some of the principles of counting. They initially learn the first few words in the number sequence and to use one to one correspondence. Later they learn to give emphasis to the last word in the count. Children seem to initially learn these things as part of their socialization and may not link them with the idea of finding out how many. Even when children learn to repeat the last word of the number sequence in response to the how many question, they may not link this with the idea of quantity. Use counting to get. Counting and subitizing initially exist along side one another in a child’s mind as distinctly different processes. Children then begin to link the list of count words with the quantities that they know through subitizing and begin to understand that the last word said at the end of the count is telling them how many items in a collection. Children thus learn the quantitative significance of the number words in the counting process. According to Fuson (1988) (cited by Nunes & Bryant, 1996) children, at about five years of age are able to use counting to quantify single sets and to get an amount of items when asked.
• After children understand numbers words and counting as a means of quantifying a single set, they develop a trust in their counting processes and learn that no matter which way they count a collection they must always get the same result. As a consequence they “trust the count” and choose to use it to solve relational problems such as to make equivalent sets.
• Children learn to connect the number words and quantity. This allows them to develop an understanding of the part whole relationships of numbers attached to particular quantities. They can see, for example, that five fish could be made up of a group of three fish and a group of two fish. This helps them to develop a more robust understanding of the numbers they use in counting. They come to trust that no matter which way a collection is arranged or partitioned, the quantity of the set will always remain the same
• Children’s understanding of number from counting, subitizing, and part whole situations comes together so that they becom able to think about numbers as representations of quantity. They are able to disembed the number from the situation and so become able to think of any five items as “five”. The number becomes a conceptual entity in it’s own right. They understand the additive composition of number and so can think of numbers as compositions of other numbers, the number five, for example, can be thought of as three and two. They can work with numbers alone without having to refer to a quantity of materials.
• Caveat:  This model has been used as the basis for an investigation involving 25 children with learning disabilities in a Western Australian school (Treacy, 2001). Tasks were developed for each of the components listed above and these were used to individually interview the children. It was found that the children showed understandings similar to those suggested by the model above. For example, there were some children who showed no evidence of understanding the quantity aspect of counting and yet could subitize to three or more.  The teachers in this school found the model and the associated tasks particularly helpful in working out what their students knew and what they needed to know in order to develop a deeper understanding of number. Further research is needed however, to establish whether this model would be helpful for teachers working with children within the ‘normal’ range of intellectual ability.
• There is a visual of this process :visual-23js5zf