Abstract
The primary goal of this chapter is to present a model of important steps in children’s construction of their numerical counting schemes because the basic hypothesis that guides our work is that children’s fraction schemes can emerge as accommodations in their numerical counting schemes. I consider a number sequence to be the recognition template of a numerical counting scheme; that is, its assimilating structure. This way of thinking of a number sequence was basic in the formulation of the reorganization hypothesis. A number sequence is a discrete numerical structure; it is a sequence of arithmetical unit items that contain records of counting acts. At all stages of construction, children use their number sequences to provide meaning for number words. A number word such as “twenty-one,” say, can refer to a sequence of arithmetical unit items from “one” up to and including “twenty one.” It is the operations that children can perform using their number sequences that distinguish among distinct stages of the number sequences. In what follows, I explain the operations that produce two prenumerical counting schemes as well as three distinctly different number sequences and, hence, three distinctly different numerical counting schemes. I also explain discrete structures that precede number sequences in development that I refer to as perceptual and figurative lots. These lot structures are produced by the operation of categorizing discrete items together, where categorizing is based on reprocessing sensory-motor items of experience using an operation called unitizing. In categorizing, when reprocessing is coordinated with re-presenting discrete items of experience, recursive unitizing emerges. Recursive unitizing is that operation which produces arithmetical units and numerical structures. I start by presenting an attentional model of unitizing and the different levels of units that this operation produces.
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Notes
- 1.
The ability to regenerate an experience of an item makes possible the falling apart of moments of life.
- 2.
We use “perceptual plurality” to refer to an awareness of the frequency of instantiation of a perceptual unit item within experiential boundaries. Our choice of “plurality” rather than “collection” is made to accentuate the child’s awareness of more than one unit item rather than a unitary whole containing the unit items.
- 3.
Quotation marks are used to indicate that we are speaking of quantity on a perceptual level.
- 4.
See Steffe et al. (1983), for examples of counters of perceptual unit items.
- 5.
An inability to create an image of a perceptual unit item would exclude experiencing perceptual unit items in their immediate absence, for then the child could only recognize the item in its immediate presence.
- 6.
Because the child is aware of figurative plurality, a child at this level can give meaning to a number word by producing visualizing images of hidden items. This extensive meaning of the number word remains indefinite, and to make it definite, the child has to actually count. Following Thompson (1982), counting is called the intensive meaning of number words.
- 7.
The ability to establish figurative unit items as substitutes for hidden perceptual unit items is one level, and the ability to run through a counting activity and produce its results in thought without motor action and without given sensory material to act on is the other level.
- 8.
In this case, an object concept is a figurative unit item; that is, a particular type of attentional pattern. An attentional pattern, when implemented, is an operation of unitizing.
- 9.
A numerical pattern can also consist of a composite of abstract unit items; that is,interiorized figurative unit items that do not contain records of counting.
- 10.
A numerical composite is a sequence of arithmetical units. But there is no unit containing the sequence except for the beginning and end of actually counting.
- 11.
The initial number sequence is an unstable number sequence that is only transitory to a more adequate number sequence.
- 12.
When the elements of a finger pattern become recorded in arithmetical units, the finger pattern is constituted as a numerical finger pattern and can be used as numerical meaning of number words.
- 13.
Note that Jason did not first count from “one” to “seventeen” in a way similar to how he counted in Protocol III.
- 14.
For a more complete analysis of the tacitly nested number sequence, see Steffe and Cobb (1988).
- 15.
A child can be aware of the types of discrete quantity at the preceding levels of the number sequence. For example, a child who has constructed only the initial number sequence can be aware of perceptual or figurative pluralities because they can engage in the operations that produce them. But it is always understood that such a child could constitute these pluralities as perceptual or figurative numerosities if they produced them using their number sequence.
- 16.
By an integration of two composite units we mean that the child, whose goal it is to find how many elements in the two composite units together, first unites the two composite units into a unit containing them, disunites each of the two composite units into their elements while maintaining an awareness of the two composite units as well as of their elements, and then counts the elements to establish their numerosity. The containing composite unit serves as background while the child is operating.
- 17.
Tomas Kieren has always been fond of saying that fractional schemes are used to find how much and counting schemes are used to find how many.
- 18.
An assimilation is generalizing if the scheme involved in assimilation is used in a situation that contains sensory material or conceptual items that are novel for the scheme but the scheme does not recognize it, and if there is an adjustment in the scheme without the activity of the scheme being implemented.
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Steffe, L.P. (2010). Operations That Produce Numerical Counting Schemes. In: Children’s Fractional Knowledge. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0591-8_3
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