Abstract
When fractions are introduced in school mathematics, they are usually introduced in the context of continuous quantity. Number sequences are essentially excluded because, as quantitative schemes, they are thought to be relevant only in discrete quantitative situations. Even though I developed number sequences in Chap. 3 in the context of discrete quantity, I can see no principled reason to keep them separate from continuous quantity. Reserving number sequences for discrete quantity stands in opposition to the concept of the real number line in higher mathematics, and, in this chapter, I argue that it also stands in opposition to the development of quantitative schemes. In articulating the reorganization hypothesis, I establish that a composite unit of specific numerosity can be used to make a split in the way that Confrey (1994) explained. This involves more than simply indicating the possibility of transferring the operations involved in compounding discrete units together to splitting continuous units. I do a deeper developmental analysis of children’s quantitative schemes in which I explore whether the operations that produce discrete quantity and the operations that produce continuous quantity can be regarded as unifying quantitative operations. If so, these quantitative operations would justify the reorganization hypothesis.
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Notes
- 1.
If a continuous item of experience is bounded, from that perspective it is also discrete. Similarly, if a discrete item of experience has an interior that would qualify it as a continuous item of experience. Of course, by an “item of experience” I refer to an implemented attentional pattern with the understanding that the experiential item is a permanent object.
- 2.
Excepting an awareness of duration would eliminate an awareness of the continuity of the scanning motion over the regenerated visual material.
- 3.
They are complementary because the former leads to length and the latter to distance.
- 4.
See Steffe (1991) for a conceptual analysis of this construction.
- 5.
Here, “segment” is not to be interpreted mathematically.
- 6.
In this case, the child is aware of a row of blocks as a segmented but connected unitary item. Nevertheless, the child may be aware also of perceptual plurality.
- 7.
By “perceptual density,” I mean an awareness of the frequency of instantiation of the block concept within definite boundaries. By “perceptual plurality” I also mean an awareness of more than one instantiation of the involved template.
- 8.
I assume that “staircase” is used for JEA rather than “zigzag path.”
- 9.
I assume that one match in Protocol III was broken into three pieces because JEA said there were seven bits.
- 10.
Number sequences develop during early childhood mathematics education, and so they do not develop independently of the children’s mathematics education. For this reason, I have placed the phrase, “spontaneous development” in quotation marks to acknowledge the contribution of children’s mathematics education in their construction of number sequences.
- 11.
When increasing the exposure time to 5 s for four items, the numbers at each age level were 21, 29, and 37, indicating that some children counted. Almost 50% of the 4-year-olds could recognize four items after an exposure of only 1 s. Recognizing four items after only a 1-s exposure is important because it is quite likely that the recognizing child would regenerate an image of the items after the exposure, which indicates that their quadriadic attentional patterns were constituted at least as figurative lots. However, the phenomenon of subitizing, instant recognition of numerosity, such as in the case of dyadic and triadic patterns, may have enabled many of these 4-year-olds to recognize the four items without their quadriatic attentional pattern being constituted as a figurative lot (von Glasersfeld 1981).
- 12.
There is a discrepancy of 10 children between the total number of children who cut the string into two pieces and the number of children reported in the subcategories.
- 13.
See discussion of the initial number sequence in Chap. 3.
- 14.
I assume that the operations of partitioning at the experiential level are simply those operations that such a child can use at the re-presentational level.
- 15.
This will be one of the major issues that are investigated in the case studies that follow.
- 16.
Here, I use “partitioning” in the sense in which Confrey uses “splitting.” Hereafter, I use “fragmenting” and “partitioning” rather than “splitting” to maintain the distinctions between partitioning and the earlier forms of fragmenting that I have identified.
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Steffe, L.P. (2010). Articulation of the Reorganization Hypothesis. In: Children’s Fractional Knowledge. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0591-8_4
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