Abstract
The basic hypothesis that guides our work is that children’s fraction schemes can emerge as accommodations in their numerical counting schemes. This hypothesis is referred to as the reorganization hypothesis because if a new scheme is constructed by using another scheme in a novel way, the new scheme can be regarded as a reorganization of the prior scheme. There are two basic ways of understanding the reorganization of a prior scheme. The first is that the child constructs the new scheme by operating on the preceding scheme using operations that can be, but may not be, a part of the operations of that scheme. In this case, the new scheme is of the same type as the preceding scheme. But it solves problems and serves purposes that the preceding scheme did not solve or did not serve. It also solves all of the problems the preceding scheme solved, but it solves them better. It is in this sense that the new scheme supersedes the preceding, more primitive, scheme.
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Notes
- 1.
Rational Number Project.
- 2.
By “continuous context” I refer to experiential episodes that contain items that are produced by moments of focused attention that are not interrupted by moments of unfocused attention, but which may be bounded by such moments of unfocused attention [cf. von Glasersfeld (1981) for the meaning of attention]. Scanning the sky from one horizon to the next on a perfectly clear day produces what I think of as an experiential continuous item as well as scanning a blank sheet of paper.
- 3.
By “composite unit” I mean a unit that is produced by uniting simple units into an encompassing unit. An example is uniting a regeneration of the chimes of a clock into a composite whole.
- 4.
The use of “concept” rather than “scheme” is intentional. The meanings of these terms will be commented on in Chap. 2.
- 5.
A connected number is constructed by the child by using the units of a numerical concept in partitioning a continuous item into parts and then uniting the parts together.
- 6.
Confrey (1994) cited sharing, folding, dividing symmetrically, and magnifying as the basis for splitting.
- 7.
See von Glasersfeld (1995a) for a discussion of the distinction between the self as center of subjective awareness.
- 8.
The operations that produce composite units often appear precociously in the case of two as unity.
- 9.
“Visualized imagination” is not restricted to visual imagery. It includes also regeneration of perceptual items in any sensory mode.
- 10.
We focus on breaking into n equal parts because of our interest in partitioning.Confrey’s analysis of splitting is not restricted in this way.
- 11.
In Chap. 6, we argue that distribution is a fundamental operation in constructing a multiplicative concept. Distribution is found in what is referred to as a coordination of two composite units.
- 12.
The unitizing operation is explained in Chap. 3.
- 13.
Triadic patterns are explained in Chap.4.
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Acknowledgment
I would like to thank Dr. Thomas Kieren and Mr. Ernst von Glasersfeld for their comments on the first four chapters.
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Steffe, L.P. (2010). A New Hypothesis Concerning Children’s Fractional Knowledge. In: Children’s Fractional Knowledge. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0591-8_1
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