Abstract
The fourth-grade teaching experiment with Joe and Patricia constituted a “replication” of the teaching experiment with Jason and Laura while they were in their fourth and fifth grades. We use scare quotes to indicate that the intent is not to repeat the experiment with Jason and Laura under the exact same conditions. Rather, the intent is to generate observations that can be used not only to corroborate the previous observations, but to refine, extend, and modify them as well. In a teaching experiment, the teacher does not establish a hypothetical learning trajectory at the outset of the experiment for the entire experiment. Rather, the teacher/researchers hypothesize what the children might learn in the next, or even in the next few, teaching episodes based on their current interactions with the children and their interpretations of it, and it is the testing of these hypotheses in the teaching episodes that, in part, constitutes the experimental aspect of the teaching experiment. Both the possibilities that are opened by the particular children and the constraints that the researchers’ experience that emanate from within the children provide new observations that can be retrospectively analyzed to generate a “replicate” case study.
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Notes
- 1.
The notation, “n-stick m ” introduced in Chap. 5, is used to denote an m-part n-stick. In this case, an eight-part 24-stick.
- 2.
A composite unit fraction scheme entails transforming three parts out of twelve parts, for example, into one composite part out of four composite parts.
- 3.
Proper fractions are commonly thought of as fractions that do not exceed the fractional whole.
- 4.
“Twice as long” means “two more” for children who are yet to construct the operation of iteration. Including the stick being iterated in the iterations is indication that each stick produced in the iterations is an instantiation of the abstracted unit used in the iterations. There is only one iterable unit and the sticks produced in the iterations can be regarded as identical one to the other.
- 5.
To judge whether a unit fraction is an iterable unit requires the observation that the child uses it to produce an improper fraction. In the case of the partitive fraction scheme, a unit fraction inherits its iterability from the iterable unit of one.
- 6.
Patricia made the 18-stick using two 9-sticks, so for Joe, each of its parts was of length equal to each part of the 9-stick.
- 7.
Note that when pulling one part out of the 99-part stick, the computer creates a beginning and an ending tick mark next to each other so the thickness appeared to be a lot more than one-ninety-ninth of the small stick.
- 8.
For a scheme to be a reversible scheme means that any result of the scheme can be taken as a situation of the scheme and that the activity of the scheme can be reversed to produce a result of the scheme, which is a possible situation.
- 9.
For these operations to be embedded in Joe’s reversible partitive fraction scheme, they must be operations used in assimilation.
- 10.
One possible explanation for the stark differences in Joe’s willingness to conceive of a fraction greater than the whole could be classroom experiences in the intervening months of which we have no knowledge. Traditional approaches to teaching fractions in the elementary grades emphasize that fractions are always parts of a whole.
- 11.
Nevertheless, it was still one out of the ninety-nine equal parts.
- 12.
Split one pizza into eight equal parts and iterate one part eighteen times.
- 13.
In Protocol XVIII, Patricia demonstrates that she has constructed an iterative fraction scheme.
- 14.
A procedure is a scheme in which the activity is only connected to rather than contained in the first part of the scheme. The first part of some procedures can contain operations that can be judged as mathematical concepts. In Patricia’s case, however, the first part of her procedure was constituted by the words “you reduce.” Her activity of dividing can be regarded as her meaning for the words.
- 15.
The hypothesis behind this statement is that one can be aware of the material on which one operates as well as of the operations that produced that material, but one is not aware of the highest level of operation until one can take the products of that operation as input for further operation.
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Olive, J., Steffe, L.P. (2010). The Partitive, the Iterative, and the Unit Composition Schemes. In: Children’s Fractional Knowledge. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0591-8_7
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