Abstract
The reorganization hypothesis – children’s fraction schemes can emerge as accommodations in their numerical counting schemes – is untenable if counting is regarded only as activity. Focusing only on the activity of counting, however, does not begin to provide a full account. When using the phrase “the explicitly nested number sequence,” I am referring to the first part of a numerical counting scheme, which consists of a number sequence, which is a sequence of abstract unit items containing the records of counting acts.1 In this case, the activity of counting is interiorized activity and it is no exaggeration to say that it is contained in the first part of the counting scheme. This number sequence is an example of what Piaget (1964) meant when he commented that “there is a structure which integrates the stimulus but which at the same time sets off the response” when speaking of a stimulus from the point of view of the child. In this case, number words such as “seven” refer to a singular unit that can be iterated seven times to fill out a composite unit containing seven counted items. So, the first thing that had to be established to justify the reorganization hypothesis was to observe how ENS children might use their numerical concepts as templates for partitioning continuous unit items. Toward that end, I start with a discussion of two protocols extracted from two teaching episodes held on the 28th of April and the 1st of May with Jason and Patricia during their third grade in school. These episodes illustrate how the two children used their numerical concepts in constructing what I referred to as the equipartitioning scheme (cf. Chap.4).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
See Chap.3, especially the sections, “Numerical Patterns and the Initial Number Sequence,” “The Tacitly Nested Number Sequence,” and “The Explicitly Nested Number Sequence.”
- 2.
Because the children counted, the elements of their composite units were arithmetical units; and so it can be said that the children assimilated the situation using their number sequences.
- 3.
Marks enables a child to position the cursor on a stick and, by clicking the mouse, place a hash mark at that place.
- 4.
Cf. Chap. 4 for a discussion of the concept of length.
- 5.
A connected number sequence can be thought of as a sequence of abstract unit items that contain records of counted segments joined end-to-end. Although it might be thought of as a number line, the term “number line” is associated with structures of the real number line that go well beyond a connected number sequence.
- 6.
When a word is in caps, it refers to an action in the computer program.
- 7.
The notation “14-stick” refers to a 14 part stick.
- 8.
To find the product of five and three, if a child mentally inserts the unit of three into each unit of five to produce five threes prior to actual activity, the involved scheme is referred to as a units-coordinating scheme (Steffe 1991). Based on the ease with which the children selected the 3-stick and the 5-stick, they seem to have abstracted “three times five is fifteen” and “five times three is fifteen” in their work on multiplication in their regular mathematics classrooms. For this reason, I made the judgment that they used their units-coordinating schemes in assimilation.
- 9.
An assimilation is generalizing if, first, the scheme is used in situations that contain sensory material that is novel for the scheme, and if there is an adjustment in the use of the scheme [cf. Steffe and Wiegel (1996) and Steffe and Thompson (2000)].
- 10.
The notation, “n-stickm” is used to denote an m-part n-stick. In this case, an eight-part 24-stick.
- 11.
In the case of an operative, numerical image, the image would be a re-presentation of an interiorized stick concept, such as a transparent segment of thread, using her concept of the connected number, eight.
- 12.
Note that the children do not use MARKS to put a mark on the stick, but draw a new stick.
- 13.
Using Pull Parts, a child can activate that action button by clicking on it and then click on one or more parts of a stick. The child can then deactivate the action button and drag copies of the parts out of the stick while leaving the stick intact.
- 14.
Unfortunately, the teacher asked her how she could tell if the 7/8-stick is exactly one-eighth, and his question closed off any further actions she may have taken with the 7/8-stick.
- 15.
After activating Parts, any number through “99” may be selected as the number of parts to be made. Then positioning the cursor anywhere on a stick and clicking the mouse partitions the stick into that number of parts. If “14” is selected, for example, 13 equally spaced hash-marks appear on the stick.
- 16.
A sharing goal involves an intention to actually break the stick apart. Laura’s use of PARTS seemed to not involve such an intention.
- 17.
See Sáenz-Ludlow (1994) for an analysis of the constructive power of a child, Michael, whom I infer had constructed this operation.
- 18.
See also Tzur (1999) for a discussion of this scheme.
- 19.
A partitive fraction scheme extends the partitive unit fraction scheme in that it can be used to produce proper fractions.
- 20.
However, he still did have to engage in the activity of partitioning in order to produce a part of the whole that he could reason with. Hence partitioning is anticipatory but not taken as given.
- 21.
At this point, there was only one blue stick beneath the original stick in Fig. 5.6 and it was beneath the yellow part of the original stick with left endpoints coinciding.
- 22.
For a goal to be a social goal, the child must be able to infer, based on the language and actions of another child, that the other child does indeed have intentions. The case of social activity is quite similar to that of a social goal in that a child might assimilate the language and actions of the other child that constitutes a mathematical activity and then reenact the assimilated activity in constituting the activity as a personal activity.
- 23.
Recall that in one out of the six parts, the child conceptually disembeds one part from the six parts while leaving the part in the six parts. While in one of the six parts, the part is distinguished within the six parts without being disembedded from the six parts.
- 24.
Comparing the disembedded parts to the partitioned whole is not indicated in Protocol XVI.
- 25.
Recall that a partitive fraction scheme extends the partitive unit fraction scheme in that it can be used to produce proper fractions.
- 26.
Four being a multiplicative concept means that four is conceived of as four times one of its units.
- 27.
“Vertical learning” refers to the reorganization of schemes at a level that is judged to be higher than the preceding level. New ways of operating are introduced that are not present at the preceding level.
- 28.
See Steffe (1994a) for a model of the interiorization of acts of counting that producethe initial number sequence.
References
Piaget J (1964) Development and learning. In: Ripple RE, Rockcastle VN (eds) Piaget rediscovered: a report of a conference on cognitive studies and curriculum development. Cornell University Press, Ithaca, NY, pp 7–19
Piaget J (1980) Opening the debate. In: Piattelli-Palmarini M (ed) Language and learning: the debate between Jean Piaget and Noam Chomsky. Harvard University Press, Cambridge, pp 23–34
Piaget J, Inhelder B, Szeminska A (1960) The child’s conception of geometry. Basic Books, New York
Sáenz-Ludlow A (1994) Michael’s fraction schemes. J Res Math Educ 25:50–85
Steffe LP (1991) Operations that generate quantity. Learn Indiv Differ 3:61–82
Steffe LP (1994a) Children's construction of meanings for arithmetical words. In: Tirosh D (ed) Implicit and explicit knowledge: an educational approach. Ablex, New Jersey, pp 131–169
Steffe LP, Thompson PW (2000) Teaching experiment methodology: underlying principles and essential elements. In: Kelly AE, Lesh RA (eds) Research design in mathematics and science education. Erlbaum, Mahwah, NJ, pp 267–307
Steffe LP, Wiegel H (1996) On the nature of a model of mathematical learning. In: Steffe LP, Nesher P, Cobb P, Goldin GA, Greer B (eds) Theories of mathematical learning. Erlbaum, Mahwah, NJ, pp 477–498
Tzur R (1999) An integrated study of children’s construction of improper fractions and the teacher’s role in promoting that learning. J Res Math Educ 30:390–416
Acknowledgment
I would like to thank the editors of the Journal of Mathematical Behavior for granting permission to publish parts of an earlier version of this chapter in this book.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Steffe, L.P. (2010). The Partitive and the Part-Whole Schemes. In: Children’s Fractional Knowledge. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0591-8_5
Download citation
DOI: https://doi.org/10.1007/978-1-4419-0591-8_5
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-0590-1
Online ISBN: 978-1-4419-0591-8
eBook Packages: Humanities, Social Sciences and LawEducation (R0)