Math II (562417), страница 21
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In other words, she constructed a sequence of values of the rates of change between successive years. We identify here a precursor of the construction of quite a new structure, the notion of the rate of change as a function taking on different values at different points in time. At this stage the synthesis phase of abstraction occurred, and a novel structure was constructed. This constructing action was not sudden but stemmed from the recognition and restructuring of an already existing structure, the table of values of two changing quantities. At the same time, it was driven by BL's awareness of the motive for the activity, namely comparing the rates of growth.
THE NESTED MODEL OF ABSTRACTION IN CONTEXT
Our analysis of BL's work exhibits a process of abstraction. In this process, structures constructed earlier in the student's learning history are recognized and reorganized into a new structure to fulfill the demands of the activity. The actions undertaken by the student include the three epistemic actions recognizing, building-with, and constructing, not as a chain but in a nested way. In other words, the action of constructing does not merely follow recognition and building-with in a linear fashion but simultaneously requires recognition of and building-with already constructed structures. We call this mechanism dynamic nesting of the epistemic actions.
These relationships among the epistemic actions naturally give rise to a model of abstraction, in which one can identify general mechanisms. The constructing of a novel structure stemming from and based on recognizing and building-with is a first effort in this direction. We take the occurrence of the three epistemic actions, nested during the construction of a new structure in the manner above (or in more complex ways), as a clear indication that a process of abstraction is occurring and is constituted by these dynamically nested epistemic actions.
We assume that BL's construction, as is true of any new construction, was rather fragile at first. We surmise that when recognized in a further activity, such structures will progressively become more consolidated. Because of this consolidation, the student will be able to recognize the structure more easily, just as BL recognized and built-with earlier structures in the present activity. The consolidation of the newly constructed structure will allow the student to recognize this structure in further activities and to build-with it with increasing ease. Hence we hypothesize that tracing the genesis of an abstraction passes through three stages: (a) a need for a new structure, (b) the constructing of a new abstract entity in which recognizing and building-with already existing structures are nested dialectically, and (c) the consolidation of the abstract entity, facilitating one's recognizing it with increased ease and building-with it in further activities.
The general mechanism described here outlines the functioning of our model of abstraction. The core of the model relates mainly to the second stage of the model, namely the constructing of a new abstract entity, and to a lesser extent, to the first one, the need for a new structure. We consider this core of the model to be established and supported by data from BL's teaching interview. Extending the model to include consolidation requires further elaboration and the analysis of more data.
To limit the complexity of this article, we have emphasized the cognitive components of the model more than the contextual ones. But the model is inherently contextual. In the remainder of this article, we briefly treat the multiple facets of context in the model. Although this treatment is predominantly theoretical, we do point out appropriate supportive data from BL's teaching interview.
The structure of the dynamically nested model itself has a contextual nature. The fact that building-with and recognizing are nested in constructing shows that construction is grounded in and concurrent with other epistemic actions. In other words, when a new structure is constructed, it already exists in a rudimentary form, and it develops through other structures that the learner has already constructed. This description of the development of abstraction echoes Davydov's theory (1972/1990), according to which the abstraction grows from an unarticulated form through a dialectical process.
In this description of the growth of abstraction, we emphasize the historical and subjective character of epistemic actions. For example, using rate of change as a function is a constructing action for BL but may be a building-with action for another learner who had constructed this concept earlier. Constructions become artifacts that may be used in further actions of recognizing and building-with; in these further epistemic actions, the use of such artifacts is part of the essence of the epistemic action. Epistemic actions may thus be nested over several activities, and the students contribute to the construction of the context in which further activities will take place. For example, in B290-B302, BL generated a modified development for the lion population by using several artifacts she had presumably constructed earlier; these artifacts include the strategy of passing to a new representation to add information and her knowledge that the values of two functions are equal when their graphs intersect.
The learner's history, embodied in the artifacts on which the learner capitalizes, forms the basis for the genesis of abstraction. However, abstraction will not occur without the need for a new structure. This need may stem from an intrinsic motivation to overcome obstacles such as contradictions, surprises, or uncertainty. Educators may purposely set such obstacles by designing appropriate series of activities. Similarly, common practices and sociomathematical norms that are accepted in the classroom are important in this connection (Yackel & Cobb, 1996). We have general information on sociomathematical norms established in BL's class (Hershkowitz & Schwarz, 1999). For example, intuitive reasons or isolated data alone did not count as acceptable evidence. Also, students' actions were driven by the students' eagerness to construct meaning. This fact is reflected in BL's urge to justify her answer to the question of whether the rate of growth of the eagle population is bigger than that of the lion population. The central step of BL's effort to justify her answer (B264, in response to I263) established conclusively that her initial intuition was correct because "the rate of change [of the eagles' population] is bigger until some point between 4 and 5" and then dips below that of the lions' population. It is this important step that led her to construct rate of change as a function. BL's need for justification was thus crucial for the process of abstraction.
A further component of context in BL's teaching interview was social interaction. Although BL constructed a mathematical structure that was new to her, she was not the only actor. For example, epistemic actions sometimes stemmed from the interviewer's probing questions (in I177-I179, repeated in I233, and yet again in I263). In fact, the interviewer played an essential role in mediating most of the epistemic actions. We point out just a few examples: When BL started to describe the rate of growth of the eagle population and raised a few ideas, the interviewer told her (I187), "You said a lot of things, and I want to understand them precisely." This utterance makes clear the interviewer's expectations concerning some norms for BL's statements: They had to be clear and well explained. Similarly, in I253 he coached her to interpret the changing numerical differences in terms of the rate of change of the population. Also, he expressed agreement or readiness to help when BL adopted a desirable track, for example, by providing her with a ready-made table of values (I235). The role of the interviewer in BL's construction of rate of growth as a function was thus not confined to uncovering BL's mental states. Constructing is mediated by human interaction and by a material tool.
More generally, the epistemic actions involved in processes of abstraction may be distributed among the participants. A teacher may bring to attention a fact or a method, leading to recognition by one participant, followed by building-with by another participant and by the collective construction of a new structure by other participants. In conformity with Vygotsky's (1934/1986) theory of human development, our hypothesis is that the individual who is participating in an activity of abstraction in which epistemic actions are mediated and distributed gradually interiorizes social interactions as well as material manipulations.
CONCLUDING REMARKS
In this article, we presented the core of a model for the genesis of abstraction. We showed how the principal components of the model, the three epistemic actions of constructing, recognizing, and building-with, emerged from the analysis of a teaching interview. We also suggested how they are dynamically nested. The epistemic actions may use artifacts that are outcomes of earlier activities, and constructing leads to artifacts available in later epistemic actions. We then showed that the nested structure of the model confers upon this model an inherent contextual nature. In addition, in the teaching interview, many of the epistemic actions were mediated by the interviewer and some, by tools. In activities in which more participants are involved, the model sustains the social distribution of abstraction: Different participants can undertake different epistemic actions.
The extension of the model in this section was based on interpretations of our empirical data, generalizations drawn from them, and hypotheses concerning the genesis of abstraction. These generalizations and hypotheses are theoretically grounded. However, more research is needed to validate and refine the full model. For example, the case of distributed abstraction among interacting peers, some being more active than others, raises theoretical and methodological issues about abstraction as a collective or an individual process (Hershkowitz, 1999). Another important issue concerns the mediation of (computer) tools in abstraction. Finally, to provide an adequate experimental basis for the nesting of epistemic actions over several activities, sequences of several activities need to be investigated. The span of activities is to be oriented not only backward but also forward in time: Apprehending a construction means observing not only from what it stemmed and which artifacts are being used but also how the newly created structures are used as artifacts in further activities. We hypothesize that the traces of a construction in later activities are intimately connected to the consolidation or the absence of consolidation following a constructing action.
The empirical study we presented in this article and other studies we currently conduct lead us to anticipate that the model will facilitate further research and that the research will guide the development of the model into a tool suitable to describe, in a comprehensive manner, processes of mathematical abstraction.
Animal Park
Ten years after its opening, the board of the Belangoo animal park ordered a survey of the development of various animal populations in order to be better able to plan for the future.
I. The variation of the park's zebra population is described in the following graph:
a) How many zebras were there in the park when it first opened?
b) How many zebras were there in the park after three years?
c) Could you describe the variation of the zebra population over the years in a different manner?
d) Do you know still other ways to describe the variation?
II. When the park opened, there were no lions. In the course of the first year, 60 lions were brought in, and then the lion population continued to grow at the constant rate of 60 lions per year.
a) How many lions were there in the park after three and a half years?
b) Compare the number of lions to the number of zebras during the first ten years.
c) What can the planners say about the two populations in the future?
III. The eagle population in the park varied according to the expression f(x) = 5x(20 - x) (x denotes the time, in years).
a) Do you think the living conditions for the eagles in the park are good?
b) Is the rate of growth of the eagle population larger or smaller than that of the lion population?
c) Is your conclusion valid for the entire first ten-year period?
d) Compare the number of eagles to the number of zebras during the first ten years.
e) What can the planners say about the two populations in the future?
f) Is there any time at which the three populations (zebras, lions, and eagles) were equal?
IV. A park is being planned that will be similar in all aspects to the existing one, except that the planners want the three populations to be exactly equal at some point in time.
a) The first planner proposed to change the living conditions of the lions so that the rate of growth of this population will change. What exactly did he propose?
b) The second planner proposed to achieve the aim by means of a change in the number of zebras present at the time the park is opened. What exactly did he propose?
We thank Ruhama Even, Anna Sfard, and several anonymous reviewers for their thoughtful comments on an earlier version of this article. They helped us to theoretically situate our research and improve its presentation.
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