Math II (562417), страница 18
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• A process of abstraction leads from initial unrefined abstract entities to a novel structure.
• The novel structure comes into existence through reorganization of abstract identities and through establishment of new internal links within the initial entities and external links among them.
In view of our experience in classrooms and our need for an operational definition, we translated these theoretical principles into the following more applicable definition:
Abstraction is an activity of vertically reorganizing previously constructed mathematics into a new mathematical structure.
We will argue that this definition integrates all five of the epistemological principles, although only three of them appear explicitly in its wording. First, the term activity is to be taken in the sense of activity theory, implying that context needs to be fully taken into account.
Next, the term previously constructed mathematics refers to two points: first, that outcomes of previous processes of abstraction may be used during the present abstraction activity and, second, that the present activity starts from an initial unrefined form of abstraction as posited by Davydov (1972/1990) as well as by Ohlsson and Lehtinen (1997). These two points show the recursive nature of abstraction.
The term reorganizing into a new structure, which implies the establishment of mathematical connections, includes highly mathematical actions like (a) making a new hypothesis and (b) inventing or reinventing a mathematical generalization, a proof, or a new strategy for solving a problem. These actions require highly theoretical thought. Although theoretical thought is necessary, empirical thought cannot and should not be excluded. As will be seen in the next section, empirical thought, like observing similarities and differences, may make an essential contribution to abstraction.
We borrowed the term vertical from the Dutch culture of Realistic Mathematics Education, in which researchers speak about vertical mathematization as opposed to horizontal mathematization (de Lange, 1996; Treffers & Goffree, 1985). Horizontal mathematization refers to relations between nonmathematical situations and mathematical ideas. Vertical mathematization is "an activity in which mathematical elements are put together, structured, organized, developed, etc. into other elements, often in more abstract or formal form than the originals" (Hershkowitz, Parzysz, & van Dormolen, 1996, p. 177). Although these authors did not exclusively associate verticality with abstraction, they did include abstraction, and they emphasized the integrative role of vertical mathematization. It is mainly this integration, which comes about through the establishment of new connections during processes of abstraction, that we wanted to describe by means of the term vertical.[2]
Finally, we return to the important role of the seemingly unimportant word new in the definition. We intentionally used this word to express that as a result of abstraction, participants in the activity perceive something that was previously inaccessible to them. Although the newly perceived feature may consist only of connections between entities (which as isolated entities were previously available), it is exactly the focus on these connections that is often most important for abstraction (Dreyfus, 1991).
The study of abstraction raises an additional challenge. Whichever definition is used, abstraction implies mental activity, which is not observable. Because we want to empirically investigate processes of abstraction, we need to devise a way to make them observable. Put another way, we need to use (theoretical) spectacles that allow us to see processes of abstraction. As has been noted above, we consider processes of abstraction as they occur during students' activities. And it is precisely this view that provides us with the desired spectacles: Activities are composed of actions, and actions are frequently observable. We answer the question "Which actions are relevant for abstraction?" with reference to Pontecorvo and Girardet (1993): Epistemic actions are mental actions by means of which knowledge is used or constructed. Epistemic actions are often revealed in suitable settings. Therefore, settings with rich social interactions are good frameworks for observing epistemic actions. In the next section, we show that we can identify three particular epistemic actions that are constituent of abstraction and provide a strong indication that a process of abstraction is taking place. In conclusion, we consider epistemic actions because they are both characteristic for abstraction and observable. In other words, they provide us with an operational description of processes of abstraction.
EPISTEMIC ACTIONS OF ABSTRACTION
This section is devoted to the experimental identification of epistemic actions involved in processes of abstraction. To this end, we analyze an activity during which, we claim, processes of abstraction occurred. Our aim in the analysis is to identify and illustrate specific epistemic actions that are constituent of processes of abstraction. We will also show that these epistemic actions occur nested in a particular manner. The epistemic actions and the manner in which they are nested are experimentally accessible; thus they constitute an operational model of abstraction.
We characterized abstraction as a process that takes place in a complex context that incorporates tasks, tools, and other artifacts; the personal histories of the participants; and the social and physical settings. Such processes may take place during work with any group of students and teachers. For simplicity, we focus in this section on a single student working on a specifically designed task in an interview situation. Our reason for choosing a single student in an interview situation as a first paradigmatic case is that in this setup the epistemic actions are relatively easy to identify. Even for this relatively simple setup, we had to separate, for clarity of presentation, aspects of abstraction that are intimately linked and present them one by one; these aspects thus appear more isolated in our writing than they are in reality. We deal with only one aspect of context, the task on which the student worked. Questions concerning other aspects of context, such as different social settings, will be briefly considered in later sections and taken up in more detail in future articles.
Experimental Setting and Task
The student (BL) who participated in the study was a ninth grader; the experiment took place toward the end of the school year during which she had participated in the functions course described in the previous section. She was asked to work on a task that was similar in spirit to tasks in the course, although the experimental task was somewhat more structured. She had access to a computer with a function-graphing program while she worked. The interviewer's task was to ask BL questions with two complementary aims: (a) to cause BL to explain what she was doing and why and (b) to induce her to reflect on what she was doing and thus possibly progress beyond the point she would have reached without the interviewer. In other words, the interviewer had some didactic intentions, and for this reason we term the interview a teaching interview. BL seemed to be at ease during the teaching interview; she was willing to explain what she meant and clearly expressed her thoughts when she was self-confident. She was ready to make conjectures when she was less sure; in these cases, she usually mentioned that she was not sure and tended to turn to the computer, hoping to get confirmation. She was quite familiar with functional representations and their interpretations; she comfortably handled graphs, tables, and building tables from graphs; she was somewhat less eager to use formulas, except for the simplest ones. She was proficient in the use of the function grapher that was available, and she easily interpreted its products.
The interview task deals with the development over time of three populations in an animal park. The animal populations were given as functions of time. Two functions were linear, one decreasing and presented graphically (zebras) and the other increasing and presented verbally as a story (lions); the third function was quadratic with a maximum and was presented algebraically (eagles). The exact task presented to BL is shown in Figure 1.
The task is structured into four parts that were presented to BL one by one, on four separate worksheets. Each of the first three parts introduces the development of one animal population during the first 10 years of the park's existence. All three parts start with brief periods of familiarization with the newly introduced population in different representations. Questions relate to the size of the populations at various times as well as to different settings to describe the populations and their development. The first part of the teaching interview contains no further questions beyond the introductory ones. In Part II, the student is asked to compare[3] the number of zebras to the number of lions during the 10-year period. At this stage in the student's mathematical education, comparing the values of two functions, for example by first finding the intersection point (e.g., by "walking on the graph"), was standard. In Part III, however, the student is asked to compare the (varying) rate of growth of the eagle population with that of the lion population. The functions are chosen so that the rate of growth of the eagles' population starts at a higher value than the constant rate of growth of the lion population and then decreases to zero. Comparing the rate of change of a linear function with that of a nonlinear function was an unusual task for the student and required the use of new concepts, notably a notion of rate of change as a varying function. The means by which BL dealt with the comparison will be central to our argument. Part III ends with the question of whether there is a point at which all three populations are equal (there is not). Finally, in Part IV, the student is asked to change the development of one of the populations so that such a point exists. This was a rather challenging task for the student and far more open than the previous tasks. Its completion requires problem-solving behavior rather than only activation of learned or new concepts.
BL was videotaped during the approximately 40 minutes she worked on the task. The entire teaching interview was transcribed and translated from Hebrew into English. We believe that the translation did not appreciably change the ideas and the spirit of the discourse, possibly with one exception.[3]
After about 20 minutes, during which BL had had the opportunity to familiarize herself with the three populations in several representations and to (mostly correctly) answer some comparison questions about them, the following dialogue took place. The 7-minute excerpt is taken from Part III of the teaching interview and was the main focus of our analysis. The excerpt is presented in its entirety to allow the reader to understand the continuity of events. B refers to the student, I refers to the interviewer, and numbers in braces refer to the time elapsed since the beginning of the teaching interview.
I177: Okay, let me ask you the following: The rate of growth of the eagle population ... {21:00}
B178: Yes?
I179: ... is it bigger than that of the lions or smaller than that of the lions? The lions, I remind you, you can look here [points to lions graph].
B180: Yes.
I181: You don't need reminders!
B182: The growth is bigger.
I183: The rate.
B184: Bigger and then it decreases.
I185: Then what?
B186: It gets to a point where it decreases. Not decreases but less, ...
I187: Let's, ...; there is a lot; the question is very subtle. Let's try, because you said a lot of things, and I want to understand them precisely. If I understood you correctly, you said that in the beginning the rate of growth ...? Once again!
B188: It is not equal; it is not, ...
I189: It is not equal, but which one is bigger?
B190: It changes all the time.
I191: [Nods.] The rate of growth of the eagles changes all the time. {22:00}
B192: [Nods.]
I193: Now, the question was to compare it [rate of growth of eagles' population] to the rate of growth of the lions' [population].
B194: In the beginning, it seems to me that it is bigger.
I195: Okay.
B196: And you see the point [points to the screen] when it meets.
I197: [Nods.]
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