Math II (562417), страница 20
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B128: [Nods.] So it is possible to put them [populations] on the computer and see.
I129: Yes.
B130: [Enters functions; obtains graph with both functions; see Figure 2b.]
I131: [Nods.]
B132: So, the lion population ... What was it [the population] before, zebras?
I133: [Nods.]
B134: So the zebras all the time became less, and the lions all the time grew.
I135: [Nods.]
B136: Should I tell you what else I see?
I137: Yes, maybe you can tell me something else about the comparison.
B138: [Points to the intersection point of the graphs.] In the fifth year, the number of lions and zebras was equal.
I139: [Nods.] How do you know that this was in the fifth year?
B140: Because, according to the ... [points to the number 5 on the horizontal axis and moves up to the point of intersection]. {17:00}
I141: According to the ... point, yes? [Laugh]
B142: The point is (5,300). There were 300 zebras and 300 lions.
Similar sequences occurred when BL was asked to compare the eagle population to the zebra population and when she was asked whether at some point all three populations were equal {III. f}.
The cases presented in this section are quite different, in terms of the nature of BL's knowledge, from the case of rate of change as a function presented in the previous section. She did not, here, need to construct new knowledge because she recognized structures that she had presumably used previously in other situations and was able to adapt them, at a structural level, to the present situation and make use of them as needed. This is the second epistemic action we associate with abstraction: recognizing.
Recognition of a familiar mathematical structure occurs when a student realizes that the structure is inherent in a given mathematical situation. Just as we used the word cognizing for describing what occurs when constructing, we propose to use the word recognizing here to make the point that this is not the first time the corresponding structure "enters the mind" of the student.
During the process, the recognizing subject may have been the single or most active participant, but conceivably she or he may simply have assisted and observed a process in which others were the main actors. This description may fit a rather passive student in a group working in a problem-solving-oriented investigative classroom and most students in classrooms in which the teacher tends to "chalk and talk." Even externally passive students may be able to recognize and meaningfully use some of what the teacher demonstrated.
Recognizing is often, though not always, at the level of empirical thought. For example, in the excerpt presented above (B134-B142), BL mainly observed, reported her observations, and classified them into categories she had formed at some earlier stage. It will become apparent in a later section that abstraction makes use of recognizing, and thus of empirical thought, but that no abstraction can take place without constructing, which requires theoretical thought.
We also emphasize the subjectivity of the recognizing process. Others (e.g., Chi, Feltovich, & Glaser, 1981; Lowe, 1993) have shown that when experts see deep structure in a problem situation or a diagram, novices often notice only surface structure. Whereas for the experts, this process is a matter of recognizing, for a suitably prepared novice, it might be an opportunity for engaging in a process of constructing a deep structure.
Building-With
In the last part of the teaching interview, somewhat more elaborate questions were presented to BL. At that time, she had already obtained all three population graphs on a single screen (see Figure 2c). She was asked to change parameters in the function describing the development of one of the populations so as to generate a point in time when all three populations would be of equal size. For example {IV. a}
I287: Now let's assume that our same planners plan a similar park, a new one, but they want to plan it so that there will be a time when the three populations are equal; and they make proposals, and I want you to help them to realize these proposals. {31:00} The first planner proposes the following: He says he wants to change the living conditions of the lions so that their rate of growth changes and that a time will occur when all three populations are equal.
B288: [Nods and looks at the screen.]
I289: How can he do this? Can you help him do this?
B290: I'll try. There is a time here [points to screen] where the two populations of the zebras and the eagles meet, which is (4, 320).
I291: Excuse me, to keep order ... [hands her the worksheet with Part IV of Figure 1].
B292: So there is (4, 320) [writes (4, 320)], so we have to find a point ... [turns to the computer].
I293: [The computer] falls asleep, sometimes. {32:00}
B294: So we have to find an appropriate point, for the lions [takes the lions' table; see I235 in the protocol]. So one can tell, just a second [thinks, turns to computer]. I make them grow by 80 each year then ... [adds the graph of y = 80x].
I295: How did you so quickly enter 80x here? I am somewhat amazed from where this came. Where did it come from?
B296: Because I wanted a point; really, I just tried and it came out; no, but really I thought here.
I297: You tried by chance?
B298: No, there was a basis. {33:00}
I299: What's the basis?
B300: I computed at which point it is because they all the time increase by the same number and at the fourth point it needs to be 320; so 320 divided by 4.
I301: That's how you found the 80?
B302: [Nods.]
Several aspects of this excerpt are of interest. First, we noticed BL's excellent problem-solving behavior. In B290 she immediately focused on the point of intersection of the zebra and eagle populations. She clearly realized that the intersection point (4, 320) is the one through which all three population graphs would have to pass because the lion population is the one whose behavior would be modified. She was thus recognizing the logical deep structure of the problem that was posed to her; she was reorganizing the available information so she could effectively deal with the particular question at hand.
But BL did much more during this short episode. She invoked and combined many structural elements related in a dialectical manner to the question: the logical structure of the problem; the knowledge that equal populations appeared as intersection points in the case of two as well as in the case of three populations; the relation, at least for the lion population, between the linear graph through the origin and the corresponding formula y = mx; the relationship between the slope of the graph and the value of the coefficient m; a (presumably dynamic) view of the family y = mx and the corresponding graphs; and the relationship between the point (4, 320) and the value 80 for the slope.
During this episode much more was involved in BL's actions than her recognizing structural elements; nevertheless, BL was not constructing a new, more elaborate structure out of the given elements; rather, she was using the given elements in different and appropriate combinations to answer the question she was asked, to progress toward one of the goals of the activity in which she was engaged. Combining structural elements to achieve a given goal is the third epistemic action of abstraction, and we call it building-with. When building-with, the student is not enriched with new, more complex structural knowledge; however, she or he uses available structural knowledge to build with it a viable solution to the problem at hand.
Building-with is most likely to occur when students are engaged in achieving a goal such as solving a problem, understanding and explaining a situation, or reflecting on a process. For these purposes, students may appeal to strategies, rules, or theorems. For example, students passing to a new representation to find the solution of a problem on functions are building-with the problem-solving strategy of passing to a new representation. Building-with has a connotation of applying: To achieve the goal, students use structures that they recognize from earlier activity as artifacts for further action. As mentioned above, the recognized structure may be the outcome of other participants' activity. The artifacts used in building-with are tools to adapt to a new situation, to a new instantiation, to a modification of an existing method, or to greater complexity.
Building-with may take place when the teacher reminds students of a resource and the students take up the idea. For example, the teacher may hint to the student to notice how the graph looks. Students may also engage in building-with when they are hypothesizing. In that case, students may appeal to numerical data or to other resources, and the idea or hint stems from these resources. If the idea is some evidence for a new mathematical structure, the building-with is nested in a more global constructing activity. These interrelationships among the three epistemic actions are discussed in the next section.
An important difference between constructing a new structure and building-with is that in constructing, one's goals for the activity are the process of constructing and the creation of the structure to be constructed, and to reach the goal (solving a problem, justifying a solution, or making an hypothesis), students must use a new mathematical structure. In building-with structures, one can attain one's goal by combining existing structures. The goals students have (or are given) and their personal histories thus strongly influence whether they are building-with or constructing. If they solve a standard problem, they are likely to alternate between recognizing and building-with previously acquired structures. If they solve a nonstandard problem, they might be constructing: finding a new (to them) phenomenon and reflecting on it, on its internal structure, and on its external relationship to things they know already. Constructing is thus not at all independent of recognizing and building-with.
The Nested Relationships Among the Three Epistemic Actions
Up to this point, the three epistemic actions have been described separately. In this section, we discuss relationships among them. We claim that constructing often includes actions of building-with and of recognizing. In other words, constructing is a combination of the three epistemic actions whereas recognizing actions are nested in the other two, and building-with actions are nested in constructing actions. To show these relationships, we scrutinize a chain of actions undertaken with a single overall goal. In other words, we analyze a whole activity. Specifically, we focus on the episode we used to exemplify constructing.
At the very beginning of this episode {III. b}, the interviewer spelled out the main goal for the activity by asking, "The rate of growth of the eagle population ..., is it bigger than that of the lions or smaller than that of the lions?" (I177-I179). The idea units[4] BL expressed initially regarding the rate of growth of the eagle population were (a) "The growth is bigger" (B182); (b) "Bigger and then decreases" (B184); (c) "It gets to a point where it decreases. Not decreases but less ..." (B186); (d) "It is not equal" (B188); and (e) "It changes all the time" (B190). These utterances seemed to be immediate, intuitive answers. They were evidence of BL's recognizing already existing structures that rely on outcomes of her previous learning and serve as artifacts for constructing the growth of the eagle population. At the same time, they constitute her initial, immediate but undeveloped, fuzzy abstract image of the rate of change of the eagle population--Davydov's (1972/1990) initial form of abstraction.
In I193 the interviewer reminded BL of the main goal of this activity, namely to answer the question "Which rate of growth is bigger?" At this stage, she still answered the question intuitively: "In the beginning it seems to me that it [rate of eagles' population growth] is bigger [than that of the lions]" (B 194). But soon she began a process of reorganizing the already existing structures to achieve the main goal. This process started with a conscious, analytic focus on the growth rate of the eagle population and terminated with the synthesis by which the notion of varying rate of change emerges.
There were four existing structures she successively recognized and analyzed to approach her goal: (a) the steepness of the graph, (b) an interpretation of the intersection point of two graphs (B 196, B204, B206, and B208), (c) the idea that different representatives stand for a function (Schwarz & Dreyfus, 1995), and (d) tables of the varying quantities of the two animal populations (as given in the protocol, I235). BL borrowed the idea unit of the steepness of the graph from her knowledge of increasing linear functions: "That the graph is closer to the y-axis" (B200)--the closer the graph is to the y-axis, the steeper it is and the bigger is its rate of change.
As we have mentioned, BL confused the comparison of the populations' rates of growth (structures she presumably had never encountered before) with a comparison of the populations' quantities (structures with which she was familiar). In other words, she recognized the intersection point of the graphs, a structure of comparison between quantities (rather than rates) and built-with it, even though it was the inappropriate structure. This building-with did not help her progress toward her goal. She was probably aware that she was not "on the right track." She hesitated in B210, continued with "maybe" in B212, and then explicitly mentioned, still in B212, quantities rather than rates.
Part of the structured knowledge BL had from her history of learning about functions includes the fact that a function has many representatives and that the operation of changing scales may be used to produce other representatives. By choosing different scales, she changed the representative of the function (B214-B218) to see more of its graph. This is another case of recognizing a structure and building-with it (in this case a new graph). When the interviewer asked her, "For what purpose did you do this?" (I221), she replied, "In order to see what happens later" (B222), and, in more detail, "They ontinue and decrease, their quantity" (B224). The fact that she built a wider picture by changing the representatives is evidence that she saw, at least implicitly, the connection between the change of quantities and the change of their rates of growth. It is also a sign of the dialectic nature of BL's thinking during the abstraction activity.
Up to this point, we could see a process of reorganization of artifacts or already existing structures through the action of recognizing and then building-with these structures some additional structure, namely the change of the populations' quantities. But this process was insufficient to answer the main question concerning the comparison between the rates of growth of the two populations. BL was aware of this insufficiency (perhaps with the help of the interviewer in I233) and tried to overcome it by again using the structures she knew and moving to a different functional setting: a table of values of the changing quantities of the two animal populations. This switch of setting finally helped her to add the additional structure of the sequences of the differences of the populations' quantities along successive years.
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