Math II (562417), страница 19
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B198: And then, it seems to me that it becomes smaller.
I199: Now, you say that in the beginning, the rate of growth of the lions is bigger--of the eagles is bigger. How do you see this? From what do you conclude this?
B200: That the graph is closer to the y-axis, the graph of the eagles.
I201: And then, at a later stage, the rate of growth of the eagles becomes smaller than that of the lions, yes?
B202: [Nods.] I203: Where does this happen?
B204: One can move on the graph ...
I205: [Nods.]
B206: ... and see at which point. Should I? {23:00}
I207: Yes.
B208: [Enters at keyboard and then moves on graph.] Here it is, this, (8, 480).
I209: Now, what happens at this point?
B210: [Hesitates.]
I211: What's its interpretation?
B212: That maybe the quantity of the ... decreases.
I213: The quantity of the eagles or of the lions? About the eagles you are speaking?
B214: Yes. Just a second; I want to enlarge the function. [Works at the keyboard.]
I215: You're enlarging? What do you make larger?
B216: The function.
I217: You're making the function larger? You're making the domain of the function larger, in fact. {24:00}
B218: [Terminates working at the keyboard and obtains the graphs of the same functions in a larger domain.]
I219: Again, now explain it to me. You are doing lots of things; you're thinking a lot, and I am trying to follow your thoughts, but I don't always succeed. You made here the scales larger, so that we can see more, right?
B220: [Nods.]
I221: For what purpose did you do this?
B222: In order to see what happens later and ... [thinks] ... see what happens later.
I223: Okay, what happens later?
B224: They continue and decrease, their quantity.
I225: The eagles?
B226: Yes.
I227: The quantity of eagles, ...
B228: Decreases.
I229: Right. Now, I want to return to the same point. What happens at this point [(8,480)]?
B230: There is the same quantity of eagles and lions, right?
I231: [Nods.]
B232: It's ... it's the lions, right? {25:00}
I233: And now, I go another step backwards, to the question I asked you. I asked you the question concerning the rate of change of the lions as compared to the rate of change of the eagles. I asked where the rate of change is bigger, whether the one of the eagles or the one of the lions, and you answered very nicely that the one of the eagles is not constant.
B234: Yes. I think one has to make a table in order to see the rate of change.
I235: [Nods.] Now, I prepared a table for you so you don't need to work so hard. [Gives her the tables for the eagles and lions.]
Time 0 1 2 3 4 5 6 7 8 9 10
No. of eagles 0 95 180 255 320 375 420 455 480 495 500
Time 0 1 2 3 4 5 6 7 8 9 10
No. of lions 0 60 120 180 240 300 360 420 480 540 600
B236: So, it changes. If there was a calculator here ... [turns to the computer].
I237: Maybe tell us what you plan to compute.
B238: Faster than I can compute?
I239: Yes, no. What do you want?
B240: The rate of change.
I241: How do you compute rate of change?
B242: This minus that [points to two consecutive numbers in the table]. Say, in each year, by how much it increases.
I243: Okay, I am willing to serve as your calculator.
B244: This is 95 [writes 95 between the initial 0 and the 95 of the first year]. Here [between the first and the second year] it is 180 minus 95.
I245: Eighty-five.
B246: [Writes at the appropriate place and points to the next one.]
I247: Seventy-five.
B248: [Writes.]
I249: Sixty-five, 55, 45.
B250: [Writes.]
I251: Now you understand how it continues, right?
B252: It decreases by 10.
I253: What's this say, this 65 here [points to the number 65 written in B250]?
B254: What? Here?
I255: Yes.
B256: That the rate of change decreases all the time.
I257: What's the interpretation of the number 65? Do you say that this is the rate of change?
B258: That each year fewer eagles joined.
I259: Okay, fine. What about the lions?
B260: The lions? They increased. There is ... [looks at the table]. They increase all the time.
I261: [Nods.]
B262: That's it.
I263: Once more, let me ask the question. I want to compare. I want to know whether the rate of change of the eagles is bigger, or that of the lions is bigger.
B264: [Thinks.] It's up to a certain point. [Points to the tables.] Here [in the lions table] all the time it [the rate of change] is 60, and here [in the eagles table] the rate of change is bigger until some point between 4 and 5. One can see this [points to the computer].
I265: How can one see this? {28:00}
B266: It's not easy to see.
I267: It's not easy to see? Why is it not easy to see?
B268: Because, it is here [points to the eagles graph], the rate of change, but it curves all the time.
In the following subsections, we analyze several excerpts from BL's teaching interview in more detail. Each excerpt has been chosen to illustrate one particular epistemic action of abstraction.
Constructing
We start from the beginning of this episode. BL was asked whether the rate of growth of the eagle population was bigger than that of the lions or smaller than that of the lions {III.b} (from here on, numbers in braces refer to the interview tasks as listed in Figure 1). At that moment, from the story she was told, she had already generated the formula y = 60x for the lion population, conjectured from the given expression f(x) = 5x(20 - x) when the eagle population increases and when it decreases, typed both expressions into the computer, and obtained their graphs on the screen (Figure 2a). When asked to compare the rates, BL first answered intuitively that "the growth [of the eagles] is bigger, ... bigger and then it decreases ...; it gets to a point when it decreases, not decreases but less ..." (B182-B186). The interviewer's probing about what she meant induced three processes that occurred in parallel.
• BL explained why she thought the rate of growth of the eagles' population was bigger than that of the lions: "The graph is closer to the y-axis, the graph of the eagles" (B200).
• BL turned to the computer for details and reassurance. For example, when asked where the rate of growth of the eagles' population becomes smaller than that of the lions, she said, "One can move on the graph ... and see at which point. Should I? Here it is, this, (8, 480)" (B204-B208; see also Figure 2a). She turned to the computer again later when she took the initiative to zoom out and investigate the long-term behavior of the populations (B214-B228); she was much more familiar with population behavior than with rates of change.
• BL's confusion between rate-of-growth of the population and size of the population, first apparent in the intuitive answer (B182-B186), became explicit when, for example, BL identified the intersection point (8,480) and interpreted that at this point "maybe the quantity of the eagles decreases" (B212). We note that the qualifying maybe expresses her uncertainty.
In these processes BL did not progress toward the goal of comparing rates of growth. In her favorite setting (graphs on the computer) she found identifying rates of change difficult--at times she confused rate with quantity--and she found comparing rates of change impossible. The structures she needed to progress were not available to her.
The interviewer decided to find out whether simply repeating questions would help BL. He started with the elementary (I229) "What happens at this point [(4, 480)]?" which BL immediately answered correctly. From then on, she did not confuse rate and quantity. The interviewer followed up by repeating the central question (I233) "I asked where the rate of change is bigger, whether the one of the eagles or the one of the lions." This repetition led to the important sequence B234 to B268. Although the interviewer did not refer to settings at all, BL proposed, in B234, to use tables of values. Using a table, she could locally compute a rate of change between two successive data points (B242); in the sequel, she used this knowledge to construct, step by step, the more complex notion of rate of change as a function taking on different values at different points in time and being amenable to comparison with the (constant) rate of change of another population. During this process, BL used structural elements at her disposal to build the new, more complex structure of a sequence of changing rates, that is, of rate as a function, the value of which can vary. She built the more complex structure from simpler structures. She clearly needed and used the number sequence 95, 85, 75, ... and her understanding of its interpretation to build up the more complex structure. Once she had built the structure, she was easily and clearly able to make the requested comparison (B264). She was even able to switch back to the graphical setting, in which, though, she still had difficulty pinpointing exactly the criterion of bigger, equal, or smaller rate of change (as she expressed in B268, which concluded this episode).
The above sequence shows clearly how BL reorganized her knowledge in response to the need to deal with varying rates of change. The reorganization was a vertical piecing together of elements that helped BL refine her notion of rate of change (rates of change can vary, and the varying values can be read from a functional table). The process included an integration: BL dealt with the many values of the rate of change as different values of a single quantity (the rate function). As a result, her conception of rate of change became deeper and more structured. Although our data are insufficient to show that this structure is novel to BL, the rather detailed information we have about the day-to-day activities of BL's class show that, at least within her mathematics classroom, she had no prior opportunity to deal with varying rates of change. We also know that, at this stage, the new structure was rather fragile for BL: She found coordinating the varying rate of change with a transition to the graphical representation difficult (B268), and we have no indication whether she would be able to apply her new knowledge to a different situation. As we will show, further stages of the process of abstraction are needed to consolidate such newly constructed knowledge.
We presented the process during which BL constructed a functional conception of rate of change as paradigmatic. Constructing in this sense is the first and most important of three epistemic actions that together constitute our proposed notion of abstraction. More generally, people may be constructing new methods, strategies, or concepts. Novelty implies construction. When a novel structure "enters the mind," it has to be cognized, or pieced together from components, usually simpler structures. According to the notion of abstraction proposed by Ohlsson and Lehtinen (1997), this constructive process that requires theoretical thought and implies vertical reorganization of knowledge is the central step of abstraction. From an activity-theory point of view, the participants who cognize a mathematical notion in this sense are assembling artifacts to produce a new structure.
We note that BL not only had reorganized her knowledge but also had become able to verbally express her reorganized knowledge. She had developed a language in which to compare rates and to explain her initial statement that the rate of growth of the eagles' population at first was greater than that of the lions' population and later was less. The reorganized structure could be and was used for explanation. Generally if the construction is an abstraction, learners develop in parallel a language for expressing their new knowledge and using it to explain or justify.
Observing the construction of structures presents a methodological problem because construction is a relatively rare event. Designing an experiment aimed at observing such events is also difficult. In fact, these events might often occur when students sit alone and think hard about mathematics. When the process is slow and incremental, the methodological problems are compounded. We consider ourselves fortunate to have encountered the above segment in an experiment that was designed to observe the use of abstractions before we knew exactly for which epistemic actions we were looking.
Recognizing
Identifying cases in which a student makes use of a construct or structure that has been constructed earlier is easier than identifying cases of construction. BL used a preconstructed structure when she
• linearly interpolated the zebra population between the points (0, 400) and (10, 200) to find (5, 300), "because it decreases in steps of 100" {I. b};
• described the development of the eagle population [with the graph on the screen in the domain 0 < x < 15 but not beyond], saying, "And the eagles, at some point, it seems, will die out or get to. ... It's possible to enlarge the function [meaning the domain] and see. ... Up to a point, to the 11th year, I think, or 10th, they grew, and from this year on they started to decrease." {III. e};
• wrote "y = 60x" on the worksheet (B118) while the interviewer was still struggling to find the right words to describe the development of the lion population {before II. a};
• volunteered information and focused on intersection points of the graphs when asked to compare populations {II. b}. In addition, she explained and interpreted her statements:
I127: Now, what we want is to compare between the zebra and the lion populations.
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