The Case Of Shawna

Metaphors as they Unfold: The Case of Shawna

Excerpts from Collapsing Dimensions, Physical Limitation, and other Student Metaphors for Limit Concepts: An Instrumentalist Investigation into Calculus Students’ Spontaneous Reasoning, doctoral dissertation by Michael Oehrtman.

Copyright 2002 - All Rights Reserved by Michael C. F. Oehrtman

The data presented to this point has been clustered according to types of metaphors and only small portions of students’ responses have been shown. In order to explore the nature of a single student’s unfolding thoughts as they wrestle to understand a concept, a brief case study will now be discussed. The following excerpts are taken from the transcript of an interview covering the definition of the derivative. The student, Shawna, was a freshman mathematics major planning on becoming a math teacher and had taken a calculus course in high school. During the beginning portions of the interview, Shawna’s descriptions did not explicitly rely on metaphors, but we include a discussion of the entire transcript in order to place her later use of metaphors in the larger context of an entire process of inquiry.

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The interview began with the problem “Let f(x)=x²+1. Explain the meaning of

.” Shawna immediately drew the graph of f (as in Figure 1 but, at this point, consisting of only the graph and axes). She recognized

as “the formula that we had to memorize” for derivatives, but couldn’t “remember how we used it.” When asked how she thought of derivatives, Shawna referred to the computational process of finding the derivative of a monomial, “I think of pulling down the exponent. I never understood what a derivative is.” The interviewer then asked whether she associated derivatives with slope, to which she replied:

Well, yeah. Because like the slope of the tangent is, you know, the derivative - like the - yeah the slope. The m is the derivative of the function, but - OK. Maybe that'll - maybe that's coming back to me. I knew that like from high school, but I never really paid attention to it. [pause] OK. Umm. I know that when you do the limit, you're like moving the slope until it gets undefined or zero or something. That kinda - that ties into it somehow. [laughs] umm. I don't know. Oh man. This is bad. I know the derivative is the slope of the tangent at a certain point, and I guess x equals 3 would be the point, right? Because [pause] f(3+h) - what would h be? h would have to be over here [points to the left of 3] then, huh? Well, h could be over here [points to the right of 3]. I don't know.

Thus, Shawna correctly remembered that the derivative is the slope of the tangent line, which appears to have triggered a memory from her high school calculus class of what was likely a standard description of secant lines approaching a limiting position at the tangent.

After this exchange, Shawna correctly identified the locations of 3+h, f(3), and f(3+h) as well as the standard referents for h and f(3+h)-f(3) on her graph (Figure 1). She then tried to make sense of dividing the length f(3+h)-f(3) by h.

Yeah. See, 3+h would be right here. That's why I kinda drew that up there. I don't know why you take… We're taking f(3+h) which is f of here, this point [points at f(3+h)] minus this point [points at height f(3) above the point at 3+h] divided by the span between them [points at h]. Why would we do that? I don't know. OK. Let me think. I don't know why we do that.

Shawna’s thinking during this period might be schematically represented as shown in Figure 2. This diagram indicates the influence of specific thoughts on others with directional arrows. In this case, Shawna took a cue from the expressions in the difference quotient to identify corresponding locations and lengths on the graph. With those referents in mind, she then asked why we would divide by h. Becoming stumped by her own question, Shawna temporarily suspended her inquiry into the difference quotient. Thus, the sequence of three boxes in this diagram represent a relatively complete portion of Shawna’s thought processes during this interview.

Figure 2. Schematic of Shawna’s initial discussion about the relationship between the difference quotient and the graph.

At this point, Shawna shifted her focus to the limit in the definition of the derivative, saying “As h approaches zero, so h is getting smaller. This [points at vertical line for f(3)] is moving towards three then, right? Yeah.” The interviewer was still interested in how she was thinking about the slope and asks her for an explanation. In the following excerpt, Shawna responds to this question, pauses, then returns to thinking about the limit. The result is her drawing of a sequence of lines through the point (3,f(3)) with successively smaller slope (see Figure 3).

Shawna:	Umm. The slope is whatever the change in y over the change in x after every unit. So like even if the y goes down one, you know, that's like a negative change. So it's the change in y over the change in x over every unit interval. That's what I think slope is. [pause] I'm trying to think of what happens as this approaches 0. We just like keep tracing this graph. Oh, I get it. As this approaches 0, my tangent keeps getting like - keeps getting smaller until it gets to 0. [draws several lines]
I:	The tangent gets - flattens out?
Shawna:	Yeah, m=0. That's what happens as it approaches.
I:	Can you tell me what you’re thinking about there?
Shawna:	Because I'm just assuming - like I know this is like I guess like a derivative, because that's how it was introduced to me, and so as h approaches 0, the derivative keeps getting, I guess smaller, or the slope keeps getting smaller and smaller until it gets 0. As this gets smaller, this line [points at would-be secant lines] gets traced in more, and so it likes come closer and closer until it's here. And so would it be 0, though? No. What would it be? Yeah. I guess so. Or, no, it wouldn't be 0. Dork! That would be like here [points at vertex]. I'm thinking about the bottom. It would - it would be the derivative at 3. Or derivative at 3 again. What am I doing?

Recall that a portion of this description was discussed in the section on closeness metaphors. Shawna treats the coordinate plane as a physical space in which the closeness of points and lines can be measured, and follows the logic that a small change in location (change in h) produces a small change in features associated with those points (positions of the lines). The result is a conflation of the two which leads her to claim that the slope (rather than h) is zero, depicted schematically in Figure 4 with arrows pointing from both “rotating tangent” and “

” to the statement “m=0, so it flattens.” Shawna realizes that this argument is not correct when she compares the statement to the graph and visually seeing that the slope is zero at the vertex of the parabola rather than at (3,f(3)). The recognition of this contradiction brings an end to this line of reasoning.

Figure 4. Shawna’s picture of multiple tangents.

Immediately following this exchange, Shawna started over trying to understand how the limit was involved in the expression

. This time, she brought in her understanding of the derivative as involving the tangent line at a point and drew the graph in Figure 5.

Shawna:	Ok. And this is my h, this little space in here [points between 3 and 3+h]. And this is my 3+h. As this get smaller [points between 3 and 3+h], the derivative - let's just put the derivative on here for a second, like a tangent line. It would be like right about here, huh? [draws tangent line at (3,f(3))] As this gets closer and closer - as h approaches zero, this line would just keep coming in more and more [mock sketches several short tangent lines between 3+h and 3] until it's like the derivative - or the, excuse me, tangent at x=3.
I:	So, you're drawing lots of little tangent lines?
Shawna:	Yeah. On each - on 3+h [draws large tangent line at (3+h, f(3+h))]. And between 3 and 3+h [draws two more tangent lines], so - yeah. So I guess that's why - I don't know.
I:	So what do you think all of this has to do with that limit?
Shawna:	That's what I'm trying to figure out. [pause]

In this excerpt, the symbols

draw Shawna’s attention to moving toward 3 along the graph. Having drawn a tangent line at (3,f(3)), she then thinks of several tangent lines on the graph. The influence of these two signs is depicted schematically in Figure 6 with arrows pointing from each to her graph drawn with multiple tangents. Symbolically, Shawna’s explanation might be represented as

, with the exception that she is really describing tangent lines rather than their slopes. When the interviewer questions her about the exact role of the limit in this explanation, Shawna is unsure, and once again, a line of inquiry ends.

Figure 6. Schematic of Shawna’s discussion of multiple tangents to the graph.

Even though Shawna had talked about slope as “the change in y over the change in x” and had identified f(3+h)-f(3) and h as such changes, she had still not recognized that the difference quotient

expressed a slope. At this point, the interviewer redrew Shawna’s picture as in Figure 7 adding the two darkened line segments (the two short vertical lines were added later by Shawna). The following excerpt begins with Shawna’s immediate, albeit tentative, recognition of the difference quotient as a slope, which leads to a burst of discovery.

Shawna:	So, that's kind of like slope? I don't know. Yeah. Because that's kind of like the x value, and that's the difference, which is the y value.
I:	Can you say more about what you mean by that's the slope?
Shawna:	Hmm. let me think. [pause] Yeah. That makes sense. Because I mean if you didn't know how to differentiate, you could do this [points at ], and if you take f(3+h) and subtract f(3) - like h could be any number like, I don't know, just a number - and if you subtract them, you get your y - your change in y. And then when you divide by this [points at h on the graph], you - what do we do? Oh, I had it there. [pause] Ok. This kind of makes sense. OK. As this approaches zero [makes motion from right vertical line to left vertical line], you divide - I still have to remember that it's going to zero. I can't JUST use this part [points at ], because I mean that wouldn't be the derivative or anything. That would just be a number. As this gets smaller [points at h on the graph], this comes down [points at f(3+h)]. OK. [pause] that kind of makes sense. Because it's a limit and it can only go so far until it reaches the point. As this comes smaller, that's your y value divided by your x value which is a slope. And so - OK. That makes sense. As you bring h towards 3, your y - your f(3+h)-f(3) gets smaller, because you're tracing down the graph. Well, that is if - if of course if the graph looks like this, but it does, so I'm going to say that [laughs]. The y value gets smaller, and this value gets smaller [points at h on the graph]. It gets smaller. So you're dividing y over x which is actually - that's the slope. And so you get so small until you can go no more and that gives you the slope at 3. Magically. I don't know. [laughs] That make sense though, because I mean, I really don't know how to explain limits like as a professor or anything or a really intelligent person because I just - that's how I understand limits to be. You know? You take something and - and I don't mean to go on that tangent. [pun intended?] You take your values and you squish them really small until you can get - until you can go no more, and magically that's the limit. I don't know why it gives you that, though. I mean I kind of do, but I don't know how you get a number out of that. You take - I couldn't explain it to too many people. As this gets smaller and this get smaller [points at the darkened vertical and horizontal segments], your - the difference between these two gets closer and closer. Say you get like here and here, and here and here [draws the two short vertical lines], and so you're getting really really close to the rise over run of this. And when you reach your limit, that's what the rise over run of this is [points at (3, f(3))] so I guess that's the tangent which is the derivative. Yeah. That does make sense. Because that's what happens on a limit. Like when you - on a graph, you get smaller and smaller until you get to the point that you want, and that's what your value is. And so I guess this would be - if you could see these two little lines down here, your tangent - or your slope - or yeah your tangent would be smaller and smaller until you finally hit this point at three which gives you like THE tangent. So if you have like a really small h like a 0.001 and you did this, and you just found the rise over run - or if you just take that divided by that - hold on. If you take – yeah, if you just take f(x) and divide by change in f(x) - like the change in y and you divide by the h, that would be like really close to the tangent, and so the smaller you go, the closer and closer to the tangent you get, and that's why you GO TO zero, because you can't divide by zero, but that's why it's the tangent. [Shawna’s emphasis throughout]

Shawna uses a collapse metaphor in this excerpt to deal with the jump from considering secant lines between two separate points and a tangent line passing through a single point: “And so you get so small until you can go no more and that gives you the slope at three. Magically.” Although her very expressions of this metaphor convey her uncertainty (she uses “magically” twice and also says explicitly “I don't know why it gives you that”), she continually returns to the idea and explores its implications. The centrality of this metaphor to her reasoning is represented in the schematic diagram in Figure 10. Notice that the collapse metaphor emerges from her converging considerations of the slope of a secant line and the motion from (3+h, f(3+h)) to (3, f(3)). After this, Shawna begins to ask questions of her new idea, most notably when she wonders “how you get a number out of that.” This question leads her to three separate conclusions all feeding back into her collapse metaphor: 1) small secants become “the tangent,” 2) approximate values of slope become the slope, and 3) “you go to zero, because you can't divide by zero.”

Figure 8. Schematic of Shawna's thinking related to her collapse schema.

This last conclusion could indicate some sense of the subtleties involved in limits, but regardless, Shawna continued to use collapse metaphors throughout the remainder of the interview. When she was reflecting back on her burst of ideas from the previous excerpt, she also reported thinking of approximation to translate her geometrical understanding into a numerical understanding. In this excerpt Shawna’s final comments (“the limit takes you as small as possible until you reach that point”) indicate that her approximation metaphor contains ideas about collapse in a manner consistent with her treatment of numerical values in the previous excerpt.

I was looking at more like this gets smaller and smaller and so like when you draw a line here [draws a secant line], it's gonna slice right through your x² graph. But as you get smaller and smaller, I was thinking no, it's gonna come further and further to the edge. And then when you get to the perfect point, you know it's going to be on the edge [points at tangent line] and I was like yeah, that's what it is. Because I was looking at that [points at

] thinking like that's a slope, but that's like really off. Like I was saying, if you just - if you didn't take the limit - and if you just did that, the smaller your h is, the more accurate the limit would be. You can like estimate if you just plug in like a number for h, but the closer it is to the number you're looking for, the better it would be. So like if I drew from here to here [points at smallest triangle], that would be a much closer limit than the one from here to here [points at largest triangle]… so I was just thinking the smaller h you got, the more accurate the limit would be, and then the more accurate the slope would be… And so then that brought me to the conclusion, I was like yeah, that's why you take the limit. Because the limit takes you as small as possible until you reach that point, so that makes sense. I never really thought about it like that before, but now I see it and I won't forget it.

In the second portion of this interview, students were asked to give an interpretation of the same limit in the context that the function represented position as a function of time,

. Shawna went through the same process as she did during the first part of the interview of identifying referents for the various expressions in this limit. With a small amount of help from the interviewer, she eventually located each of the times (t=3, t=3+h, and

) and distances (p(3), p(3+h), and p(3+h-p(3)) on a straight line. representing a road along which a car was traveling (see Figure 9). Shawna’s initial inability to figure out what the quotient,

, represented was similar to her thought process while interpreting this limit in the context of the graph of a function. In the new car context, she made comments like “So that would be like the ratio of how they convert together - how one affects the other? Maybe?” Finally, once again, it was remembering that the expression

is related to derivatives and that derivatives give velocity, she is able to recognize the quotient as “a good estimate of” velocity.

Figure 9. Shawna’s picture of a road showing distances and times from the difference quotient,

Shawna:	The distance traveled in h time. I think. Yeah. It would be. It would be the distance traveled in - but that would be the same thing as here divided by that [points at the region between the marks on the line]. So I don't know. But you have to think about as h goes to zero, so as h gets smaller, we travel less. Hmm. This is hard. Not thinking about a pretty little tangent line and stuff. [laughs] OK. OK. [pause] This would be a good estimate of - I don't want to say - what is this in terms of? Like the derivative? What would that be called? The – like p'(t)? That wouldn't be like the velocity of it?
I:	So you're looking at - trying to interpret what the derivative of p would be?
Shawna:	Yeah. That would be velocity I think. Yeah, because second derivative is acceleration. So like this - if you just divided these two, that would be a good measure - I mean like not good. It depends on how close your h is. But it would be like an estimate of what the velocity is. But the lower your h - the smaller your h gets, the closer you get to a real point with a real velocity, so you would - you have smaller and smaller numbers to divide until you got to - until you made h zero and you got your velocity at t=3. So velocity is [pause] - what is velocity? Velocity is speed? Right? So - yeah. That would be your speed and, because you traveled so much distance in some amount of time, but - I mean, that's like an estimate, because it's not gonna be exact, but the closer and closer you get to a real point, that's gonna be your speed. OK. That kinda makes sense. So this would determine how fast you were going at a certain time t. Yeah. Because the closer you get - the lower your numbers get, the more accurate your rise over run division would be to what it really is at I guess. So yeah, that makes sense. [pause] What else can I add to that?

         This explanation is given primarily in terms of approximation, using phrases such as “it would be like an estimate of what the velocity is,” “that's like an estimate, because it's not gonna be exact,” and “the lower your numbers get, the more accurate your rise over run division would be to what it really is.” Shawna also treats the limit for this approximation with a collapse metaphor as she did in the graphical context. Here she describes the collapse as “the smaller your h gets, the closer you get to a real point with a real velocity, so you would - you have smaller and smaller numbers to divide until you got to - until you made h zero and you got your velocity at t=3.”

Other excerpts from the dissertation
Warning: these refer to the Shawna piece but are somewhat out of context.

On the Usefulness of “Mathematically Incorrect” Metaphors

            Recent research on experts’ reasoning indicates that treating nonstandard interpretations simply as misconceptions may miss many important aspects of the learning process. Roth & Bowen (2001) presented research scientists with graphs that were unfamiliar but intended for undergraduate instruction in their own field. While interpreting the information contained in the graphs, these scientists made errors strikingly similar to those of students such as conflating interpretations of slope and height, ignoring changes in one quantity as a result of change in another, and improperly attributing physical properties of a graph with related features of the represented phenomenon (iconic translation). In similar studies, experienced professors have been shown to extend analogy beyond the applicable structure of a situation (Roth & Bowen, 1999; Roth, Tobin, & Shaw, 1997) and struggle with concepts and experience frustration and anxiety during problem solving (Carlson & Bloom, under review).

            When observed in students, such errors and behaviors are typically treated with deficit arguments, but these studies suggest that powerful reasoning is still possible. Carlson & Bloom’s study of research mathematicians’ problem solving strategies suggests that struggle, frustration, and anxiety are an integral part of larger metacognitive and affective cycles, leading to greater control and motivation. Scientists in the studies of Roth et al. were able to use even nonstandard interpretations to produce and test hypotheses that moved their thinking forward in positive ways.

            Such active ways of engaging with ideas is similar to the students’ productive uses of metaphors observed in this study. Shawna, in her interview about the definition of the derivative, went through at least three cycles of developing an idea, applying it to the situation, and eventually rejecting or revising it through testing for its entailments. These ideas included trying to make sense of various algebraic expressions in terms of lengths on the graph, imagining lines through a single point on the graph rotating to a limiting position, and lines tangent to the graph at various points sliding along the curve to the point of interest. In these cycles she begins to experiment with putting various lengths in relationship to one another, makes points move and line segments change length, and compares the slopes at different parts of the graph to her intuition about the slope at the point in which she is interested.

            This process is crucial for Shawna’s developing thinking, and arguably, without the availability of these “misconceptions,” she may have never arrived at a point to be able to make the progress she did. Her fourth iteration of the cycle of developing and applying an idea involved the collapse metaphor with an unspoken (at the time) influence of approximation when translating to a numerical domain. While the idea of secant lines collapsing to a tangent is mathematically incorrect, it is very productive for Shawna, helping her understand some powerful mathematical ideas about the definition of the derivative. Minimally, it provides her with a way to connect the secant lines with the tangent through attention to the appropriate limiting process. More than this, however, it becomes a tool with which she is able to ask some interesting questions and develop good connections. As shown in the schematic of this portion of her reasoning (Error! Reference source not found. in Chapter 5), she uses the collapse metaphor as a central point around which to organize her reasoning. Through this process, she is able to see that the slopes of the secants become closer to the slope of the tangent, and she wonders about and finds the connection between graphical, algebraic, and numerical representations of the definition. Later, she is even able to connect these ideas to her interpretation of the derivative in the physical context of the changing position of a car through her collapse metaphor.

On Critical Evaluation and Modification

            An important aspect of instrumentalism is the testability of a tool used in inquiry. Whether or not students actively engaged in critically evaluating the metaphors they were using was important for their ability to recognize problems and make helpful modifications. The case study of Shawna’s work on the definition of the derivative illustrates this claim. She continually monitored the statements she was making for consistency between one another and with her intuitions. This allowed her to abort lines of reasoning that were becoming problematic and look for other alternatives. When she started to make progress, she kept checking herself (repeatedly saying things like “Yeah. That does make sense.”) and asking questions of the metaphors she was using (such as “I don't know how you get a number out of that” and going on to investigate the implications numerically.)

On Functional Aspects of Research

            Students’ reported structural organization of limit concepts often did not account for their actual use of those ideas. When they were asked to explain what limits are, they typically described a graphical setting accompanied by language about motion. Determining what concepts about limits a student possesses and how they are connected is a very structurally oriented approach to their understanding. In this study, when they were asked to use limit concepts to think about something new or approach a difficult problem, motion language tended to remain at the level of language and did not enter their descriptions referring to their thinking about anything actually moving. Instead, other metaphors surfaced.

            Such results suggest that research cannot fully uncover the nature of students’ metaphors by examining only their surface language and responses to direct questions about their conceptualizations of the topic. Not only does this methodology miss the different structures that might appear in such problem solving contexts, but it also lacks the important characterizations of how those metaphors are actually applied, of the questions the metaphors are used to ask and the resulting answers, and of the changes the conceptual tools undergo in the process. One must look at richer data on these functional aspects in addition to their structure and logic.

            This research found students using specific metaphors as organizers of ideas and touchstones for reasoning. For example, the case study on Shawna’s understanding of the definition of the derivative showed how she repeatedly returned to the idea of a collapse from two points determining a secant line to a tangent line through a single point in graphical, algebraic, numerical, and physical contexts. She used this metaphor in asking questions such as wondering “how you get a number out of that,” referring to the graphical version of the collapse metaphor. She answered this question using a numerical version, and continually checked her ongoing work against her ideas about various versions.   These ideas became central to her developing understanding of the definition of the derivative and eventually even limits, holding together the thought of a collapse happening “magically” with the reasoning “that's why you go to zero, because you can't divide by zero.” Similarly, students who appealed to a physical limitation to argue that the volume of a solid of revolution is finite if the radius goes to zero were mathematically incorrect. This compelling imagery, however, appeared to become a major aspect of their understanding of solids of revolution, the meaning of volume, and limits.