What students' learning of representations tells us about constructivism

Journal of Mathematical Behavior 19 (2000) 481 – 502

What students’ learning of representations tells us about constructivism Andrew Elby Physics Department, University of Maryland, College Park, MD 20742, USA

Abstract This article pulls into the empirical realm a longstanding theoretical debate about the prior knowledge students bring to bear when learning scientific concepts and representations. Misconceptions constructivists view the prior knowledge as stable alternate conceptions that apply robustly across multiple contexts. By contrast, fine-grained constructivists believe that much of students’ intuitive knowledge consists of unarticulated, loosely connected knowledge elements, the activation of which depends sensitively on context. By focusing on students’ intuitive knowledge about representations, and by fleshing out the two constructivist frameworks, I show that they lead to empirically different sets of predictions. Pilot studies demonstrate the feasibility of a full-fledged experimental program to decide which flavor of constructivism describes students more adequately. D 2001 Elsevier Science Inc. All rights reserved. Keywords: Constructivism; Misconceptions; p-prims; Conceptual change; Representations

1. Introduction Students’ interpretations of graphs and other representations can shed empirical light on a longstanding theoretical debate about science learning. To spell out my claim, I must distinguish between two flavors of constructivism. According to misconceptions constructivists, students walk into the classroom with alternate conceptions and theories (McCloskey, 1983b; Strike & Posner, 1985). By contrast, fine-grained constructivists believe that much of students’ intuitive knowledge consists of loosely connected, often inarticulate minigeneralizations and other knowledge elements, the activation of which depends heavily on context (Hammer, 1996a; Smith, diSessa, & Roschelle, 1993/1994; Tirosh, Stavy, & Cohen, 1998). I will show that (1) these two frameworks, when fleshed out, lead to different sets of 0732-3123/00/$ – see front matter D 2001 Elsevier Science Inc. All rights reserved. PII: S 0 7 3 2 - 3 1 2 3 ( 0 1 ) 0 0 0 5 4 - 2

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predictions about student behavior, and that (2) pilot studies show the feasibility of a fullfledged experimental program to decide which flavor of constructivism describes students more adequately, and also give us reason to take fine-grained constructivism seriously. So, this article operationalizes a debate usually conducted on a theoretical plane. First, I explicate the two flavors of constructivism, underscoring the difference between representational ‘‘misconceptions’’ and finer-grained, context-dependent intuitive knowledge elements. Unlike diSessa (1993), who lays out a detailed account of students’ intuitive knowledge about physics, I provide only a sketchy overview of students’ intuitive knowledge about representations. But I begin to build a fuller account by spelling out the nature and activation tendencies of one crucial representational knowledge element. Finally, using this element, I show that the fine-grained and misconceptions frameworks make empirically different sets of predictions about students’ behavior in certain circumstances. Pilot studies demonstrate a method of putting those different sets of predictions to the test.

2. Two flavors of constructivism In this section, I tease apart misconceptions constructivism and fine-grained constructivism, focusing initially on students’ intuitive knowledge about physics, which is a heavily researched topic (diSessa, 1982; Halloun & Hestenes, 1985; McCloskey, Carramazza, & Green, 1980; McDermott, 1984). Then, I explore a less-traveled terrain, students’ intuitive knowledge about representations. 2.1. Two flavors of students’ intuitive knowledge about physics For my purposes, a ‘‘constructivist’’ is someone who believes the following: ‘‘Learners do not walk into the classroom as blank slates ready to be filled with knowledge. Instead, as students construct a new understanding, their prior knowledge plays a crucial role.’’ Within this broad framework, however, different camps tell different stories about the structure of this prior knowledge and the mechanism of conceptual change. According to misconceptions constructivists such as McCloskey (1983a) and Strike and Posner (1985), students’ prior knowledge consists largely of noncanonical conceptions and theories. For instance, according to McCloskey (1983b), students’ intuitive knowledge about mechanics resembles the ‘‘impetus theory’’ believed by natural philosophers in the middle ages. This alternate theory contains the misconception ‘‘motion requires force,’’ according to which an object in motion requires a force to keep it moving.1 By assuming that the misconception exists as a comparatively stable knowledge element inside peoples’ heads, we can explain why students mistakenly think that a desk being pushed across the floor at constant velocity

1 By contrast, according to Newtonian physics, a net force is required to initiate or change motion, but not to maintain motion at constant velocity.

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feels a net forward force. And to explain why some of these same students hold the apparently contradictory belief that a ball thrown in outer space keeps drifting indefinitely,2 we can flesh out a story of competing conceptions (Maloney & Siegler, 1993) or of a transition stage during which the student fluctuates between her original misconception and the new conception she is learning (Thornton, 1995). In this way, the misconceptions framework can accommodate inconsistencies in students’ reasoning. However, since misconceptions are assumed to be somewhat theory-like, or are at least described in general terms that are not linked to particular contexts, the misconceptions framework cannot make predictions about the contexts in which fluctuations are most likely to occur. Within misconceptions constructivism, the process of conceptual change resembles the mechanism by which scientific communities are claimed to alter their theories (Strike & Posner, 1985). When confronted with evidence that contradicts her old conceptions, and when made aware of the difference between her old theory and the scientifically accepted theory, the student becomes ready to accept the new theory. In brief, the student’s old conceptions are confronted and then replaced. Some misconceptions theorists, responding to research on conceptual change, allow for the possibility that some misconceptions have internal cognitive structure and can be compiled on the spot (Carey, 1992; Strike & Posner, 1992). However, even in these newer formulations, the misconception remains the primary unit used to describe and analyze students’ conceptual reasoning. Furthermore, although the modified misconceptions framework allows for learning mechanisms besides ‘‘confront and replace,’’ the misconceptions are not considered to be part of the raw material out of which the student builds a new understanding. Fine-grained constructivists (Smith et al., 1993/1994; Tirosh et al., 1998), by contrast, believe that much of students’ intuitive knowledge takes the form of inarticulate minigeneralizations from experience, as Hammer and Elby (in press) explain: In this framework, students’ reasoning about the desk and the ball can be understood in terms of the context-specific activation of the following fine-grained resources. Maintaining agency 3 is an element of cognitive structure useful for understanding any continuing effect maintained by a continuing cause, such as a light bulb needing a continuous supply of energy to stay lit. Actuating agency is another resource, an element of cognitive structure involved in understanding an effect initiated by a cause, when the effect outlasts the cause, such as the strike of a hammer causing a bell to ring. The desk scenario tends to activate Maintaining agency, and hence, the idea that a continued net forward force is needed to keep the desk

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I have observed this phenomenon in my high school physics students. Along the same lines, Steinberg and Sabella (1997) show that many students, in response to a multiple-choice item and a free-response item probing the same misconception, give inconsistent responses. Other evidence (diSessa, 1993; Tytler, 1998) also suggests that students’ reasoning is inconsistent in ways that a misconceptions account can accommodate only by introducing competing conceptions or fluctuations, as discussed in the text. 3 diSessa (1993) called this continuing push. However, the word push in that name may be misleading as the agency need not take the form of a force. We will also use the name actuating agency instead of diSessa’s force as mover.

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moving forward. By contrast, the ball question tends to activate Actuating agency, and the idea that the ball’s motion can outlast the force exerted by the thrower.4 Unlike the misconception Motion requires force, the finer-grained cognitive resources Maintaining agency and Actuating agency are not ‘‘incorrect.’’ Neither are they correct. They are resources that can be activated under various circumstances, sometimes appropriately, sometimes not. Furthermore, whereas the misconception is an element of cognitive structure specifically tied to motion and force, the finer-grained resources also apply to light bulbs, bells, and numerous other situations. In this sense, these resources are finer-grained but more general than misconceptions (though some finer-grained resources might be tied more tightly to a particular setting).

Within the fine-grained framework, conceptual change is not a matter of replacing bad minigeneralizations with good ones. Instead, it is partly a matter of tweaking those minigeneralizations into a more articulate, unified, coherent structure. For instance, when constructing an understanding of Newtonian mechanics, actuating agency can serve as an intuitive grounding of Newton’s first and second laws, according to which a force is needed to initiate or change motion but not to maintain motion (at constant velocity). In Newtonian mechanics, maintaining agency (merely) contributes to an informal heuristic for reasoning about situations involving strong friction or other dissipative forces. But actuating agency and maintaining agency both play a role in a physicist’s reasoning. As novices become experts, few if any minigeneralizations ‘‘die’’ completely. They are restructured, not replaced. In summary, misconceptions constructivism and fine-grained constructivism disagree not only about the form of students’ intuitive knowledge, but also about the mechanism of learning and conceptual change. Consequently, these two flavors of constructivism invite different instructional practices, as Hammer (1996a) discusses. For both theoretical and practical reasons, we must decide which kind of constructivism better accounts for students’ behavior in various situations. 2.2. Two flavors of students’ intuitive knowledge about representations To see how the distinction between misconceptions and fine-grained constructivism plays out in the context of representations, consider this velocity vs. time graph (Fig. 1) representing a car’s motion. Novices sometimes think the car is not moving. Within a misconceptions framework, this is taken to show that students are misreading the velocity graph as a position graph, a mistake the student is likely to make on other velocity graphs (see Leinhardt, Zaslavsky, & Stein, 1990; McDermott, Rosenquist, & van Zee, 1987). By contrast, within a finegrained framework, the flat horizontal line can be taken to activate stillness, an element of

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On this view, the ‘‘internal force’’ students often invoke in their explanations is not part of a stable, preexisting misconception, but rather something they conceive of on the spot, if the context requires them to explain the ball’s continued motion.

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Fig. 1. A velocity vs. time graph for a car.

cognitive structure associated with lack of motion. In this story, if stillness gets cued when the student is thinking about the car itself, she is likely to conclude that it is motionless. By contrast, if she is consciously thinking of the car’s speedometer needle when stillness gets activated (perhaps due to a teacher’s intervention), then she is more likely to interpret the graph as indicating steady motion. So, the fine-grained account predicts some context-dependent inconsistencies in whether the student interprets the graph as if it indicates position instead of velocity. As noted above, misconceptions constructivism accommodates inconsistencies in students’ reasoning. Therefore, in the absence of a detailed story about these contextual dependencies, the fine-grained and misconceptions stories do not make empirically distinguishable predictions. They agree that students will sometimes read the velocity graph as if it were a position graph, and that conscious reflection about what the graph represents can help students make fewer such mistakes. A specification of contextual dependencies is what distinguishes finegrained from misconceptions constructivism. Without going into detail, I will now propose some other intuitive knowledge elements that students might bring to bear when interpreting visual representations. These speculations play no role in my later arguments, but illustrate the fine-grained constructivist framework. Constancy, triggered by straight lines on graphs (flat or sloped) and presumably by other visual cues, corresponds to the idea that something about the situation does not change. For instance, the activation of constancy might cause a novice musician to interpret a long horizontal line on the staff as indicating that she should hold whatever note she is playing.5 Sudden change, cued by steep segments on a graph or by borders on a map, corresponds to dramatic change. In this framework, a misconception can emerge in a particular context, but perhaps not in other contexts, by the (mis)activation of various fine-grained intuitive knowledge elements — elements which in other contexts might contribute to productive interpretations. 5

By the way, graphical expertise may consist in part of having constancy rather than stillness cued by a horizontal line on a graph, since constancy is not tied to position, while stillness is.

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3. A fine-grained intuitive knowledge element: WYSIWYG 3.1. What you see is what you get A middle-school student, informed that Fig. 2 is a speed vs. time graph of a bicyclist, speculates about what is happening between t1 and t2. Experienced teachers and researchers, even if they have not seen this particular example in the graphing misconceptions literature (Janvier & the Universite´ du Que´bec a` Montre´al, 1987; Leinhardt et al., 1990; McDermott et al., 1987), can often anticipate that some students think the bicycle is going over a hill. This iconic, or naively realistic interpretation — the ‘‘hill’’ on the graph is taken to represent an actual hill — fits into a pattern that teachers recognize. In my view, that is because the hill mistake and similar iconic interpretations spring, in part, from the activation of a cognitive structure, specifically, an intuitive knowledge element I call what-you-see-is-what-you-get (WYSIWYG). WYSIWYG : x means x: For instance, on a child’s drawing of her family, the bigger people can be interpreted as representing the grown-ups; ‘‘bigger means bigger.’’6 Or, in the above example, ‘‘hill means hill.’’ On a colorized image of Jupiter, a blue halo can be interpreted, inappropriately, as a blue atmosphere; blue means blue. On a street sign showing a thick curvy line, WYSIWYG contributes to the quick — and in this case, productive — conclusion that the road curves. Two more examples will clarify the kinds of interpretations I take to be triggered by the activation of WYSIWYG. Consider this map (Fig. 3) of the central California coastline. The right-hand region is green, while the left-hand region is blue. If WYSIWYG is activated while a student focuses on the green, he is likely to think that California is solid green. Similarly, a WYSIWYG-triggered interpretation of the boundary line leads to the conclusion that it is indeed a boundary; ‘‘boundary means boundary.’’ In Fig. 4, some people may quickly interpret the central dot as the source of the arrows, even when they do not know that the diagram portrays a charge and its electric field lines. If WYSIWYG gets triggered while the student focuses on that ‘‘source,’’ she is likely to interpret it as the source of whatever the arrows represent as moving. So, WYSIWYG is one of the intuitive knowledge elements contributing to a ‘‘naive’’ interpretation of a visual representation or an aspect thereof.7 As the above examples show, WYSIWYG can contribute to both productive (‘‘boundary means boundary’’ on map) and flawed (‘‘blue means blue’’ atmosphere) interpretations. Like

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See Bruce Sherin’s article in this issue (Sherin, 2001) for more about how students use relative sizes and distances on spatial representations to indicate the corresponding spatial relationships in the real world. 7 In this paper, I will not address several interesting questions about WYSIWYG. For instance, does WYSIWYG do its cognitive work by spawning a bunch of specific implications, such as ‘‘bigger means bigger’’ and ‘‘blue means blue’’? Or, is WYSIWYG itself a larger cognitive structure arising from a collection of case-specific minigeneralizations? Does this process go both ways, reinforced by a feedback loop? Fortunately, my central argument does not depend on the answers to these detailed questions.

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Fig. 2. A velocity vs. time graph for a bicycle.

most intuitive knowledge elements in a fine-grained framework, it is neither correct nor incorrect. Young children learn, probably unconsciously, that WYSIWYG is not always productive. For instance, when learning to read, a 5-year-old learns that certain squiggles on the page refer to things in the world; ‘‘cat’’ refers to a furry critter that looks nothing like the word ‘‘cat.’’ Similarly, when looking at a color-coded relief map, many children know to interpret the different colors as heights, not as actual colors of the terrain. In other words, children learn to interpret more abstract, less iconic representations — representations for which WYSIWYG must be put in the background. 3.2. Compelling visual attributes Because it remains productive, WYSIWYG does not die. For example, bigger really does mean bigger in many representations. To spell out my claim about which contexts cue WYSIWYG most strongly, I must introduce a new concept: the compelling visual attribute.

Fig. 3. Grayscale version of a color map of the middle California coastline.

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In some visual representations, one feature (or a particular Gestalt involving multiple features) quickly draws your attention. That is the compelling visual attribute. Experiments can determine which visual attributes are most compelling in which circumstances. For instance, the motion of a subject’s eyes can be measured during the first few tenths of a second after a new visual representation is presented. In addition, a welldeveloped line of research about the human visual system establishes that the layers of neurons behind the ‘‘light detectors’’ in our retinas are hard-wired to ‘‘see’’ certain features such as edges, corners, and motion (Churchland & Sejnowski, 1992). These visual attributes are detected even before the information reaches the visual cortex in the brain. Edges, corners, and motion probably constitute compelling visual attributes in some circumstances and likely contribute to other compelling visual attributes in other circumstances. Therefore, we can reasonably infer that the neuroscience of vision can contribute to our understanding of compelling visual attributes.8 Which visual attributes are most compelling probably depends on context. For instance, in Fig. 2, the ‘‘hill’’ may be an especially compelling visual attribute to a student who has just read about a bicycle going over a hill.

3.3. When is WYSIWYG most likely to be cued? My claim is that, even though WYSIWYG is not cued strongly in all contexts, it is cued strongly with respect to the compelling visual attribute of a representation: WYSIWYG activation claim: In a visual representation, the compelling visual attribute tends to cue WYSIWYG.

Some old and new examples illustrate what the WYSIWYG activation claim means. The examples also contribute to an argument for the claim’s plausibility, an argument I will present immediately after the examples themselves. Consider Fig. 3, the California coastline. Given that the human vision system is hard-wired to detect edges (see Section 3.2), the compelling visual attribute is likely to be the boundary. So, according to my activation claim, the boundary on the map (disproportionately) gets interpreted as representing a boundary in real life. This interpretation, of course, is correct. By contrast, even young children do not interpret the green and blue as showing the actual colors of California and the ocean. In other words, WYSIWYG is not applied to all aspects of the representation, but it is applied to the compelling visual attribute, the boundary. Crucially, because the application of WYSIWYG to the compelling visual attribute leads to a productive interpretation, that pattern of activation tends to get reinforced, as argued below. Fig. 4 (dot and arrows) may provide a less clear-cut but more typical example of WYSIWYG activation. It may turn out that subjects, quickly and without conscious thought, 8

Put another way, some of the ‘‘primitives’’ in the space of intuitive representational knowledge may connect closely to sensory ‘‘primitives’’ hard-wired into our visual processing systems. This connection deserves study. Nothing in my main argument rides on these speculations.

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Fig. 4. Positive charge-emanating electric field lines.

perceive the overall Gestalt of the figure to be an outward flow from the central dot. If so, the WYSIWYG activation claim says that the diagram is likely to be interpreted as showing an outward flow from the center, which is, indeed, a productive way of viewing the relationship between a charge and its electric field lines. By contrast, it is not productive to view the arrows as representing actual arrows. So once again, assuming subjects perceive ‘‘outward flow’’ as the compelling visual attribute, the activation of WYSIWYG with respect to the compelling visual attribute leads to a productive interpretation. I give one last example. This ‘‘picture’’ of a galaxy (Fig. 5) is not a photograph, but rather, a visual display of digitally stored data. In a 1996 high school physics class, students were told to look for a supernova in this galaxy. The students knew that supernovae are extremely compact, bright

Fig. 5. A display of a digital image of a galaxy containing a supernova.

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objects. For somebody in this context, one of the most compelling visual attributes is probably the small bright blob on the left side of the galaxy;9 the student’s eye may dart to that spot even before she begins consciously searching for the supernova. And indeed, the small bright blob on the image represents a small bright blob in space, a supernova. By contrast, the apparent edges of the galaxy on the image do not reliably indicate the edges of the actual galaxy; adjusting the ‘‘MIN’’ setting of the software causes the displayed image to shrink or expand (Friedman & diSessa, 1999).10 So, once again, WYSIWYG leads to a useful interpretation of the compelling visual attribute, but not to a useful interpretation of other aspects of the visual representation. I now discuss the speculative developmental story underlying my WYSIWYG activation claim, starting with a one-paragraph summary and then going into more detail. I propose that WYSIWYG, or particular instantiations of WYSIWYG, develops extremely early. As David Hammer (personal communication, October 14, 1999) puts it, the default, if you see something, is to see what you see! Couched more carefully, the default attitude toward something you can easily ‘‘see’’ is that you see it directly and unproblematically — what you see is what you get. Consider a visual attribute that is particularly useful for interpreting the world. Its usefulness causes — or at least favors — the development of quick and direct interpretative strategies (often involving WYSIWYG) that are effective and that call attention to themselves, making them compelling. As a result, WYSIWYG becomes strongly connected to compelling visual attributes. According to this story, biological evolution produced edges as a hard-wired compelling visual attribute partly because they are so useful in functional tasks such as knowing the boundary of an object. This usefulness favors development of the quick, direct interpretation of those edges as the boundaries of objects — ‘‘edge means edge.’’ Because it is paired with a quick, direct interpretation, the ‘‘edge’’ visual attribute becomes more compelling, that is, more likely to grab attention. Similar reasoning applies to soft-wired perception mechanisms, including representational resources and the connections between them that implement interpretive strategies. The most useful visual attributes become involved in quick and direct interpretive strategies, which are often strongly — and productively — connected to WYSIWYG. By their very nature, these quick-and-direct strategies disproportionately grab attention, making the underlying visual attribute more compelling. Even when other interpretations of the visual scene might be available, compelling attributes are so often useful that they compete effectively for attention and carry along the WYSIWYG interpretive stance. In sum, the usefulness of a visual attribute causes the development of quick and effective interpretive strategies that preferentially call attention to themselves (are compelling) and for which WYSIWYG is an appropriate default stance. 9

The big blob in the center of the galaxy may also be a compelling attribute. Each displayed pixel corresponds to a brightness level (called the ‘‘brightness count’’) recorded by the digital camera that took the image. Any pixel corresponding to a brightness count less than the MIN setting gets displayed as black. For this reason, adjusting the MIN setting causes previously gray pixels to appear black, or vice versa, increasing or decreasing the apparent width of the galaxy. 10

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The strong cueing of WYSIWYG by a compelling visual attribute may be reinforced by additional ‘‘natural selection’’ at the societal (as opposed to biological or individual) level. A representation for which WYSIWYG leads to an unproductive interpretation of the compelling visual attribute will fool people, making the representation less useful — and presumably, less used by creators of representations — than it would otherwise be. Partly for this reason, as illustrated by the California coastline and the electric field, consumers of representations experience WYSIWYG as a productive attitude toward compelling visual attributes. In net, repeated experience with representations selected to make productive use of compelling visual attributes reinforces the links between compelling visual attributes and WYSIWYG. I am not claiming that children consciously learn all of this. If a connectionist-style network of fine-grained interpretational knowledge elements accurately depicts students’ quick, unreflective reactions to visual representations, then WYSIWYG gets cued — or does not get cued — quickly and automatically. For instance, according to this model, within a moment of seeing the California coastline representation, people just think ‘‘boundary!’’ without consciously pondering the applicability of WYSIWYG. Equally unconscious, in all likelihood, is the learning process by which WYSIWYG and compelling visual attributes become strongly paired. Plausibility arguments aside, the truth or falsehood of the WYSIWYG activation claim is ultimately an empirical matter. In the next section, I put the activation claim to the test.

4. How do the two flavors of constructivism disagree? A deeper story needs to be told about WYSIWYG, compelling visual attributes, and the activation claim that compelling visual attributes disproportionately cue WYSIWYG. For the purposes of my central argument, however, I have gone far enough. The WYSIWYG activation claim, which is one small part of a fine-grained constructivist account of intuitive representational knowledge, allows me to make predictions about students’ behavior in certain contexts, predictions that go beyond those of misconceptions constructivism. I am not arguing for the completeness of my fine-grained explanations. Instead, I am using the WYSIWYG activation claim as an illustration of the kind of story that separates fine-grained constructivism from misconceptions constructivism. To highlight why the two flavors of constructivism generate different sets of predictions, I first sketch the general form of my argument. Misconceptions constructivism views conceptual change as a switch from one set of stable conceptions to another. Within this framework, nothing can be said about the fine structure of the transition, or about the ‘‘fluctuations’’ between conceptions. By contrast, the context-dependent cueing of finegrained constructivism allows stories to be told about the fine structure of these transitions, stories that predict particular patterns of nonrandomness in the fluctuations. So, fine-grained constructivism generates predictions about students’ behavior in cases where misconceptions constructivism predicts nothing other than random fluctuations. If enough of those fine-grained predictions turn out to be correct, then we have reason to

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prefer fine-grained constructivism. By contrast, if fine-grained constructivists cannot predict and detect patterns of nonrandomness in the fluctuations, then we have reason to prefer misconceptions constructivism. In this section, I describe two analyses of pilot-study data about which fine-grained constructivism — specifically, the WYSIWYG activation claim — makes a prediction, while misconceptions constructivism makes no prediction. My main point is methodological: experimental data of this type, collected over a sufficiently diverse collection of experiments, can eventually favor one flavor of constructivism at the expense of the other. 4.1. Analysis 1: Final exam question in a summer class about representations In the MaRC project 1997 summer class about representations (see diSessa & Sherin, 2001, this issue), the final exam included this item: Cars A and B start at the same position and move according to the graph of speed vs. time [Fig. 6]. a. Is car A going forward or backward? What about car B? b. What happens at time T1? Circle the correct response. i. Car B is ahead. ii. Car A is ahead. iii. Neither car is ahead; car B and car A cross each other.

I call Parts (a) and (b) the direction question and the crossing question, respectively. What do the two flavors of constructivism predict about students’ answers? Within the misconceptions framework (see Section 2.2), many mistakes are expected to stem from reading the velocity graph as a position graph, a manifestation of the height vs. slope confusion (Leinhardt et al., 1990).11 Therefore, if someone has confronted and replaced that misreading, they will answer both direction and crossing correctly. If the replacement has not occurred, the student will answer both questions incorrectly. Finally, students in a transitional state could make ‘‘random errors’’ about direction and crossing. Random errors could stem from other causes, as footnote 11 discusses. But nothing more can be said. Misconceptions are generally described as applying robustly across multiple contexts. For this reason, misconceptions constructivism cannot predict whether more errors will occur on direction or on crossing. A fine-grained constructivist who accepts my WYSIWYG activation claim has more to say. On the direction question, WYSIWYG applied to the two oppositely sloped lines on the graph leads to the incorrect conclusion that the cars go in different directions. On the crossing 11

The misconceptions framework allows for errors that do not arise from a misconception. Examples include knowledge-gap errors, such as lack of awareness that the area under a velocity vs. time graph represents displacement; perceptual errors, such as seeing the area under curve A (between t = 0 and t = T1) as no bigger than the area under curve B; and ‘‘careless’’ errors in processing the information. But the misconceptions framework says nothing about which contexts and which tasks are more likely to evoke these types of errors. In this scenario, the only prediction put forth by that framework is the expectation that some (many?) students will exhibit the height vs. slope confusion, reading the velocity graph as if it were a position graph.

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Fig. 6. Graph for MaRC summer exam question.

question, WYSIWYG applied to the intersection of the graphs leads to the incorrect conclusion that the cars cross at time T1. Given that our optical systems are hard-wired to detect corners (Churchland & Sejnowski, 1992), and given the sparseness of distinct features on the graph, the intersection is likely to be the compelling visual attribute. So, according to the WYSIWYG activation claim, students are more likely to cue WYSIWYG — and therefore, to get the wrong answer — on the crossing question. By contrast, as just explained, misconceptions constructivism gives us no reason to expect more wrong answers on one question than on the other. Fine-grained constructivism makes a specific prediction concerning a distribution of incorrect answers. Misconceptions constructivism makes no such prediction. This empirical difference is the main point of my article. As it turns out, partly because the class spent little time on velocity graphs, only one student out of nine got both direction and crossing right. Two students got both direction and crossing wrong, choosing (iii) on Part (b). The other six students got direction correct, but incorrectly concluded that cars A and B cross at time T1. The disproportionate number of errors on crossing counts as evidence for the fine-grained account.12 Of course, to turn this pilot study into a more solid result, we would need to confirm that the intersection is the compelling visual attribute, and we would need a larger sample size. Again, my point is that such experiments are feasible. 4.2. Analysis 2: Homework problem given in high school physics This analysis from September 1998 nearly duplicates the one just described, but with different students and context. The subjects were eleventh-grade physics students at a science/ technology ‘‘magnet’’ public high school in Virginia. After completing a series of microcomputer-based laboratories using motion detectors, 71 students completed a homework 12

If we focus on students who got exactly one of those two questions wrong, and assume that those errors are randomly distributed like coin flips, then the probability that all six such students would err in the direction predicted by fine-grained constructivism is P = 1/64 = .016.

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assignment about position and velocity graphs. The assignment included this item from the ‘‘Tools for Scientific Thinking’’ physics labs (Thornton, 1987): Both the velocity graphs below, 1 and 2, show the motion of two objects, A and B [Fig. 7]. Answer the following questions separately for 1 and 2. Explain your answers when necessary. (a) Is one faster than the other? If so, which one is faster? (A or B) (b) What does the intersection mean? (c) Can one tell which object is ‘‘ahead’’? (define ‘‘ahead’’) (d) Does either object reverse direction? Explain.

I coded students’ written responses concerning graph 2. The crossing question is Part (b). No question directly asks about the direction of motion of each object. But in Part (d), most students explicitly indicated whether they thought objects A and B move in the same direction or in opposite directions, thereby providing an unambiguous answer to the direction question.13 As discussed above, fine-grained constructivism, but not misconceptions constructivism, makes a prediction about students who incorrectly answer exactly one of those two questions. The fine-grained account expects more mistakes about crossing than about direction. Of the 67 students who answered unambiguously, 49 got both questions correct, 9 got both questions wrong, and 9 got one right and one wrong. Of the nine students who incorrectly answered exactly one of those questions, seven missed the crossing question and two missed the direction question,14 in agreement with the results of the other pilot study (Section 4.1). A misconceptions advocate might claim that a fine-grained constructivist framework cannot explain why many students (nine, in this experiment) answered both crossing and direction as if they were consistently interpreting the velocity graph as a position graph. But fine-grained constructivism does not require students to display pervasive inconsistency.15

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Typical student responses included ‘‘Both cars go the same direction, but A decelerates’’ and ‘‘No, both cars always have positive velocity.’’ In some cases, a student’s answer to part (c), such as ‘‘A starts ahead, but B catches up and passes it’’ clarified an otherwise ambiguous answer to part (d), such as ‘‘No, both cars the go the same way the whole time.’’ However, for 4 of the 71 students, an answer to the direction question could not be unambiguously inferred. Part (c), when coded separately, did not yield data that could be used to support one flavor of constructivism at the expense of the other. For instance, many students’ answers were consistent with — and even explicitly referred to — their part (b) answers. A misconceptions constructivist could argue that this consistency stems from a robust misconception, whereas a fine-grained constructivist could argue that the knowledge elements activated by part (b) are still turned on when the student addresses part (c) a few seconds later, and that some students consciously seek to answer neighboring questions consistently. Along the same lines, many students’ part (c) answers were consistent with their part (d) responses. Again, a misconceptions advocate would claim that this consistency stems from a misconception, whereas a fine-grained advocate could claim that (c) and (d) ‘‘go together’’ partly because neither involves a compelling visual attribute. My point is that the cleanest way to drive a wedge between the two flavors of constructivism is to analyze (b) and (d). 14 If the careless errors among those nine students were randomly distributed like coin flips, then the probability that seven or more students would err in the direction predicted by the fine-grained account is P = .090. 15 For instance, in an analysis of students’ conceptions about the origin of species, Samarapugnavan and Wiers (1997) take students’ internal consistency as evidence against a diSessa-style fine-grained account of students’ preconceptions.

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Fig. 7. Velocity graphs for a high school homework question.

On the contrary, fine-grained constructivism gives us reason to expect that students will often be consistent. For instance, as now argued, it can explain why some of those nine students got both crossing and direction wrong. 





First, if a student consciously thinks of the graph as indicating position, then the same representational and metacognitive resources that allow her to interpret position graphs correctly will lead her to get crossing and direction wrong. Second, several different clusters of intuitive representational knowledge elements lead to consistently wrong answers about crossing and direction. For instance, the student might ‘‘see’’ (i) the graphs as ‘‘pictures’’ of the paths of the cars; or (ii) the slope as indicating the amount and direction of motion; or (iii) the height as indicating distance traveled. In a fine-grained story, these three interpretations could stem from the activation of different (though overlapping) sets of intuitive representational knowledge elements. In experts, (ii) and (iii) are tightly and consciously linked; when an expert interprets the height of a graph as representing position, she automatically interprets its slope as representing velocity. In novices, by contrast, (ii) could get triggered without (iii), or vice versa. The activation of (i), (ii), or (iii) can explain why a student who does not hold a stable, robust misconception might answer both crossing and direction incorrectly. Third, when answering a series of related questions, some students monitor — consciously or unconsciously — the coherence of the story they are constructing (see Schoenfeld, 1992). For instance, suppose a student says that the cars cross when the velocity graphs cross. On subsequent questions, the student’s consistency monitoring might lead him to give extra weight to intuitive knowledge elements that seem consistent with his earlier conclusion. The student’s consistency stems not from a preexisting misconception, but from a productive metacognitive constraint on the process by which he constructs answers out of his intuitive knowledge.

Again, as mentioned in Section 2.2, fine-grained constructivists do not deny that a cluster of intuitive knowledge elements can come together to form a misconception. But

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in a fine-grained framework, the misconception is not assumed always to be stable and robust across multiple contexts. Rather, it is sometimes expected to be emergent knowledge that arises in some contexts but not in others. Importantly, the misconception arises from finer-grained knowledge elements that, in other contexts, serve a useful function. In this way, the fine-grained framework reinterprets rather than denies the phenomenology of ‘‘misconceptions.’’

5. Another method for distinguishing the two flavors of constructivism In Section 4, the two pilot studies employed the same methods, namely, the coding of students’ written work produced in a classroom setting. A full-fledged experimental program to decide which flavor of constructivism is best, however, must triangulate among multiple methods in order to produce convincing results. With that in mind, I now show how clinical interview transcripts can be used to argue for one flavor of constructivism at the expense of the other. In a clinical setting, Jeff Friedman interviewed pairs of high school students to uncover the prior knowledge they bring to bear when interpreting visual representations of digitized astronomical images (Friedman & diSessa, 1999). Sections of his transcripts focus on students’ interpretation of slice graphs. The Slice Tool is a component of the software used to view and manipulate images. Specifically, the student uses the mouse to draw a line (‘‘slice’’) across the displayed astronomical image on the screen. The computer then draws a graph showing brightness as a function of distance along that line, as illustrated by Fig. 8. The vertical axis shows brightness counts, as recorded by the digital camera that took the image. The distance on the horizontal axis is measured in pixels.

Fig. 8. Slice graph of moon crater with a mountain in the middle.

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During Friedman’s interviews, students used slice graphs and other information to address questions about craters and peaks on the moon. I now lay out the empirical distinction between fine-grained and misconceptions constructivism regarding students’ interpretation of slice graphs. When slicing the moon, students often seem to misinterpret the slice graph as showing altitude (height) instead of brightness (Friedman & diSessa, 1999). According to misconceptions constructivists, this phenomenon is easy to explain: Students are indeed misinterpreting the slice graph as an altitude graph. Within this framework, some students are expected to interpret the slice graph as showing altitude, some are expected to interpret it as showing brightness, and others are expected to fluctuate between the two interpretations. Crucially, misconceptions constructivism makes no predictions about which particular questions or contextual cues are most likely to induce fluctuations in a given direction. By contrast, a fine-grained constructivist has a story to tell about the so-called fluctuations. Some slice graphs contain a compelling visual attribute, such as a sharp peak or deep valley. According to the WYSIWYG activation claim, students are more likely to interpret a visually compelling ‘‘peak’’ or ‘‘valley’’ of the slice graph as an actual peak or valley than they are to apply a WYSIWYG interpretation to other aspects of the representation. In other words, a student is disproportionately likely to misinterpret the graph as showing altitude when he is focused upon the most visually compelling ‘‘peaks’’ or ‘‘valleys’’ of the slice graph. So, once again, misconceptions constructivism predicts nothing more than fluctuations, while finegrained constructivism predicts a particular pattern in students’ reasoning. The following episode from Friedman’s transcripts illustrates the kind of data that count as evidence for fine-grained constructivism. Two students, L and H, are discussing how to decide which is higher, a crater wall or a mountain peak. After working directly with a printout of the image for several minutes they decide that using a slice graph might be helpful. L explains why: Interviewer:

L: H: Interviewer: H: L: H: L:

Can you think of any; uh, if you were on the computer, can you think of any other, anything else you could do to; anything else you could do to uh, to find, to compare the two? Would it be helpful in any way if you were on the computer? I’m sure it would, but I can’t really think right now of how I would go about doing that. Um. Did the slice graph have anything to do with like the height, or was it just distance? Why don’t the two of you discuss that. Never mind. No, no, I know what you’re talking about . . . Cause like I forgot what the, what the thing . . . distance and count (talking over each other). But the light counts, but then you would be able to figure something because the shadow; because I think the shadow would have a lot to do with it because the sun’s obviously hitting, you know, these at the same

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point so if this is taller, then this is going to have a bigger shadow, you know. But if we did use the slice graph, and we sliced this or whatever, we could see how um, what, how long this distance is at that certain light intensity, and then how long this distance is at that light intensity. I mean, I feel the light intensity is like around the same, so we can just see like until the light intensity gets back to all this the, what the distance of that is, and then if that’s bigger. So the slice graph would probably work then, I think. [emphases added] Although H initially thinks the slice graph might have something ‘‘to do with . . . the height,’’ L clearly and repeatedly states that it shows light intensity. She explains how slice graphs of the shadows created by the crater wall and the mountain peak could be used to determine which shadow covers more distance, and hence, which object is higher. For the next several minutes, the two students deal with the logistical details of creating and reading the slice graph. L never wavers from her correct contention that slice graphs show light intensity vs. distance. But when she gets one of the needed slice graphs in front of her, it contains a wide and deep ‘‘valley’’ corresponding to the shadow. As a result, she briefly switches to an altitude interpretation of that feature before catching herself: L:

Well, let’s figure out which counts are, which counts, like; See how this gets really low right here so it would be like right here, so the counts are lower where it’s lower down I guess, and then the counts are higher um. Oh, but that’s light intensity. So the light intensity for this. Like you can see how it’s brighter right there, you know? [emphasis added]

The first italicized phrase suggests that WYSIWYG got triggered by a compelling visual attribute, the lowest valley on the slice graph. Consequently, L thinks lower on the graph means lower on the moon; ‘‘lower means lower.’’ But then, when she considers a less visually compelling part of the graph, a place where the ‘‘counts are higher,’’ WYSIWYG gets cued less strongly, and some of L’s other knowledge takes over. She says, ‘‘Oh, but that’s light intensity,’’ indicating that, in the previous few moments, she was interpreting the slice graph as showing something other than light intensity. Realizing her mistake, she now returns to a light-intensity interpretation: ‘‘Like you can see how it’s brighter right there, you know?’’ In response to this story, a critic could argue as follows: Critic:

L’s statements occurred in the context of trying to use slice graphs to locate the highest peaks. Because she was looking for peaks and valleys, that’s exactly what L saw when a plausible ‘‘valley’’ presented itself on the slice graph. It’s just a matter of seeing what you’re looking for, not a matter of a compelling visual attribute cueing a naive interpretation.

This critique gives us reason to doubt that my WYSIWYG activation claim fully explains L’s behavior. But it does not refute my claim that L’s statements are best explained within a fine-grained constructivist framework rather than a misconceptions framework. A misconceptions constructivist could claim that L’s brief, isolated reversion to an altitude interpretation was just a random fluctuation. The above critique, however, does not take this line.

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Rather, it says that a contextual factor — namely, L’s goal — predisposed her towards an iconic ‘‘valley’’ interpretation of the dip; and presumably, once the goal was fulfilled, intuitive knowledge elements associated with other interpretations could assert themselves more strongly. In summary, the critique’s focus on context-dependent activation of interpretive resources places it squarely within the fine-grained camp. Since WYSIWYG and the associated activation claim are just small pieces of a fine-grained theory of intuitive representational knowledge, other knowledge elements and cognitive processes — such as the tendency to see what you are looking for — undoubtedly play a role in explaining Friedman’s data. Again, the point of this article is not to argue for the completeness of my particular fine-grained explanations, but rather, to show that fine-grained constructivism generates predictions that go beyond those made by misconceptions constructivism, and that the data give us reason to take fine-grained constructivism seriously.

6. Conclusion Educators can have different takes on the nature of the debate between misconceptions constructivism and fine-grained constructivism: 





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‘‘It is just semantic.’’ Perhaps people who talk about ‘‘misconceptions’’ and people who talk about ‘‘preconceptions,’’ or ‘‘intuitive resources,’’ or ‘‘alternative theories,’’ are all talking about the same cognitive structures. They disagree only about word choice. ‘‘It is about valuing students’ ideas.’’ Upon discovering that students have misconceptions, some teachers become exasperated about their students’ wrongheaded ideas, while other teachers get excited about the ability of their students to reason in terms of ideas they did not learn in the classroom (see Hammer, 1997). Because these two sets of teachers place different value on students’ ideas, they often favor different pedagogical strategies for dealing with misconceptions. Vigorous debates can therefore arise. But the debates are not necessarily about the cognitive structures underlying the misconceptions. ‘‘It is purely theoretical.’’ When Einstein proposed his special theory of relativity, mathematically encoded in the Lorentz transformation equations, Lorentz had already derived those equations from a complicated ether model of electromagnetic propagation. So, the disagreement between the two theories was not empirical; it was ontological. The two theories attributed different properties to space and time, but generated empirically indistinguishable predictions (within a certain restricted domain of phenomena).16 Similarly, it is possible to view the fine-grained and misconceptions constructivists as disagreeing only about what cognitive structures to attribute to students, not about empirical predictions.

Furthermore, just as new epicycles could ‘‘rescue’’ the Ptolemaic model of the solar system, modifications to the ether model could rescue it from subsequent empirical findings, up to a point.

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This article shows that none of these three takes fully captures the disagreement between fine-grained and misconceptions constructivists. First, they posit different cognitive structures, not just different words to label the same structures; a stable, context-independent belief that is either correct or incorrect is a different thing from a fine-grained minigeneralization, the activation and appropriateness of which depends on context. Second, a teacher can respect and value a student’s intuitive knowledge, no matter what form she thinks it takes. Third, the different cognitive structures posited by the two flavors of constructivism are not empty theoretical baggage; they lead to empirical differences, as shown in Section 4. In summary, misconceptions constructivism and fine-grained constructivism, when taken seriously, do not disagree only about word choice, or about attitudes toward students, or about cognitive structures. They make different sets of predictions about students’ interpretations of representations. Therefore, by fleshing out a fine-grained theory of intuitive representational knowledge, and by experimentally exploring numerous situations in which the fine-grained account makes a specific prediction while the misconceptions account makes no prediction (or makes a different prediction), we can gain insight into which flavor of constructivism best describes students’ knowledge. To make this argument, I first proposed the existence of WYSIWYG, an intuitive knowledge element about representations, according to which x means x. I argued that particularly useful visual attributes tend to become connected to quick and direct interpretations (many of which involve WYSIWYG) that call attention to themselves (are compelling). As a result, compelling visual attributes end up with strong connections to WYSIWYG. Because it leads to naively iconic interpretations, WYSIWYG gets cued less strongly and less frequently as students gain experience with abstract representations. But the link between a compelling visual attribute and its corresponding WYSIWYG interpretation does not die off. In other words, the compelling visual attribute tends to cue WYSIWYG.

Using this WYSIWYG activation claim, I generated predictions about students’ interpretation of graphs, predictions that go beyond those made by misconceptions constructivism. Pilot studies established the feasibility of testing these kinds of predictions, and also gave us reason to take fine-grained constructivism seriously. I briefly review why, in general, the empirical disagreement arises. Misconceptions constructivism views conceptual change as a switch from one set of stable, robust conceptions to another. Therefore, nothing (other than random fluctuations) can be predicted about the fine structure of the transition. By contrast, the context-dependent cueing inherent to fine-grained constructivism leads to hypotheses about the fine structure of these transitions, hypotheses that predict particular patterns of nonrandomness in the fluctuations. I close by pointing out an instructional implication of favoring one flavor of constructivism over the other. As discussed in Section 2, fine-grained knowledge elements that are unproductive in some contexts can be productive in others; those elements are neither ‘‘right’’ nor ‘‘wrong.’’ Therefore, teachers can view the knowledge elements as useful raw material out of which students can construct more sophisticated understandings. By contrast, since misconceptions are cross-contextually stable and inconsistent with

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expert knowledge, teachers cannot view them as contributing to expert understanding (Hammer, 1996a, 1996b).

Acknowledgments I would like to thank Andy diSessa and David Hammer for excellent feedback and editing suggestions. This work was supported by NSF grant DGE-9714474 (Andrew Elby, PI). The ideas expressed here are those of the author, not necessarily those of the NSF.

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