What students' learning of representations tells us about constructivism

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<ul><li><p>What students learning of representations tells us</p><p>about constructivism</p><p>Andrew Elby</p><p>Physics Department, University of Maryland, College Park, MD 20742, USA</p><p>Abstract</p><p>This article pulls into the empirical realm a longstanding theoretical debate about the prior</p><p>knowledge students bring to bear when learning scientific concepts and representations.</p><p>Misconceptions constructivists view the prior knowledge as stable alternate conceptions that apply</p><p>robustly across multiple contexts. By contrast, fine-grained constructivists believe that much of</p><p>students intuitive knowledge consists of unarticulated, loosely connected knowledge elements, the</p><p>activation of which depends sensitively on context. By focusing on students intuitive knowledge</p><p>about representations, and by fleshing out the two constructivist frameworks, I show that they lead to</p><p>empirically different sets of predictions. Pilot studies demonstrate the feasibility of a full-fledged</p><p>experimental program to decide which flavor of constructivism describes students more adequately.</p><p>D 2001 Elsevier Science Inc. All rights reserved.</p><p>Keywords: Constructivism; Misconceptions; p-prims; Conceptual change; Representations</p><p>1. Introduction</p><p>Students interpretations of graphs and other representations can shed empirical light on a</p><p>longstanding theoretical debate about science learning. To spell out my claim, I must</p><p>distinguish between two flavors of constructivism. According to misconceptions constructi-</p><p>vists, students walk into the classroom with alternate conceptions and theories (McCloskey,</p><p>1983b; Strike &amp; Posner, 1985). By contrast, fine-grained constructivists believe that much of</p><p>students intuitive knowledge consists of loosely connected, often inarticulate minigeneral-</p><p>izations and other knowledge elements, the activation of which depends heavily on context</p><p>(Hammer, 1996a; Smith, diSessa, &amp; Roschelle, 1993/1994; Tirosh, Stavy, &amp; Cohen, 1998). I</p><p>will show that (1) these two frameworks, when fleshed out, lead to different sets of</p><p>0732-3123/00/$ see front matter D 2001 Elsevier Science Inc. All rights reserved.</p><p>PII: S0732 -3123 (01 )00054 -2</p><p>Journal of Mathematical Behavior</p><p>19 (2000) 481502</p></li><li><p>predictions about student behavior, and that (2) pilot studies show the feasibility of a full-</p><p>fledged experimental program to decide which flavor of constructivism describes students</p><p>more adequately, and also give us reason to take fine-grained constructivism seriously. So,</p><p>this article operationalizes a debate usually conducted on a theoretical plane.</p><p>First, I explicate the two flavors of constructivism, underscoring the difference between</p><p>representational misconceptions and finer-grained, context-dependent intuitive knowl-</p><p>edge elements. Unlike diSessa (1993), who lays out a detailed account of students intuitive</p><p>knowledge about physics, I provide only a sketchy overview of students intuitive</p><p>knowledge about representations. But I begin to build a fuller account by spelling out</p><p>the nature and activation tendencies of one crucial representational knowledge element.</p><p>Finally, using this element, I show that the fine-grained and misconceptions frameworks</p><p>make empirically different sets of predictions about students behavior in certain circum-</p><p>stances. Pilot studies demonstrate a method of putting those different sets of predictions to</p><p>the test.</p><p>2. Two flavors of constructivism</p><p>In this section, I tease apart misconceptions constructivism and fine-grained constructi-</p><p>vism, focusing initially on students intuitive knowledge about physics, which is a heavily</p><p>researched topic (diSessa, 1982; Halloun &amp; Hestenes, 1985; McCloskey, Carramazza, &amp;</p><p>Green, 1980; McDermott, 1984). Then, I explore a less-traveled terrain, students intuitive</p><p>knowledge about representations.</p><p>2.1. Two flavors of students intuitive knowledge about physics</p><p>For my purposes, a constructivist is someone who believes the following: Learners do</p><p>not walk into the classroom as blank slates ready to be filled with knowledge. Instead, as</p><p>students construct a new understanding, their prior knowledge plays a crucial role.</p><p>Within this broad framework, however, different camps tell different stories about the</p><p>structure of this prior knowledge and the mechanism of conceptual change. According to</p><p>misconceptions constructivists such as McCloskey (1983a) and Strike and Posner (1985),</p><p>students prior knowledge consists largely of noncanonical conceptions and theories. For</p><p>instance, according to McCloskey (1983b), students intuitive knowledge about mechanics</p><p>resembles the impetus theory believed by natural philosophers in the middle ages. This</p><p>alternate theory contains the misconception motion requires force, according to which an</p><p>object in motion requires a force to keep it moving.1 By assuming that the misconception</p><p>exists as a comparatively stable knowledge element inside peoples heads, we can explain</p><p>why students mistakenly think that a desk being pushed across the floor at constant velocity</p><p>1 By contrast, according to Newtonian physics, a net force is required to initiate or change motion, but not to</p><p>maintain motion at constant velocity.</p><p>A. Elby / Journal of Mathematical Behavior 19 (2000) 481502482</p></li><li><p>feels a net forward force. And to explain why some of these same students hold the</p><p>apparently contradictory belief that a ball thrown in outer space keeps drifting indefinitely,2</p><p>we can flesh out a story of competing conceptions (Maloney &amp; Siegler, 1993) or of a</p><p>transition stage during which the student fluctuates between her original misconception and</p><p>the new conception she is learning (Thornton, 1995). In this way, the misconceptions</p><p>framework can accommodate inconsistencies in students reasoning. However, since mis-</p><p>conceptions are assumed to be somewhat theory-like, or are at least described in general terms</p><p>that are not linked to particular contexts, the misconceptions framework cannot make</p><p>predictions about the contexts in which fluctuations are most likely to occur.</p><p>Within misconceptions constructivism, the process of conceptual change resembles the</p><p>mechanism by which scientific communities are claimed to alter their theories (Strike &amp;</p><p>Posner, 1985). When confronted with evidence that contradicts her old conceptions, and</p><p>when made aware of the difference between her old theory and the scientifically accepted</p><p>theory, the student becomes ready to accept the new theory. In brief, the students old</p><p>conceptions are confronted and then replaced.</p><p>Some misconceptions theorists, responding to research on conceptual change, allow for</p><p>the possibility that some misconceptions have internal cognitive structure and can be</p><p>compiled on the spot (Carey, 1992; Strike &amp; Posner, 1992). However, even in these</p><p>newer formulations, the misconception remains the primary unit used to describe and</p><p>analyze students conceptual reasoning. Furthermore, although the modified misconcep-</p><p>tions framework allows for learning mechanisms besides confront and replace, the</p><p>misconceptions are not considered to be part of the raw material out of which the student</p><p>builds a new understanding.</p><p>Fine-grained constructivists (Smith et al., 1993/1994; Tirosh et al., 1998), by contrast,</p><p>believe that much of students intuitive knowledge takes the form of inarticulate minigener-</p><p>alizations from experience, as Hammer and Elby (in press) explain:</p><p>In this framework, students reasoning about the desk and the ball can be understood in terms</p><p>of the context-specific activation of the following fine-grained resources. Maintaining</p><p>agency 3 is an element of cognitive structure useful for understanding any continuing effect</p><p>maintained by a continuing cause, such as a light bulb needing a continuous supply of energy</p><p>to stay lit. Actuating agency is another resource, an element of cognitive structure involved in</p><p>understanding an effect initiated by a cause, when the effect outlasts the cause, such as the</p><p>strike of a hammer causing a bell to ring. The desk scenario tends to activate Maintaining</p><p>agency, and hence, the idea that a continued net forward force is needed to keep the desk</p><p>2 I have observed this phenomenon in my high school physics students. Along the same lines, Steinberg and</p><p>Sabella (1997) show that many students, in response to a multiple-choice item and a free-response item probing</p><p>the same misconception, give inconsistent responses. Other evidence (diSessa, 1993; Tytler, 1998) also suggests</p><p>that students reasoning is inconsistent in ways that a misconceptions account can accommodate only by</p><p>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</p><p>agency need not take the form of a force. We will also use the name actuating agency instead of diSessas force</p><p>as mover.</p><p>A. Elby / Journal of Mathematical Behavior 19 (2000) 481502 483</p></li><li><p>moving forward. By contrast, the ball question tends to activate Actuating agency, and the</p><p>idea that the balls motion can outlast the force exerted by the thrower.4 Unlike the</p><p>misconception Motion requires force, the finer-grained cognitive resources Maintaining</p><p>agency and Actuating agency are not incorrect. Neither are they correct. They are resources</p><p>that can be activated under various circumstances, sometimes appropriately, sometimes not.</p><p>Furthermore, whereas the misconception is an element of cognitive structure specifically tied</p><p>to motion and force, the finer-grained resources also apply to light bulbs, bells, and numerous</p><p>other situations. In this sense, these resources are finer-grained but more general than</p><p>misconceptions (though some finer-grained resources might be tied more tightly to a</p><p>particular setting).</p><p>Within the fine-grained framework, conceptual change is not a matter of replacing bad</p><p>minigeneralizations with good ones. Instead, it is partly a matter of tweaking those</p><p>minigeneralizations into a more articulate, unified, coherent structure. For instance, when</p><p>constructing an understanding of Newtonian mechanics, actuating agency can serve as an</p><p>intuitive grounding of Newtons first and second laws, according to which a force is</p><p>needed to initiate or change motion but not to maintain motion (at constant velocity). In</p><p>Newtonian mechanics, maintaining agency (merely) contributes to an informal heuristic for</p><p>reasoning about situations involving strong friction or other dissipative forces. But</p><p>actuating agency and maintaining agency both play a role in a physicists reasoning. As</p><p>novices become experts, few if any minigeneralizations die completely. They are</p><p>restructured, not replaced.</p><p>In summary, misconceptions constructivism and fine-grained constructivism disagree not</p><p>only about the form of students intuitive knowledge, but also about the mechanism of</p><p>learning and conceptual change. Consequently, these two flavors of constructivism invite</p><p>different instructional practices, as Hammer (1996a) discusses. For both theoretical and</p><p>practical reasons, we must decide which kind of constructivism better accounts for students</p><p>behavior in various situations.</p><p>2.2. Two flavors of students intuitive knowledge about representations</p><p>To see how the distinction between misconceptions and fine-grained constructivism plays</p><p>out in the context of representations, consider this velocity vs. time graph (Fig. 1)</p><p>representing a cars motion.</p><p>Novices sometimes think the car is not moving. Within a misconceptions framework,</p><p>this is taken to show that students are misreading the velocity graph as a position graph, a</p><p>mistake the student is likely to make on other velocity graphs (see Leinhardt, Zaslavsky,</p><p>&amp; Stein, 1990; McDermott, Rosenquist, &amp; van Zee, 1987). By contrast, within a fine-</p><p>grained framework, the flat horizontal line can be taken to activate stillness, an element of</p><p>4 On this view, the internal force students often invoke in their explanations is not part of a stable,</p><p>preexisting misconception, but rather something they conceive of on the spot, if the context requires them to</p><p>explain the balls continued motion.</p><p>A. Elby / Journal of Mathematical Behavior 19 (2000) 481502484</p></li><li><p>cognitive structure associated with lack of motion. In this story, if stillness gets cued</p><p>when the student is thinking about the car itself, she is likely to conclude that it is</p><p>motionless. By contrast, if she is consciously thinking of the cars speedometer needle</p><p>when stillness gets activated (perhaps due to a teachers intervention), then she is more</p><p>likely to interpret the graph as indicating steady motion. So, the fine-grained account</p><p>predicts some context-dependent inconsistencies in whether the student interprets the</p><p>graph as if it indicates position instead of velocity. As noted above, misconceptions</p><p>constructivism accommodates inconsistencies in students reasoning. Therefore, in the</p><p>absence of a detailed story about these contextual dependencies, the fine-grained and</p><p>misconceptions stories do not make empirically distinguishable predictions. They agree</p><p>that students will sometimes read the velocity graph as if it were a position graph, and</p><p>that conscious reflection about what the graph represents can help students make fewer</p><p>such mistakes. A specification of contextual dependencies is what distinguishes fine-</p><p>grained from misconceptions constructivism.</p><p>Without going into detail, I will now propose some other intuitive knowledge elements that</p><p>students might bring to bear when interpreting visual representations. These speculations play</p><p>no role in my later arguments, but illustrate the fine-grained constructivist framework.</p><p>Constancy, triggered by straight lines on graphs (flat or sloped) and presumably by other</p><p>visual cues, corresponds to the idea that something about the situation does not change. For</p><p>instance, the activation of constancy might cause a novice musician to interpret a long</p><p>horizontal line on the staff as indicating that she should hold whatever note she is playing.5</p><p>Sudden change, cued by steep segments on a graph or by borders on a map, corresponds to</p><p>dramatic change.</p><p>In this framework, a misconception can emerge in a particular context, but perhaps not in</p><p>other contexts, by the (mis)activation of various fine-grained intuitive knowledge elements </p><p>elements which in other contexts might contribute to productive interpretations.</p><p>Fig. 1. A velocity vs. time graph for a car.</p><p>5 By the way, graphical expertise may consist in part of having constancy rather than stillness cued by a</p><p>horizontal line on a graph, since constancy is not tied to position, while stillness is.</p><p>A. Elby / Journal of Mathematical Behavior 19 (2000) 481502 485</p></li><li><p>3. A fine-grained intuitive knowledge element: WYSIWYG</p><p>3.1. What you see is what you get</p><p>A middle-school student, informed that Fig. 2 is a speed vs. time graph of a bicyclist,</p><p>speculates about what is happening between t1 and t2.</p><p>Experienced teachers and researchers, even if they have not seen this particular example in</p><p>the graphing misconceptions literature (Janvier &amp; the Universite du Quebec a` Montreal, 1987;</p><p>Leinhardt et al.,...</p></li></ul>

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