What students' learning of representations tells us about constructivism

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 constructi-

vists, 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 minigeneral-

izations 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: S0732 -3123 (01 )00054 -2

Journal of Mathematical Behavior

19 (2000) 481–502

predictions about student behavior, and that (2) pilot studies show the feasibility of a full-

fledged 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 knowl-

edge 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 circum-

stances. 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 constructi-

vism, 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.

A. Elby / Journal of Mathematical Behavior 19 (2000) 481–502482

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 mis-

conceptions 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 misconcep-

tions 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 minigener-

alizations 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

2 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.

A. Elby / Journal of Mathematical Behavior 19 (2000) 481–502 483

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 fine-

grained framework, the flat horizontal line can be taken to activate stillness, an element of

4 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.


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 fine-

grained 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.

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

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.


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 Quebec a Montreal, 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

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.

6 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.


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

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


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 circum-

stances. 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 well-

developed 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.


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.

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


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 prefer-

entially 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.10 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.


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. TheWYSIWYG activation

claim, which is one small part of a fine-grained constructivist account of intuitive representa-

tional 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 fine-

grained 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


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.


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 cueWYSIWYG— 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 micro-

computer-based laboratories using motion detectors, 71 students completed a homework

Fig. 6. Graph for MaRC summer exam question.

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.


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

13 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.


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 repre-

senting 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

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


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 knowl-

edge 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.


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 con-

structivism 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 constructi-

vism 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 fine-

grained 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 print-

out of the image for several minutes they decide that using a slice graph might be helpful. L

explains why:

Interviewer: 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?

L: I’m sure it would, but I can’t really think right now of how I would go

about doing that. Um.

H: Did the slice graph have anything to do with like the height, or

was it just distance?

Interviewer: Why don’t the two of you discuss that.

H: Never mind.

L: No, no, I know what you’re talking about . . .H: Cause like I forgot what the, what the thing . . . distance and count

(talking over each other).

L: 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


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 . . . theheight,’’ 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 miscon-

ceptions constructivist could claim that L’s brief, isolated reversion to an altitude interpre-

tation was just a random fluctuation. The above critique, however, does not take this line.


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 inter-

pretive 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 con-

structivism, 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:

� ‘‘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.

16 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.


This article shows that none of these three takes fully captures the disagreement between

fine-grained and misconceptions constructivists. First, they posit different cognitive struc-

tures, 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 ofWYSIWYG, 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’ interpre-

tation 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 understand-

ings. By contrast, since misconceptions are cross-contextually stable and inconsistent with


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.

References

Carey, S. (1992). The origin and evolution of everyday concepts. In: R. N. Giere (Ed.), Cognitive models of

science (vol. XV, pp. 89–128). Minneapolis: University of Minnesota Press.

Churchland, P. S., & Sejnowski, T. J. (1992). The computational brain. Cambridge, MA: MIT Press.

diSessa, A. (1982). Unlearning Aristotelian physics: a study of knowledge-based learning. Cognitive Science, 6,

37–75.

diSessa, A. (1993). Towards an epistemology of physics. Cognition and Instruction, 10 (2/3), 105–225.

diSessa, A., & Sherin, B. (2001). Meta-representation: an introduction. Journal of Mathematical Behavior, 19 (4),

385–398.

Friedman, J. S., & diSessa, A. A. (1999). What students should know about technology: the case of scientific

visualization. Journal of Science Education and Technology, 8 (3), 175–195.

Halloun, I. A., & Hestenes, D. (1985). The initial knowledge state of college physics students. American Journal

of Physics, 53 (11), 1043–1056.

Hammer, D. (1996a). Misconceptions or p-prims: how may alternative perspectives of cognitive structure influ-

ence instructional perceptions and intentions? Journal of the Learning Sciences, 5 (2), 97–127.

Hammer, D. (1996b). More than misconceptions: multiple perspectives on student knowledge and reasoning, and

an appropriate role for education research. American Journal of Physics, 64 (10), 1316–1325.

Hammer, D. (1997). Discovery learning and discovery teaching. Cognition and Instruction, 15 (4), 485–529.

Hammer, D., & Elby, A. (in press). On the form of a personal epistemology. In B. K. Hofer, & P. R.

Pintrich (Eds.), Personal epistemology: the psychology of beliefs about knowledge and knowing. Mahwah,

NJ: Erlbaum.

Janvier, C. & Universite du Quebec a Montreal, Centre interdisciplinaire de recherche sur l’apprentissage et le

developpement en education. (1987). Problems of representation in the teaching and learning of mathematics.

Hillsdale, NJ: Hillsdale, NJ: Erlbaum.

Leinhardt, G., Zaslavsky, O., & Stein, M. M. (1990). Functions, graphs, and graphing: tasks, learning and

teaching. Review of Educational Research, 60, 1–64.

Maloney, D. P., & Siegler, R. S. (1993). Conceptual competition in physics learning. International Journal of

Science Education, 15 (3), 283–296.

McCloskey, M. (1983a). Intuitive physics. Scientific American, 249, 122.

McCloskey, M. (1983b). Naive theories of motion. In: D. Gentner, & A. Stevens (Eds.), Mental models (pp. 299–

324). Hillsdale, NJ: Erlbaum.

McCloskey, M., Carramazza, A., & Green, B. (1980). Curvilinear motion in the absence of external forces: naive

beliefs about the motion of objects. Science, 210, 1139–1141.

McDermott, L. C. (1984). Research on conceptual understanding in mechanics. Physics Today, 37, 24–32.

McDermott, L. C., Rosenquist, M. L., & van Zee, E. H. (1987). Student difficulties in connecting graphs and

physics: examples from kinematics. American Journal of Physics, 55, 505–513.


Samarapugnavan, A., & Wiers, R. W. (1997). Children’s thoughts on the origin of species: a study of explanatory

coherence. Cognitive Science, 21 (2), 147–177.

Schoenfeld, A. H. (1992). Learning to think mathematically: problem solving, metacognition, and sense

making in mathematics. In: D. Grouws (Ed.), Handbook for research on mathematics teaching and learning

(pp. 334–370). New York: Macmillan.

Sherin, B. (2001). How students invent representations of motion: a genetic account. Journal of Mathematical

Behavior, 19 (4), 399–441.

Smith, J., diSessa, A., & Roschelle, J. (1993/1994). Misconceptions reconceived: a constructivist analysis of

knowledge in transition. Journal of the Learning Sciences, 3 (2), 115–163.

Steinberg, R. N., & Sabella, M. S. (1997). Performance on multiple-choice diagnostics and complementary exam

problems. Physics Teacher, 35 (3), 150–155.

Strike, K. A., & Posner, G. J. (1985). A conceptual change view of learning and understanding. In: L. H. T. West,

& A. L. Pines (Eds.), Cognitive structure and conceptual change (pp. 211–231). New York: Academic Press.

Strike, K. A., & Posner, G. J. (1992). A revisionist theory of conceptual change. In: R. A. Duschl, & R. J.

Hamilton (Eds.), Philosophy of science, cognitive psychology, and educational theory and practice (pp. 147–

176). Albany: State University of New York Press.

Thornton, R. (1987). Tools for scientific thinking: microcomputer-based laboratories for physics teaching. Physics

Education, 22, 230–238.

Thornton, R. (1995). Conceptual dynamics: changing student views of force and motion. In: C. Tarsitani,

C. Bernardini, & M. Vincentini (Eds.), Thinking physics for teaching. London: Plenum.

Tirosh, D., Stavy, R., & Cohen, S. (1998). Cognitive conflict and intuitive rules. International Journal of Science

Education, 20 (10), 1257–1269.

Tytler, R. (1998). The nature of students’ informal science conceptions. International Journal of Science Educa-

tion, 20 (8), 901–927.


Documents

What students' learning of representations tells us about constructivism