Subjects: Ninety three undergraduate students were the subjects for this experiment. The average age of these students was 19.8. 71.4 % of them were female and 28.6 % of them were male. 40.5 % of the students had taken college Physics, 19.0% of them had taken high school Physics, and 40.5% of them had never taken any physics.
Stimuli: An airplane flying above the ground was drawn on a piece of 8.5in×11in paper. A brief instruction on the top of the paper informs the subjects that the plane was 2000 feet above the ground and flying at the speed of 100 mph; a wheel suddenly dropped off.
Procedure and Design: Before being given the stimuli, all subjects were instructed that 1) the experiment is not a test; 2) the graph represents a real situation in the world; 3) try to visualize the situation when solving the problem. The subjects were then given the stimulus and were asked to do separate two tasks: 1) draw the path of the wheel as it falls; 2) draw the path of the wheel after it hits the ground. Upon finishing their drawing of the wheel's rolling on the ground, the subjects then completed this experiment.
Of the 93 subjects tested, 17 had drawn a straight-down path of the wheel; 37 had drawn forward paths in the direction of the plane's flying; and 39 had drawn backward paths opposite the direction of the plane's flying. Figure 12 illustrates these results.
Of the 17 straight-down responses, 8, in doing Task 2), drew a forward path on the ground after the wheel had landed, 5 drew a backward path, and 4 drew either no path at all or a path of a wheel's bouncing up and down. The 5 backward responses, which we had not seen in the last experiment, were categorized together with 8 forward responses as horizontal responses. Since there is no horizontal motion in either the responses that were staying still or those that bounced up and down, we call them vertical responses. Therefore, with respect to Task 2), we have 13 horizontal responses and 4 vertical responses. Figure 13 shows the frequency distribution of the vertical and horizontal responses to Task 2) of the 17 subjects. Comparing between the horizontal and the vertical responses, a Chi-square test gives X²=4.76, p<0.05, indicating that significantly more subjects responded with a horizontal path on the ground to Task 2).
Table 5 summarizes the responses to Task 2) by the subjects who had given either forward or backward responses to Task 1). Among the 39 subjects who gave backward responses to Task 1), 23 of them (59%) continued to give backward responses when asked to draw the path of the wheel after it landed on the ground; However, 13 of these subjects (33%) gave responses clearly indicating that the wheel would then move forward after it landed on the ground; Only 3 of these subjects (8%) chose that the wheel would stay where it landed. In comparison, 31 of the 37 subjects (84%) who gave forward responses to Task 1) believed that the wheel would go forward after it landed on the ground; Only 2 of the 37 subjects (5%) gave backward responses; and 4 of them (11%) indicated the wheel would stay where it landed. The wheel's sudden change of direction upon hitting the ground, in both the forward and the backward responses, is worth our attention. Interestingly, 13 out of 39 (33%) subjects who gave backward responses believed the wheel would reverse its direction upon hitting the ground. By contrast, only 2 out of 37 (5%) subjects who gave forward responses believed the wheel would reverse its direction upon hitting the ground. This proportional difference is significant (z=3.0576, p<0.001). A 2×3 Chi-square test comparing the forward responses and the backward responses in term of the paths on the ground also produced a significant effect (X2=10.29, p<0.01), showing, significant differences exist in how the wheel would roll after it landed on the ground between the forward responses and the backward responses.
This simple experiment shows first of all that for the horizontal motion of the wheel, though it can be misconceived in direction, its representation has never been destroyed by the more salient representations of the dynamic discontinuity and the gravitational motion. Recall, in the last experiment, the horizontal motion was totally absent in the responses of 16 subjects, an extreme reaction of these subjects to the dynamic discontinuity information in the stimulus, but then reemerged in the responses of the same subjects when gravity is kept in abeyance. In the present experiment, we saw that even when there is no abeyance of gravity, the absence of the horizontal motion in the responses is only temporary because it recovers later on the ground. This has been shown not only in the responses of those who drew straight-down paths of the wheel, but also in the forward and the backward responses. For the subjects who had drawn the straight-down responses to Task 1), significantly more of them gave horizontal responses to Task 2); For the subjects who had drawn forward and backward responses to Task 1), significantly more backward responses, in responding to Task 2), reversed the wheel's direction of motion on the ground than did the forward responses. All these indicate that the horizontal motion of the wheel, particularly the forward motion, had never been without its representation in the cognitive system of the subjects, even when they were drawing a straight-down path or a backward falling path.
Explaining the Backward Responses:
According to the analysis in the last experiment, a cognitive over-sensitivity to the dynamic discontinuity information in the pendulum problem had caused most subjects to include a spatial discontinuity in the path of the pendulum bob at position O. In the conception of some subjects, the dynamic discontinuity information in the given event had such a strong saliency that the representation of horizontal motion was totally subdued while the operation of gravity was being conceived. In the responses of most subjects, however, the horizontal motion, though with the direction sharply altered for the exactly same reason, would persist and coexist with the gravitational motion.
While the same basic conclusion also applies to the straight-down responses in the present experiment, the responses of the subjects now appear to be further influenced by their subjective experience of riding on a fast-moving vehicle. In addition to the effect of the dynamic discontinuity information, the decisions of the subjects on how the wheel would fall were further influenced by the conception of a backward motion cognitively attributed to the wheel as it drops off the plane.
The apparent explanation is that the subjects had formed the representation of a backward motion purely through watching from inside a moving vehicle an object falling off the vehicle. Remembering the object's path of motion with reference only to themselves (possiblly at a very young age) and the vehicle, the subjects had learned to conceive an object's falling from a fast-moving vehicle as traveling backward and subsequently generalized such a conception to cases even when they are no longer in the vehicle.
An alternative explanation may be possible. In the conception of the subjects, the backward motion, though appearing to be quite different from that of the straight-down motion, shares with it in fact a common feature --- a dynamic change in the mechanical state of the wheel has occurred --- the force which had been pulling the wheel forward suddenly stopped. In other words, a dynamic discontinuity occurred in the horizontal direction in the form of the sudden termination of a force that had been pulling the wheel forward. Therefore, what makes different a backward response from a vertical response in the conception of the subjects is that now the dynamic discontinuity occurs in the backward direction, rather than in the direction of gravity. Since the cognitive system is very sensitive to a dynamic discontinuity in the mechanical state of an object, it over-reacted to such a discontinuity by conceiving the wheel as moving backward.
Salient vs. Latent Dynamic Representations:
We have seen that the subjects, in responding to the mechanical situations given to them, greatly overstated the effect of a dynamic discontinuity and distinctively invented a backward motion for the wheel of the airplane. As in the last experiment, the representation of the forward motion by most subjects was found to be unmistakably distorted and even totally absent in a number of responses. However, as we have also found, the representation of the forward motion had never disappeared from the minds of the subjects as they were drawing straight-down or backward responses because it was found to be completely recoverable in the same subjects.
Recall, in the last experiment, the responses of the subjects varied according to their level of physics education. The results suggested that physics education appears to play the role of correcting and fine-tuning the subjects' commonsense physical concepts, which enable the subjects to respond more objectively, i.e., physically accurately, to the stimulus events. On the other hand, without (or with little) exposure to physics, the subjects would more likely conceive the pendulum bob as falling straight-down in their responses to the Drop stimulus in Figure 9(a). In other words, the representation of a straight-down response to the Drop stimulus might be cognitively of the most primitive type in comparison to those of the other responses found in the experiment, since it was given by the subjects who had the least amount of training in physics. The representations of these subjects, who had drawn straight-down or backward responses and then rediscovered the forward motion in their later responses, are therefore of great significance as they might reveal to us some important features of the conceptual organization in the cognitive system for representing mechanical events.
First of all, the results found with respect to the straight-down responses suggest that the subjects hold two somewhat 'independent' representations, one for the gravitational motion and one for the horizontal motion, rather than a single and indecomposable representation for the entire falling path of the bob or the wheel. Secondly, the representation of gravitational motion is more saliently present in the cognitive system, because the subjects, as they were drawing a straight-down path, focused exclusively on the representation of the gravitational motion. The representation of the forward motion, on the other hand, remained only cognitively latent, i.e., it was stored away as the subjects were drawing their straight-down responses. Interestingly, as we have noticed, in both experiments, no subject had automatically drawn a forward traveling path of the bob or the wheel on the ground after they completed their straight-down responses, they gave out their forward responses only when they were asked to.
According to the physical understanding of the events in our experiments, the gravitational motion and the forward motion occur completely simultaneously. An objective (i.e., physical) characterization of the bob or the wheel's traveling trajectory requires precise mechanical descriptions of these two individual motions as well as the coherency and synchronicity between them in space. We have seen that this is not at all the case for the representations of our subjects. Instead of precisely capturing all the mechanical details in the event, the cognitive system was found to spontaneously attend to the most salient and immediate information in the event and store other information only as latent representations, which could be evinced only when the subjects were forced to retrieve them and when the most salient information no longer occupied their minds. In other words, rather than representing the event according to its exact mechanical relationships, the cognitive system reconstructed the event into a representational structure very different from the real event itself and appeared to organize the relevant information of an event at two different conceptual levels --- the salient and the latent.
As in Experiment 3, the responses of the subjects in Experiment 4 also suggest a salient-latent distinction in the subjects' representations. The recovery of the forward motion in many responses to Task 2 indicates that many subjects had stored their representation of the forward motion as latent while making the responses of a backward trajectory. The representation of a backward trajectory is then a salient vector representation.
We can regard the salient and latent representations as forming a simplified cognitive model within the subjects' mental space. Corresponding to such a cognitive model is a two-level information 'hierarchy'organized according to the saliency of the information. The salient representations allow the cognitive system to hold the exigent information of an event, while the latent representations supplement to the salient representations to complete the conception for the entire event. In other words, in conceiving a physical event, the cognitive system pictures only the information in the salient representations while being mostly oblivious of the latent representations trailing behind the salient representations. Nonetheless, the event is represented in its entirety because the latent representation is closely bonded to the salient representation by a cognitive constraint we now turn to.
The Nexus of Dynamic Representations:
Obviously, the salient representation and the latent representation are not completely independent, as they inseparably contribute to the representation of a single mechanical event --- the falling of the pendulum bob or the wheel. For the subjects who gave straight-down responses, the representation of the entire event appears to be simply a sequential conjunction of the salient and the latent representations. Such a result seems to be very consistent to our intuition that the latent representation, though weak and latent as it is compared to the salient representation, nonetheless contains information which is an integral part of the event and, as a result, must be somehow closely linked to the salient representation. But how?
It is common physical knowledge that different episodes of a mechanical event are linked causally under mechanical laws. As psychologists, we often implicitly assume that our mental activities are also governed by cognitive laws or principles. In these two experiments, we found that a salient representation can weaken or even make latent another representation within the cognitive system and the latent representation can re-emerge following the salient representation. To use the language we have developed, cognitive representations of mechanical information appear to follow a principle consisting of the following two components; 1) the representation of an object in a mechanical state of motion can be interrupted and made latent by the more salient representation of a dynamic discontinuity occuring in the mechanical state of the object; 2) the representation of an object in a mechanical state of motion, though being driven out of view upon encountering the representation of a dynamic discontinuity in the mechanical state, persists nonetheless with its original mechanical content in latent form.
The first component captures why the horizontal motion in many responses was suddenly cut off as the dynamic discontinuity occurred. The second component explains why the subjects can rediscover the horizontal motion even though they had totally ignored it previously. More specifically, the first component explains why the vertical motion had come before the forward motion in the subjects' responses as the cognitive system's differential representation between different types of dynamic information and discriminatory reaction to different dynamic information according to the corresponding content saliency. Following the second component, although the representation of the horizontal motion may change in its representational form (between salient and latent) and distort in its temporal relationships with respect to the physical event, it is cognitively persistent and 'causally' bonded to the representation of the entire event. The second component is in fact a weaker version of the Continuity Representation we have discussed in the last chapter. The difference is that it is continuous at a later time. To distinguish it from the stronger Continuity Representation in the last chapter, I will refer to it as the Persistence Representation.
Therefore, what has made the vertical motion jump before the horizontal motion in the subjects' responses is the information saliency of a dynamic discontinuity to which the cognitive system is sensitive to and what has made a latent representation straggle behind a salient representation is the persistence of the representation for dynamic motion. Clearly, rather than representing the pendulum event mechanically, we found that the cognitive system had dismembered the event by the level of information saliency and reorganized it into an assembly of two distinct and individual representations. With its characteristic potency and immediacy, a salient representation captures the information which appears to be the most important to the subjects; with its distinctive frailty and quiescence, a latent representation firmly retains the information which, though being of lesser saliency, is nonetheless an integral and indispensable part of the event.
Vector Representation of Composite Dynamic Motion:
In both experiments, however, most subjects did not respond to the stimluli by drawing a straight-down path. In Experiment 3, 80% of the subjects had drawn responses with a component of the forward motion; in Experiment 4, 82% of the subjects had drawn responses containing either a component of the forward motion or a component of the backward motion. These results suggest that the majority of subjects had in fact represented as salient two motion components simultaneously, instead of sequentially as in the case of a straight-down response, and cognitively combined two motion representations into a single representation of a nonstraight-down trajectory, either in the forward direction or in the backward direction.
In the responses of Experiment 3, we find two motion components, the vertical motion and the forward motion; in the responses of Experiment 4, we find three motion components, the vertical motion, the forward motion, and the backward motion. In both experiments, we find evidence that the subjects were cognitively sensitive to these individual motion components. For example, the responses of the subjects to the Drop stimulus in Figure 9(a) differ uniquely with respect to the departing angle A, revealing a strong and overall representational sensitivity of the subjects to the onset of the vertical motion; As already found, a straight-down response is in fact nothing but the subjects' re-creation of the stimulus event through cognitively reorganizing the relationship between the representation of the vertical motion and the representation of the forward motion; In Experiment 4, both the forward motion and the bakcward motion, which had been absent in the subjects' responses to Task 1, were found to be recoverable in their responses to Task 2, indicating that the subjects had conceived both the forward motion and the backward motion as latent representations and stored them away individually as they were responding to Task 1.
If the subjects had simultaneously represented two individual motions in the stimulus events as salient, the nonstraight-down responses have to be the outcome of a representation which is cognitively combined from the representations of the two individual motions. In the following, we will use the term vector representation to denote the combination of two or more individual motion representations. Following the convention in physics and mathematics, the word vector not only connotes the magnitude and the direction of which it describes, but also intimates the compositive quality of a vector itself. Referring to Experiment 3, we can then say that most subjects had responded to the Drop stimulus with a salient vector representation which is combination of a representation for the vertical motion and another representation for the forward motion. In Experiments 4, since three motion components were found to have been represented by the subjects, another salient vector representation which is a combination of the representations for the backward motion and the vertical motion was also found to be held by a number of subjects.
A vector representation can also be a combination between a salient representation and a latent representation. For example, as shown, two individual motion components were represented in the vector representation of a straight-down response by the subjects but organized sequentially in a salient-latent bi-level conceptual space. A vector representation can be further a combination of two salient representations with a latent representation. For example, three individual motion components were represented by the subjects in Experiment 4 who had given a backward response in Task 1 and a response of forward motion in Task 2; the backward response was the result of a salient vector representation, which is a combination of two salient representations for the vertical motion and the backward motion, respectively; but the representation for the forward motion had remained latent in the presence of the salent vector representation of a backward response in Task 1 and emerged only after the salient representation had ended. Therefore, characteristic of any response of a subject in these two experiments is a vector representation with relation to two or three individual representations of which the vector representation is composed.
With the idea of vector representation, we can now propose that it is in the spatiotemporal ordering or organization of the individual motion representations, not in the individual representations themselves, where we found the differences with which the subjects respond to the same stimulus events, i.e., form their specific vector representations of these events. In addition, what determines the different orderings or organizations of the individual representations are the relative representational strengths or saliencies of these motion components. In other words, it is by imputing different representational saliencies to the individual motions that a subject cognitively composes his or her vector representation with its unique spatiotemporal structure between the individual motion representations.
Let's examine such a conclusion in more detail with respect to the Drop stimulus event in Experiment 3. Given such a stimulus event, the subjects, having recognized the forward motion and the vertical motion of the event, faced the immediate problem of how to integrate conceptually these two motions into a complete event representation. As already mentioned, it is the representational saliencies attributed to each of the two individual motions at this point which would determine how a subject respond to the problem. According to the experimental results, most subjects were found to have represented a strong discontinuity effect and disproportionatedly exaggerated the magnitude of the vertical motion. Some subjects had conceived the discontinuity as so strong that their responses were completely dominated by the representation of the vertical motion, the representation of the forward motion for the greater part remained only in a state of latency. For others, however, the discontinuity did not have such a strong effect, the representation for the forward motion virtually remained intact, and the effect of the discontinuity only gradually joined in to complete the representation of the event.
What a subject would do if neither of the motions could be made latent? That is, how would a subject conceive the event if both of the motions in the event were represented by the subject as equally salient? Expectedly, the subjects would have to take consideration simultaneously of both motions in conceiving a response. Further, since there was only one single bob that was in motion, the two motions would have to be combined into one single motion. In other words, since both motions were represented as persisting with equal strength and able to do so only in one single object, it is inevitable and necessary that a combination of the two motion representations be cognitively performed. Thus, underlying a combination of two salient represenations of motion is in fact the representations for the persistency or continuity of these motions, which we have already discussed extensively.
With respect to all the responses found in these two experiments, we therefore find two different spatiotemporal patterns or structures by which two or three individual representations were organized into a vector representation --- the sequential and the simultaneous. When the structure was sequential, as in the case of straight-down response, the salient representation of the vertical motion makes latent and precedes the representation of the forward motion. Particularly, it is only by virtue of making the forward representation latent we find the cognitive system able to reorder the stimulus event sequentially or in time and hence conceive a straight-down response. On the other hand, when the structure is simultaneous between two motion representations as in the case of a nonstraight-down response in Experiment 3, since no motion could be made latent by the other in the representations of the subjects, no distortion in time had occurred concerning the existing motion components in the responses of the subjects; The two motions were represented simultaneously with respect to the pendulum bob, i.e., they were combined into the salient vector representation of a motion which is the composite and synchrony between a vertical motion and a forward motion in space.
In addition, corresponding to the sequential and simultaneous structures of vector representations, we find the distortion of a response occurring either in time, as in the case of a straight-down response, or in space, as the case of a non-straight-down responses with A>0. A distortion in time is therefore characteristic of the sequential structure and a distortion in space is characteristic of the simultaneous structure. As time is one dimensional and space is two dimensional, much greater variations can be expected in those responses resulting from the simultaneous structure. This is indeed the case, as can be easily observed. Not only the departing angle A changes in the entire 90 degree range, the shape of trajectories in these responses were also extremely various; In comparison, the responses resulting from the sequential structure, i.e., the straight-down responses, look extremely homogenous. For this reason, a precise characterization of a vector representation resulting from the simultaneous structure is much more difficult. Only future research would reveal to us its exact nature.
Finally, for the vector representation of those subjects in Experiment 4 who had given a backward response in Task 1 and a response of forward motion in Task 2, it can be characterized as both simultaneous and sequential. We see that while the representation of a backward response is a simultaneous combination of two salient motion representations, it had made latent the representation of the forward motion and, as a result, only sequentially bonded to the representation of a backward response.