Subjects: Seventy nine university undergraduate students participated in this study. These students were from a General Psychology class. The testing was done in small groups, ranging from 1 to 4 students in each group. The average age of the subjects was 19.5. 40.5 percent of these students were male and 59.5 of them were female. 20.3 percent of these subjects had taken college physics, 50.6 percent of them had taken high school physics, and 29.1 percent of them had never taken a physics course.
Stimuli: The two drawings in Figure 9 were the stimuli in this experiment. The first drawing is the similar to that used in (Caramazza, McCloskey, and Green,1981) in their original experiment mentioned earlier; The left-most position of the pendulum bob is marked by the letter 'A' in the stimulus, the right-most position by the letter "C", and the mid-point by the letter "O"; One line was drawn at the bottom of the graph representing the ground, with the word "Ground" beneath the line, and another line was drawn above the pendulum representing the ceiling, with the word "Ceiling" above it. In the second drawing, everything remained the same as in the first one except a table, drawn shaded to make it look like the texture of wood, is now placed beneath the pendulum. The path of the pendulum bob was drawn to be almost skiming the table. An instruction for each drawing was written on the top of the drawing on a single 8.5in×11in size paper. The instruction in the second stimulus emphasized to the subjects that the pendulum bob swings skimming the surface of the table without quite touching it.
Procedure: The experiment was carried out in a small quiet conference room with an occupancy capacity of 10 to 12 people. At the beginning of the experiment, the subjects were told by the experimenter that: 1) the experiment is not a test; 2) the graphs they will be working on represent real situations in the world; 3) use their visualization; 4) do their task strictly independently. They were given Stimulus one, the original stimulus, to do first; The instruction of this stimulus was "the diagram shows a pendulum swinging between positions A and C. The string is cut right at the moment when the pendulum bob reaches the mid-point O while traveling from A to C. Draw a line indicating the path of the pendulum bob after the string is cut". Upon finishing the first stimulus, stimulus two, one with the table, was given to them; Besides the information as in Stimulus one, the instruction in the second stimulus told the subjects that "When the pendulum is swinging, the pendulum bob touches the table but without being stopped by the table. That is, it skims the surface without slowing". Upon finishing their drawing on this stimulus, the experiment was done.
Design: The design of this experiment was quite simple: the Original stimulus forms the Drop condition of the experiment, so called because the bob can drop some distance to the floor. The Table stimulus forms the Table condition of the experiment. Counterbalance of these conditions was not part of the design because we wanted to see the effect of the table on the responses of the subjects who had given straight-down responses to the original stimulus, i.e., if the horizontal motion that was lost in the responses of some subjects could be recovered in the their responses to the second stimulus.
For the Drop stimulus of Figure 9(a), the responses of all the subjects were classified into four different categories by the values of the departing angle A in them, which is defined as the angle between the pendulum bob's direction of motion on leaving point O and the horizontal direction. The responses were measured with a protractor to guarantee that the accuracy for each category is within ±2-degree angle. Specifically, straight-down responses, which correspond to having a departing angle A=90, belong to the first category; Those responses which have the departing angle A>45 belong to the second category; The responses which have A<45 are of the third category; The last category of responses have A=0. The results are shown Figure 10. It can be seen, of all the 79 subjects, 16 (20%) had drawn straight-down responses, i.e., with A=90; 19 (24%) had accurately preserved the horizontal motion in their drawing at mid-point O, i.e., having A=0; 27 (34%) had drawn responses with A>45 and 17 (22%) with A<45.
Interestingly, of the 16 subjects who drew straight-down responses, 5 had not taken any physics, 8 had taken high school physics, and only 3 had college physics. In comparison, of the 19 subjects who drew responses with A=0, only 3 had not taken any physics, 6 had taken high school physics, and 10 had taken college physics. These results appear to support the idea that as the level of physics education gets higher, more subjects would tend to choose the physically correct answer. Under a 2×3 Chi-square test, the difference between the straight-down responses and the correct responses (i.e., with a departing angle A=0) in term of the level of physics education is significant (X2=7.233, p<0.05). In consideration of the low quality in high school education, no body probably would expect a high school student to fully and accurately master the physical concepts required to solve the problem in this experiment. Then, a 2×2 Chi-square test (i.e., treating high school level physics and no physics alike), we found a highly significant difference between the subjects in term of the level of physics education (X2=7.05, p<0.01).
According to physics, the correct answer requires A=0. Since overall there were 43 responses (including those that are straight-down) with A>45 and 36 responses (including those that have A=0) with A<45, the entire distribution of the responses is skewed more to the straight-down side or the physically incorrect side than to the physically correct side. The 19 correct responses by the measure of the departing angle (i.e., A=0) is only 24% of the entire sample. Therefore, significantly more responses are in fact incorrect in term of the angle A (X2=27.0, p<0.001). This result is consistent to that found in (Caramazza, McCloskey, and Green, 1981), where only 25% of the subjects tested gave correct responses.
For the group of subjects who had given straight-down responses to the stimulus in Figure 9(a), how did they respond to the Table stimulus in Figure 9(b)? It has turned out that among the 16 subjects who had drawn a straight-down path under the Drop condition, 11 of them drew a horizontal path of the bob under the Table condition, so indicating that the pendulum bob would now roll on the table; only 3 of them drew that the bob would simply stay at location O on the table; 2 of them drew a straight-down path in spite of the table. Figure 11 shows these results. A Chi-square test produced (X²=9.13, p<0.02), showing that significantly more subjects had decided the ball would roll on the table. Note, all the responses in the other three categories (i.e., the other 63 subjects) had drawn horizontal responses in the table stimulus.
According to physics, there are two forces participating in the dynamic operation of the pendulum and causing the pendulum bob to swing between points A and C. The first force is gravity. In a swinging pendulum, gravity acts on the bob through its two components, one continuously pulling the bob in the centrifugal direction and the other acting on the bob either in or opposite its direction of motion. The string reacts to the centrifugal pull of gravity with the second force in the pendulum, the tension in the string, which serves to balance the centrifugal component of gravity by pulling the swinging bob in the centripetal direction. It is however the second component of gravity that causes the pendulum to swing between points A and C. At point A or C, this component has the greatest magnitude and exerts the strongest push on the bob towards center of the pendulum. As the bob is moving away from points A or C, this component gradually decreases and becomes zero as the bob reaches point O. Thus, at point O, the downward pull of gravity and the tension in the string is in complete balance and no horizontal component (i.e., the second component) of the gravity is there to act on the bob; At the same time, the velocity of the bob has reached its maximum because of cumulative effect of the second component of gravity on the bob before point O. If the string is cut at this moment, the tension of the string that balances gravity and upholds the bob would suddenly disappear, which would allow gravity to pull the bob down. However, the horizontal motion that is already with the bob would not disappear and thus the bob should continue to move horizontally after the string is cut. Therefore, the trajectory of the bob after the string is cut should be the result of two motions, the horizontal motion which is already in place when the string is being cut and the downward motion occuring only as the tension of the string is removed. In addition, because the downward motion can only increase in magnitude gradually (following the second law of Newton), the bob has to travel in a smooth curve at point O, i.e., with A=0. A smooth projectile is therefore the physically correct answer.
It is obviously not very reasonable to expect the subjects who had not taken much physics to see the problem in such a physically accurate fashion. One alternative is to adopt the 'motion-implies-force' model (Clement, 1982). Then, the horizontal motion of the pendulum bob can be regarded as a 'force'. Moreover, the fact that in the responses of all the subjects the bob would eventually reach the ground indicates that gravity is a firmly-rooted concept, as has also been shown by Clement. Thus, two 'forces', the horizontal motion and gravity, can be seen as the two basic pieces of dynamic information that the subjects practically use to solve this problem. Different drawings of the subjects can then be explained as the results of the subjects' cognitive representations and processing of these two pieces of information. Consistent to such a simple model are the findings that the tension in the string was found to be either ignored (McDermott, 1984; Sjberg and Lie, 1981) or considered of secondary importance (Clement, 1982) by most subjects when they were asked to indicate all the forces in the pendulum problem. However, such an oversimplified model neglects some important information present in the responses of our subjects and ultimately would not help us explain the results of this experiment, as will become evident in the following discussion.
Sensitivity to the Dynamic Discontinuity Information:
In order to more adequately and precisely explain the responses of the subjects and penetrate more deeply their conceptions for this problem, we have to go beyond both what physics objectively informs us and what can be directly reported by the subjects. A detailed examination of the subjects' responses with reference to the accurate description of Newtonian physics must be the path of our investigation. To do so, I now break down, according to physics, the pendulum problem in Figure 9(a) into four physical components: 1) the horizontal motion will continue in the same direction after the string is cut; 2) cutting the string destroys the balance between gravity and the tension in the string; 3) without the force holding the bob, gravity would pull it down; 4) the downward motion caused by gravity increases only gradually in magnitude, starting from 0 velocity at point O to the maximum velocity when reaching the ground.
In this experiment, all the responses show that the bob would fall to the ground, which indicates that the subjects were well aware of 2) and 3), though they might not have explicitly recognized the tension in the string as a force, as shown in (McDermott, 1984; Sjberg and Lie, 1981). 63 subjects (80% of the entire sample) drew responses that were not straight-down, which indicates that the majority of subjects had taken consideration of 1), i.e., the persistence of the horizontal motion, in solving the problem. Then, it has to be the information in 4) which most subjects have failed to correctly conceive.
Notice, in almost all the responses, the bob leaves position O in directions which are not the continuation of the bob's motion immediately before position O, i.e., discontinuities were found in the path of the bob's motion, as shown by A>0. In other words, in most responses, the direction of the bob's motion path was suddenly altered in the direction of gravity at position O. Also, notice the fact that the amount of discontinuity, as measured by the value of A, varies a great deal, from a few-degree angle to a sharp turn of 90 angle (as in the case of a straight-down response), which indicates that the straight-down responses are actually the extreme cases of all the responses for which A is greater than zero. Therefore, it appears that it is the failure to accurately conceive the effect of the interruption of a dynamic equilibrium which underlies most responses to the Drop stimulus of the experiment. In most responses, i.e., in all the responses that have A>0, this effect has been over-estimated, as a correct response requires the departing angle A=0.
The interruption of a dynamic equilibrium can be regarded as a particular type of dynamic information --- the sudden removal of the upholding force (tension) in the string and thereby the dynamic action of gravity on the bob --- which we define here as a dynamic discontinuity. The fact that most subjects have drawn their responses with A>0, i.e., motion paths with spatial discontinuities, confirms our hypothsis that the cognitive system is overly sensitive to the information of a dynamic discontinuity.
Analysis of the responses to the Table stimulus, i.e., Figure 9(b), further confirms this conclusion. Essentially, it shows that it is not because the horizontal motion of the pendulum bob was not being represented by the subjects, but rather, its representation was overriden by the more salient and 'urgent' representation of a dynamic discontinuity. This has clearly been shown in the responses of those subjects who had given straight-down responses to the stimulus in Figure 9(a), but gave horizontal responses to the stimulus in Figure 9(b). Recall, when gravity was counteracted by the table in the stimulus of Figure 9(b), the horizontal motion was recovered in the responses of the same subjects who had given responses to the stimulus of Figure 9(a) without any horizontal motion. Apparently, when there was no need for the subjects to deal with the more urgent information, i.e., the effect of a dynamic discontinuity, their representation of the horizontal motion naturally continued.
It is interesting to compare our results here to what (diSessa and White, 1981) has discovered, which they call 'force as a mover' effect in a quite different context. They found that novice students have a false intuition that a force causes motion in the direction of the force and that all the previous motion would be nullified by the force. Here, we have seen that the representation of the previous motion by the subjects might not have been nullified. It is very likely that the previous representation of the horizontal motion was only temporarily stored away as the more salient information of a dynamic discontinuity was being conceived. We will investigate this possibility in our next experiment.
Physically speaking, gravity has always been present in the pendulum problem. But it is only when the string is cut that it starts to pull the pendulum bob down. This is because, when the string is intact, the tenstion in the string and the gravity are balanced and the bob is in a state of dynamic equilibrium. In the physical sense, cutting the string, which is to remove the force that balances gravity, is then equivalent to 'applying' gravity to the bob at the moment when the string is cut and enables it to actually influence the mechanical state of the bob. We have seen that the subjects were very sensitive to the gravity that is 'applied' this way, but, they are very inaccurate at calculating the effect of this force, especially when another 'force', the horizontal motion, is also present. In the sense of diSessa and White, the 'force as a mover' is the gravity 'applied', which had wiped out the previous horizontal motion of the bob, as shown by the straight-down responses of our subjects.
The Mechanical State Model:
A more general model for describing the pendulum problem and understanding how the cognitive system conceives it is one by making use of the notion of mechanical state. Recall, a mechanical state of an object contains all the mechanical information about the object. Spatiotemporal information only describes the object with respect to its kinematic state, which, according to physics, can be depicted by various kinematic properties like displacement, speed, direction of motion, change of speed (acceleration), etc.. Dynamic information describes the dynamic features of an object; particularly, in addition to the mass of the object, the force relationships of this object to other objects are also part of dynamic information. Together, spatiotemporal information and dynamic information fully portray the mechanical state of an object.
In the case of the pendulum problem, the swinging pendulum bob at the time of passing position O can be described as having a particular amount of mass (a dynamic feature), being in motion with a particular velocity (a kinematic property), being in a gravitational field (a dynamic relation to the earth), and being upheld by the string (a dynamic relation to another object). According to physics, a change of the mechanical state of an object can only be caused by the dynamic relations of this object to other objects. For the pendulum bob, its mechanical state can be changed only through a change in its relation to the string (being cutting, for example) or the Earth (gravity being altered; an unlikely circumstance, but not impossible).
Following our earlier discussion, it is not gravity but the gravity that is suddenly given the opportunity to dynamically influence the mechanical state of the pendulum bob upon the destruction of the balancing force in the string which seems to have a prominent representational status in the cognitive system. In order for the subjects to be sensitive to such a specific piece of information, it obviously requires them to represent, in the swinging pendulum, not only the gravity which is continuously pulling the bob, but also the force in the string, i.e., the dynamic equilibrium between these two forces. This is to say that the subjects must have at least a rough representation of all the dynamic relations the pendulum bob has during its swinging state, even though they may deny having any awareness of the force in the string, as several researchers have found. The spatial discontinuity found in most subjects' responses is the result of an acute representational sensitivity of the cognitive system to the dynamic discontinuity in the mechanical state of the bob. But, it is not possible for the subjects to represent a dynamic discontinuity without first of all having a representation of the dynamic equilibrium in which the dynamic discontinuity occurs.
The notion of dynamic discontinuity captures both the removal of the force in the string and the immediate dynamic action of gravity on the bob because the removal and the emergence of the two forces are physically inseparable in the problem and happen at exactly the same moment--- the moment of discontinuity. The notion of mechanical state, on the other hand, describes an object's state in terms of space, time, and dynamic relationships to other objects. Together, these two notions have enabled us to more precisely discuss the results of this experiment.
In the following, I will refer to a dynamic discontinuity of the kind we have encountered in this experiment as negative, for the simple reason that with this kind of dynamic discontinuity the mechanical state of an object is influenced not as a result of the application of a new force to the object but the removal of an old force already having a dynamic relationship to it.
Dynamic Discontinuity and Strength of Representation:
Following the above discussion, the negative dynamic discontinuity in this experiment is where gravity actually starts to pull the pendulum bob down as the force of the string is removed. It is natural to expect the gravitational motion following the dynamic discontinuity be represented by the subjects with respect to the force dynamics of the discontinuity. In other words, with a dynamic discontinuity immediately preceedinng a gravitational motion, the cognitive system must distinguish it from a gravitational motion unrelated to any dynamic discontinuity, because the gravitational motion in the former case is causally connected to the dynamic discontinuity, as it is in this experiment. The sensitivity of the cognitive system we found in this experiment in fact demonstrates that the subjects were well aware of such a causal connection when they incorporated the dynamic discontinuity information into their representation of the gravitational motion.
It is the strength of the representation which seems to be the cognitive attribute by which the cognitive system associate the gravitational motion to the dynamic discontinuity. With a preceeding dynamic discontinuity, a representation of the gravitational motion displays a greater strength overcoming the representation of the horizontal motion, as the experimental results indicate. This should not be a surprise since the discontinuity information to which the cognitive system is sensitive is in fact the information about the changes of dynamic relations in the mechanical state of the bob, as we have just discussed.