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# PHYSICS

## 7.1 A Mathematical Model of Mechanical Motion

The 'motion implies force' conception directly implies that people do not see an object in motion merely as spatiotemporal information, empty of any dynamic content. On the contrary, dynamic information as directly related to the mass of an object, plays a key role in people's conception of a moving object, as our earlier experimental results have shown. In other words, a moving object is mechanically conceived as a moving mass that is able to influence or potentially influence the mechanical state of another object (hence, the 'motion implies force' conception). An adequate description of such a concept is, therefore, possible only by the use of both spatiotemporal information and dynamic information. An objective characterization of the mechanical state of a moving object, in both spatiotemporal and dynamic terms, now seems to be necessary for our further discussion of this commensense concept of force and the conceptions of dynamic discontinuities we have dealt with in the last chapter.

The momentum concept in physics, as defined by mv(t), can be used as a mathematical model here to describe the mechanical state of a moving object at a specific time instant, t. At this instant, the spatiotemporal information in such an event is characterized by the velocity v(t), i.e., the instantaneous speed and the instantaneous direction of motion. However, similar to the discussions about Equation (5), the object mass m determines the Dynamic Magnitude of the momentum mv(t) by weighting the velocity v(t). Furthermore, the weighting is not simply mathematical, it is the consolidation of spatiotemporal information into mechanical information by bringing dynamic content back into the complete representation of the mechanical state.

Our early experimental results have shown that the cognitive system is quite sensitive to the dynamic strength of a moving object, as evidenced by the sensitivity of the cognitive system to the mass of an object. Therefore, quite consistent to the 'motion implies force' conception, at a particular time instant, a moving object is conceived not only spatiotemporally (i.e., about the kinematic information of v(t)) but also dynamically (i.e., about the dynamic information characterized by m in mv).

However, the 'motion implies force' conception that people have is not just the dynamic representation of an object (i.e., MASS) in motion at one time instant that is characterized by mv(t). The fact that it is conceived as a 'force' indicates that its mechanical ability to potentially influence other objects is also part of the conception. As shown in Experiment 2, at any point (a specific time instant) in the trajectory of a curved response, the lead ball is conceived as continuously changing its direction according to its past mechanical history. Therefore, the conception of an object in motion also depends upon the conception of the mechanical history of the object before that time instant. All this indicates that, although mv(t) physically describes and mathematically represents the instantaneous mechanical state of an object, i.e., at the time instant t, it is not able to capture the commonsense conception of an object in motion, because characterization of such a conception further requires at least representations of previous mechanical states. A wider extent of the mechanical state information in time is thus needed for the characterization of the commonsense conception of an object in motion.

Let sn=mv(tn) represents the instantaneous mechanical state of an object at time tn. All the previous instantaneous mechanical states can be similarly represented by sn-1=mv(tn-1), sn-2=mv(tn-2), ... sn-N=mv(tn-N). Note, each state contains dynamic information as well as spatiotemporal information, but only about the instantaneous mechanical states at particular time instants. Our need to physically describe the entire mechanical event urges us to consider all these instantaneous states together as a whole by means of possible mechanical relationships between them.

One way to characterize a mechanical event of single-object free motion in term of the instantaneous mechanical states of the event is to describe the most recent instantaneous mechanical state sn in term of all the previous mechanical states, sn-1, sn-2 ... sn-N. Mathematically, the idea that the most recent mechanical state of an object being characterized by the object's previous mechanical states can be expressed by an autoregressive model

pn= a1sn-1 + a2sn-2 + + aN sn-N ............(7)

through the least-mean-square parameters a1, a2, ...aN. That is, at each most recent time instant tn, parameters a1, a2, ...aN are optimally estimated from mechanical information about the event up to tn by minimizing the mean-square error between sn, the most recent mechanical state, and pn, the prediction of the most recent mechanical state from all the previous mechanical states. Under the best estimation of these parameters, a1, a2, ...aN, the most recent mechanical state sn can then be optimally characterized by the previous mechanical states sn-1, sn-2 ... sn-N through these parameters.

It is known that this model can describe highly sophisticated mechanical motions quite precisely and has commonly been applied in the design of antibalistic missile radar systems. Further, the precision of this model is dependent upon the order of the model N and the sampling interval between any two instantaneous mechanical states. For a specific description of the power and the precision of this model, please refer to (Franks, 1969; Hykin, 1986; Oppenheim and Schafer, 1975).

Under such a mathematical description, a single mechanical object in motion can now be characterized in its entirety, i.e., in term of the mechanical state of the entire event. Such a characterization has been made possible through relationships or connections between the instantaneous mechanical states of the event, as captured by the least-mean-square parameters a1, a2, ...aN. Furthermore, like a single instantaneous mechanical state that is both kinematic and dynamic, this description of the mechanical event is also both kinematic and dynamic because it is formed by both kinematic and dynamic information from the instantaneous mechanical states.

Equation (7) can be regarded as a computational model for free motion of a single object. If we apply this model to the cognitive system, i.e., assuming that it represents single object motion in a similar fashion, some changes have to be made to this model.

Consider the process of drawing a curved response outside the spiral tube by a subject in Experiment 2. Mainly, at each moment during the drawing, the subject had to cognitively determine the instantaneous motion of the ball at this moment based on its previous motion before this moment; As we have discovered, the subject's drawing also depends upon the mass and the speed of the ball. In other words, the subject had to conceptually predict the most recent instantaneous mechanical state of the ball from his/her conception of all the previous mechanical states of the ball, particularly, in terms of his/her conceptions of the mass and the speed of the ball. Such a predictive relationship is exactly what the model in Equation (7) captures. However, three important differences exist between the human conception of the mechanical event and the computational model in Equation (7): 1) a conceived mechanical state Sn may deviate or be quite different from a mechanical state sn of an actual event; 2) more random variation exists in the representation of a mechanical state than the mechanical state itself, i.e., Sn is a random variable; 3) Since the physical instantaneous mechanical states, i.e., ... sn, sn-1, sn-2 ... sn-N ... , are no longer relevant as we consider only people's conception of the mechanical event, the cognitively predicted state of the event at time instant t is also the conceived state at t, i.e., Pn=Sn; Thus, the model takes the form

Sn = a1Sn-1 + a2Sn-2 + + aN Sn-N ..............(8)

This model describes how the cognitive system represents a mechanical motion of a single object in term of its representations of specific mechanical states in the event under the constraint of the least-mean-square parameters. It has the power to capture what occurred in Experiment 2 that, at each point on the path of a curved response, the subject had to decide how the ball would travel based on how the ball had been traveling before that point. Referring to Equation (8), the subject determined what Sn was like from how he/she represented the event with the previous mechanical states Sn-1, Sn-2, Sn-N and the relationships between them under the least-mean-square principle.

In Equation (8), because the subject's representation of the most recent mechanical state Sn is conceptually predictable from Sn-1, Sn-2, Sn-N, parameters a1, a2, ...aN can be regarded as the realization of a conceptual rule that governs the interrelationships among the representations of the instantaneous mechanical states of the event.

Let ST=(Sn-1, Sn-2, Sn-N) and AT=( a1, a2, aN ), the condition for the predictability of Sn from Sn-1, Sn-2, Sn-N can be found by minimizing the prediction error using the least-mean-square method (Hykin, 1986) as

E(SSn) = E(SST) A..............(9)

This equation shows that representing Sn from Sn-1, Sn-2, Sn-N requires the correlation between Sn and Sn-1, Sn-2, Sn-N. Moreover, the correlation depends on the parameters, a1, a2, ...aN, which are particular to the mechanical event. Since the matrix E(SST) is positive definite (Hykin, 1986), these parameters can in fact be found as

A = E(SST)-1 E(SSn)..............(10)

Therefore, Equations (8)-(10) form a conceptual model which describes how the cognitive system conceives the mechanical motion of a single object as in the case of the responses in Experiment 2. In particular, this model can explain a left-curved response in that experiment in term of a conceptual rule as the least-mean-square minimization in relation to the strong mechanical magnitude of the event. This model coincides with the History Minimization Principle as applied to motion proposed in(Leyton, 1992), but bears the further advantage that the minimization scheme has been made computationally and representationally specific. In addition to its power of sufficiently explaining the left-curve responses in Experiment 2, this model is also capable of explaining the right-curved responses, particularly the significance of the right-curved responses under the Slow condition.

Sometimes, there is no correlation between Sn and Sn-1, Sn-2, Sn-N , for example, when an external force suddenly influences the object at time tn, in which case the conception of the most recent mechanical state Sn deviates drastically from conceptions of all the previous mechanical states of the object, since the cognitive system is very sensitive to the dynamic discontinuity in the mechanical state of an object, as we have found in Experiment 3, Experiment 4, and Experiment 5. Such sensitivity has made the conception of the mechanical state at time tn, i.e., Sn, suddenly deviate from the mechanical state that is conceptually predictable from Sn-1, Sn-2, Sn-N. Mathematically, such a case can be characterized by E(SSn)=0 in Equation (10), i.e., Sn is no longer correlated to the conceived mechanical states of the event before the dynamic discontinuity. As a result, we would have A=0 in Equation (10), showing that the cognitive system represents the object as having left an old mechanical state and entered into a new mechanical state --- a mechanical discontinuity occurring in-between them. Therefore, the least-mean-square model as developed can not only describe the conception of a mechanical state of a single object in free motion (with no dynamic discontinuities), but also reveal to us an interesting correspondence between a represented mechanical discontinuity and the least-square parameters of the model.

Mathematically, though A=0 means the degeneration (i.e., loss of power) of the model, physically, it means that an external force has to be treated as new information, i.e., it brings innovation to the mechanical event, to borrow a term from Information Theory. In addition, it is interesting to notice that new information occurs at the time of a dynamic discontinuity, to which the cognitive system was found to be extremely sensitive.

Therefore, we can conclude that the least-mean-square model is able to mathematically capture the essential features, with respect to either the information of a mechanical state in continuation or a dynamic discontinuity, in the representations of the cognitive system, which were responsible for the responses of the subjects in our experiments.

## 7.2 Conceiving Mechanical Motion --- Commonsense Conception of Force

It is commonly believed that there are two types of commonsense force conceptions, the resistive and the motive (diSessa, 1993; Halloun and Hestenes, 1984a, 1984b; Leslie, 1994; Minstrell, 1982; Talmy, 1988). For example, two most important p-prims proposed by diSessa are spontaneous resistance, 'which resides in the patient of action', and force as a mover, 'which pushes an object from rest causes it to move in the direction of the push'; Halloun and Hestenes found that students generally relate motion or changes of motion to either an external force or an internal force (or an intrinsic resistance); Based on the latest findings from infant research, Leslie discussed the concept of motive force with respect to mechanical motion and mechanical agency; Talmy called the distinction between the Agonistic and the Antagonistic entities as a basic force-dynamic distinction underlying all force dynamics in language.

Following our earlier discussion, we can see that the concept of resistive force is closely related to the primitive of substantial force and the object concept. That is to say, in knowing mechanical objects, we already have a conception of resistive force because it is nothing but the resistive component in MASS, the concept of the substantial force of an object. Now, I argue that the concept of motive force is only slightly different from the concept of resistive force. The distinction lies only in the relative mechanical states of the objects in a mechanical event. Conceptually, while the resistive force of an object resists, with its own substantial force or mass in the state of rest, being mechanically changed into a state of motion by another object, the motive force of an object also resists, with its own substantial force or mass in the state of motion, but, being mechanically changed into a state of rest. Therefore, the concept of resistive force and the concept of motive force is distinguishable by the relative kinematic states of the two objects involved, i.e., the object that has more motion (higher speed) would be conceived as having more motive force and than the object that has less motion (lower speed), if both objects have the same amount of mass. For example, in a mechanical impact, while the object at rest (or in lesser motion) resists being changed into a mechanical state of motion (or higher motion), the object in motion (or higher motion) actively resists being stopped and changed into a mechanical state of rest (or lesser motion).

In addition, both the motive force and the resistive force are related to the basic mechanical tendency of an object to continue its mechanical state, which is in fact the essence of the first law of Newton. The cognitive representation of such a basic property of motion, which can be regarded as the subjective counterpart of the first law of Newton will be discussed in the next section. Here, I only want to point out that the cognitive representation of this active-resisting mass in motion (but not in an impact!) is in fact nothing but the 'motion implies force conception (Clement, 1982), as we have frequently referred to in the previous sections. Therefore, essentially, the resistive force and the motive force can all be described as related to an object's mechanical tendency to continue its mechanical state, whether it is in motion or at rest. Conceptually, however, such a mechanical tendency to continue a mechanical state can be represented either as with an object in an impact situation, in which the object influences another object as it persists in its mechanical state, or simply as with an object at rest or in motion, in which the object simply continues its mechanical state without resistance. Thus, as conceived by the cognitive system, in the first case, the mechanical tendency of an object to continue its mechanical state manifests as FORCE in the impact, while, in the second case, the mechanical tendency of the object to continue its mechanical state represents force only potentially, i.e., as the dynamic capability of the object to influence another object in an impact or the "motion implies force". The representation of such a capability of an object in motion has no correspondence to any physical force in the mechanical world, therefore, it is purely cognitive construction. It is in dynamic representations of this kind that we see the fundamental differences between the commonsense conception of force and the physical force of Newtonian mechanics.

The commonsense conceptions of force, P, can therefore be defined as a cognitive attribution to an object of a certain degree of dynamic capability, corresponding to the substantial force and the mechanical state of the object, which enables the object either to mechanically influence (or potentially influence) another object or to be mechanically influenced (or potentially influenced) by another object, as characterized by the changes of the mechanical states of the objects involved.

Thus, P, in its semantic content, can be either resistive or motive and either actual or potential; in representation, it has to be both kinematic and dynamic. To study human conception of force is to investigate how dynamic representations are related to and constructed from both the kinematic and dynamic attributes in these representations and the origins of these attributes in the objective descriptions of Newtonian physics.

Finally, while FORCE P conceptually represents what the object can do or could do mechanically, either in the motive or the resistive sense, the concept MASS conceptually emphasizes the object itself in term of its substantial quality. Yet, the ability of an object, as represented by the cognitive system, to move another object or to resist being moved by another object is directly correlated to the amount of SUBSTANTIAL FORCE or MASS of the object. Thus, to conceive mass is also to indirectly conceive force. Conversely, it is not possible to conceive force without simultaneously also conceiving the 'carrier' of the force --- mass.

The reader may already have noticed that the picture given here is much closer to the Aristotelian physics than to the Impetus Theory. The weakness of the Impetus Theory is that it connects mechanical motion to some unmeasurable entity called impetus. Insofar as impetus can not be further defined, the study of dynamic representations in psychology would most likely reach a dead end, if such an approach were followed.

## 7.3 The Lawfulness of Commonsense force Conceptions

A key concept proposed in this book is that of mechanical state. Recall, a mechanical state describes an object in term of all its mechanical properties, i.e., contains all the mechanical information about the object; The kinematic information describes the spatiotemporal properties of the object, whereas the dynamic information the dynamic properties. Thus, the mechanical state of an object describes not only the kinematic change of the object but also the dynamic quality and attributes of the object, i.e., its mass and its dynamic relations to other objects. The change of a mechanical state can be of two different types depending exclusively on the dynamic relations the object has with other objects. In cases of events with no dynamic discontinuities or sudden change of any dynamic relation therein, the mechanical state of the object can be characterized as stable, since all the changes that occur in the kinematic state are gradual as the object remaining in the same dynamic context throughout; In cases where there is a dynamic discontinuity or a sudden change of any dynamic relation, the change of the mechanical state of the object is generally more distinguishable since different dynamic contexts now rule the mechanical states of the object before and after the dynamic discontinuity, which often make the mechanical states qualitatively different around the dynamic discontinuity. Therefore, what separates two different mechanical states in time is a mechanical or, more precisely, dynamic discontinuity.

In Chapter 3, the mechanical state of an object, such as those in Experiment 2, can be simply characterized dynamically by the object mass and kinematically by the velocity of the object and the dynamic relations (such as that between the ball and the table) of the object remain totally unchanged throughout the mechanical event. We have found that the subjects responded to the mechanical magnitude of the events as characterized by both of these two mechanical variables. In this section, we will continue the discussion about how the cognitive system conceives a mechanical state of an object.

In Experiment 2, a positive relationship has been found between the number of left-curved responses and the mechanical magnitude of the ball; a negative relationship has been found between the amount of curve to the right in the responses and the mechanical magnitude of the ball.

According to the first law of Newton, any curve in motion is the effect of a Newtonian force, thus, only a mechanical state of straight motion or being-at-rest is totally free from any force. By contrast, we found that a curved motion was not related to any external force in the representations of the cognitive system, i.e., curved motion is conceived by the cognitive system as a feature of the motion itself. The violation by the curved responses of the first law of Newton shows that the cognitive system has its own rule to follow in conceiving a mechanical state of an object in free motion. It has been shown that the Impetus Theory can not give us an answer as to what this rule is because the responses were found to differ with respect to the mass and the speed of the ball. A more elaborate explanation is required to account for the curved responses.

The important fact in the results of this experiment is that it is to the magnitude of the mechanical state of the ball to which the cognitive system is sensitive. Responding to the highest magnitude, significantly more subjects have drawn a left-curved path, i.e., in the curving direction of the tube. Corresponding to the lowest magnitude, significantly more amount of curve in the right-curved responses has been found. These results suggest that the conception of curved motion is directly related to the conception of the mechanical state of the object, particularly that of its mechanical magnitude as characterized by the mass and the speed of the object. Particularly, the significance of the right-curved responses reveal to us how the cognitive system represents the more deviation of the ball's rolling path when its mechanical magnitude is weaker; on the other hand, the significance of left-curved responses shows that the cognitive system represents the ball's rolling as more continuous along its own past direction of motion when the ball is traveling with a greater mechanical magnitude. So, it seems, the conception of the mechanical magnitude of the mechanical state of the ball has a direct influence on how the cognitive system determines the kinematic behavior of the ball and, therefore, can be regarded as a key component in the mechanical constraint in the cognitive system, the function of which is to modulate conceptually how much the ball's traveling path should continue its own path. Under this constraint, the representation of a stronger mechanical magnitude would be immediately translated in the cognitive system into a representation of a stronger tendency of the ball to continue its own state of motion. On the contrary, with the mechanical magnitude extremely weak, no such a tendency would be represented about the mechanical state of the ball and the ball would be represented as rolling more randomly with larger variations, which is responsible for the deviations of the right-curved responses under the low magnitude (i.e., PP/Slow) condition.

The translation, which is the operation of the mechanical constraint in the cognitive system, from the conception of mechanical magnitude to the conception of a relevant degree of the mechanical tendency to continue the mechanical state, can be further explained by means of the mathematical model introduced in Section 1. For example, in this model, the information about the mechanical magnitude is well present in the representations of the instantaneous individual mechanical states Sn-1, Sn-2, Sn-N because they contain both the representations of the velocities of the ball at each previous moments (thus the amount of curve) and the ball's mass. If such information is used to decide to what degree the mean-square error should be minimized, which, mathematically, can be explained as to what degree the next mechanical state Sn would continue from the previous mechanical states, the operation of the mechanical constraint can be expressed quite simply by an inverse linear relationship between the level of the mechanical magnitude and the level of minimization, i.e., the higher is the mechanical magnitude the lower should be the mean-square-error. With this model, the representation of the stronger mechanical tendency for the continuation of the previous mechanical state under the representation of a stronger mechanical magnitude of the mechanical state can then be expressed by a simple numerical relationship between the mean-square-error and the mechanical magnitude of the previous mechanical states, which is exactly what we need to describe how the cognitive system comes up with the left-curved responses in the experiment, i.e., the 'rule' by which the cognitive system follows to choose a curved response. Conversely, when the mechanical magnitude of a mechanical state is very weak, as in the case of a slow ping-pong ball, a large minimization error would accordingly follow from exactly the same formula of the inverse linear relationship between the mechanical magnitude and the level of minimization, which would allow greater deviations to appear in the responses. Therefore, this model enables us to explain not only the left-curved responses, but also the right-curved responses.

In the following, the representation of the mechanical constraint in the cognitive system as just discussed will be referred to as the representation of continuity of the mechanical state, or simply, the Continuity Representation. The Continuity Representation captures a very important aspect of how the cognitive system conceives motion as dynamic by taking consideration of the conception of mass into the conception of motion and explains the conception of the tendency of an object to continue its mechanical state by the cognitive system as an outcome of its conception of dynamic motion. With this representation, we are now able to explain the curved responses or curved motion as conceived by the cognitive system when no external force can be identified.

However, in Experiment 2, though many curved responses occurred, there were always responses that were straight, and, the proportions between the straight responses and the curved responses changed from condition to condition in the experiment. This shows that the Continuity Representation can not yet fully explain the cognitive operations of the subjects tested in that experiment. To do so, we must consider the cognitive system to have another mechanical constraint: an object should move straight (especially when traveling at high speed) unless a strong tendency to curve or lack of mechanical strength to prevent it from going straight exists in the mechanical state of the object. Strong evidence suggests that such a belief for straight motion does exist in most people.

First of all, at least some of the straight responses can be considered as Newtonian answers by subjects who have a good master of Newtonian physics. However, the fact that the proportions between the straight responses and the curved responses change from condition to condition indicates that not all straight responses were the result of Newtonian conceptualization. It is the straight responses of those subjects who made no use of Newtonian concepts and, as a result, whose responses had therefore changed from condition to condition (for example, from PP/Fast to Lead/Fast) which have revealed to us that the second mechanical constraint must have been also in the operation of the cognitive system. I believe that this constant is more basic than the Continuity Representation because it represents how the ball should travel when there is neither a tendency for it to curve to the left as the result of a strong mechanical magnitude nor random movement deviation as the result of a very weak mechanical magnitude. Under the Fast condition in Experiment 2, significantly more ping-pong-ball responses were straight than lead-ball responses, which suggests that when a tendency to curve to the left is lacking (due to light Mass) in the representation of the mechanical state, the cognitive system would conceive the ball as going straight. Interestingly, under the Slow condition, there were less straight ping-pong-ball responses than straight lead-ball responses, showing that when the mechanical magnitude is too weak more responses would drift from the straight path and curve to both sides of the measurement positions. But, since from the Lead/Slow condition to the PP/Slow condition the mechanical magnitude was reduced, while the number of the left-curved responses would, on one hand, decrease as the result of the conception by some subjects of the weaker mechanical magnitude and, on the other hand, increase as more responses of the subjects would likely randomly drift and curve to both sides as the result also of the conception of the weaker mechanical magnitude. This is exactly why significance has been found only in the right-curved responses, but not in the left-curved responses, under the Slow condition. Such an analysis shows that the second cognitive constraint, the belief of traveling straight, is indeed more basic than the Continuity Representation which governs both the left-curved responses and the right-curved responses.

In addition, McCloskey commented that his subjects who gave the curved responses in the original spiral tube experiment believed the ball would eventually go straight. I have questioned a number of subjects who had given curved responses in my own experiment, every single one of these subjects questioned replied that the ball would eventually go straight. Therefore, it is likely that even for the subjects who have given curved responses a different belief was also represented when they were drawing their responses. This suggests to us that, if we set up an experimental situation in which the tendency of the ball to curve was destroyed, there would be no need for the subjects to draw the curved responses (i.e., to accommodate the curved history in the ball's motion) before their straight responses, i.e., they would draw straight responses. This was exactly what happened in Experiment 4, in the responses of the subjects, we can see, since ball B does not have a mechanical tendency to curve, it goes straight as an object normally travels in a straight path at high speed.

Thus, it seems that the responses of the subjects to the mechanical event of a ball coming out the curved tube are in fact the results of a conceptual reconciliation in the cognitive system between two mechanical representations that the subjects have. One is the Continuity Representation that a ball coming out the spiral tube would either curve to the left when the mechanical magnitude is strong or deviate from the straight path when it is very weak. Another is the more basic and background-like representation that a ball that travels in space normally travels in a straight path, for which I will call the Rectilinear Representation, to distinguish it from a Newtonian concept of freely straight motion. The degree of domination of the Continuity Representation over the Rectilinear Representation in the cognitive system of a subject would determine, for this particular subject and in a particular experimental condition, the left-curved response the subject gives. We can further make the distinction that the Continuity Representation is specific, i.e., it depends upon a specific environmental input and only comes as a response to a specific mechanical situation; whereas, the Rectilinear Representation is general, i.e., it is a basic belief not dependent upon any specific input. The Rectilinear Representation is general can be seen not only in the fact that a curved response can be regarded as a deviation from a straight one, but also in the claim of the subjects that a curved response should eventually become straight. Following such a claim, a tendency to move in a straight path, as represented by the subjects, must exist even when the subjects were drawing a curved response and continue to exist even when the drawing was reaching the end of the curved response where a straight path, as the subjects believed, would come into existence. But such a straight path at this point would not be a Newtonian straight motion at all because it comes completely from the mind of the subjects. For this reason, the Rectilinear Representation should not be confused with a Newtonian Representation of straight motion. But, to save space, a more detailed discussion about the relationship between the Continuity Representation and the Rectilinear Representation will have to be postponed for a separate publication, where a more precise and quantitative model will given.

To summarize, the responses in Experiment 2 can be explained by three different types of representations in the cognitive system, the Continuity Representation, the Rectilinear Representation, and the Newtonian Representation. While the Continuity Representation is specific and the Rectilinear Representation is general, both of which are natural, the Newtonian Representation is scientific.

Finally, the Impetus Theory explains that the ball would eventually go straight because impetus dissipates. So, changing from a large curve into a small one on the ball's path is explained as the curved impetus is shrinking in magnitude. But, this theory can not explain the relationship between the straight responses and the curved responses in terms of the object mass and speed as we have done here.

In the last section, we have briefly mentioned that change of a mechanical state of an object can be of two different types depending on whether or not a dynamic discontinuity exists in the dynamic context of the object. A dynamic discontinuity has been defined earlier as a sudden change of a dynamic relation of an object to another object in a particular dynamic context. According to Newtonian mechanics, every object exists in a particular dynamic context in space and time, which essentially decides the kinematic behavior of the object. The dynamic context of an object consists of all the dynamic relations the object has with other objects. What precisely characterizes dynamic relations between objects is the Newtonian force which is described either as a dynamic process of energy transmission and distribution or as existing statically in a dynamic equilibrium in the form of potential or kinetic energy. With a single object, whether it is in a (static or kinetic) dynamic equilibrium or participating in a steady process of energy transmission and distribution, it can be simply described mechanically by its mechanical state. Therefore, for an object, between one dynamic equilibrium and another dynamic equilibrium or between a dynamic equilibrium and a dynamic process of energy transmission is there always --- a dynamic discontinuity --- the ending of an old mechanical state and the beginning of a new one. We have already found that the cognitive system is extremely sensitive to dynamic discontinuities. In this section, we will continue the discussion about how the cognitive system represents dynamic discontinuities.

The cognitive sensitivity to dynamic discontinuities has been shown through the exaggerated drawing of mechanical discontinuities of the ball in the responses of subjects in Experiment 3 and Experiment 4. In Experiment 4, the significant difference in the representations of the subjects between the movement of ball A outside the tube and the movement of ball B outside the tube also shows a qualitative different representations of the mechanical state of ball A in the tube and the mechanical state of ball B outside the tube as the effect of a dynamic discontinuity --- the impact. In these experiments, the fact that the changes in the mechanical states, as shown by discontinuities of mechanical states depicted in the responses, are all qualitative is quite consistent to the experimental results of diSessa and White (diSessa and White, 1981), in which subjects were shown to believe that a new force can wipe out all the previous motion an object has had. Other supporting evidence can also be found in (Clement, 1982; diSessa, 1982; Viennot, 1979).

Closer examination of these experiments reveal to us two important facts about the conception of mechanical discontinuities by the subjects: 1) all the mechanical discontinuities as indicated in the responses of these experiments occur at the moment in time when, and, at the point in space where the dynamic discontinuities happen; 2) the directions of change or a mechanical discontinuity in the mechanical state of the object as indicated by the responses in every experiment are, without exception, highly in accordance to the direction of the dynamic discontinuity of the experiment. In Experiments 3, for example, most subjects have drawn in their responses paths that have a non-zero angle A in the direction of gravity, which shows that for most subjects a sudden movement or a mechanical discontinuity was represented as occurring in the direction of the dynamic discontinuity. The straight-down responses particularly demonstrate how some subjects represented such a discontinuity to the extreme. In the responses of Experiment 4, ball B, upon an impact by ball A, moves in the same instantaneous direction of ball A's motion at the point of impact, which is also exactly the direction of the dynamic discontinuity in the event. The same observations also apply to the experimental results of other researchers mentioned above.

Based on these observations, several further conclusions can be drawn about the representations of the cognitive system with respect to a dynamic discontinuity. First, the cognitive system represents the spatiotemporal properties of a dynamic discontinuity with the greatest accuracy. As already mentioned, a dynamic discontinuity happens at single moment in time, at a single point in space, and pointing to a unique direction. All this information, as having been found in the responses of the subjects, were precisely represented by the cognitive system. In other words, the representation of a mechanical discontinuity by the cognitive system coincides exactly in space and time with the dynamic discontinuity in the mechanical event. By contrast, the magnitude of the dynamic discontinuity, i.e., the magnitude of the external force applied to the object at the moment of the dynamic discontinuity, seems to be represented by the cognitive system very imprecisely --- almost always conceptually exaggerated. This exaggerated representation of the magnitude of a dynamic discontinuity can be considered as responsible for the discontinuous and the straight-down responses in the pendulum problem.

If we wish to treat the representations of a dynamic discontinuity as a whole, i.e., as a single representation of the collective of all the spatiotemporal and the magnitude information, we can use the term Discontinuity Representation to denote such an representation. The high saliency found of the Discontinuity Representation in the cognitive system, as in the experiments, can now be seen more clearly as the cognitive system's highly precise representations of the spatiotemporal information of mechanical discontinuities in the Discontinuity Representation on one hand and the highly exaggerated representation of the magnitude of the dynamic discontinuity information on the other. Because a Discontinuity Representation always appears in relation to a Continuity Representation of the mechanical state of an object, it can therefore be regarded as forming a boundary of the mechanical state of the object with the distinctive representation of the location of the mechanical discontinuity in space and time and the exaggerated representation of the magnitude of the dynamic discontinuity. As such, it signifies in the cognitive system the ending of an old mechanical state and the beginning of a new one. This is why, as having been observed in the pendulum problem, the Discontinuity Representation can override the Continuity Representation in the cognitive system and thus impose discontinuities in the mechanical state in the subjects' responses. The Discontinuity Representation is probably the most salient representation concerning mechanical information.

The Continuity Representation and the Discontinuity Representation are, therefore, opposite in the mechanical situation they each separately encodes, independent of each other in information content, and uniquely characterized by their particular cognitive predilections. However, they complement each other in the complete representation of a mechanical event and help the cognitive system organize the mechanical world into meaningful and interconnected segments.

Underlying these two important mechanical representations in the cognitive system are the keen awareness in human cognition about the acting of force and the persistence of motion, though not being represented in the Newtonian terms.

Free of dynamic influence from outside, an object persists in motion and maintains its own unique mechanical state in accordance to its mass and velocity, which is the essence of first law of Newton. The Continuity Representation tunes into the same essential information and translates the persistence of motion as a mechanical continuity constraint on the representation of an object in motion, therefore, can be regarded as the subjective equivalent to the first law of Newton characteristic of human evolution and experience. However, such an experientially and evolutionarily derived representation of motion circumvents the objective and the idealistic concept of free motion characterized by the first law of Newton, incorporates the concept of mass into the concept of motion, and allows curve as a normal dynamic state of motion subject to the same rule of continuity. On the other hand, under the influence of a newly emerged force from outside, the mechanical state of an object is made to change from its own unique state of motion under the government of the second law of Newton. The discontinuity representation emphasizes the presence of an external force by highlighting the space and time of its arrival and by exaggerating its effect and magnitude.

Therefore, though being qualitatively different from the Newtonian laws in term of quantitative precision, the Continuity Representation and the Discontinuity Representation nevertheless possess some of the key information contained in the first law and the second law of Newton as well as the basic relationship between them, namely, complementarity. But, while the two Newtonian laws are complementary in term of the Newtonian force, the Continuity and Discontinuity Representations are complementary in term of dynamic discontinuities.

The bond between the Continuity and the Discontinuity Representations can be further seen in each of their connections to the conception of mechanical state in the impact situation. If the mechanical magnitude of an object has a key role to play in the Continuity Representation, as demonstrated in the experiments, then the representation of the mechanical magnitude of a dynamic discontinuity, or the magnitude of Discontinuity Representation, should be also directly related to the conception of the mechanical magnitude of a single object because an impact, which is a mechanical discontinuity of the positive kind, can be conceptually seen as the mutual blocking of dynamic motion between two objects, both of which however are strictly in compliance with the Continuity Representation. Therefore, the stronger the Continuity Representation of the single objects prior to the impact, the stronger the magnitude of the Discontinuity Representation. To put it simply, in Discontinuity Representation lies the root of the actual commonsense concept of force, and, in the connection between the Continuity Representation and the Discontinuity Representation lies the connection between the "motion implies force" conception and the 'actual' commonsense force conception. More detailed discussion on these issues will be dealt with elsewhere.

Hence, the Continuity Representation and the Discontinuity Representation can explain the conceptual connections among several important types of mechanical information about the dynamic component in the human cognitive system. Together, they are able to cognitively account for the diverse responses of the subjects under various mechanical conditions. Therefore, they form a general principle or model for dynamic cognition in a language of Newtonian physics and Information Processing. Though by no means complete, this principle captured some important aspects of dynamic cognition which so far have been totally ignored. Its development shows to us the power of a new approach to the problem of dynamic representations, which is the natural outcome of the Newtonian concepts meeting the modern information concept.