What cues do billiard experts use? Conceptual and ...€¦ · conceptual and action-based knowledge, billiards lends itself to such an analysis since every shot is premeditated and

What cues do billiard experts use?

Conceptual and perceptual judgments of spin

Heiko Hecht

Universität Bielefeld (ZiF)

Dennis Proffitt

University of Virginia

Correspondence:

Heiko Hecht

ZiF der Universität Bielefeld

Wellenberg 1

33615 Bielefeld, Germany

E-mail: [email protected]

Running Head: Dynamics of Spin in Billiards

Dynamics of Spin in Billiards2

Abstract

Complex dynamic events are often misunderstood at the perceptual and intuitive level. In spite

of these difficulties, experts, such as athletes or billiard players, learn to make very precise

dynamic judgments. This study explored the basis of such judgmental abilities using the

example of rotational motions (spin) as they occur in billiards. Five experiments were carried

out to evaluate two paradigms that might explain how observers may judge the dynamics of

spin: perceptual attunement and learning of procedural heuristics that apply to one salient

dimension of information. Conceptual knowledge about the effects of spin on trajectories of

billiard balls was found to be erroneous. Almost all novices and more than half of the experts

tested had mistaken beliefs about one particular type of spin (english). These beliefs were

mirrored in animated contexts where subjects had to observe computer-generated events of

spinning balls that had possible and impossible effects. Visual performance was also assessed

in an ecological setting. It is concluded that novices are not perceptually attuned to the general

dynamics of spin and that experts do not acquire such attunement. The heuristics approach, on

the other hand, seems to explain the systematic errors that novices make. However, it

thoroughly fails to explain why expert observers are able to judge multi-dimensional with an

accuracy that equals the precision with which shots can be executed. Perceptual abilities appear

to be more situation-specific than assumed by either of the two approaches.


Introduction

The degree to which a human observer can use visual information to judge dynamic

properties of moving objects is the object of research in the field of dynamic event perception.

Since the ecological approach to perception (Gibson, 1979) made the topic popular and

inspired such seminal studies as Runeson's (1977) work on collision dynamics, three rather

heterogeneous approaches can be discerned in this field: First, there are studies that are

concerned with improvement of ball skill (see Abernethy, 1987 for an overview). They have

produced a large body of practical knowledge for training purposes but tend to be rather

descriptive, as for example Abernethy's (1993) concept of the minimally essential information

for event perception and action. Second and rather isolated from this approach, two conflicting

principles have been suggested attempting to explain how visual information of moving objects

is utilized to arrive at judgments about the dynamics of these objects. To date, the latter

approaches of direct attunement and of perceptual heuristics have not been put to a test

determining their generality, mainly because novices have typically been used as observers. In

five experiments involving expert billiard players the present study reveals the importance for

dynamic event perception theories to be able to explain expert performance. Both existing

approaches will be shown to have serious shortcomings when it comes to expert judgments.

Existing perceptual theories have often been limited, if not in scope at least in empirical

testing, to novice performance. The events studied were usually either simple enough to

document good perceptual understanding of novices (e.g. collision dynamics) or they were so

complex that they were perceptually impenetrable (gyroscopes). There are, however, some

exceptions to this tendency, such as Beek and Turvey's (1992) study of temporal parameters in

cascade juggling. Another limitation of dynamic event perception studies has been that they

often neglect the cognitive skills involved in controlling physical systems, according to a

scrutiny of the field by Pittenger (1991).

To minimize such shortcomings and to put the two main contenders in the arena of

dynamic event perception to a thorough test, the domain of billiards was chosen for the present

analysis. It has the advantages of being an ecological situation and involving a relatively small

number of variables. The effects of spin in billiards are less complex than in other ball sports

but sufficiently complex to amaze the novice. Motions are confined to occur on a 2-D surface

and air resistance is negligible (unlike for example in squash or baseball). Moreover, this

domain, which is poorly understood by novices but in which experts excel, encompasses

cognitive as well as purely perceptual elements. If it is possible at all to differentiate between

conceptual and action-based knowledge, billiards lends itself to such an analysis since every

shot is premeditated and analysis of each shot gone wrong is crucial (Byrne, 1987).

Therefore, it seems to be an ideal domain to put existing theories to a test.

In what follows, the two theories in question, or rather approaches, are outlined. The

first advocates direct perceptual attunement to variables that specify the dynamics of the


situation (kinematic specification of dynamics). The second approach posits that experts

acquire a set of procedural heuristical rules that do not need to be based on the relevant dynamic

variables, but may use some other static or kinematic cues (perceptual heuristics approach).

Although dynamic event perception could, in principle, be addressed by other cognitive

theories, such attempts have not been made. For instance, such an endeavor might be

conceivable within an inference theoretical framework (Rock, 1983) suggesting that perception

is the conscious result of a hierarchy of unconscious inferences.

Judging dynamic events

Dynamic events are those events that physicists group together in the field of classical

dynamics and which they describe in terms of masses, forces, conservation of momentum, and

energy. The throw of a baseball would be an example of such an event. It is thrown forward

with a particular force; and while in motion, the constant force of gravity acts on it, creating a

parabolic trajectory. The ball's potential and kinetic energy are transformed into other forms of

energy (heat and deformation) on impact. The process of judging such dynamic events is

based on perceptual processes that differ from those used to judge non-dynamic events (e.g.

recognizing chess configurations). The latter could be judged based on some static pattern-

matching procedures, whereas the former have to rely on more complex knowledge. Dynamic

events evoke impressions that cannot be easily inferred from the objects' static properties.

Causality, impetus, or weight are examples for such impressions (for reviews see Gilden,

1991; Pittenger, 1991; and Proffitt & Kaiser, 1995). The two approaches in question differ

mainly in terms of how observers obtain the information required for such dynamic judgments.

Kinematic specification of dynamics (KSD)

Dynamic event perception has been a major focus of interest among Gibsonians, and

within this tradition, Runeson and Frykholm (1981, 1983) proposed a model that links the

dynamics of an event directly to kinematic information that is visually given. This is possible

whenever the kinematics specify the dynamics of a situation. Kinematics can be thought of as

geometrical descriptions of motion (changes in the optic array) solely based on length and time

parameters, but not on mass. The lawful relation between kinematics (e.g. a particular

acceleration pattern) and a causal impression (e.g. that an object was thrown) has been

subsumed under the principle of kinematic specification of dynamics (KSD). Runeson and

Frykholm demonstrated that the visual system can achieve excellent dynamic evaluations of

motion contexts on the basis of kinematic information: For example, observers could

accurately estimate the weight of a box that was picked up by an actor when all they saw was a

set of point-lights that were placed on the actor's joints and on some locations on the box (see

also Bingham, 1987, 1993).

Extreme interpretations of the principle notwithstanding (Gilden, 1991) KSD proposes

that it is possible, not necessary, for observers to pick up the dynamics of kinematically


specified events. That is, as long as the proximal stimuli on the retina contain enough

information to reconstruct the dynamic properties of the observed event, observers can learn to

do so. In order to test the validity of the KSD principle, one needs to provide the observer

with sufficient opportunity to judge and manipulate complex events. KSD would only be

refuted if extensive training failed to enable the observer to become perceptually attuned to the

relevant kinematic invariant. However, most of the empirical data in the event perception

literature is based on novice performance and thus is fraught with serious doubts about the state

of attunement. More studies with experts, for whom maximal attunement can be assumed, are

needed. This is particularly important since KSD does not provide any ancillary hypotheses

about when attunement has been achieved. Attunement is merely thought to be accomplished

by a process of perceptual learning and differentiation (E. J. Gibson, 1969). With experience,

perceptual invariants can be extracted more reliably and with greater differentiation.

Perceptual heuristics (PH)

In sharp contrast to KSD, a heuristical account of dynamic event perception has been

proposed (Gilden & Proffitt, 1989; Proffitt & Gilden, 1989; Gilden, 1991), suggesting that

observers use heuristical rules that do not necessarily penetrate awareness. Only one rule can

be used at a time; and only if the rule that observers happen to use is adequate for the situation

are judgments accurate. In all other cases, if the rule does not apply to the situation or if more

than one rule determine the event, observer's judgments are poor. The term heuristic was

introduced to dynamic event perception by Todd and Warren (1982) but has assumed a

meaning which is quite different from the definition by Tversky and Kahneman (1974).

Perceptual heuristics typically are not consciously accessible and can thus only be tested

indirectly.

The PH approach suggests that performance is poor whenever the motion context is

multi-dimensional or when observers attend to an irrelevant dimension. The dimensionality of

a motion context is a function of how many attributes or descriptors (e.g. position, mass,

shape, size, orientation) of the moving object influence its motion (Proffitt and Gilden, 1989).

Any given heuristic deals with one and only one object descriptor. Depending on the number

of descriptor involved, two motion contexts can be distinguished, extended body motions and

particle motions. Extended body motions are multi-dimensional because more than one

category of information influences the motion trajectories, thus more than one heuristic is

needed to describe the event. A wheel rolling down an inclined plane is an example for such an

event: Mass distribution within the wheel, length, and slope of the incline are the categories of

information necessary to determine how long it takes the wheel to reach the bottom of the

ramp. Objects within particle motion contexts can be adequately described in terms of the

motion of their centers of gravity, that is by a single descriptor. An object falling straight down

is an example for such a one-dimensional motion context. Proffitt and Gilden have collected

strong evidence that novice observers are very good at judging the dynamics of such particle


motions as long as they do not misconstrue them as being multi-dimensional. However, our

perceptual judgments of multi-dimensional motions are very limited. Be it gyroscopes,

balances, colliding objects, volume displacements, or rolling wheels, untrained observers do

not seem to be able to integrate the relevant variables accurately. Rather, they behave as if they

pick the most salient dimension and apply a heuristic that is based on it, thereby neglecting all

other less salient factors. Most people behave as if they could only apply one heuristic at a

time. With respect to wheel motions, observers are able to roughly differentiate between

constant and acclerated motions but they have a hard time coupling rotation and translation

(Hecht, 1993). This means that billiard experts, who have to differentiate between sliding and

rolling motions have to acquire this skill.

Expertise in judging multi-dimensional dynamic events

As different as the two approaches may be, both make rather general claims about our

understanding of dynamic events. Studies of expertise seem ideally suited to put both

approaches to a strong test. It is in the domain of expertise where the predictions of the

approaches differ the most: KSD predicts strong perceptual performance whereas PH predicts

performance that can eventually be reduced to a set of simple heuristical rules.

Here we are concerned with experts in the realm of dynamic event perception, that is

most athletes would fit the bill but not chess players. Such experts have to acquire skills at a

level of accuracy that often escapes the physicist's analysis (Bayes and Scott, 1963, Gel'fand

& Tsetlin, 1962). From a psychological point of view, it is unresolved how experts acquire

expertise with such complex dynamic events. There seems to be agreement among researchers

that experts are more likely than novices to exploit advance cues like arm movement or racket

position (Abernethy, 1987, Bard & Fleury, 1976; Jones & Miles, 1978; Neumaier, 1983;

Tyldesley, 1981); but there is less agreement concerning experts' superior abilities of picking

up visual information during the motion phase of the event. Experts seem to spend more time

looking at critical portions of the motion, usually the first third of the ball's trajectory. Also,

they are able to see or extract the ball's implied final state earlier than novices (Abernethy,

1990; McLeod, 1987; Ripoll & Fleurance, 1988).

The present studies focused on novices' and experts' conceptual knowledge of the

dynamics of spin as well as on their perceptual abilities. Billiards was chosen as the domain of

expertise because, in principle, it does not require attunement to spin or good conceptual

knowledge. It is by no means established what constitutes expertise in billiards, and unlike

most ball sports, billiards does not rely on fast perception-action sequences. Thus, if anyone,

billiard players would have had the opportunity to develop perceptual heuristics while

premeditating their shots. Alternatively, they could have become perceptually attuned to the

critical variable of spin, which - as will be shown - is not picked up by novices.

Overview to Experiments


Experiment 1 assessed observers' conceptual knowledge of the dynamics of spin.

Novices had misconceptions about the effect of spin on a billiard ball. Surprisingly, experts

also had misconceptions about a rather fundamental aspect of spin (side english). Experiment

2 tested whether the misconceptions were mirrored in perceptual tasks involving one-

dimensional motion contexts. It was found to be the case when computer animated sequences

involving different types and amounts of spin had to be judged. To investigate whether visual

information about the interaction of cue-stick and cue ball is used to achieve the high

performance which is typical for billiard experts, Experiment 3 was carried out on a physical

pool table. Shots with spin components had to be judged either based on visual information of

the spinning ball or based on the interaction of cue-stick and cue ball. Experiments 4 and 5

dealt with more complex multi-dimensional situations. Expert performance poses serious

problems to PH and KSD.

Experiment 1: Conceptual understanding of spin

In order to assess the extent of experts' knowledge about spin, it is important to

establish if they had to acquire it in opposition to or in congruence with common-sense

understanding. This experiment therefore sought to determine the conceptual understanding

that novices and experts have about effects of spin on ball trajectories.

As depicted in Figure 1, any given spin of a ball that is moving across a planar surface

(along the z-axis) can be decomposed into three orthogonal parts (Walker, 1985; Whitehead &

Curzon, 1983). In keeping with conventions of billiards, spin around a vertical axis (y) is

called english or side english. Spin around a horizontal axis perpendicular to the motion (x) is

called follow or draw, and spin around the axis of motion (z) is called massé. Given an initial

straight sliding motion of the ball along z, english, follow or draw will not change its linear

trajectory, but massé will. Clockwise massé will make the ball curve to the right, while

counterclockwise massé makes it curve to the left.

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Insert Figure 1 about here

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Method

Subjects. Four-hundred University of Virginia undergraduates participated in a

mass-testing session in order to partially fulfill a research option in an introductory psychology

class. Ninety-one subjects had to be dropped from the analysis because they failed to fill out

the questionnaire or gave uninterpretable answers. Separately, sixteen world-class experts

were interviewed and asked the same questions that were posed to novices.


Stimuli and Design. Students were presented with a drawing of a ball. It was

indicated graphically and in writing that a particular spin was added to the ball's forward

velocity. Six different trials were presented between subjects. They differed only in the type

of spin that was meant to be on the ball: left english (46 subjects), right english (49), clockwise

massé (45), counterclockwise massé (57), follow (54), and draw (58). Experts received

verbal instructions while looking at an actual billiard ball, which was placed at the head spot of

a pool table. Spin around the vertical axis and forward momentum of the ball were specified

verbally.

Procedure. As Figure 2 exemplifies for the case of left english, the task consisted of

completing the drawing by adding the trajectory that the ball would follow given its spin and

forward sliding motion. In addition to drawing the path, subjects had to circle a number

corresponding to the amount of deviation from a straight forward path. On a separate page

they also had to check whether they intended to draw a straight or a curved path. Experts were

asked either the left-english or the right-english question in random order. They were first

asked whether the ball would travel on a straight path or whether it would curve. If they

decided on curvature they were to specify roughly how far the ball would deviate from the

straight within a table-length.

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Results and Discussion

Trials that were answered inconsistently or incompletely were dropped from the

analysis. In such trials, subjects typically failed to draw a trajectory and/or to circle a number.

Sometimes the trajectory was incompatible with the direction indicated by the number.

The results for novices showed that massé was judged correctly in 83.3 % of the cases;

it was predicted to go straight in 6.9 % and to curve against the spin in 9.8 % of the cases.

Follow and draw were judged correctly by 45.5 % of the subjects; they indicated that it would

curve to the left in 28.6 % of the cases and to the right in 25.9 %. English was believed to

make the ball curve with the spin by 40 % of the subjects in this condition, to curve against the

spin by 49.5 % and to go straight by only 10.5 %.

Four pilot subjects, who had previously been asked the same questions interactively,

never made erroneous predictions for follow, draw, and massé. Thus, the instructions for

follow and draw might have been unclear, and more students might have had accurate beliefs

about the effect of those spin types. This is supported by the fact that 91 subjects were unable

to produce any interpretable results.


A follow-up experiment was performed on those subjects who gave wrong answers in

the original testing. Eighteen subjects were picked at random, 6 in each class of spin. They

were then presented individually with their questionnaire and asked to communicate verbally to

the experimenter how they interpreted the instructions. It turned out that all 12 subjects who

had answered the massé and follow/draw questions "wrong" had misunderstood the type of

spin that was meant to be on the ball. When asked how a ball with the intended spin would

behave, they all answered that massé would make the ball curve with the spin and that follow

and draw would make it go straight. The 6 subjects in the english conditions, on the other

hand, interpreted the instructions in accordance with the experimenter's intention.

In sum, novices seem to have accurate explicit knowledge about the effects of massé,

follow, and draw. However, they exhibit systematically wrong conceptions about the effects

of english. 89.5% of the subjects believed incorrectly that english causes the ball to follow a

curved path.

A quite different picture emerged for the experts. The majority believed that english

would make the ball curve against the direction of the spin. Nine out of 16 experts stated that

the ball would curve against the sense of the spin by a small amount. They did not even

change their mind after the experimenter re-emphasized the vertical spin axis. The other 7

experts correctly guessed a straight path. Interestingly 5 of the 7 players who gave correct

answers reported to have read somewhere that the ball would go straight in this situation.

Thus, experts' conceptual bias was exactly the opposite of what was found with novices.

Experiment 2a: General one-dimensional spin (novices)

The biases that were found in Experiment 1 might be merely conceptual in nature. To

test whether more accurate knowledge about spin might be accessible perceptually or

intuitively, a purely visual task was designed. Physically possible and impossible computer

animations of billiard shots were presented. Subjects had to make naturalness judgments,

which they could base on their visual impression or intuition.

To analyze components of spin the "possible" shots had pure spin around only one given axis

at a time and could thus be classified as one-dimensional events. Such shots rarely occur in a

realistic setting and sometimes might even be impossible to accomplish with a cue stick. Thus,

the simulations sought to test general knowledge about basic effects of spin rather than

situation-specific knowledge as needed during the game of billiards. According to the principle

of KSD, experts might have acquired such knowledge since a few general laws of spin are

much more parsimonious than numerous situation-specific invariants. The PH approach on the

other hand would be less likely to predict generalized knowledge.

In order to determine whether novice observers appreciate the fact that only massé spin

(Figure 1) can make a ball curve, a variety of spin-trajectory combinations were presented in

random order. That is, for a given spin both a natural or unnatural trajectories could be


assumed by a given ball. It was expected that follow and draw would be associated with

straight trajectories whereas massé (correctly) and that english (incorrectly) would be

associated with curved trajectories.

Method

Subjects. Sixteen University of Virginia undergraduates, eight females and eight

males, participated in order to partially fulfill a research option in an introductory psychology

class. None of the subjects had participated in Experiment 1.

Stimuli and Apparatus. As shown in Figure 3, the displays consisted of a portion

of a billiard table viewed from above. The rail was visible on one side of the monitor, next to it

a gray occluder, which covered half the ball (when at rest) and all cue-action. The rest of the

display simulated green cloth. All stimuli were created on a Sun 3/60 Color Graphics

Workstation and then recorded frame by frame onto 3/4 inch videotape using a RGB Videolink

1400 encoder. They were then displayed on a 24 x 18 cm Sony Trinitron color monitor. The

balls had two differently colored halves in order to provide clear information about their

orientation. The ball was always positioned such that the direction of rotation became

apparent. That is, for english and massé trials the ball started out as drawn in Figure 3; for

follow and draw trials the equator of the ball was parallel to the occluder (rotated by 90o). The

ball would then move across the table on different trajectories. Note that in all events the ball

had sufficient forward motion to make it slide across the surface. A qualitatively different

motion would occur as soon as a ball starts to roll. Once the ball had left the visible area it did

not reappear, that is no rebounds were involved.

Design. Five trajectory types (straight, two with different amounts of leftward

curvature, and two with rightward curvature) were paired with six types of spin (left and right

english, follow, draw, clockwise and counterclockwise massé) resulting in 30 different

stimuli. Trajectory and type of spin were within-subjects factors. Naturalness ratings served

as dependent variable.

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Procedure. The monitor was turned on its back such that the screen was parallel to

the floor suggesting a straight-down view onto a billiard table. The visual angle was

approximately 18o. Between subjects, the monitor was turned such that the ball's motion

started from the left, right, top, or bottom to counterbalance for possible directional biases.

Subjects were tested individually in a dark room. They were instructed that a "magic"

cue stick would strike the ball underneath the gray occluder and move it across the table. It


would also make the ball rotate. The task was to judge whether a ball was likely to follow the

path that it did given the motion plus rotation that the magic cue stick had put on it. Thus,

observers were asked to judge whether the combination of translation and rotation that they

saw was natural, and not whether a given shot could be produced. They were also alerted to

the fact, that on a real billiard table balls would often rotate a lot faster. It was necessary to

slow down the rate of spin compared to a typical canonical event. The rates of spin used were

between two and 8 times slower than in realistic events that would produce curvatures similar

to the ones that were shown. Imitating realistic rates of spin would have created enough blur at

a play-back rate of 30 frames per second to make the direction of spin undetectable.

Each stimulus was presented twice before a judgment had to be made by circling a

number of rating scale on a piece of paper. The scale ranged from 1 (virtually impossible) to 6

(very natural) in steps of 1. Subjects received 15 practice trials that contained one example of

each trajectory-spin combination to anchor the naturalness scale.

Results

Novice performance was comparable to their conceptual knowledge about the effects of

spin. As visible in Figure 4, follow and draw looked significantly more natural when the ball

traveled on a straight trajectory as compared to impossible curved paths [t(15) = 4.90, p =

.0002]. Massé shots that curved appropriately with the sense of the spin (with) received higher

ratings than impossible ones that curved the opposite way (against) or that went in a straight

path (straight) [t(15) = 6.84, p = .0001]. Balls with english looked more natural when they

curved with the spin compared to a straight path or curvature against the spin [t(15) = 3.38, p =

.004]. A left-english ball (see Figure 2) was typically judged to curve toward the right. That

is, observers preferred the impossible event of trajectory curvature over the canonical straight

path.

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Discussion

The availability of visual information about the spin of billiard balls did not facilitate

observers' abilities to distinguish canonical events from impossible ones. The misconceptions

about english spin, which were found in Experiment 1, were paralleled in judgments that were

based on perceptual information. Often, a visual presentation of physics problems facilitates

performance over mere conceptual reasoning (Kaiser, Proffitt, Whelan, & Hecht, 1992), for

example when judging trajectories of pendulum bobs that are severed at different phases during

their motion.

The failure to exploit perceptual information in the case of spin could be interpreted in

two ways: From a KSD point of view one could argue that novices are not perceptually


attuned to the dynamics of spin, since attunement should have lead to general knowledge about

the different dynamic effects of spin. From a PH point of view one could say that novices use

a heuristic that is appropriate for follow, draw, and massé but erroneous for english. The

heuristic "a billiard ball always curves with its spin, regardless of spin axis" would fit the data.

In follow and draw shots the effect of spin is parallel to the motion vector and thus, according

to this heuristic, does not change the trajectory of the ball. In massé and english shots the

heuristic would predict curvatures as preferred by novice observers. Here the PH explanation

appears to be superior to a KSD explanation involving a mere lack of attunement. The latter

would suggest that unattuned observers perform at chance on english trials; but they make

systematic errors preferring curvature with english spin.

Experiment 2b: General one-dimensional spin (experts)

Experiment 2a was replicated with expert billiard players. If it is possible to become

attuned to the general dynamics of spin-motion interaction, then experienced billiard players

would be expected to be flawless in the computer animation task. This hypothesis is based on

the assumption that experts use visual information about spin when acquiring their expertise.

According to PH, on the other hand, observers, including billiard experts, should only be able

to use one perceptual heuristic at a time. Thus, their responses should be systematically

biased.

Method

Subjects. Eight expert billiard players volunteered to participate in the study. They

had all participated in numerous Nine-ball tournaments in the north-eastern US and were

mostly professional or semi-professional players. Their average age was 32.3 years. They had

played pool or billiards for 17.3 years at an average of approximately 12 hrs weekly.

Stimuli and design. The same stimuli, apparatus, and design were used as in

Experiment 2a.

Procedure. The same procedure as in Experiment 2a was used, except that experts

were not debriefed until they had also been run in Experiment 3c. They were run in a quiet

corner of one of the billiard halls where they typically practiced and participated in

tournaments.

Results

As shown in Figure 5, within each class of spin, experts judged the canonical events to

look most natural. However, they were by no means perfect. In particular, they allowed

english balls to curve a little against the direction of spin. Post-hoc tests showed that such

balls, even though impossible, were rated significantly more natural than impossible massé


[t(7) = 3.97, p = .006] trials. The same held marginally in a comparison to impossible follow

and draw trials [t(7) = 2.35, p = .051].

As expected, experts clearly judged straight follow and draw events to be more natural

than curving ones [t(7) = 6.82, p<.0002]. They also picked massé balls that curved with their

spin as more natural then impossible ones [t(7) = 4.22, p<.004]. Experts judged balls with

side english to be most natural when they went in a straight path [t(7) = 2.74, p<.03].

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Insert Figures 5 and 6 about here

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To compare judgments of experts and novices, the naturalness ratings were converted

into scores that were standardized for each subject to a mean of 3.5 and a standard deviation of

1. Thus, possible interindividual differences in the use of the naturalness scale were

eliminated. These scores were then subjected to a repeated measures ANOVA with level of

expertise, spin type, and spin direction as independent factors. Experts gave somewhat higher

ratings than novices to canonical follow and draw events [F(1,45) = 6.48, p = .014]. No

difference was found for massé. The most striking expert-novice differences occurred for

english trials, as visible in Figure 6. Experts gave the canonical straight path much higher

ratings than novices [F(1,45) = 40.41, p < .0001]. Also, balls that were curving with the

sense of the english looked less natural to experts [F(1,45) = 11.96, p = .0012] then they did

to novices, whereas balls curving against the english were rated more natural [F(1,45) =18.19,

p < .0001].

Discussion

Experts did not show evidence of good perceptual attunement to the general dynamics

of spin. In particular, they were able to detect impossible trajectories more easily for follow,

draw, and massé than for english. This performance corresponds to the conceptual and

perceptual difficulties that novices and experts have with spin. For balls with english spin,

experts' naturalness judgments appeared to differ from those of an ideal observer and also from

the accuracy with which experts can execute shots. Experiment 5 explicitly addresses this

issue.

The range of stimuli that were used did not constitute a strong test of PH or KSD, since

only a subset of the possible stimulus space was tested. For all simulated shots the axis of spin

was perfectly horizontal or vertical, which is a rather unusual situation, especially as far as

english shots are concerned. However, Experiment 2 has shown that observers' judgments

based on visual information about spin are not perfect, even when the context only involves 1-

dimensional but general effects of spin.

Experiment 3a: Judging english in natural context (novices)


The purpose of this experiment was twofold. First, the problems associated with

simulated events were cured by replicating Experiment 2, with necessary modifications, on an

actual billiard table. Side english shots were used, which are advanced shots used in rebound

situations when the desired path is obstructed. English changes the rebound angle and thus

enables the player to use a different rebound location on the rail. Observers had to judge the

motion of such balls based on information about the ball before it hit the rail and was reflected.

Second and more importantly, two viewing conditions were introduced, one in which the

observer saw only the cue-stick interacting with the cue-ball, and one in which he/she saw only

the spinning and sliding billiard ball. Expert knowledge could, in principle, be based on laws

of spin or on information about the location where the cue-ball needs to be struck. That is,

experts might map cue impact locations on the cue ball into outcomes without ever

understanding spin.

A typical shot in billiards can be understood as consisting of 3 stages. (1) The cue

stick hits the cue ball at a particular point of impact on its surface. (2) Spin and horizontal

velocity (v) are imparted to the ball; the amount of spin is a function of how much off-center

the ball is hit, and the direction of v is roughly parallel to the cue1. (3) Amount and type of

spin determine how the ball is reflected off the rail or how it behaves after colliding with

another ball.

Stage 3 is completely determined by stage 2, and stage 2 in turn is determined by stage

1. That is, the information contained in stage 1 as well as that contained in stage 2 are

sufficient to predict the event of stage 3. However, it is likely that the expert does not pick up

all the information contained in every stage. Rather, it seems plausible that only information

that is salient and functional for the desired outcome is used. Additional redundant information

may be picked up incidentally or ignored completely. Thus, expert knowledge could be

categorized as follows.

First, mechanistic knowledge could link stage 1 directly with stage 3. For example, if

the cue ball is hit at a location straight above its equator, then it will follow the object ball after

the collision. This would be a simple procedural heuristic that could serve as a guideline to

execute shots in billiards. Second, knowledge could link stage 1 with 2 and then stage 2 with

3. Here, performance would be perceptually mediated by knowledge about spin and its

effects.

To assess the importance of mechanistic and spin-related information, the availability of

this information was manipulated by occluding stage 1 or stage 2 respectively. Observers had

to base their judgments (predicting the trajectory) exclusively on the information available in

spin (stage 1 occluded) or in location of impact (stage 2 occluded). If the perception of spin is

not a necessary part of the experts' skills, then they should perform particularly well when they

can see the cue-stick striking the ball (stage 1). It is expected that novices do not appreciate the

1 This corresponds to implicit assumptions made by Walker (1985) and by Griffing(1987).


effects of spin or, at best, in qualitative terms. Since experts need to acquire accurate

knowledge about the effects of english, that is, knowledge that deviates from common sense

intuitions, the outcome of this experiment should not only allow us to draw conclusions about

the role of perceptual and procedural knowledge in expertise with spinning objects, but it

should also enable us to gain some clarity about the extent to which visual learning of non-

intuitive dynamic relations occurs.

Method

Subjects. Eight University of Virginia undergraduates, four females and four males,

participated in order to partially fulfill a research option in an introductory psychology class.

Subjects had played billiards at most two or three times and were unfamiliar with any of the

previous studies.

Stimuli and Apparatus. A standard 9-foot pool table (254 x 127 cm) was used for

the experiment, as depicted in Figure 7. A striped billiard ball (e.g. ball 12) was positioned

between the head spot and the center of the head rail. A rectangular occluder (tunnel) was put

between ball and head spot such that the ball was free to move underneath it.

-------------------------------------------


-------------------------------------------

Design. On each trial, the ball was aimed at the center of the foot rail with differing

amounts of side english, which would cause the ball to rebound within an angle of about 60o

to either side of the line of aim. Two spin types (left and right english) were paired with three

amounts of spin. A shot without spin was added that rebounded straight back along the line of

aim, resulting in 7 unique events.

Two within-subjects conditions were run separately. In the cue condition (stage 1)

observers could only see the cue-stick striking the cue-ball. They had to make a judgment or

guess where the ball would hit the second rail after it had rebounded from the center of the foot

rail. In the spin condition (stage 2) they could only see the ball (but not the cue-stick) as it

came out of the occluder until it almost hit the foot rail. Each event was repeated three times

per condition. Thus, 21 shots were presented in random order for each of the two conditions.

Procedure. Subjects were run individually at a 9-foot pool table in a game-room at

the University of Virginia. In the cue condition, once the ball and the occluder were

positioned, the experimenter explained that he would always aim for the center of the foot rail,

which was marked with a piece of bright tape. The ball would hit the rail and then rebound and

hit a second rail. The task was to predict which second rail the ball would hit after the

rebound. Also, subjects were asked to guess where on the rail the ball would touch, by


walking around the table and pointing to that location. The two side rails and the head rail were

pointed out as possible locations for the rebounding ball.

In the cue-condition, subjects were instructed to close their eyes as soon as the ball

disappeared behind the occluder after being struck. For cases where they had failed to shut

their eyes in time or to get a good view on the strike, they were asked to notify the

experimenter so he could insert the trial back into the random list. Between two and five

practice trials ensured that observers felt comfortable shutting their eyes in time. They were

instructed to keep their eyes shut until the experimenter had measured the exact position where

the ball had struck the rail. After the ball was removed from the table observers guessed the

location, which was then recorded. Subjects were not allowed to touch the table and listened to

non-intrusive classical music through a walkman in order to drown out acoustic cues, such as

the noise when the ball happened to fall into a pocket, or vibrations when it hit a rail. The time

between strike and the go-ahead for the subject to make her or his guess was kept constant at

approximately 10 s.

During the spin condition of the experiment, observers held up a piece of cardboard that

prevented them from seeing the cue-ball or the cue-stick. They watched the ball exiting the

occluder and were asked to shut their eyes when the ball was about one inch in front of the

marker on the foot rail. Again, two or more practice trials were carried out. Otherwise, the

procedure remained unchanged.

Whenever the experimenter missed the center of the foot rail by 1 inch or more or failed

to put the desired spin on the ball (when the intended rebound angle was missed by more than

15o), the trial was aborted and inserted back into the random list. This happened on average

about three times per condition. Cue and spin conditions were run as counterbalanced blocks.

A short break was taken between the two conditions.

Results

Novices performed very poorly at this task. In cases where a considerable amount of

spin was on the ball, they were able to guess the correct rail in 51.0% of the trials, chance level

being at 50 % as long as observers make the assumption that spin has an effect at all and thus

diverts the ball to the left or to the right. This assumption was induced by the task of making

differential predictions about the rebound. Trials in which the ball had no spin were not

included in this count, they rebounded straight back. On these trials, performance was 65.3 %

correct in the spin condition and 36.5 % in the cue condition [t(7) = 2.59, p<.04].

A repeated measures ANOVA was conducted with the following dependent variable:

the absolute error between judged the actual point of impact on the second rail. The rebound

angle was computed in degrees with the right corner pocket at the foot rail being 0o and the left

corner pocket on the same rail being 180o. The actual range of rebounding balls was between

35o and 145o (see Figure 7). According to this measure, the spin condition was not

significantly better than the cue condition [F(1,7) = 3.94, p<.088]. On average, subjects came


within 5o of the actual location on the second rail in only 13.7% of the trials (spin: 17.3%, cue:

10.1%).

Discussion

Novices found this task hard to do and often stated that they were making wild

guesses. Most of them reported that they were changing their strategies several times during

the experiment, unsure whether left english should produce a rebound towards the left or the

right side rail.

It can be concluded that the effects that english spin has on rebound angles of the

billiard ball are by no means obvious or perceptually available. The question arises to what

extent these effects can be learned and if so, how quickly.

Experiment 3b: Judging English in natural context (trained novices)

Experiment 3a was replicated as a learning study with feedback in order to determine

how easily information about the effect of english on rebound angles could be learned.

Method

Subjects. Eight University of Virginia undergraduates, four females and four males,

participated. They fulfilled the criteria that were set for novices in Experiment 2a.

Stimuli and Design. The same equipment, stimuli, and controls as in Experiment

2a were used. However, subjects received a short training before the actual experiment.

Procedure. The experimenter positioned a striped ball close to the head rail in the

center of the pool table. He then asked the subject to observe carefully and try to "figure out

what was going on". He presented two series of 7 shots ranging from extreme right english to

extreme left english while always aiming straight at the marker (center of foot rail). Such series

were repeated until the subject reported to have an understanding of what was going on. It

took on average three or four series. The concepts of spin and contact point with cue-ball were

deliberately withheld from subjects, in order to keep them from developing any biases.

The experiment proceeded as described under Experiment 3a, with the exception that

observers received feedback about the actual position where the cue-ball hit the second rail.

Feedback was given after each trail after the prediction had been recorded.

Results

Trained novices performed fairly well at this task. On average, they were able to guess

the correct rail (for trials with clear side english) in 88.5% of the trials. In the spin condition,

they guessed correctly 89.6% of the cases and in the cue condition 87.5%. The absolute error


between actual impact point and subjects' predictions (ANOVA) was not significantly different

between spin and cue conditions. On average, subjects came within 5o of the actual location

on the second rail in 46.1% of the trials (spin: 44.6%, cue: 47.6%). This performance was

significantly better than that of untrained novices [F(1,14) = 31.37, p<.0001].

Trained novices continued to improve slightly throughout the course of the experiment.

The signed difference between actual and predicted rebound angle correlated negatively with

the order of the trials [r=-.15, p<.008]. This correlation was only significant for the spin

condition [r=-.22, p<.004], but not for the cue condition [r=-.08].

Discussion

The brief training showed dramatic effects. Subjects hardly ever missed the correct

rail, and in one half of the trials they came fairly close to the actual point of impact. When

asked for a rationale onto which they based their predictions, most subjects came forth with the

appropriate rules of thumb. For example, when the ball was spinning clockwise (left english),

it would rebound to the left. However, some observers reported to use mistaken heuristics

when it came to make quantitative judgments. They thought, for instance, that balls that travel

slower would produce less of an effect, which is not the case.

It can be concluded that it takes very little training to pick up qualitative aspects of the

spin-rebound interaction. Observers seem to have formed heuristics during practice and

elaborated or changed them very little throughout the experiment. The fact that these would-be

heuristics can be learned quickly, however, does not imply that experts rely on similar

heuristics when executing billiard shots.

Experiment 3c: Judging english in natural context (experts)

Experiment 3a was replicated with expert billiard players. If mechanistic mapping of

impact points on the cue-ball into desired outcomes of a shot is the basis of their expertise,

experts should perform well in the cue condition but worse in the spin condition. They should

never err in qualitative terms, but they should also be able to predict within about 5o where the

rebounding ball would hit the second rail. The range of 5o was chosen since it roughly

corresponded to the accuracy with which the experts who participated in this study were able to

execute rebounding english shots. On the other hand, if visual understanding of spin is

functional in acquiring expertise, they should do equally well in the spin only condition.

Method

Subjects. The same eight experts as in Experiment 2b participated voluntarily.

Experiment 2b was always run first.


Stimuli and Design. Standard pool tables of two different sizes (234 x 117 cm and

254 x 127 cm) had to be used, depending on the pool hall in which the experts were tested.

Otherwise, all conditions, stimuli, and controls were the same as in Experiment 2a.

Procedure. Before the experiment, experts were asked to "get a feel for the table", in

particular for the foot rail, by shooting a few balls across the table and into the rail. After the

expert thought that he had acquainted himself with the table, the procedure was identical to that

of Experiment 3a. Experts did not receive any feedback about their performance until after they

were done with all experiments.

Results

Experts did very well in both the cue and the spin conditions. They picked the correct

rail in 100 % of the trials (with side english). They came within 5o of the actual impact point

on the second rail in 56.6% of the cases (spin: 56.0%, cue 57.1%). As shown in Figure 8,

their performance did not differ significantly from that of trained novices. However, a repeated

measures ANOVA on absolute errors revealed that they performed much better than untrained

novices [F(1,14) = 48.23, p<.0001].

Whenever expert predictions differed from the actual rebound angle, it seemed to be

due to a systematic over- or under-estimation of the effect that english would have. This can be

seen in Table 1. The correlations between actual and predicted angles were significantly higher

for experts (ranging from .94 to .98) than they were for trained novices (ranging from .31 to

.93) based on a Mann-Withney U-test [U(8,8) = 64, p<.0001].

----------------------------------------------------------

Insert Table 1 and Figure 8 about here

----------------------------------------------------------

Discussion

Even though experts were not able to predict rebound angles with higher absolute error

than trained novices, the former made more systematic errors. They tended to underestimate

the effects of english. Visual performance (spin condition) was almost as good as expert

ability to carry out the shots they had to judge. When asked to execute a shot that would hit the

marker on the foot rail and then rebound into the left side pocket, experts often missed the

pocket by about 5o and took three or four trials before they could pocket the ball. That is, the

task was extremely difficult, likely because typical english shots require less english that the

situation that was chosen here.

Experts are able to use the information contained in the spin of the ball just as well as

the information contained in the impact point of the cue-stick on the cue-ball. The lack of

differences between cue and spin conditions suggests that expert knowledge in billiards might

be mediated by perceptual understanding of spin. Given an optimal viewing situation, which


would correspond the position of the person executing the shot, the spin condition might have

produced even better results. Observers had an unusual viewing angle, since they were

positioned near a head corner of the table.

In sum, Experiment 3 has established that billiard experts do not merely use

mechanistic rules about cue impact locations to master their sport. Their impressive visual

abilities suggest that visual understanding of spin is highly beneficial for their skills.

However, this appears to be hard to reconcile with the earlier findings that experts have

systematic conceptual (Experiment 1) and perceptual (Experiment 2) biases when judging

spin.

Experiment 4a: General multi-dimensional spin (novices)

So far, since experts have demonstrated excellent visual knowledge in the realistic

setting, it cannot be determined whether this knowledge can be thought of as heuristical or

direct in nature. More complex situations are required to differentiate between the approaches.

Since shots in billiards are typically multi-dimensional events, a fairer test would consist in

testing novices and experts in a generalized version of such complex events. Thus, Experiment

2 was modified to include two and three simultaneous components of spin (Figure 1).

Angular and translational velocities were held constant to avoid confounds even though on a

real billiard table friction would eventually slow down the ball's translation and rotation.

Experiment 4 sought to assess whether observers would use different strategies when

judging multi-dimensional motions. According to PH novices and experts should make

systematic judgment errors that indicate the application of a heuristic that only refers to one axis

of spin, neglecting effects caused by spin around the other axes. KSD would predict

erroneous judgments for novices, for lack of attunement, and good performance for experts.

As in Experiment 2, a variety of possible and impossible spin-trajectory combinations were

presented in random order.

Method

Subjects. Eight graduate students at the University of Munich, four females and for

males, were paid for their participation.

Stimuli and Design. Computer-generated stimuli were used which included trials

with spin complexities that were in the range of some of the most complicated shots in

billiards. Three events had spin around one axis only (pure left-english, pure follow, and pure

clockwise massé); two around two axes (left-english paired with clockwise massé, and left-

english paired with counterclockwise massé); and two had spin around three axes

simultaneously (left-english with counterclockwise massé plus follow, and left-english with

clockwise massé plus follow). In a within-subjects design, each of these 7 spin types was


coupled with 3 different trajectories that were straight, curving left, or curving right (an

intermediate curvature between the two values depicted in Figure 3 was chosen). These

pairings resulted in a total of 21 unique stimuli.

Procedure. Subjects were asked to make a naturalness rating after each stimulus.

Stimuli were presented in random order. Viewing conditions and procedure were identical to

those of Experiment 2a. A 20 x 15 cm SABA M25 color monitor was turned on its back such

that the screen was parallel to the floor suggesting a straight-down view onto a billiard table.

Viewing distance was approximately 60 cm which amounted to a visual angle of 18o. Subjects

were tested individually in a dark room. They were asked to judge whether the combination of

translation and rotation that they saw was natural, given that it was produced with a "magic"

cue stick. Subjects received 7 practice trials that contained one example of each spin type to

anchor the naturalness scale.

Results

As expected, novices' performance in this task based on perceptual information of spin

depended on the complexity of the spin. For one-dimensional trials, results were similar to

those obtained in Experiment 2a. Observers even gave english trials that curved with the spin

(down) marginally higher naturalness ratings (M=4.81) than clockwise massé trials curving

with the spin (M=3.31) [F(1,15) = 4.15, p<.0596]. However, english combined with massé

showed a different picture. As visible in Figure 9, when left-english and clockwise massé

were combined observers preferred curvature to the right over other trajectories [F(1,14) =

14.64, p<.0019], whereas when left-english and counterclockwise massé were combined, they

still judged curvature to the right as most natural [F(1,14) = 13.91, p<.0022]. That is, the

only relevant component of the spin (massé) did not seem to have a bearing on naturalness

judgments, whereas the english component did. When three spin-dimensions were involved,

the ability to make naturalness judgments completely broke down. Between none of the ratings

of three-dimensional spin trials could any significant differences be found.

-------------------------------------------


-------------------------------------------

Discussion

When only one spin axis is involved, novice observers seem to base their naturalness

ratings on the heuristic that the ball curves with the spin. As soon as a third dimension (spin

axis) is included their judgments break down completely. Performance is particularly revealing

when two-dimensional spin is on the ball. According to PH observers cannot integrate two

different heuristics. Thus they have to decide between the massé heuristic, which predicts

curvature to the right for clockwise and to the left for counterclockwise massé, and the english


heuristic, which mistakenly predicts curvature to the right in all cases. The massé outcomes are

labeled "C" (correct) and the erroneous english predictions are labeled "EH" (english heuristic)

in Figure 9. The data are in agreement with the exclusive use of the dominant but incorrect

english heuristic. In the critical case where EH and C contradict each other (left english with

counterclockwise massé) subjects judge balls that curve to the right to be most natural.

Once again, novices are not found to be perceptually attuned to the generalized

dynamics of spin. They seem to apply a one-dimensional heuristic that is accurate for pure

follow and massé but mistaken for trials with a discernible english component. Once the

motion becomes more complex (three components of spin) the ability to make naturalness

judgments breaks down altogether, probably because in those cases observers are unable to

detect any prevalent direction of rotational motion.

Experiment 4b: General multi-dimensional spin (experts)

Experiment 4a was replicated with expert billiard players. The experts chosen were

proficient in three-cushion billiards. Three cushion billiard is played with three balls on a table

that has no pockets. In this sport, the goal is to execute as many valid shots in a row as

possible. A valid shot is scored if the cue ball hits both object balls, and if before hitting the

second object ball, the cue ball has touched at least three rails. For two reasons, three-cushion

experts should be even more susceptible to detecting the effects of spin than pool players.

First, whereas a pool cue ball is typically white, it is marked with two black dots in three-

cushion billiards, which makes the ball's rotation visible. A pilot study with two experts

revealed that they were able to pick up kinematic information about spin from the dotted ball

just as well as from a striped ball. Second, since the ball has to rebound at least three times

from a rail, even a very small amount of english typically has dramatic effects. Thus,

knowledge about the effects of spin is even more important than it is in pool.

If these experts have become attuned to general aspects of the dynamics of spin, they

should do very well even when confronted with the most complex events. If, on the other

hand, their knowledge is situation-specific they are expected to produce errors.

Method

Subjects. Eight expert billiard players volunteered to participate in this study. They

were semi-professional players of an average age of 33.9 years, their experience with billiards

amounted to 11.5 years at an average of approximately 13.4 hrs per week.

Stimuli and Design. The same stimuli, design, and equipment as in Experiment 4a

were used.


Procedure. The same procedure as in Experiment 4a was used. Subjects were run

in a quiet room at a billiard club. In all other respects viewing conditions remained unchanged.

Results

One-dimensional events were judged similar to those presented to pool experts in

Experiment 2b. Effects of massé, follow, and draw were judged with great accuracy, whereas

english that resulted in straight (correct) trajectories did not receive significantly higher

naturalness ratings than english resulting in impossible curvatures against the sense of the spin

[F(1,14) = 3.44, p<.0849]. Correct follow events [F(1,14) = 127.93, p<.0001] and correct

massé events [F(1,14) = 53.35, p<.0001] were distinguished perfectly from unnatural ones.

Unlike novices, who mistakenly seemed to judge two-dimensional events based on

their english component, experts based their judgments on the correct aspect of two-

dimensional spinning motions, the massé component. As shown in Figure 10, experts judged

events that curved towards the correct side (pooled over clockwise and counterclockwise

massé) to be more natural than events that followed the opposite curve [F(1,14) = 5.53,

p<.0338]. However, the effects were comparatively small, they roughly remained within one

point of the 6-point rating scale.

When experts had to judge three-dimensional spinning events, they still performed

better than chance. Events that correctly curved according to the massé component of the spin

were judged to look more natural than those that curved into the opposite direction [F(1,14) =

11.46, p<.0044]. This can be seen in Figure 11. All analyses were carried out with

unstandardized scores since they did not significantly differ from the standardized ones.

-----------------------------------------------------

Insert Figures 10, and 11 about here

-----------------------------------------------------

In sum, experts' judgments on the basis of visual information about spinning billiard

balls failed to appreciate the null-effect of english in one-dimensional events, but were

astonishingly accurate in two-, and three-dimensional motion contexts. In all cases, experts

performed considerably better than novices and they based their decisions on the correct

component of spin. However, the effects were small, and experts' judgments lost a large

amount of clarity once the motion context had more than one dimension (F = 127.93 for follow

events vs. F = 5.53 for english combined with massé).

Discussion

Confirming the results of Experiment 2b, experts were confused about a fundamental

aspect of spin, the effect of pure english on the ball's trajectory. Here, english associated with

curvatures in the sense of the spin were judged to be as natural as english associated with

straight trajectories. Taken together, the results suggest that experts have not acquired general

perceptual knowledge about spin that is accurate enough to serve as a vehicle for their


expertise. They do not show evidence for perceptual attunement to the underlying dynamics of

spin. In particular, they were able to detect impossible trajectories for one-dimensional events

more easily for follow, draw, and massé than for english. Two- and three-dimensional events,

which should have been even harder to judge, yielded good performance. Thus, experts have

acquired visual knowledge about the effects of spin which is superior to that of novices.

However, their judgments were not accurate and general enough to conclude that they have

internalized the laws that govern the dynamics of spin.

Experiment 5: Comparing perceptual and motor accuracy in multi-dimensional

natural events (experts)

Experiment 5 sought to assess the domain-specific knowledge that experts acquire

when executing complex shots. That is, do experts fail to make accurate judgments about the

generalized effects of spin because their perceptual knowledge is precise but situation-specific

or because the motor-learning that is involved in their skill is in general more accurate than the

perceptual knowledge? In the latter case, if experts are more accurate when executing a shot

compared to the case where they perceptually judge a shot, one would have to conclude that

expertise at the level of a billiard player involves motor-learning based on the knowledge of

results (Schmidt, 1975) and not based on perceptual learning. In the former case, on the other

hand, if experts are equally good at executing or making visual judgments about complex

shots, one could conclude that expertise is mediated by situation-specific perceptual learning.

Method

Subjects. Eight male professional and semi-professional billiard players participated

voluntarily in this study. They had all played in numerous three-cushion tournaments in

Germany and Europe. On average they had played three-cushion billiards for 15.8 years with

approximately 12 hours of weekly practice. Three of them had already participated in

Experiment 4b.

Stimuli and Design. The same pocketless billiard table (108 x 216 cm) was used

for all subjects. As in Experiment 3c, experts had the opportunity to familiarize themselves

with the table before the experiment. Two conditions were administered within subjects. In

the execution condition, the expert was asked to execute a shot from the head spot with a

particular type of spin such that the ball rebounded at the center of the head rail and hit a second

rail at a spot determined by the experimenter. This point (aim) varied randomly between the

third and fourth diamonds on the left and right side of the table. The experts were asked to

produce shots with six different types of spin (high and low left english, high and low right

english, clockwise and counterclockwise massé). For each of the shot types two different

rebound positions were designated resulting in 12 unique shots. Each shot was repeated once.


The second condition (observation) consisted of the task to watch the same 12 shots

twice in random order, executed with larger error margins by the experimenter. The experts

could only observe the spinning ball after it was struck and before it hit the foot rail. The ball

was a standard three-cushion ball marked with two black dots.

Procedure. The conditions were balanced such that half the experts started with

observation while the other half started with execution. The procedure for the observation

condition was identical to that of Experiment 3c. The ball was placed behind the occluder such

that observers could not see the cue-stick but only the spinning ball as it appeared from

underneath the occluder. Subjects closed their eyes before the ball hit the foot rail. They were

then to predict where the ball had hit the left or right side rail after rebound. The actual contact

point with the side rail was measured before the experts opened their eyes, then their predicted

value was measured. These values were converted into rebound angles measured from the

center of the head rail.

Results

No differences between the conditions were found. The average absolute deviation of

the predicted rebound angle from the actual one was 3.55o for the observation condition and

2.71o for the execution condition. A repeated measures 2-way ANOVA revealed no significant

differences between these values [F(1,7) = 2.96, p<.088].

Discussion

The fact that observers were equally good at determining the rebound angle of a billiard

ball when watching the rotating ball and when executing a shot allows several conclusions.

First, in the context of an ecological setting the visual information specified by the rotating ball

can be extracted by an experienced observer. That is, this visual information could play a vital

role during the acquisition of expertise. Second, the information is situation-specific since

observers were unable to extract similarly accurate information from the computer-generated

displays that were used in Experiment 4b. Thus, it is likely that visual information plays an

important role in the acquisition of billiard expertise. However, what is learned visually is not

an attunement to the dynamics of spin, rather visual patterns produced by the rotating ball are

matched with outcomes in a situation-specific way.

General discussion

The dynamics of spin are not understood easily. Almost all novices and half of the

experts had erroneous conceptions about the effects of english on ball trajectories (Experiment

1). These misconceptions were mirrored in visual judgments of animated sequences

(Experiments 2 and 4). Nonetheless, experts have learned to master very complex shots.


These motor abilities are reflected in visual judgments experts make in ecological contexts. As

the following paragraphs will elaborate, the discrepancy between a lack of general

understanding and high visual performance cannot easily be explained by existing theories of

dynamic event perception. Approaches that emphasize situation-specific learning or attunement

need to be developed.

Why was expert performance poor in computer animations?

At first sight it appears startling that experts did not perform better in the simplest of

animations, when only one component of spin was involved. Why should they make mistakes

at all? One could argue that the animations were unrealistic enough to put observers in a

problem-solving mode rather than into a perceptual one. This, however, appears to be unlikely

since experts did reasonably well with more complex events. A problem-solving strategy

using one or more heuristics should have lead to similar mistakes in complex situations.

Moreover, a potential lack of realism in the animation should have been most disruptive for the

more complex simulated types of spin. It can thus be ruled out that the quality of the

animations is responsible for the results. It is more likely that experts do not posses visual

knowledge about the fact that english has no influence on the ball's trajectory. The conclusion

seems to be warranted that experts, and novices, do not have conceptual or perceptual

knowledge about the dynamics underlying the effects of spin.

The problem with pure english

How can this conclusion be reconciled with expert performance? Should experts not

make performance errors whenever english is involved? All experts who thought that left

english would cause the ball to curve agreed that the ball would curve to the left and often said

that it would first go a little to the other side before starting to curve. This behavior of the ball

is in fact possible, but massé and squirt are responsible for such a behavior of the ball. The

ball squirts to the right (often not more than 1o) and then curves if it is hit from above with a

slanted cue-stick. When five of the experts who thought that english makes the ball curve

demonstrated an "english" shot, they hit the ball with an elevated cue-stick and thus put some

massé on the ball in addition to the desired english. That is, these experts misattributed a

correctly observed effect (curvature) to a spin component that was not responsible for it

(english). It is not implausible that experts were able to maintain such a misconception because

they might consistently add a constant massé component to every english shot they execute. A

left-english shot hit with an elevated cue would curve left, a right-english shot hit with the same

cue-angle would curve right. Because of this symmetry, the true cause of the curvature could

have been obscured and misattributed to english. Thus, expert performance is possible in spite

of erroneous visual knowledge.


Perceptual attunement. As long as attunement is interpreted as attunement to the

underlying dynamics of spin, there is no evidence for such attunement. Although the visual

abilities of experts are extraordinary, they cannot be described as attunement to the laws of spin

dynamics. Even though the KSD principle does not seem to require perfect attunement

(Runeson, 1989) expert knowledge should transfer to slightly different contexts such as used

in the computer-animations of Experiments 2b and 4b. For one, a general attunement would be

very parsimonious. Also, the KSD principle entails that knowledge which has been acquired is

generalized to some degree. The kinematics of a ball with english spin is sufficient to specify

its spin and motion vectors and thus the ball's behavior. The ball should move in a straight

line. Likewise, the kinematics of massé specify curvature. Thus, a fair test of the principle

would be to assess whether experts have picked up these general relationships. Since they

have not, one can conclude that the principle of KSD is not sufficient to describe expert

learning.

In an attempt to salvage the approach, Runeson (1989) has suggested the concept of an

incomplete invariant, which would deal with the problem of lacking generality. If observers

are only attuned to a subset of the critical information they can be said to use an incomplete

invariant. In the case of expert billiard players separate incomplete attunement would have to

be assumed for various situations that might occur in a billiard game. Unfortunately, once the

concept of incomplete invariants is brought into play, it becomes virtually impossible to think

of empirical findings that cannot be described in terms of KSD. This would render the KSD

approach immune to falsification (Hecht, 1996) and thus theoretically questionable.

Perceptual Heuristics

Novices behave as if they were consistently using one heuristic. This heuristic has

been identified in Experiments 2a and 4a as stating that the ball always curves with the sense of

the spin. For massé and follow/draw shots where spin and direction of motion coincide, the

trajectory is not affected and the heuristic leads to correct predictions. For english, however,

the heuristic is wrong and leads to errors. If more than one dimension of spin is involved,

novices stick to the heuristic but they appear to focus on the english component of the spin

(english heuristic). They go with the mistaken english heuristic even when the massé

component specifies a curvature in the opposite direction.

Another indication that novices might use heuristics is the lack of systematic differences

between their conceptual and perceptual performance. The fact that not intuitively available

relationships between spin and ball trajectory could be learned quickly on a qualitative level

(Experiment 3b) also lends support to the notion of heuristic judgments. These results may be

very similar to findings that important visual cues to sex chickens were learned rapidly by

novices (Biederman & Shiffrar, 1987) as long as simple visual features could be described.

Novice performance and learning can be described in heuristical terms.


Expert data, on the other hand, are not easily described by the use of heuristics.

Conceptual errors that some experts make about the effects of english spin are mirrored in one-

dimensional simulated spin events but cannot be found in more complex situations. The good

performance on complex animations rules out the use of a simple-minded rule or heuristic (e.g.

"english makes the ball curve against the spin"). Finally, the extraordinary visual abilities in

the ecological situation with complex spin are in strong contradiction to the idea that billiard

experts might use one-dimensional heuristics when judging these balls. As well as PH may

describe novice performance, the approach does not apply to expert performance.

In order to salvage the heuristics approach one could argue that experts have learned to

integrate information from more than one dimension. This could be done by introducing a set

of rules that allow experts to combine heuristics, which would require other superordinate

heuristics that prescribe how heuristics are integrated. They could consist of additive or of

weighted multiplicative rules. However, it is unlikely that expertise is captured by such a

model. First, if expert knowledge were organized in terms of heuristics, it is hard to

understand why experts should have developed visual knowledge of spin rather than

knowledge in terms of mappings between the contact points of cue-stick and cue-ball into

outcomes of the shot. Second, the existence of such meta-heuristic rules would be hard, if not

impossible, to verify, especially if they do not need to enter awareness. The issue of

awareness of heuristics becomes problematic, because consciously accessible heuristics (or

conceptual knowledge in more neutral terms) often disagree with implicitly used heuristics.

For instance, many expert billiard players adamantly defend mistaken heuristics that are in

conflict with their tacit knowledge that comes to bear when they execute shots.

A striking example can be found in Willie Mosconi's (1959) book on pocket billiards.

Being one of the greatest billiard players of all time (Byrne, 1987) he put forth the classic but

mistaken explanation that, when banking a ball (reflection at a rail) one should always aim at a

point halfway between the object-ball and the pocket. For instance, if the object ball is

positioned close to the left side rail and needs to be shot into the middle pocket of the right side

rail, Mosconi suggests the following procedure. Find the spot on the left side rail that points

orthogonally to the object-ball. Then bisect the distance between this point and the middle

pocket on that rail, and this is where to aim. This rule does not take the distance between the

object-ball and the left side rail into account. However, this distance is critical. Therefore

Mosconi's prescription is flawed. It seems that he could not explain what he was doing in

terms of the contact points between balls, but his success proved that he was doing it perfectly

well. Thus, consciously accessible heuristics may clearly be wrong without impairing

performance.

Conclusion


The experiments have shown that experts do not use the prima facie simplest strategy to

achieve results: They are not mapping impact points on the cue ball into outcomes. In a way,

such a mapping would be the minimal essential information required to carry out the desired

action (Abernethy, 1993). Experts also acquire skills that go far beyond an intuitive

understanding of spin. This lack of intuitiveness is mirrored in the history of billiards. Spin

was presumably introduced to the game as late as 1807 by Captain Mingaud, who was first to

put a leather tip on his cue-stick and amazed his peers with the results (Byrne, 1987).

The role of perceiving spin is functional in acquiring billiard expertise, since experts are

able to predict outcomes practically perfectly solely based on observing the spinning ball. The

visual knowledge about spin that experts have acquired presumably plays the most important

role when they create their own knowledge of results (KR). An infinitesimal difference in spin

can change collisions and trajectories significantly. By observing the spin on a ball, experts

receive feedback about the accuracy of their shot considerably earlier than were they to wait for

the outcome of the shot. This contention is supported by the fact that pool teachers recommend

practicing with a striped cue ball. Striped cue balls are even sold especially for that purpose.

Interestingly, the two experts how also taught billiards professionally were aware that english

does not affect a ball's trajectory.

The idea that spin might be crucial in obtaining advance KR is supported by findings

that show the ability to use efferent motor signals to predict performance . In a different

context, Stimpel (1933) asked trained actors to throw a ball at a target. They were able to

predict the outcome of their throw (the light was turned off as soon as the ball left the hand)

with the same degree of accuracy as they were to hit the target.

The perceptual skill level and conceptual knowledge about the underlying mechanics

appear to exist in isolation. Similarly, separate mechanisms for perceptual-motor skill and

more abstract knowledge representation have been found in studies on aiming skill acquisition

during childhood (Krist, Fieberg, & Wilkening, 1993). The discrepancies found here between

performance and explicit knowledge of billiard experts strengthen the notion of separate

knowledge systems.

Given that both PH and KSD appear to be inadequate to explain expert performance,

how would a more adequate theory have to look? The need for a more comprehensive theory

of dynamic event perception seems obvious. A theory of dynamic event perception would

have to acknowledge the coexistence of errors in simple situations and of extraordinary

accuracy in complex situations. The physics of spin, and for that matter of most other dynamic

events, are so complicated that general perceptual attunement cannot be expected. Maybe a

notion of rule-governed evaluations paired with highly situation-specific perceptual abilities

might eventually lead to an explanation of this strange coexistence.


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Table 1

Correlations between Actual and Predicted Rebound Angles

Condition

Group Cue Spin

__________________________________________________________________________

Experts .95 .96

Untrained Novices -.18 .28

Trained Novices .64 .70


Figure Captions and Figures

Figure 1. In billiards, the following types of spin on a moving ball can be distinguished. If

the ball is moving across the table along the axis labeled z, spin around its vertical axis (y) is

called english or side english. Depending on where the ball was struck, it is called left or right

english. Spin around the z-axis is called massé, and it is imparted by striking down on the

ball. Spin around the x-axis is follow if the ball was struck above, and draw if it was struck

below its center.

Figure 2. A stimulus used in Experiment 1. In this particular case, the ball had been struck

with left english resulting in spin and a forward sliding motion. The question was whether the

ball would curve or move in a straight path.

Figure 3. Schematic of the displays used in Experiments 2a and 2b. A section of a pool table

is depicted from a bird's eye perspective. The spinning ball could move along one of the five

trajectories depicted here.

Figure 4. Novice performance in Experiment 2a. A naturalness scale form 1 (least natural) to

6 (most natural) was used. The abscissa labels indicate the shape of the spinning ball's

trajectory. The first bar of each panel indicates the correct solution (straight for follow, draw,

and english; curving with the spin for massé). The bottom panel reveals novices' erroneous

judgments of english.

Figure 5. Expert performance in Experiment 2b. The panels are set up in analogy to Figure 4.

The bottom panel reveals experts' biases that english could curve to some degree against the

direction of spin.

Figure 6. Expert and novice performance compared between Experiments 2a and 2b. The

canonical straight path was judged to be most natural by experts, but not by novices.

Figure 7. Schematic drawing of a billiard table used in Experiment 3 (a, b, and c). The

experimenter would stand behind the head rail and hit the resting ball along the line of aim into

the foot rail. The subject stood behind the right side rail at the head of the table to the

experimenter's right.

Figure 8. Performance of novices, trained novices (learning), and experts in Experiment 3.

The bottom part of each bar indicates how often the predicted rebound angle came within 5o of

the actual angle, separately for cue and spin conditions. The top part of the bar reflects the

same measure, but allowing a range of 15o.


Figure 9. Novice performance (Experiment 4a) for two components of spin shows that

judgments seem to be based on a mistaken rule ('left english makes the ball curve to the right').

The massé, which is exclusively responsible for the ball's curvature, is barely taken into

account. Correct solutions are labeled "C"; solutions according to the mistaken heuristic that

english makes the ball curve with the spin are labeled "EH".

Figure 10. Experts (Experiment 4b) are able to pick up the effect of massé in trials with two

axis of spin. However, incorrect events receive naturalness scores that are almost as high as

those for correct events. "C" stands for correct solution, "EH" for the heuristic that english

causes the ball to curve against the sense of the spin.

Figure 11. Even in trials that contain three simultaneous axes of spin experts are able to

discern the effects of the massé component, however, they are by no means perfect.

Correct solutions are labeled "C".


Figure 1

Y

X

Z


Figure 2

6 5 4 3 2 1 0 1 2 3 4 5 6


Figure 3

Head spot

Head rail

Foot railS

ide

rail

Ball

Line

of a

im

Occluder

Monitor

Presented trajectories


Figure 4

6

with against straight1

2

3

4

5

6

Follow / Draw

straight left right

1

2

3

4

5

6

Novices

Massé

straight with against1

2

3

4

5

6

English

Nat

ura

lnes

sN

atu

raln

ess

Nat

ura

lnes

s


Figure 5

Follow / Draw

Massé

English

Nat

ura

lnes

sN

atu

raln

ess

Experts

Nat

ura

lnes

s

straight left right1

2

3

4

5

6

with against straight1

2

3

4

5

6


2

3

4

5

6


Figure 6


2

3

4

5

6

ExpertsNovices

English

Nat

ura

lnes

s (s

tan

dar

diz

ed)


Figure 7

Head spot

Head rail

Foot rail

Sid

e ra

il

Occluder

Ball

Lin

e o

f ai

m

Range of

rebound angles


Figure 8

cue spin cue spin cue spin0

20

40

60

80

100

Within 5 deg

Within 15 deg

Condition

Per

cen

t C

orr

ect

Novice Trained Expert


Figure 9

s t r a i g h t l e f t r i g h t1

2

3

4

5

6

Left English + Clockwise MasséLeft English + Counterclockwise Massé

NovicesN

atu

raln

es

s

C

C

EHEH


Figure 10

straight left right1

2

3

4

5

6

Left English + Clockwise Massé

Left English + Counterclockwise Massé

ExpertsN

atu

raln

ess

C

C

EH

EH


Figure 11

s t ra igh t l e f t r i g h t1

2

3

4

5

6

Left English + Clockwise Massé + Follow

Left English + Counterclockwise Massé + Follow

ExpertsN

atu

raln

es

s

C

C


Acknowledgements

Bennett Bertenthal, Linda Bunker, David Gilden, Jackie Johnson, Mary Kaiser, Michael

Kubovy, and Art Schulman provided valuable advice. Special thanks belongs to Kegler's

Pool League in Charlottesville, the Billardsportverein München, and all the billiard

players who volunteered their time and expertise. Maryam Ahmed, Steve Degroot, Jutta

Halstenberg, Rob Marino, and Gail Morris helped conducting the experiments; Stephen

Jacquot provided programming assistance.

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