Thinking and Problem Solving Running head: THINKING AND PROBLEM SOLVING IN CHESS Thinking and Problem Solving in Chess: The Development of a Prototypical Model to Account for Performance on Chess Problem Solving Tasks BY ERIC J. FLEISC.HMAN A MASTER'S THESIS SUBMITTED TO THE GRADUATE FACULTY OF RICHARD A. CONOLLY COLLEGE, LONG ISLAND UNIVERSITY, BROOKLYN CAMPUS IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS IN PSYCHOLOGY MAY, 1998 MAJOR DEPARTMENT: SPONSORING COMMITTEE: PSYCHOLOGY Committee/Chair Dr. Gary Kose, Ph.D. CERTIFIED BY Reader Dr. Jerold Gold, Ph.D. Department Chair Dr. Jerold Gold, Ph.D. DATE: 5/12/98
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Thinking and Problem Solving
Running head: THINKING AND PROBLEM SOLVING IN CHESS
Thinking and Problem Solving in Chess:
The Development of a Prototypical Model to Account
for Performance on Chess Problem Solving Tasks
BY
ERIC J. FLEISC.HMAN
A MASTER'S THESIS SUBMITTED TO THE GRADUATE FACULTY
OF RICHARD A. CONOLLY COLLEGE, LONG
ISLAND UNIVERSITY, BROOKLYN CAMPUS
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF ARTS IN PSYCHOLOGY
MAY, 1998
MAJOR DEPARTMENT: SPONSORING COMMITTEE:
PSYCHOLOGY
Committee/Chair
Dr. Gary Kose, Ph.D.
CERTIFIED BY
Reader
Dr. Jerold Gold, Ph.D.
Department Chair Dr.
Jerold Gold, Ph.D.
DATE: 5/12/98
Thinking and Problem Solving i
Acknowledgments
I would like to express my deep appreciation and gratitude to Dr. Gary Kose for
his patient and insightful guidance and assistance in helping me to complete this most
difficult project. His understanding and acknowledgment of my particular interest in this
project has provided me with a perpetual source of cognitive inspiration. Moreover, at
the inception of this project. Dr. Kose gave me the encouragement and support to embark
on it. After initial frustrations and a lengthy intervening period, he suggested some
modifications on the original proposal. These modifications and the trials and tribulations
accompanying them have finally proven worthwhile.
I would also like to thank Dr. Robert Fudin, whose teaching and words of wisdom
have remained etched in the farthest reaches of my imagination. Ultimately, this project
would not have been possible without an imagination coupled with the sometimes harsh
realities of statistics and theory of which Dr. Fudin has been most helpful.
Thinking and Problem Solving ii
Abstract
There has been a growing interest in constructing models and theories of internal
representations in order to understand how the human brain represents spatial dimensions.
Game playing is a medium which is well suited for this purpose. Of the many games
available for research, chess is the most popular. Utilizing Eisenstadt and Kareev's
(1975a) thesis as a starting point, this paper hypothesizes that thinking and problem
solving is inherent in chess, which provides analogical material for the study of internal
representations and spatial dimensions. Based on this hypothesis, this paper presents a
model to address the following questions: (1) what is the internal representation chess,?
(2) what is the structure of that internal representation,? and (3) how is that internal
representation used in solving chess problems?
Thinking and Problem Solving
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS .............................................................................................. i
ABSTRACT .................................................................................................................. ii
LIST OF TABLES
LIST OF FIGURES
CHAPTER
I. INTRODUCTION................................................................................................. 1
Literature Review .................................................................................... 1
Internal Representation of Chess .............................................................. 1
II THE STRUCTURE OF INTERNAL REPRESENTATIONS ............................ 10
Visual and Spatial Imagery ..................................................................... 10
Problem Solving and Chess .................................................................... 17
III. THE DEVELOPMENT OF A PROTOTYPICAL MODEL TO ACCOUNT FOR PERFORMANCE ON CHESS PROBLEM SOLVING TASKS .......................................................................................... 22
Presentation of Chess Problems ............................................................. 23
12. Problem 2 - exemplar - variation 1: post-triangulation of king
position - 13. Kd3 - c2 ............................................................................... 43
Thinking and Problem Solving 1
CHAPTER I
INTRODUCTION
LITERATURE REVIEW
It has been hypothesized that the thinking and problem solving inherent in
chess, provides analogical material for the study of internal representations and
spatial dimensions (see Eisenstadt & Kareev, 1975a). If this is true, then three
important questions need to be asked: (1) what is the internal representation of
chess,? (2) what is the structure of that internal representation,? and (3) how is that
internal representation used in solving chess problems? The present paper will
utilize and adapt Eisenstadt and Kareev's hypothesis regarding thinking and
problem solving, as a way to study unique aspects of thinking and problem
solving.
Internal Representation of Chess
Eisenstadt and Kareev (197 5 a) have argued that internal representations in
chess retain both conceptual and perceptual components. Intuitively, it is apparent
that internal representations are involved in playing chess. Studies of memory for
chess situations suggest that chunking and scanning are involved when thinking
about chess. Therefore, Eisenstadt and Kareev (1975a; 1975b) assume that chess
masters' performance in chess will correlate with a high level of chunking and
scanning. The following memory studies will address this issue.
Thinking and Problem Solving 2
De Groot (1965; reviewed in Eisenstadt & Kareev, 1975b) and Chase and
Simon (1973; reviewed,in Eisenstadt & Kareev, 1975b) in their studies of memory
for chess positions, found that chess experts were better than beginners in
reconstructing meaningful board positions from memory but were not as good at
constructing random board configurations. De Groot, therefore, assumed that
chess players divide the board into various chunks or configurations which
correspond to patterns stored in long-term memory. Simon and Gilmartin (1973)
estimated the number of these chunks to be about 50,000. De Groot believed that
higher skilled players know more of these patterns and therefore can extract more
chunks out of a position. De Groot also studied specific chess representations of
master level chess players to demonstrate that they develop a 'dynamic perception'
of chess arrangements. He found that master chess players are able to obtain
information from a position with great speed and process it in terms of their
knowledge and understanding.
Based upon their research, Chase and Simon (see Eisenstadt & Kareev,
1975b) concurred with De Groot regarding the chunking of chess pieces. Chase
and Simon further hypothesized that since chess masters are better at chunking,
and chunking ability depends upon memory, then masters will be better at
chunking and remembering familiar positions.
Eisenstadt and Kareev (1975b) examined the effects of context on memory
for chess. In one experiment, they used a board reconstruction task to demonstrate
Thinking and Problem Solving 3
the effect of context on the subjective organization of board configurations.
Subjects were asked to reconstruct the board after rotation and piece color
reversal. However, instead of chess, they studied memory for configurations on
"Go" and "Gomuko." According to Eisenstadt & Kareev, the results obtained
supported their prediction that there would be an interaction between the problem
posed and the type of pieces to be remembered. They concluded that individuals
encode external elements of games in a manner similar to the way they play the
game.
Gobet and Simon (1996) suggest The Multiple Template Hypothesis, as an
alternative explanation for how chunking occurs. This hypothesis states that:
previously stored multiple templates are used to remember chess positions. Chess players have seen thousands of positions; and for expert players, most positions they see readily remind them of positions they have seen before. They have information about the positions that arise when the Ruy Lopez opening is played or the King's Indian Defense. A Grandmaster or Master holds in memory literally thousands of such patterns, each of which specifies the locations of 10 or 12 pieces, with revisable defaults for others (P-31).
Consequently, they suggested that chess players are like experts in other recall
tasks in the way they utilize long-term memory. Gobet & Simon (1996) believed
that long-term memory structures or templates are used as a way of explaining
chunking. Nevertheless, these studies only show that chess is represented in
meaningful chunks, and do not demonstrate what type of representation or
structure is utilized. Nor do these studies say anything about how the game is
actually played.
Thinking and Problem Solving 4
Eye movement studies or scanning studies have been used to understand
how chess is played (Fogiel, 1990). Chase and Simon (1973) noted that eye scans
seem to chunk the board. Pieces belonging to the same chunk are likely to be
scanned together by relations of attack and defense, whereas, pieces belonging to
different chunks are less likely to be related in this way. However, Chase and
Simon (1973) do admit that:
although eye movements give us a record of how the board is scanned . . .
they don't tell us precisely which pieces are observed (especially in
peripheral vision) and in what order; they only tell us the general area being
aimed at by the fovea" (pp. 56-57).
In 1969, Simon and Barenfeld (see Eisenstadt & Kareev, 1975b) designed a
program which simulated how board positions are scanned by chess players. They
used the relationships between pieces to direct the scanning of subjects. What they
discovered was that they could simulate the initial eye movements of Masters
studying chess positions for the first time. Eisenstadt and Kareev (1975b), believe
that these board positions are composed of meaningful chunks which are based
upon prior chess knowledge. They also suggest that scanning behavior is also
determined by chess knowledge. According to Eisenstadt and Kareev (1975b),
Gestalt principles of perception such as "proximity, continuity, and similarity"
(Koffka, 1935), should play important roles in scanning.
Eisenstadt and Kareev (1975b) tested scanning behavior in the board games
Go and Gomoku using the Gestalt principles of proximity and continuity. They
hypothesized that subjects would lose more games on the longer of the two
Thinking and Problem Solving 5
main axes or diagonals. For their demonstration, they used different geometrically
shaped boards, performed forty-five degree board rotations, and removed board
lines replacing them with dots. What they found was that proximity and
continuity of stimuli affect the scanning behavior of subjects in all board tasks,
not only along the diagonals. In addition, the results indicated that there was a
higher ratio of losses incurred after proximity and continuity of goodness is
disturbed. Thus, Eisenstadt and Kareev claim that the results of their experiment
support their prediction. This claim is supported by experiments from Church
(1974; 1977; reviewed in Sternberg & Frensch, 1991), which tested linear scans of
chess players along relevant rows, diagonals, or columns. What they found was
that scanning speeds were greatly increased when pieces were vertically or
horizontally aligned, as opposed to aligned along a diagonal.
In a later experiment, Eisenstadt & Kareev (1975b) studied the scanning
behavior of players during actual games. They decided to restrict viewing of the
board through a movable one-by-one window. This allowed them to gather verbal
information from the subjects as well as observe what they were attending to,
since they didn't have peripheral vision. In addition, since subjects were scanning
their own games, observations were recorded for the influence of long-term
memory on scanning ability.
From the scanning behavior of subjects in these experiments, with
particular emphasis on the last experiment, Eisenstadt and Kareev found four
Thinking and Problem Solving 6
major types of scanning behavior. The first type of scan observed was the
'confirmatory scan.' This scan suggested that a subject has a hypothesis about
certain pieces or squares on the board, and looked at these squares in order to test
the hypothesis. The second type of scan recorded was the 'exploratory scan.' In
this scan the subject did not have a specific hypothesis, but looked at various
squares to see what they contained. This scan often created new hypotheses
replacing the exploratory scan with a new confirmatory scan. The third type of
scan observed by Eisenstadt and Kareev was the 'revival scan.' This time the
individual again looked at a square which he recently examined to confirm its
contents. This type of scan was essentially a 'rehearsal' mechanism. The
distinction between a revival scan, and a confirmatory scan depended on the
subject's degree of certainty regarding the existence of a particular configuration.
The fourth type of scan used by subjects was the 'imaginary scan.' In this scan the
subject planned a move, and pointed to any of the squares where he imagined
placing a piece.
According to Eisenstadt and Kareev, these four types of scanning are
important for the development of any model of scanning and internal
representation. Therefore, a complete model of scanning behavior must include a
precise knowledge of the contents of the board at any given moment. Otherwise, it
would be very difficult to differentiate between exploratory and confirmatory
scans.
Thinking and Problem Solving 7
Utilizing their experimental results, Eisenstadt and Kareev constructed a
model of scanning and internal representation based upon the aforementioned
constraints and guidelines. The major components of this model consisted of three
parts: (1) a "fovea," (2) a working memory, and (3) a long-term memory. The
fovea referred to the focal point of attention, working memory consisted of the
execution processes and data for future processing, while long-term memory was
composed of pattern recognition rules.
Execution processes in working memory were designed for either scanning
the board, examining the contents of working memory, or adding new items to
working memory. Eisenstadt and Kareev claimed that data processes in this
model were analogous to descriptions of different features of board positions.
Long-term memory, contrariwise, consisted of pattern rules which illustrated
different board configurations, processing rules, and pattern rules for developing
internal representations of the board in working memory.
Processing rules in this model operate based upon either high or low
priority. Priority is a way of representing the importance for a particular game.
Eisenstadt and Kareev's processing rules consist of the following: a top-down
confirmation rule, a bottom-up discovery rule, a bottom-up suggestion rule, and an
exploration rule. Consequently, these rules are repeatedly applied to working
memory.
Eisenstadt and Kareev viewed their model as satisfying the aforementioned
Thinking and Problem Solving 8
constraints and guidelines for scanning and internal representation. They argue
that internal representations are built into the model, since the pattern rules are
specifically designed for each game. Consequently, they propose that the Gestalt
principles of proximity, continuity and similarity, as well as the four types of scans
are incorporated into their model.
Recently, top-down processing models like Eisenstadt & Kareev's have
received criticism (Weisstein & Harris, 1974; Palmer, 1975b; cited in Pinker,
1984). Pinker (1984) questioned the kind of knowledge being used for recognition
purposes, and has several concerns. He is concerned about the possibility of top-
down processing models doing the following: ". . . altering the order in which
memory representations are matched against the input, searching for particular
features or parts in expected places, lowering the goodness-of-fit threshold for
expected objects generating and fitting templates, filling in expected parts" (pp. 3-
4).
Consequently, top-down processing models such as Eisenstadt & Kareev's
deserve close scrutiny. Despite the seeming efficiency of execution processes,
top-down processing models do not inform us about the nature of internal
representations of actual players in game-type situations. Nor do they tell us
whether the internal representation is propositional or pictorial.
The main points of this section may be summarized as follows. Regarding
memory for chess ability, chess masters are better than novices at reconstructing
Thinking and Problem Solving 9
meaningful board positions. They accomplish this through chunking familiar
positions that are stored as templates in long-term memory. These templates are
specific to the game being played and the particular problem being addressed. The
process of chunking is enacted through the use of scanning. Four major types of
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