MODELS FOR THE CLASSIFICATION .
OF PROBLEMS AND THE: ~
PREDICTION OE GROUP PROBLEM? :4 . ‘
SOLVING‘FROM”? . ~
INDIVIDUALRESULTS ,

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
James Henry Davis
- 1 961

 

 

 

This is to certify that the

thesis entitled
Models for the Classification of Problems

and the Prediction of Group Problem-

Solving from Individual Results
presented by

James H. Davis

has been accepted towards fulfillment
of the requirements for

Ph .D . degree in Psychology

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L I B R A R Y
Michigan State

University

 

 

 

 

 

 

 

 

 

 

 

 

 

MODELS FOR THE CLASSIFICATION OF PROBLEMS
AND THE PREDICTION OF GROUP PROBLEMASOLVING
FROM INDIVIDUAL RESULTS

BY

James H. Davis

 

AN ABSTRACT OF A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Psychology

1961

 

If? APE/74;}?
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ABSTRACT
MODELS FOR THE CLASSIFICATION OF PROBLEMS

AND THE PREDICTION OF GROUP PROBLEM-SOLVING
FROM INDIVIDUAL RESULTS

by James H. Davis

Eureka problems (word puzzles having unique
solutions), were worked on by individuals, and by 23
Egg groups in which unrestricted, face-to-face inter-
action was permitted between members who had no

tradition of working together. The first objective of

the study was to introduce an ordering of problems and

a method measuring that order. Second, the investiga-

tion sought to predict group problem solving performance
from a knowledge of the problem solving behavior of
persons working as individuals. Major emphasis was
placed upon the development of a model that dealt
concurrently with group product and emergent group
structure. Third3 the frequently-noted superiority of
problem solving groups over individuals working on the
same problems was critically examined.

A model for the distribution of solution times of

individual solvers was developed from the idea that the

solution of a problem is composed of several steps or

 

 

 

 

 

James H. Davis

stages, and solution occurs upon completion of the

k-th stage. Problems can be classified as to the
number of stages, which is assessed from the distribu-

tion of individual solution times. The Classification

Model (statistically a waiting-time model) predicted
solution times to be a gamma distribution. The argu-
ment proceeded logically to the gamma distribution from
such assumptions as that the probability of a stage
solution is constant over time until solution occurs,
any one time interval is taken so small that one and
only one stage may be solved within it, and the stages

are independent and equally difficult.

Two models for predicting group from individual
performance were proposed, viz., the Hierarchical and
Equalitarian Models. These models predicted the

distribution of group solution times to be a simple

transformation of the gamma distribution that fitted

the individual solution times. The Hierarchical Model

assumed that group members organized themselves into a
hierarchy with the more successful members consuming
more than their share of the group's time. The

Equalitarian Model assumed each member took his share

of the group's working time, whether he contributed

to solution or not.

Data were gathered from individuals working alone

 

 

 

_..__._——;
__. ._._——..._ -

,_.._._—

 

I I: 17-... li'lII I! I .

 

 

James H. Davis

and from 29 322 groups of four. These data were:

(a) correctness of solution, and (b) the time
consumed on each problem. Each group and each indivi-
dual were tested on a sample problem and three experi-
mental problems, the experimental problems being given
in different orders for different subjects. In the
groups, not only was solution time recorded, but also
a count of the frequency with which each member talked,
their choices for future problem-solving partners,
observation of member contributions and any unusual
events.

The parameter k of the gamma distribution (inter-
preted as the number of problem stages) was estimated
by the method of moments from the sample observations.
The theoretical curves thus determined were found to
fit the distributions of individual solution times for
all three problems. The Equalitarian but not the
Hierarchical Model was found to predict the distribution
of group solution times in each case. Implicit support
for the social psychological assumptions of the
Equalitarian Model was found through an analysis of the
partner-choice data. The analysis of the communication
frequencies of group members, however, was indecisive.

The problem solving behavior of individuals was

pooled mathematically and such concocted groups were

 

 

 

 

 

 

James H. Davis

found to perform significantly better than the real

groups on two of the three problems. This finding

was interpreted to indicate that member interaction

actually inhibited problem solving, at least under the

conditions of this investigation.

9mm W

Frank Restle, Major Professor

 

 

 

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MODELS FOR THE CLASSIFICATION OF PROBLEMS
AND THE PREDICTION OF GROUP PROBLEM-SOLVING
FROM INDIVIDUAL RESULTS

By

Ti
James foDavis

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Psychology

1961

 

 

 

 

:l

 

 

 

 

 

ACKNOWLEDGEMENT

The author is deeply grateful for the guidance and
intellectual stimulation of his teacher, Dr. Frank Restle,
who was never too busy to listen and comment.

Important suggestions and valuable criticisms were
offered by the members of the committee, Dr. Milton Rokeach,
Dr. Charles Wrigley, and Dr. Terence Allen; the author is
greatly in their debt.

ii

 

 

 

DEDICATION

To my wife

 

2)

 

 

 

 

 

 

 

TABLE OF CONTENTS

CHAPTER

I.
II.
III.
IV.

V.

PAGE

INTRODUCTION

C O O O I O O O O C C O . C I J D O C O C O O O C O O I O O 1
TIEORY O O O O I O O I O O O C O O C O O O O O O O O O O C O O O O I O O O O O 20
ExPERImNTAI-l PROCEDURE 0 O C O 0 O O O O O O O O O O I O O O O 36

RESULTS

0.0IOOOCOOOCOCOOOOOIOOOOOOI00...... 52

DISCUSSION AND CONCLUSIONS

0.00.00.00.00... 102

iv

 

 

 

LIST OF TABLES

TABLE TITLE PAGE
h.1 Individuals Vs. Groups on each

experimental problem, Ta 720 seconds.. 53
h.2 Mean solution times in seconds for

Groups and Individuals ............... 5h
U.3 Method of moments estimates of k and,l

for each problem. (Individual

solvers).............................. 60

h.h Summary of Kolmogorov-Smirnov one-

 

sample, goodness-of-fit tests for
theoretical curves determined by
estimates of k ....................... 6h
h.5 Estimates of k and.d.with reading time
included and with reading time
removed............................... 67
4.5 Summary of Kolmogorov-Smirnov goodness-
of-fit tests for curves determined by
A
ki' (i = 3, 5, 8) 72
h.7 Kolmogorov-Smirnov goodness-of-fit tests
of predicted group solutions (Lorge-
Solomon Model) with the observed data. 78
4.8 Kolmogorov-Smirnov one-sample, goodness-
of-fit tests for observed and
predicted group solutions under the

HierarChical MOdeloooooooooooooooooooo 85

 

 

 

ca

1.0-.

 

 

 

h.9 Kolmogorov-Smirnov one-sample, goodness-
of-fit tests for observed and
predicted group solutions under the
Equalitarian Model ................... 88

b.10 Mean uncertainty, E, of member

companion choices - means and standard

deviations for Problems 3, 5, and 8... 95
n.11 Analysis of variance where criterion

variable is sample uncertainty,

computed from the sociomatrices of

choices. Analysis of variance where

criterion variable is sample

communication uncertainty............. 97

n.12 Uncertainty, g, of member communications-

means and standard deviations for

PrOblemS 3, 5, and 8oooooooooooooooooo 99

vi

 

 

 

 

LIST OF FIGURES

 

FIGURE PAGE
2.1 Graph of the function
1 T(k'1)e-T.....-............ 27
W—
U.1 Frequency distributions of solution timesr

for individual solvers and non-solvers. 55
h.2 Frequency distributions of group

solutions, solvers and non—solvers..... 56
h.3 Cumulative relative frequency distribution

of solution times for individual

solvers, Problem 3..................... 61
h.h Cumulative relative frequency distribution i

of solution times for individual

solvers, Problem 5..................... 52
h.5 Cumulative relative frequency distribution

of solution times for individual

solvers, Problem 8..................... 63
“.5 Cumulative relative frequency distribution

of solution times for individual

 

solvers (reading time removed),
Problem 3.0.0.....O...‘.............O.. 69

”-7 Cumulative relative frequency distribution

of solution times for individual
solvers (reading time removed),

PrOblem 5.coco...00000000000000.0000... 70 :

 

 

 

 

 

 

 

 

4.13

Cumulative relative frequency distribution

of solution times for individual

solvers (reading time removed),

Problem 8.............................. 71
Cumulative relative frequency distribution

of group and individual solution times,

and predictions from the Lorge-Solomon

Model, Problem 3....................... 75
Cumulative relative frequency distributions

of group and individual solution times,

and predictions from the Lorge-Solomon

Model, Problem 5....................... 76
Cumulative relative frequency distributions

of group and individual solution times,

and predictions from the Lorge-Solomon

Model, Problem 8....................... 77
Predictions of group performance under

the Hierarchical and Equalitarian

Models, Problem 3...................... 82
Predictions of group performance under

the Hierarchical and Equalitarian

Models, Problem 5...................... 83
Predictions of group performance under

the Hierarchical and Equalitarian
Models, Problem 8...................... 8b

Atypical SOCiomatriXOoooooeoooooooooooooo 91

viii

 

 

 

 

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LIST OF APPENDICES
APPENDIX TITLE

PAGE
1

Use of Pearson's Tables of the . _ ‘
Incomplete Gamma. Function. to o o I o o o o o 0 0'0". 117

Proof that g(r;o(/\,k) a g(o(r;A,k)....... 119

ix

 

 

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__.._ ...——-—

 

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CHAPTER I

INTRODUCTION

In his review of human problem solving, Duncan
(1959) summarized his conclusions thus: "The field of
problem solving is poorly integrated. The reasons for
this seem to be the use of a 53223 varietz 22 EEEEE 32
provide problems, the freguent use 23 unanalzzed 22g

*
non—dimensionalized variables, the lack of an agreed-

 

 

upon taxonomy of behavioral processes, and to some
extent the failure to relate data to other data or to
theory. Problem solving particularly needs research
to determine the simple laws between dimensionalized
independent variables and performance" (1959, p. h26).
Duncan was, of course, referring particularly to
investigations of the problem solving process £23 is.
But there are two sub-areas of research and theory
that are of particular interest to the social psychol-
ogist. First, there are studies of group problem
solving in which members of a small group collectively
address themselves to some problem or task; of major
interest are those cases where the group's performance
has been contrasted with individual effort, (Lorge,

Fox, Davitz, and Brenner 1958).

 

 

 

 

*
Emphasis supplied.

__ ..._._._- -——-

 

 

 

 

2.
Second, small group researchers have often used problems
and other tasks to focus the group's effort, while their
main interest was in manipulating other situational vari-
ables and observing their effects on group structure or
various group processes, e.g., creating an out-group
threat and observing the increased cohesion of the group
under study.

It is the writer's contention that the poor organi-
zation and integration of studies, reported by Duncan,
creates a difficulty for those who use problems in the
course of investigating other phenomena, e.g., communi-
cation structures in small, task-oriented groups. At
the heart of many difficulties is the lack of knowledge
about the task or problem itself: How may problems be
ordered or classified to provide some continuity across
situations and provide a basis for comparison between
studies employing different problems in the study of the
same phenomenon?

The lack of a systematic treatment of the problem
itself precludes the kind of social-psychological theory
that deals simultaneously with the nature of the problem,
the group product (task performance), and the social
behavior of the individuals in the group. Even regard-
ing the group network studies, which deal with the

effects of imposed communication structure on group

 

 

 

 

 

performance, Glanzer and Glaser comment, "Perhaps the

most surprising thing about the entire area has been

the fact that despite the highly formal origins of these
studies, the organized body of theory has not yet ap-
peared" (19576, p. 35).

With this overview in mind, the purpose of this
dissertation can be described as an attempt, first, to
introduce an ordering of problems and a method for
measuring this order, and second, to predict group
problem solving performance from a knowledge of the
problem solving behavior of persons working as indivi-
duals. Major emphasis is placed upon the development
of a model that deals concurrently with group product
(problem solving performance) and the emergent structure
of the group. In addition, the frequently noted
superiority of problem solving groups over individuals
working on the same problems (Shaw, 1932) is critically

examined.
RESTRICTIONS 0F SCOPE

Before examining previous research it is well to
specify the restrictions of this inquiry. Only verbal
problems will be considered, for these generate inter-
personal interactions and lead to social organization.
Only problems with definite answers, which the subject

can solve without special outside knowledge, are useful

 

 

 

 

 

 

 

 

in the present experiments. These problems are called

"Eureka problems," (Large 23 31., 1958, p. 355). Only
23.222 groups with unrestricted face-to-face contact
will be considered. Groups may restrict their communi-
cations or choose leaders, and a structure may emerge,
but the present study does not involve imposing a
communication network or social organization upon the
group.

CLASSIFICATION OF WORD PROBLEMS:

STUDIES FOCUSING ON THE PROBLEM AND ITS ELEMENTS

Despite the need for studying the problems them-
se1Ves, implied above by Duncan, the literature
contains little that represents a systematic attack on
the behavioral effects of the particular problem--
especially word problems. Studies focusing on problems
BEE 33 have been mainly concerned with (a) criteria for
selecting problems, (b) presenting problems by different
methods, or (c) varying the elements of problems. These
will be considered in turn.
a. Criteria for selecting problems. Ray (1955) has
developed several criteria for tasks suitable for pro-
blem solving studies. For example, one should choose
problems which may be scored along a continuum; the
problem should restrict subjects to few hypotheses

about its solution, etc. Unfortunately, Ray did not

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5.

consider verbal problems at all. Marks (1951)
suggested that problems should be plausible (stimulate
interest), complex and difficult (elicit solving
behavior), structured (allow quantification and the
recasting of the problem in different contexts), and
solvable.

The selection of problem criteria does not really
meet the issue of problem classification as set forth
in this study. Not only does "meeting the criteria"
depend heavily on subjective decisions by the investi-
gator, but it avoids the question by restricting the
kinds of problems that may be used. For instance,
Marks' criteria would exclude many word puzzles,
although these have been among the more important
tasks for studying social interaction. However, the
problems selected for this study appear to meet the
major criteria outlined by Ray and Marks, viz.,
continuum, restricted hypotheses, plausible, complex,
structured, and solvable.

b. Variations in Method of presentation of problems.
There have been several studies in which the experi-
menter varied a problem's "concreteness" (Saugstad,
195?; Large, Tuckman, Aikman, Spiegel, and Moss, 1955a,
1955b, Cobb and BrenneiSe, 1952; and Gibb, 1956).
Large £3 21. (1955a, 1955b) presented the "mined road

problem" at seven different levels of reality (verbal,

 

 

 

 

 

 

 

 

 

 

6.
photographic, miniture scale model, real presentation,
or various amounts of the model and real situation).
Level of reality appeared to have little effect on
problem-solving performances. These investigators were

not concerned with ordering problems, but rather

 

explored the effects of administering the same problem
under different conditions. Style of problem presenta-
tion, therefore, does not bear directly upon the
question of problem classification.

0. The effects of manipulating the elements of a

 

problem. Katz (in Duncan, 1959) found that increasing
the degree of disorder in which digits were presented
increased the errors made in computing sums. Benedetti
(1956) searched for those aspects of Luchins water jar
problems which might be contributing to rigidity; he
added one, two, or three jars to the original problems
with noticeable effect. The increased freedom of
choice seemed to destroy much of the rigidity noted
with the series of problems. Judson and Cofer (1956)
presented a group of four words to subjects who were
to choose the word that was out of place; each con-
tained two ambiguous words and two unambiguous words.
The subjects tended to choose on the basis of the
first appearing unambiguous word. Dominance of the
first occurring ambiguous word was increased by

increasing the number of ambiguous words between the

 

two unambiguous words.

Battig (1957), using a task that met Ray's
criteria, had his subjects play a word game in which
they were to guess a word, given only the number of
letters it contained. A significant difference in
variability of performance between words was found,
much of which seemed attributable to the specific
letters contained in the words. There was limited
evidence that word length and frequency of usage were
important variables.

Admittedly, the studies dealing with the effects
of varying the elements of a problem were not concerned
with ordering problems and introducing a method for
measuring this order. Various manipulations of the
number of pieces, number of alternatives, etc., are not
represented in this investigation. But this last group
of studies does illustrate the importance of a particu-
lar problem and the behavioral effects of slight varia-
tions in the problem. Presumably the addition of
problem elements would increase a problem's complexity
or heterogeneity. The necessity of a solver's choosing
from among the added elements would add another stage
to the problem-solving process. Duncan has said, "In
contrast to the experiments on methods of problem
presentation, studies of variation among problem ele-

ments consistently reported at least some significant

 

 

 

 

 

 

 

 

 

 

 

8.

effects, occasionally powerful effects, on problem
solving performance. Thus, performance on a problem
may or may not be influenced by contextual variables,
such as methods of presentation that do not change
relationships among elements of a problem. But

changes of a problem's internal structure usually
influence performance, even in cases where the problem
remains, in some physical sense, the same." (1959,p.h10)
Duncan's recent review of human problem solving includes
an important summary of studies of tangental interest

to the issue of problem classification.
PROBLEMS AND GROUP SOLVING

It is no surprise that in studies employing some
kind of task the particular task seems to make a great
deal of difference in group behavior. In this

"...generalizations

connection, Lorge gt 3;. have said,
about problem solving and about group superiority seem
to depend upon the nature of the tasks" (1958, p. 356).
The same authors generally criticize the research on

group versus individual problem solving on the grounds

that investigators have paid insufficient attention to

the particular task; the usual tasks are petty, unreal,
etc. But most of all, the diversity of the tasks used
by different investigators has been so great that it is

difficult to draw conclusions about the superiority of

 

 

 

 

 

groups over individuals.

An ordering of tasks within a problem class would
seem to be a first step in studying task-inspired
influences on group-individual differences. The present
study explores an approach to the ordering of problems,
the number of steps or stages to arrive at a solution.

In a recent theoretical study, Roby and Lanzetta
(1958) have emphasized the importance of the particular
task to group functioning. They suggest that "objective"
task properties be given equal status with "modal" task
properties. An objective property is one for which a
single, definite, value may be specified through
either measurement or control. A modal property
"...reflects 'typical' behavior of individuals or task
variables but is subject to variation due to group
characteristics and their interaction effects with
other task properties" (Roby and Lanzetta, 1958, p. 89).
These authors present a rather complex descriptive
paradigm of the interaction of "group-task systems."
Although, they do not present a true classification
scheme for problems, Roby and Lanzetta point out that
a severe limitation on such a design has been the
general neglect of task parameters in small group
research.

The classification and prediction models, to be

presented subsequently, take cognizance of Roby and

 

 

 

 

 

 

 

 

 

 

 

 

lO.
Lanzetta's recommendations. The Classification Model
represents an attempt to deal with at least one kind
of task parameter (stages to solution). The prediction
models relate group pr0perties with task properties
while focusing on the product of group interaction--

problem solution.
GROUPS VERSUS INDIVIDUALS

Formal models of small group behavior appear to
have been very slow in developing. Hays and Bush (l95h)
have expressed surprise that so few mathematical models
have been constructed for use with group interaction,
inasmuchas the utility of such an approach has been
demonstrated in learning theory, etc. These authors

speculated that it is the great complexity of social

 

interaction which has hampered the development of

mathematical models. They also pointed out that in the
early stages of model construction the particular task
is crucial. Many suitable tasks could be appropriated

from experimental psychology. These should be (a) tasks

 

in which the number of behavior possibilities is limited
(important for model building) and (b) tasks whose

behavior-producing capacity is well known for individual

_ _ __ ___.-_

subjects.

Among the models which have recently been suggested

are those of Hays and Bush (195%), and Large and

__ ____.. __.____...————-

 

 

 

 

 

 

 

 

11.
Solomon (1955).

The work of Hays and Bush draws directly upon that
of Bush and Mosteller (1953). Two models for group
action are proposed and an experimental test of the
predictions, using three-man groups and a two—choice,
probability-type learning task. The "Group-Actor Model"
assumes that the group acts as if its guesses are those
of an individual, i.e., individuals and groups were
taken to be entirely interchangeable actors in this
situation. The "Voting Model" assumes that no basic
change occurred between the individual guessing alone
and the guessing that takes place in a group, i.e.,
three men behave as if they were independent voters.
Neither model was rejected when compared with group
data. In fact, the observed group results fell between
the predictions of the two models and the authors
interpreted the models as establishing limits for the
behavior of such groups.

The authors lamented that they did not collect
individual data which would have allowed clearer inter-
pretation. Unfortunate, too, is the fact that the
Hays-Bush models appear to be restricted rather
definitely to the two-choice learning task they employed.
Nevertheless, their work demonstrates the feasibility of
such an approach and the possibility of making use of

existing models from other fields in the study of small

 

 

 

 

 

12.
groups. Their results forced them to shift the focus
of attention from the adaptive to the integrative
aspects of interaction--something not anticipated until
they began to speculate about the requirements of a
more suitable model.

Lorge and Solomon (1955) reinterpreted Shaw's 1932
results with a group problem solving model which utilizes
results obtained with individual solvers. The Lorge-
Solomon model predicts the solution probability of an
r-man group to be the probability of one or more solu-
tions by the r individuals working alone. Thus the
model relies upon a simple pooling effect, and inter-
action is assumed to have no effect on the probability
that a group will solve a problem. These authors
proposed their model as an alternative interpretation
of the results obtained by Shaw (1932).

Generally speaking, groups are considered to be
superior to individuals in solving problems. The evi-
dence for such a position is much less than one might
suppose, considering the extent to which the view is
held. The preeminent position of the group in solving
problems is reflected by the following quotation from
Kelley and Thibaut: “...group problem solving involves
individual problem solving, 222 £222 £253. The 'extra'
does not refer to a 'group mind' or a magical 'plus'

which arises out of groups. Rather, it refers to the

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

13.
simple fact that the thinking done by a group member
occurs in a different context from that done by the
‘isolated' thinker. Because it occurs in communica-
tion and interaction with other persons, the products
of the group member's thought may be unpredictable
from the observation of solutions obtained in isola-
tion" (195b, p. 738).

Watson (1928) conducted one of the early studies
to compare groups and individuals working at the same
problem. Individuals and groups of three to ten mem-
bers were to produce small words from long ones. The
average number of words produced by the group was
considerably higher than that produced by the best indi-
vidual. Watson then examined his groups more closely;
he added together all of the different words of indivi-
dual subjects and compared the results with their
subsequent behavior in a group. The real groups were
found to be clearly inferior to the "summated" groups.
These results suggested that real groups did not take
into account the resources of all members.

The most frequently cited study in the area of
group problem solving is that of Shaw (1932). In the
first half of the experiment, individuals and groups
of four worked on three similar eureka problems. In
the second half, subjects were given three rather

different problems-~unscrambling words to form the

 

 

 

l___,___,____~——————w—»r

 

 

 

 

 

 

 

 

 

 

 

 

 

14.
last sentence of a prose passage, unscrambling words to

create the last lines of a sonnet, and discovering the

shortest route for two school buses, given certain
conditions.

Shaw interpreted her results as indicating a clear
superiority in favor of groups, and accounted for this
on the grounds that groups tended to reject incorrect
solutions by members and to guard against errors.
HoweVer, Shaw grouped all of the problems together when
comparing the proportion of correct solutions among
groups with that among individuals. Lorge and his co-
workers have closely scrutinized the Shaw experiment.
"For the first period, on the so-called Eureka problems,
three of the 21 individuals and three of the five groups
solved the first problem; no individual and three groups
solved the second problem; and two individuals and two
groups solved the third problem. No individual solved
more than one problem, but just three groups made the
eight group solutions. Two groups and 16 individuals
never solved any of the three puzzles. For the
second period problems, three of 17 individuals and
four of the five groups solved the first problem
completely; a fifth group and seven other individuals
made just one error. No individual and no group
solved the other two problems. Group superiority

rests only on the eight solutions by groups in contrast

 

 

 

 

__ ._.—-

 

 

 

 

 

 

 

 

 

 

 

15.

with the five by individuals. In general, interpreters
of the Shaw experiment have disregarded not only the
similarity among the three problems but also the fact
that the solutions were based on the sum overall prob-
lems rather than on the number of identical solutions
by groups. For instance, when only the solutions by
individuals and by groups for the first problem are
compared, there is no statistically significant
difference. Shaw neither discussed the fact that two
of the groups never solve any of the three puzzles, nor
the relative efficiency of three solutions among 21
individuals versus three solutions for five groups of
four members each, i.e., 20 individuals altogether"
(1958, p. 355).

Marquart (1955) advanced criticisms similar to
those of Lorge. Essentially replicating the Shaw
experiment, Marquart obtained comparable results with
groups and individuals. She analyzed her data so as
to take account of the effect of grouping individuals,
and she then discovered the group-individual discre-
pancy to be negligible. In fact, individuals had a
slight advantage over groups. (Note the similarity
between Marquart's results and the earlier findings
of Watson (1928).

Thorndike (1938) obtained the usual group

superiority in comparing individuals with groups on

 

 

 

 

 

 

 

 

16.

four verbal problems. The problems were presented so
that some subjects could make only a limited number of
reSponses to them; other subjects were allowed an un-
limited number of reSponses. Thorndike hypothesized
that as the range of responses increased the superi—
ority of groups over individuals would increase. In
general, the results tended to confirm his hypothesis.
Unfortunately, Thorndike made no analysis that would
throw light on the question of group superiority
based only on the pooling of individual solutions.

Husband (19u0) found pairs superior to individuals
in solving word puzzles and arithmetic problems.
However, groups needed to solve the problems in one-
half the time required by individuals in order to
"justify" themselves. When the data were analyzed in
terms of how efficiently groups and individuals used
their time, he concluded that the time saved by the
pairs was never more than a third.

Taylor and Faust (1952) compared group and indi-
vidual performance on a game of Twenty Questions which
resembled a eureka problem. The proportion of solu-
tions favored groups over individuals. However, in
terms of man-hours required for a solution, individuals
were superior.

Faust (1959) reported two group-individual experi-

ments, identical except for a slightly different

 

 

 

 

 

 

 

 

 

 

 

17.

composition of subjects. In both experiments he found
that four—man groups averaged more problems solved,
both verbal and spatial, than did individuals. When
these observed groups were compared with "nominal"
groups, generated from the mathmatical model of Lorge
and Solomon (1955), a significant difference was found
on only one of the problems used with the second experi-
ment. In this case the real group was superior to the
"nominal" group: Thus, the results of Faust's well-
executed study are generally in line with previous
experimental evidence.

Faust pointed out that the Lorge-Solomon Model A
provides a much-needed baseline against which to compare
the effects of social interaction in real groups. (The
same proposition is advanced later in the present
study).

The preceding discussion suggests that more groups
are likely to solve a problem than individuals. That
the interpretation is not as simple as it first appears
is indicated by the fact that the group superiority
disappears when individuals are artificially pooled in
a manner approximating the group aggregation. On some
efficiency measures, time for example, the individuals
turned out to be superior when pooled in this way.
Watson (1928) even suggested that group interaction,

instead of providing an extra boost, resulted in

 

 

 

 

 

 

 

 

 

 

 

 

18.

restraint.

A recent review by Large 33 21. (1958) provides a
convenient summary of group-individual experiments. One
should consult this review for studies not limited to
Ed £22 groups and eureka problems.

The above studies suggest that small, problem-
solving groups have not received the attention they
deserve in regard to formal models or the precise
organization of group—individual solving data. The
recommendation by Hays and Bush that existing models
from other fields may be modified for use with small
groups seems quite appropriate, considering the impor-
tance of the work of Bush and Mosteller (1955) to the

developments in the next chapter.
SUMMARY

The study of group problem solving requires (a)
an apt and relevant description of the problems used
and how they are solved, (b) a systematic way of repre-
senting both the pooling of individual work and social
interaction in group work, and (c) a sufficient volume
of data to permit an exact comparison between groups
(and individuals. A review of previous work indicates

that (a) There is no suitable framework for describing

problems. (b) The pooling of individual accomplishments

is clarified by the Large-Solomon model, which however

 

 

 

 

.___...'— -—-

 

 

 

 

 

 

 

19.
does not deal with social interaction. Some possibi-
lities of social interaction are bracketed but not
specified by the work of Hays and Bush on group learning.
(0) Experimental comparisons of group and individual
problem-solving are neither numerous nor, in the main,
done on a sufficiently large scale to give a clear indi—
cation of whether, and to what degree, groups can out-

perform individuals.

 

 

 

 

 

 

 

 

 

 

CHAPTER II

THEORY

The problem solving process, which begins with the
presentation of the problem, is terminated when the sub-
ject arrives at a correct solution, when he arrives at
an incorrect solution he believes correct, or when the
experimenter arbitarily ends the experimental session.
Although the proportion of correct solutions is a
standard means of describing experimental results in the
area of problem solving, the inefficiency of such an
approach is revealed by the question: What proportion
of correct solutions occurs after what time interval?
The choice of different time intervals (two seconds,
three minutes, 12 hours, etc.) can yield different

proportions. A distribution of solution times over

 

s0me interval is more desirable. Consequently, enough
time should be allowed so that the subjects themselves
terminate the process. These more nearly complete
data are advantageous for: (a) classifying problems,
and (b) constructing a model for prediction of group
problem solving.

In this chapter attention is focused on the
distribution of the times when the problem solving
process terminates, and the correctness of answers,

When subjects work individually and when they work in

20.

 

 

 

 

 

 

 

 

 

 

 

 

 

21.

small 29 Egg groups. When groups of subjects work to-
gether on the problem it is also possible to observe
their statements, questions, etc., and to study the
social structure which emerges during the problem
solving process.

The central issue is the prediction of group per-
formance from individual performance. It does not seem
feasible to attempt to predict the performance of each
individual or group separately, because one can collect
only a small amount of information about each subject
aside from his actual problem solving performance.
Hence the problem may be approached through a stochastic
model which attempts to account for the distribution of
observations gathered. In this way the complete data,
including mean, variance, and shape of the obtained
distribution, can be brought to bear on the theoretical
issues.

CLASSIFICATION:
A MODEL FOR THE DISTRIBUTION OF SOLUTION TIMES

Under standard conditions a particular problem
should consistently yield a characteristic distribution
of solution times for samples of subjects drawn from
the same parent population. The following discussion
derives a theoretical distribution, using the approach

of Bush and Mosteller (1955, Chap. 1h). Consider first

 

 

 

 

 

 

 

 

 

 

 

22.

that time is divided into short, equal intervals of
duration h. In any interval the subject may either
solve the problem or not. Imagine that there exists a
probability p that the problem is solved in the N-th
interval, given that it has not been solved previously.
The probability p is taken to be constant over intervals
and each interval represents an independent event.

The probability that the problem is solved in the
first interval is p. The probability that the problem
is first solved in interval 2 is (1-p)p, which is the
joint probability that it is not solved in the first
interval and is solved in the second. The probability
that the problem is solved in the N-th interval is
(1-p)N-1p, the probability that it is not solved in any
of the first N-l intervals and then is solved in the
N-th interval. Since each interval is of length h the
solution time T, when the problem is solved in the N-th
time interval, is
(1) T a Nh
Let the probability of solution at time T be called
P(T). In this simple one-stage process,

(2) HT) = p<1~p>N‘1

which is the geometric distribution. This is a posi-
tively skewed distribution with mean 1/p and variance
(l-p)/p2-

The problem solving processes dealt with here may

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

23.

be expected to approximate the situation given above,
except that solution of the problem may require the
completion of more than one stage. Now consider that
there are k stages to be accomplished, and that solu-
tion of the problem is recorded when all k stages have
been completed. Assume further that each of the k
stages has probability p of being completed in a given
time interval of duration h, and assume that h is
chosen so small that two stages are never completed in
the same interval.

The probability that the k-th stage is Completed
at trial N is given by the negative binomial (Pascal)
distribution,

N-l k N
(3) P(N) = k-l p (1-P)

 

 

An intuitive justification of this formula is given by
Bush and Mosteller, and its derivation is also discussed
by Feller (1957, pp. 155-157). (Note that (3) reduces
to (2) when k = 1). It can be shown that the expected
number of intervals, E(N) is

()4) E(N) = k/p

and, since each interval is of duration h, the expected
time to solution is

(5) E(T) = h.E(N) a hk/p .
Furthermore, the variance of the negative binomial
distribution is

<6) o’zm) . ku-pvpz .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2b.

The variance of the time scores is obtained by multi-
plying the variance of N by hz, which gives
(7) C72(T) = h2k<1-p)/p2
Other statistical prOperties of the Pascal distribution
are discussed in standard sources on probability theory
and mathematical statistics (Fisher, 1950; Feller, 1957).

The above model is not quite satisfactory because,
in the problem solving process, time is measured as a
continuous variable rather than as a succession of
finite intervals. The above argument is applied to
continuous time by letting the duration of an interval,
h, approach zero as a limit while keeping the "rate,"
or probability per unit time p/h, constant. Let the

first interval be designated 0, the second 1, etc.;

 

since T 2 Nb, the geometric distribution of (2) becomes
(8) P(T) = (1-p)Np = (1—p)T(hp .
Let p/h = Abe a constant so that p = Ah and

(9) P(T) = (1—Ah)T/hlh

If one takes the limit of p(T) as hs-) 0, the term Ilh

goes to 0 and, consequently P(T) approaches 0. There-

 

fore let

(10) f(T; A) .. lim (1-lh T/h )Lh
h--> _E‘o ' "‘ " " '

= K11“; (1.. AMT/h

h--) o

. A T/h - AT
and it can be shown that (1- h) approaches e

0

Hence,

(11) f(T; A) = Ae- AT .

_..._.-___._..—————

 

 

 

 

 

 

 

 

25.

If each individual stage has the distribution

 

f(T3 A.) = .Ae- T, the distribution of solution times
to complete k stages in the gamma distributionA
- T k-l
(12) gm A. k) . A e (AT) .
(k-l)

The gamma distribution g(T; A., k) may be readily
obtained from f(T; A.) by the use of moment generating
functions, but such a development is beyond the scope
of this thesis.

From formula (12),

(13) E(T) = k/A
and
(11.) (72(1“) = k/ A2 .

Figure 2.1 shows the graph of gamma distributions
with k = 2 and k = U. As k increases the graph of the
function becomes more nearly symmetrical.

The parameter k depends on the structure of the
problem and how it is solved by the subjects; k must
be estimated from the data.

An estimate of k can easily be obtained using
only the first two moments of the distribution. Recall
that equations (13)

(1b) gave the mean and variance as

E(T) = kM

g2(T) . k/ A2

 

 

 

 

 

 

 

26.

Therefore,

(15) hm] 2 = 1:2 AZ - 1.
O'2(T) k //12

Replacing the population expectation and variance by

estimates from the data, T and 5T2 respectively, one

has

A _ 2
(16) k . (T) .
PREDICTION or GROUP FROM INDIVIDUAL PERFORMANCE:

LORGE AND SOLOMON MODEL

Lorge and Solomon (1955) have suggested a non-

interactional ability model, which however does not

take into account the distribution of solutions over

some experimental time interval. Their model provides

the main example of an approach to the prediction

problem. Lorge and Solomon submit the hypothesis that

group superiority on eureka problems is not due to

social interaction, but simply to the abilities of the

members. "Such an hypothesis may be expressed in terms

of two ability models: (A) group superiority is a

function only of the ability of one or more of its

members to solve the problem without taking account of

the interpersonal rejection and acceptance of sugges-

tions among its members; (B) group superiority is a

function only of the pooled abilities of

 

 

 

 

 

 

 

 

)‘
\
J

 

 

 

 

 

 

 

 

 

 

 

 

 

. o _.: 1 v3
.H: :1va H doflocafl out. mo ~3an .. H.N oaswfim
m a e m e m N H o
\ ~ _ _ L? [P _ L P
NH
Va .mm .
mm
7. cs rm.
2
Ax:
I».
r O.H

 

 

 

 

 

 

 

 

______________________________________TIJIIIIIIIIIIIIIlIIlllllllllllllllllllllllwI)

28.
its members. The latter model, B, implies that any
problem may be composed of, and solved in, two or more
stages. Model B reduces to Model A for one-stage

problems." (1955, p. 140)

 

Consider first the Lorge and Solomon Model A.

PG = probability that a group of size r solves
the problem;
PI a probability that an individual solves the
problem.
Now l-PI is the probability that an individual will not i H
solve the problem, and the probability that none of a

group of r persons will solve is (l-PI)r

. Consequent-

ly the probability that a group of size r will solve

 

the problem is

 

r l
(17) PG = l-(l-PI) . 1,
Equivalently

l/r ;
(18) PI = 1-(l-PG) .

Model B: Recall that a problem may be solved in two
or more independent stages. Equations (17) and (18)

may then be generalized to the k-stage case. Let P .

 

 

Ia
be the probability that an individual will solve
stage j. H
k r] k
(19) PG = TI [l-(l-PIj) , PI .‘TI' Fla 1
jul i=1
i
I

 

 

 

 

 

 

 

29.

where Model A applies at each stage j. If PIj = P is
assumed to be the same for all stages,
(20) PG = [1-(1-P11/k)r] k

(The argument may be extended to the case where
PIj is not constant from stage to stage).

The Large-Solomon model does not take into account
the distribution of solution times, and the results may

depend upon the time given the subjects.
PREDICTION: COMBINATION OF CONTRIBUTIONS MODEL

The Lorge-Solomon approach may permit prediction
of the proportion of groups which solve a problem, but,
as just indicated, such a preportion depends upon the
time given the solvers. A satisfactory resolution of
the theoretical problem involves predicting the distri-
bution of group solution times from the distribution of
individual solution times.

The problem will first be taken up for the idealized
case in which all individual subjects and all groups
solve the problem. It is necessary later to consider
the consequences of having nonsolvers in the experiment,
but this discussion will be postponed until later.

To evaluate the possibility of group facilitation
or interference, one must first decide the performance
which would result if the members of the group proceed

without any important effect on one another's success.

 

 

 

 

 

 

 

 

 

 

 

 

30.

In analyzing this situation it is most convenient
to proceed at once to the model of continuous time.
Recall that in the case of individual solvers the
probability of solution per unit time was A.. If one
has a group of r independent SOIVers contributing to
the group, then the probability per unit time is r). .
This approach can be used, despite its great simplicity,
because it is assumed in going to continuous time that
the time interval h is very small. As h approaches
zero the probability that two different subjects will
both make a contribution at the same time becomes
negligible. Hence, there exist none of the diffi-
culties, discussed by Lorge and Solomon, of taking

account of the possibility that a group may contain

 

more than one Solver. Since the concern here is with

solution times, even if a group has more than one

 

solver, one may neglect the infintesimal possibility

 

that both solve at exactly the same time. Supposing,
in the same Spirit as the Large-Solomon model, that the
group solves the problem immediately after the first

subject solves, groups will differ from individuals

only in the probability-per-unit-time parameter, which

in groups is rzi .

The above argument obviously applies to the one-
stage problem. However, it should be clear that it

applies equally well to the k-stage problem, for the ‘

 

 

 

 

 

 

 

31.

number of subjects in the group can in no way change
the number of stages in the problem. The prediction
of group performance from individual performance, in
this idealized case, is very simple. Let the gamma
distribution for probability-per-unit-time of a) and k
stages be called g(T;,\ , k).

Hypothesis: If performance of individual solvers
is described by g(T¥,l, k), then performance of groups
of r solvers will be described by g(T; rA, k).

As was mentioned at the beginning of this chapter,
subjects can arrive at wrong answers. In the case of
individuals working alone one can make quite adequate
estimates of the prOportion of subjects who will not
arrive at the correct answer. When there is, say, one
such person in a group of four, the group will very
probably arrive at the correct answer. In fact, the
nature of the experimental problems being what they
are, it is unlikely that a whole group of size four will
arrive at a wrong answer. However, it seems evident,
that a group member, who alone would arrive at a wrong
answer (or fail to arrive at an answer at all), is not
likely to aid his group in arriving at the correct
answer.

One may formulate two hypotheses about the effects
of these wrong answers and no answers, and these two

hypotheses have different implications for the

 

 

 

 

 

 

 

 

32.
distribution of success of groups. One asked of the
data will be which of these two hypotheses is the more
reasonable.

The first hypothesis is that non-solvers are non-
functional in their groups. If so, a group with two
solvers and two non-solvers will solve the problem
just as rapidly as a group of two solvers alone. This
would happen if solvers suppress non—solvers, forming
an intellectual "hierarchy." The second hypothesis,
which seems to square better with informal observation,
is that the non-solvers consume some share of the
groups working time, though they do not contribute to
solution. This would reflect an equalitarian group
structure. In this case, a group with two solvers and
two non-solvers should take longer than a group made
up solely of two solvers. If it is assumed that non-
solvers consume their share of the group's time, the
following proposition is suggested. Let A be the
number of solvers and B be the number of non-solvers in
a group, where A + B a r is the number of persons in the
group. The probability of solution-per-unit-time for
each of the A solvers will be [A/ (A + B)]A . Thus, if
there are A solvers in the group and l}/ (A + B“ A_is
the rate at which each contributes, the group rate is
A ~ [A/ (A + B)1)\ = [A2/ (A + 3)]A .

In other words, if the distribution of solution

 

 

 

 

#4.!- _——-——-"A ’ ' 4 4 .,u

 

 

 

 

 

 

 

 

33.

times of individual solvers is described by g(T;A.s k),
then the two hypotheses are:

prothesis l, Hierarchical Model: Non-solvers have
no effect, and the group's distribution of solution
times will be described by g(T; A,K , k);

Hypothesis 3, Equalitarian Model: Non-solvers
consume time, and the group‘s distribution of solution
times will be described by g(T; [Az/ (A + B)] A . k)-

Except by extremely detailed and accurate protocols
of group discussion, solvers are indistinguishable from
non-solvers within a group. However, suppose that pro-
portion 2 of the individuals solve the problem correctly,
and the remainder, arrive at a false conclusion or no
answer and are non-solvers. Since the groups are
formed randomly, the probability that a group will have
exactly A solvers and r-A - B non-solvers is given by
the binomial distribution,

(21) P (A) AZ) aA (1-a)r'A .

The performance of a collection of groups can now
be predicted. Hypothesis 1, that non-solvers have no
effects, gives the proposition:

The distribution of solution times of a collection
of groups will be described by (Hierarchical Model)

r
(22) me =2 [P (A) gm AA, k)] .
A-O

Similarly, Hypothesis 2, that non-solvers consume

time, leads to the proposition:

 

 

 

 

 

 

 

The distribution of solution times for a collection

of groups will be described by (Equalitarian Model)
(23) wEm a if [p (A) g(T; [AZ/(A + B)]/\ ,1.) .
A=0

sThese last two distributions do not have any very
simple explicit forms. However, all the parameters
(a, A., and k) can be estimated from the performance of
individuals (by the method of moments). Then it is
possible to compute the above distributions simply by
taking the gamma distributions from a table (Pearson,
1922) for various values of k and T, and computing the
weighted average, following the above equations through.
With groups of size r = u, this process is not overly
laborious. The predicted distributions can be compared
with the obtained results.

If, as in the present study, the fitted gamma
distribution is very close to the obtained distribution
of individual solution times, the predictions can be
made directly from the distribution of solution times
obtained from the individual solvers without going
through the fitted gamma distribution.

A final note on the use of estimates in making
predictions is in order. Some practical problems are
associated with the use of k rather than the parameter
itself. The sampling distributions of these estimates

are not known and it is not possible to set up confidence

 

 

 

 

 

 

 

intervals in the customary fashion. Large samples

(such as those used in the study to be reported in
Chapter IV) will be of some assistance with this prob-
lem. If predictions are made from Q instead of k, but
the predictions are treated as fixed values, one fails
to take account of the sampling variations of the pre-
dictions. Hence, any regular statistical test is overly

stringent.
SUMMARY

The distribution of group solution times is pre-
dicted from ).(individual solver rate of solution),

k (stage-structure of the problem), and the distribution
of A (the manner in which groups are formed). The
parameters k and ).are estimated from the first two
moments of the sample of individual solvers. The
proportion of all individuals who solved a problem, 3,
can be used with the binomial formula to give P (A).

The intuitively more difficult problem of predicting
the distribution of group solution-times actually turns
out to be easier than the problem of predicting the
probability of group solution. Conceptually, this
represents an improvement over the Large-Solomon

formulation.

 

 

 

 

 

 

 

 

 

 

 

CHAPTER III

EXPERIMENTAL PROCEDURE

The quantitative models stated in Chapter II were
submitted to experimental test. A large number of
college students were given several Eureka problems to
solve, and their times to completion and correctness of
solution were observed. These data are the basis for
evaluating the gamma-distribution hypothesis, and serve
as the source of estimates of .X, k, and a. A number of
gg Egg groups of four students each were also given the
same problems under similar conditions except that the
members of group cooperated in solving the problems.
Comparison of group with individual performance were
used to choose between the Hierarchical and Equalitarian
models. Data on communication and social structure

within the working groups were also collected.
SUBJECTS

The subjects were 267 volunteers from an introduc-
tory psychology course. The sign-up sheets had spaces
for four subjects at each experimental session. When
four subjects signed up and appeared on time, they were
tested as a group. Whenever a group could not be formed,
as when only two or three subjects signed up, a subject

failed to appear, or some other accident occurred, the

36.

 

 

 

 

 

 

 

 

 

37.

subjects present were tested as individuals. This method

of assignment to the Group and Individual Conditions

was not random,
who appeared on
on the behavior
sign up, or who

likely that the

but the assignment of any particular S
time at the laboratory depended solely
of other students who did or did not
did or did not report on time. It is

Group and Individual Conditions were

composed of comparable samples of the population. This

recruiting method was used to maximize the number of

four-person groups obtained with the least waste of

subjects. Subjects at any one experimental session,

group or individuals, were of the same sex. There were

twelve groups of men and ten groups of women. There

were 98 men and 81 women tested as individuals.

APPARATUS AND MATERIALS

The room in which the experiment was conducted was

a ho ft. x 22 ft. seminar—type classroom. For the

experiment, two tables were removed from their usual

semi-circular arrangement and placed to one side. Four

straight-backed chairs were placed around one of them;

the other table, behind which E was seated during

experimental sessions, was located six to eight feet

away. At the experimenter's table was a large faced

clock (from which elapsed time could easily be obtained),

four cumulative frequency counters, and an electric

 

 

 

 

 

 

 

 

38o

timer which emitted a soft click every six seconds.

During a group session, the four 85 were seated
around the table and a carboard plaque, bearing the
identifying letter A, B, C, or D, was placed in front
of each. Individual sessions were conducted with 55
seated at the group table and/or other tables located
about six feet away, depending on the number of indi-
vidual 55 available for any one session.

The four eureka-type problems used were labeled
"Sample," 3, 5, and 8. The magnitude of the numbers
was meaningless and.served only to bolster problem
security; if 85 perceived the problems on which they
labored as only three of a larger series, they would
possibly be less inclined to relate them to naive class-
mates. Each problem was mimeographed on a separate
sheet of paper.

The first or sample puzzle, used for practice, was
developed for this study in colaboration with
Dr. F. Restle, although it has no doubt appeared else-

where.
SAMPLE

A drunk is walking down the sidewalk; his condition is
such that he is unable to proceed without stumbling and
staggering. He decides that he wants to cross to the
other side of the street. Unfortunately, he is so
inebriated that he staggers back and forth, from one
side of the street to the other--wasting a great deal
of effort. Finally, he is able to stay on the side of

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

39.

the street he had been aiming for. He continues his
journey on that side of the street.

Among the following, which is the best guess as to the
number of times he crossed the street in either direction?

1. h h.8
2. 5 506
3. 2

Puzzle #3 (hereafter referred to as the "Rope
Problem") was taken from Dudeney (1958) and subse-
quently modified.

3.

My dungeon did not lie beneath the moat, but was in one
of the most high parts of the castle. So stout was the
door, and so well looked and secured withal, that escape
that way was not to be found. By hard work I did remove
one of the bars from the narrow window, and was able to
crush my body through the opening; but the distance to
the courtyard below was so exceedingly great that it was
certain death to drop thereto. I did find in the corner
of the cell a rOpe that had been left there and lay hid
in the darkness. But this rope had not length enough,
and to drop in safety from the end was nowise possible.
So I made haste to divide the rope in half and to tie
the two parts thereof together again. It was then full
long and did reach the ground, and I went down in safety.
How could this have been? (Answer: Divide the rope
lengthwise and tie the ends of the unraveled rope
together).

Puzzle #5 (hereafter referred to as the Double

Problem) was adapted frOm one given by Kaufman (l95b).

5.

Here is a simple question. It happens to be couched in
complex phraseology, but that shouldn't disturb you if
you can find a way to reduce it to the fundamentals. It
can be done by rephrasing the question in such simple
terms that the answer is immediately apparent.

 

 

 

 

 

 

 

#0.

Now go ahead:

"If the puzzle you solved before you solved this one,

was harder than the puzzle you solved after you solved

the puzzle you solved before you solved this one, was

the puzzle you solved before you solved this one hard .
than this one?" (Answer: Yes; problem says it was 3
harder in the beginning and the central portion is a

self-defeating qualification).

Answer
Yes No Why?

Puzzle #8 (hereafter called the Dust Problem) was
constructed by the writer, using the familiar Luchins

water jar problem as a guide.

8.

As a prOSpector you have been very successful. You
have a container that holds exactly 163 ounces of gold
dust. You must discharge a debt by paying exactly 77
ounces of gold fudy (no more, no less) to a friend.

You have containers that hold exactly lh ounces, 25
ounces, and 11 ounces. How would you remove the exact
amount from the safe, using only these four containers?
You must use all four containers. (Answer: Subtract
the following from the 163 ounces container:

2(25) + 1% + 2(11) = 86 or 25 + 2(1b) + 3(11) = 86).

 

The above problems were developed on the basis of
a pilot study. This preliminary investigation suggested
problems which seemed (a) to yield a high probability
of solution within 12 minutes and (b) to represent,
intuitively, differing numbers of stages. The number
of stages to solution of a eureka problem may be
likened, in the ideal case, to the number of moves to

solution of a chess problem. The chess player may or

 

 

 

 

 

 

#1.
may not make an appropriate or an inappropriate move
in any one interval of time. Obviously a eureka prob-
lem is less specifiable than a chess problem and hence
less readily analyzed into sub-strategies or stages.
In fact, it is a goal of this investigation to provide
a framework within which the eureka problem may be
analyzed into moves or stages.

Consider, intuitively, the Rope Problem. Once a
subject hits upon the idea that the rope may be divided
lengthwise, escape from the dungeon is possible. On
this basis the Rope Problem would consist of one stage;
only one strategy is involved. In the Double Problem,
on the other hand, the subject must note that the prob-
lem states the previous puzzle was harder; then he must
realize that the subsequent verbiage refers only to the
present puzzle which is less difficult. The answer is
self-evident with the completion of these two or three
stages. The Gold Dust Problem requires perhaps three
or four stages to completion. The subject must realize
that he is to subtract each container in turn, leaving
the original with the correct amount of gold dust. The
second stage is reached when the subject decides the
number of times the 25 oz. container is to be sub-
tracted. Stages three and four would be the number of
times the 1b oz. and 11 oz. containers, resPectively,

are filled and emptied.

 

 

 

 

 

 

 

#2.

Note that these stage-estimates are intuitive.
Future investigation may show, for example, that the
Rope Problem requires two stages: one for the length-
wise unraveling of the rape and another for the joining
of the ends together. Or it may even be discovered that
reading and comprehending the nature of the problem
requires one or more stages. Whatever future research
reveals the nature of these stages to be, the present
investigation is concerned only with empirically esti-
mating their number for a particular problem.

The questionnaire used to evaluate group choice
structure arising from collective effort is given

below:

QUESTIONNAIRE

 

Your Letter

 

Age

 

'Sex

Your answers to the following questions will be kept in

the strictest confidence.

Consider each question carefully before you answer. Your

reSponse should refer only 32 the immediately preceding

 

 

 

 

 

 

 

 

I43.

problem! Problems differ a great deal.

1. In a few weeks, new groups may be assembled for work
on similar problems; participants would be paid and con-
siderable prize money would be awarded to the best-per-
forming group. Which members of this group would you
like to have with you, if you were to decide to partici-
pate in such a contest? Consider performance on the
immediately preceding problem only. Indicate your-choice
(s) by a check mark(s) before the appropriate letter.

You may decide to check none, one, two, or three of the
other members of the group.

 

Think carefully.

Person A

 

Person B

-————

Person 0

 

Person D

 

a. As a separate issue, rank the other members of this
group-~one through four on the basis of how much you
feel that they contributed toward the solution of the

immediatelz preceding problem. In your ranking, include
individual X (a fictitious person) whom you know to be

an "average solver" of problems like the one you have
just considered. Rank 1 would be the person contrlbu-
ting the most; rank 2 the next greatest contributor, etc.

Person A

Person B

Person C

Person D

Person X (An average solver

of such a problem)

PROCEDURE

In both the Group and Individual Conditions the Sample
Problem was given first, with a time limit of five

minutes. The experimenter answered any questions and

 

 

 

 

 

 

 

 

 

nu.

then proceeded to the first of the three test problems;
twelve minutes were allowed for each of these problems.
For any experimental session, the order of presentation
of the three problems was randomized. After E had dis—
tributed a copy of the problem to every S in the room,

he retired to the table where the timing apparatus was

located and gave the signal to commence work. The

experimenter then recorded his observations.

1. Group Condition.

Only one four-person group was tested at an experimental
session. When the 55 had been seated around the square
table, E placed the cardboard plaque, bearing the
identifying letter, in front of each. The experimenter
then gave the following verbal instructions to the

group:

"I would like to thank you for coming here this after-
noon (evening). In a few moments I will give you a kind
of word puzzle or word problem on which you are to work
as a group. Talk freely among yourselves and try to

'think out loud' as much as possible."

"Try to work as rapidly as you can and still do the best
work possible. It is very important, if this experiment
is to be successful at all, that I have an accurate

record of how long it took you, as a group, to solve

 

 

..I|.. .

 

 

 

 

 

b5.

the problem. Please do not arrive at an answer you feel
is correct and then wait until the alloted time has

elapsed before you write down the group's answer.

"When you, as a group, have decided on a solution, one
of you (anyone will do) should write it down immediately
and then jot down the time remaining as given by this

clock. You read the clock this way ...

"You will have three problems in all. A maximum of 12
minutes will be alloted for each problem. I will tell
you when to begin and when to stap, if that is necessary.
Are there any questions-~I will not be able to answer

questions after we begin."

After collecting the practice problem, the E answered
any further questions and corrected the procedures if

necessary. The E continued with the following:

 

"Here is a copy of one of the 'real' problems: you

must stop and start when I signal: Begin."

Following each problem, E advised the group, "Here is a
very short questionnaire, pertaining to the problem you
have just finished; you will have three minutes in which
to complete it. I know that you may be reluctant to
answer at first, but I would like to have your honest

opinion anyway."

 

 

 

 

 

 

 

 

 

#6.
E closed the session as follows: "I would like to
thank you for your time and cooperation. It is very
important to the success of this study that you do not
reveal to others the nature of these problems: some of
your classmates may receive some of the same problems
when they serve as subjects later on. In fact, the
best-performing group will win a prize of $8.00--about

$2.00 per individual."

The exPerimenter emphasized certain aSpects of the
proceedings with informal asides. For example, it was
stressed that the questionnaire referred only to the
immediately preceding problem, and just before handing
out the Double Problem it was casually pointed out that
each problem was complete in itself and referred only

to itself.

One measure of problem-solving activity was the fre-
quency with which each S talked. During the inter-
action, E recorded the frequency with which each
member communicated.to another member or to the group
as a whole. The recording was accomplished by means
of four hand-tally counters-~one for each member of the
group. The clock and timer shielded this activity from
the Ss and none of them gave any indication they were
aware of the fact that they were being observed in this

Way. Every time a member addressed a communication to

 

 

 

 

 

 

 

 

 

—:——=-_—wv

1+7.
another member or to the group he was counted as having
"talked." Every six seconds, at the soft click of the
timing device, E registered a count on one (or more)
counters depending on which members were talking at
that time. If no one was talking, no entry was made.
This yielded a systematic time-sample of verbal activi-
ty of the Ss sufficient to indicate which one talked
most often and longest, prerequisite to the descrip-
tion of a communication structure. The six-second
interval was chosen as the shortest which could be
recorded with precision by E under the conditions of
observation and the other activities he had to main-
tain. Ideally, the total speech of a S would be re-
corded as a deflection on a kymograph and the area
circumscribed by the base line and deflection would be

taken as the communication measure.

 

 

Another important datum was, who did most to solve the .
problem? Right after completion of the problem, E i
made an intuitive estimate as to the major contributor I
to the solution. Furthermore, any unusual events which

occur during problem-solving may lead to deviations

from the overall theoretical predictions. Consequent-

ly, observations of unusual happenings were recorded-—

e.g., angry exchanges, pre-group friendships, etc.

 

 

 

 

 

 

 

 

 

#8.

2. Individual Condition

One, two, or three Ss were tested individually at an
experimental session and were seated in the same
general area as that occupied by groups. The experi-

menter gave the following verbal instructions:

"I would like to thank you for coming here this after—
noon (evening). In a few moments I will give you a
kind of word puzzle on which you are to work indivi-
dually. Please do not talk among yourselves now or at

any time during the course of the experiment.

"Try to work as rapidly as you can and still do the
best work possible. It is 3331 important, if this
experiment is to be successful at all, that I have an
accurate record of how long it took you to solve the
problem. Please do not arrive at an answer you feel
is correct and then wait until the alloted time has

elapsed before you write it down.

"When you have decided on a solution, you should write
it down immediately and then jot down the time remain-
ing as given by this clock. You read the clock this

way. 0 0

"You will have three problems in all. A maximum of 12

minutes will be alloted for each problem. I will tell

 

 

 

 

 

 

 

 

 

 

 

1:9.
you when to begin and when to step, if that is necessary.
Are there any questions--I will not be able to answer

questions after we begin."

After collecting the practice problem, E answered any
further questions and corrected the procedures if
necessary. E continued with the following: "Here is

a copy of one of the lreal' problems; you must stop and

start when I signal...begin."

Following each problem, E advised the individuals,
collectively: "I would like for you to list the names
of three of your acquaintances who, in your opinion,
would do well on the kind of problem you have just
finished. It is preferred that those named be students
at MSU--but that is not necessary; they could be any-

one at all."

The E closed the Session as follows: "I would like to
thank you for your time and cooperation. It is very
important to the success of this study that you do not
reveal to others the nature of these problems; some of
your classmates may receive some of the same problems
when they serve as subjects later on. In fact, the

best-performing individual will win a prize of $2.00.”

Again, E emphasized certain aspects of the proceedings

With informal asides. For example, it was emphasized

 

 

 

1
l
I
I

 

 

 

 

 

 

 

 

50.
that a particular name-listing applied only to the
immediately preceding problem. And just before handing
out the Double Problem it was pointed out that each
problem was complete in itself and referred only to

itself.

Each S received the same problem. After he had written
down the solution, he recorded the time in minutes and
seconds that had elapsed; the time was obtained from

the large-faced clock.

When it became apparent that all 55 had finished, E
waited a couple of minutes longer, to avoid creating
a sense of urgency in the Ss finishing late, and then

stopped the clock. Following the collection of papers,

 

the 55 received a blank sheet of paper and were
requested to name three of their acquaintances, any 2

three at all, they felt might do well on the immediate-

 

ly preceding problem. This activity was designed as a E
counterbalance to the questionnaire of the Group Condi-

tion. The same data were collected for each problem.
SUMMARY

Each group and each individual was tested on the
sample problem and three experimental problems, the

eXperimental problems being given in different orders

for different 55.

 

II I: .I Ill.lll||||..

 

 

 

 

51.

Solution time was recorded for each individual on
each problem. In the groups, a count of frequency with
which each member talked, their choices of future
partners in such problem-solving work, observations of
contributions of each, and any unusual events, were also

recorded.

 

 

 

 

 

 

 

 

 

CHAPTER IV

RESULTS

In direct comparison, group problem solving was
found to be clearly superior to individual problem
solving. As is shown in Table n.1, a higher proportion
of groups than individuals solved each of the three
problems correctly. Although more groups solve the
problems, groups and individuals do not seem to differ
systematically on mean time to a right or wrong answer;
see Table h.2. In fact, standard t-tests applied to
the differences between individual and group mean
times on each problem do not show significant differences.
However, mean solution time does differ from problem
for both groups and individuals.

The distributions of answer-times (right or wrong)
throughout the lZ-minute experimental sessions are
represented by the frequency polygons in Figures b.l
(Individual Condition) and h.2 (Group Condition). To
facilitate graphical representation, subsequent dis-
cussion will deal with cumulative relative frequency
distributions.

The statistical models discussed in the previous
chapter use the idea that stages of problems are
solved at randomly distributed times, so that total
time to solution should have a gamma distribution, and

the parameter k of the distribution should be the

52.

 

 

 

 

 

 

 

 

53-

Table h.1 Individuals Vs. Groups on each experimental

problem, T a 720 seconds.

 

 

 

Individual Group Individual
Condition Condition Vs.
Group**
Prob— No.of No.of Prop.of No.of No.of Prop.of two tail
lems Cases* 501- Solvers Cases 501- Solvers tests)
vers vers
3 178 133 .735 22 21 .955 Z=2.h8 (p 4.013)
5 177 89 .503 22 17 .773 Z-2.7o(p L .007)
8 163 71 .h36 22 17 .773 Z-B.30(pL .oo1)

 

 

 

 

 

 

 

 

*The relative frequency of correct answers for the
Individual condition were based on different numbers of
subjects. Some few subjects were discarded because they
did not comply with the directions, found it necessary
to leave early, etc. There was no evidence to indicate
that any systematic bias affected the results.

** The statistical tests reported in Table h.1
make use of normal curve theory where

P1 ‘ P2

%
[p(l-p) (N1+N2)/N1N2]
and P is the proportion answering correctly among N

1
indiv duals; p2 is the proportion of correct answers
among N2 groups.

 

 

 

 

 

 

 

5h.

Table h.2 Mean 1 Standard Deviation of Solution Times
in Seconds for Groups and Individuals.

 

 

 

 

 

Outcome froblem Individuals Groups
Correct 5 325.8 1 15h.l 330.0 1 216.2
Solution 8 “14.9 1 166.7 h19.h 1 219.9
Wrong or 3 2h9.1 i 151.5 (Only one case)
No solution 5 276.8 .+_ 169.0 376.0 1 227.7

8 h62.5 1 3h6.2 h86.0 1 217.2

3 208.0 1 127.h 172.3 + 1h2.0
A11

5 301.5 1 163.0 3h0.h i 21h.1
Subjects

8 hhl.8 1 282.6 h3h.5 1 215.0

 

 

 

 

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57.
number of stages in the problem.
CLASSIFICATION: THE MODEL FOR THE
DISTRIBUTION OF SOLUTION TIMES

Bach problem was classified by its distribution of
individual solution times. Only the distribution of
solvers was used for this purpose; non-solvers (those
who failed to solve or gave a wrong answer) were
excluded from this analysis, since a person who fails
to solve the problem may have blundered on any one of
the k stages; it is not possible to determine from the
data the stage on which this might have occurred. Con-
sequently, data composed of persons who failed at
different and unknown stages cannot be expected to
provide an estimate of the number of stages in the prob-
lem.

Hypothetically, this distribution of solution
times is a gamma distribution, and k is the number of
stages in the problem. The parameter k was estimated

by the method of moments, equation (16).

A
k = (71")2 ,
2
S
T

An estimate ofll, the rate parameter, was obtained by

the method of moments making use of equations (13 and

 

 

 

 

 

 

 

 

 

 

 

58.
1h),

 

Table 4.3 shows the estimated number of stages and
the estimates of rate of solution of each stage (1,) for
each problem. As might be expected, £1 did not turn out
to be an integer for any one of the three problems.
Theoretically, k should be an integer, but it is not
surprising that the estimates, which are subject to
error, do not happen to be integers.

The obtained estimates of k for the three problems
in order were similar to the intuitive estimates
established from pre-experimental inspection of the
problems. Moreover, the small number of stages esti-
mated is in fair accord with intuition. For example
the estimate of n.47 stages (Table h.3) for the Double
problem is similar to the three stage pre-exPerimental
Speculation.

The sampling distribution of k is unknown under
the estimation method used. To provide a check on
the estimates, the theoretical curve, determined by a
particular Q1, was fitted to the observed distribution
of solutions for each problem. The appropriate cumu-
lative relative frequency curves are shown in Figures
2.3, b.h, and h.5. The obtained distributions are

strikingly similar to gamma distributions, as

 

 

 

 

 

 

 

 

 

 

59.

hypothesized. The theoretical curves were taken from
tables of the incomplete gamma function (Pearson, 1922).
Transformations of the time scale, etc., necessary for
the use of the tables are discussed in Appendix 1.

Goodness of fit was tested by the Kolmogorov-
Smirnov one—sample test. (See Walker and Lev, 1953;
Siegel, 1956). For each problem, the statistical hypo-
thesis tested was whether the sample of individual
solvers could reasonably have been thought to come from
a population having the theoretical distribution. This
hypothesis was regarded as tenable for each problem.
That is to say, the theoretical curves determined by
£5 a 2.8, 25 = n.5, and £8 - 6.2 gave acceptable fits

to the data for problems 3, 5, and 8. See Table h.h.

 

Reading 2322. The eXperimental time interval for
any one problem has heretofore been treated as a uni- '
form block of time. However, one can distinguish
between the relatively—invariant time required to read I
a problem and the highly variable time to solve it. It )
would be desirable to remove the reading time.

It was assumed as a rough estimate, that the '
first solver would solve the problem about 10 seconds i
after reading time was completed. This gave estimates “
of reading time for the three problems of 50, b0, and
90 seconds respectively. When these solution-times

were subtracted from all individual time-scores, the '

 

I I: I‘ll II'I‘I.

 

I'll):

 

 

 

60.

Table h.3 Method of moments estimates of k andlk for
each problem. (Individual solvers)

 

 

A _ ‘ _
Problem k a (T) 2/ 5T2 A a T / st

3 (Rape) 2-83 .1u6

5 (Double) #.U7 .137

8 (Gold 6.21 .150
Dust)

 

 

 

 

 

 

 

 

 

 

61.

 

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Table h.h Summary of Kolmogorov—Smirnov one-sample,
goodness-of—fit tests for theoretical
curves determined by estimates of k.
Problem No. of A (maximum) Sig. Values
Solvers k D of D: .05
theoret.-obs.|) level
3 (Rape) 133 2.8 .110 .118
5 (Double) 89 “'05 0086 elul"
8 (Gold 71 6.2 .082 .161

Dust)

 

 

65.
resulting net solution times also fit gamma distributions,
with Q' = 1.5, 3.b, and 3.8 reapectively, where Q' is
the estimate from net solution times. Note that this
is a sharp drop from the estimates of k obtained before;

2.8, n.5, and 6.2.

 

Subsequent to the completion of the main experiment,
students in a regular class meeting of elementary
psychology at Miami University (Ohio) were given the
same three problems with essentially the same instruc-
tions, except that they were directed to record the
time when they finished reading the problem, as well
as when they had written an answer. The mean reading
times for problems 3, 5, and 8, as recorded by the
Miami students, were 62.7, 61.0, and h1.9 sec. (See
Table h.5).

The Miami University students were probably quite
similar to the Michigan State University students, in
reading time of such simple problems. If the Miami
reading times are subtracted from the times-to-solution
in Table u.2 above, new estimates of the number of
stages in these problems can be obtained. These new
estimates, with reading time removed, are 1.3, 3.0, and
5.0 for problems 3, 5, and 8 re5pective1y. The writer's
judgments of number of stages were 1, 2 or 3, and 3 or u
(see Chapter III above).

The Miami students were also asked to estimate the

 

 

 

 

 

 

66.

number of stages in each of the problems, discounting
reading time. The definition of a stage was essential-
ly as given above in this dissertation, and the subjects
made their judgments after having solved the problems in

question. The mean judgments of number of stages, for

 

problems, 3, 5, and 8, were 2.8, n.0, and 6.3. These
mean judgments differ from the best estimates (using the
gamma-distribution theory) by 1.5, 1.0, and 1.3 stages,
with the students' judgments being higher than the ,
theoretical estimates, a very striking correlation. It

should be borne in mind that the theoretical estimates

come from the distributions of solution times, which

 

were entirely unknown to the judges; and the theoreti-
cal estimates in no way used the judgments of the
students. Hence, the two estimates were entirely inde-
pendent and could have been completely unrelated.
Hereafter "solution time" will refer to net time,
with reading time (estimated by the rough method of

the original study) subtracted. Using the k'.

1 suggested

by table h.5, new theoretical functions were selected
from Pearson's tables (1922) and fitted to the new
distributions of individual solution times. By in-
spection of Figures u.6, n.7, and b.8, it is evident
that high agreement exists between the data and the
theoretical curves. Results of the Kolmogorov-

Smirnov one sample goodness to fit test appear in Table h.6.

 

 

 

 

 

67.

Table 11.5 Estimates of k and A with reading time
included and with reading time removed,
where reading time was estimated by two
different methods.

 

 

Reading Time Included Reading Time Removed

 

 

 

A A Original Esti- Second Esti-
Problem k A mate removed made Removed
/\ A
as A} flu An
3 2.8 .1116 1.6 .108 1.3 .099
5 11.5 .137 3J4 .120 3.0 .111

8 6.2 .150 3.8 .117 5.0 .1314

 

 

 

 

 

 

680

PREDICTION: THE LORGE AND SOLOMON MODEL

As indicated in earlier discussion, the Lorge and
Solomon Model does not deal with the question of the
time interval allowed. In assessing the predictive

usefullness of their model it was, therefore, necessary

 

to apply it at various arbitrarily selected points in
time.

The probability of an individual solution at every
fifth time unit was taken to be the cumulative relative
frequency of solutions by T a 50, 100, 150,..., 720.
(Transformations of the time scale to take reading time
into account were unnecessary since they do not affect
the calculations.)

One first considers the case of a one-stage prob—
lem (Model A). Of the 178 individuals, .118 had solved
Problem 3 by T a 100. Applying Model A (equation (18))
to this observed value of PI', the predicted cumulative
relative frequency for the group is
PG' s l - (1 - .118)“ = .395 .

By T . 150 the probability of a correct solution had
become .332 and the predicted group value is

PG' . 1 - (1 - .332)” e .800 .

The same procedure was repeated for all the points
selected on the individual cumulative relative frequency

distribution. (PI' is the sample estimate of PI).

 

 

 

 

 

 

 

 

 

 

 

59.

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72-

Table h.6 Summary of Kolmogorov—Smirnov goodness—
of-fit tests for curves determined by

8'1 (1 . 3, 5, 8).

 

 

 

 

Problem No. of A D Sig. Values
Cases k' (maximum) theoret. of D: .05
obs. level
3 133 1.6 .092 .118
89 3.h .080 .1uu
71 3.8 .098 .161

 

 

 

 

 

 

 

 

 

73.

Under the assumption of two equally probable
stages, equation (21), Model B, was used to predict the
group probabilities. For example, on Problem 3 the
probability of an individual solution by T a 100 was
taken to be .118 and by T a 150 this probability was
.332, just as before. For k a 2, the group predictions
were

PG' . [1—(1-(.118)%)u] 2 . .66n and
PG' - [1-(1-(.332)%)u] 2 - .927 .

Figures n.9, “.10, and b.11 show the cumulative
relative frequencies observed for groups and individuals
(at every 50 seconds) along with the values predicted by
the Lorge and Solomon Model (k - l and k = 2). Pre-
dictions of group performance are poor on each problem,
and the predictions become worse with the larger value
of k. Only for Problem 3 (The Rope Problem) would one
regard as tenable the hypothesis that the observed dis-
tribution of group solution times was drawn from the
theoretical distribution represented by the Lorge-
Solomon predictions, 0n Problem 3, this null hypothesis
was tenable for both R = 1 and k = 2. However, the pre-
dictions were everywhere uncomfortably larger than the
observed proportions of group solutions. A summary of
the Kolmogorov-Smirnov significance tests on each prob-
lem appears in Table h.7.

The estimates of R from the classification section

 

 

 

 

 

 

 

 

 

 

7b.
are decidedly inappropriate with the Lorge-Solomon
Model, regardless of their usefulness in choosing
descriptive functions. Inspection of predicted and
observed curves (Figures n.9, 4.10, and b.11) suggests
that k must be taken less than one to provide good
agreement between the two throughout the experimental
interval. Any k<.1 is uninterpretable psychologically;
so it is concluded that the Lorge-Solomon Model as
applied here is unsuitable.

In essence, the Large-Solomon Model predicts the
probability of a group solution to be the probability
of solution by any of four persons working separately.
Groups in this study were found to fall 2212! this
standard. That is, the real groups did not do as well
as was expected on the basis of a simple pooling of

individual results by means of the Lorge—Solomon Model.
PREDICTION: THE COMBINATION-OF-CONTRIBUTIONS MODELS

Two combination-of—contributions Models were pro-
posed in Chapter II: (1) The Hierarchical Model; and
(2) The Equalitarian Model.

The Hierarchical Model

 

Suppose that non-solvers are non-functional in
their groups, as when the "solvers" in the group suppress
the contributions of non-solvers and form a functional

hierarchy. A group with two solvers and two non-solvers

 

 

 

 

 

 

 

 

 

 

 

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78. ‘ 1

Table h.7 Kolmogorov-Smirnov goodness-of-fit tests

of predicted group solutions (Lorge-
Solomon Model) with the observed data.

 

 

 

 

D
Problem No. of (maximuml theoret.- Sig.Va1ues of D for

 

Cases obs.| ) 25 df: .05 .01
k s 1 k = 2 level level
3 22 .211 .205 .270 .320
5 22 .314 .385 .270 .320 .

8 22 .306 .521 .270 .320

 

 

 

 

 

 

 

 

 

 

 

79.
would be expected to solve a problem just as rapidly as
a group of two solvers alone. Then the probability of
a group solution over time may be written as

u
(21») wHw) =- z [Prm gnu Mm] .
A=0

 

 

where A is the number of solvers in the group.

(The more general form was given in Chapter II, equa-

 

tion (22). Here T' time after reading is completed, has
been substituted for T, and r s h). Computationally
more convenient is

h
(25) wH(T') - Z [Pr(A) g(AT';A,k)] .
A=0

It is proved in Appendix 2 that the distribution
g(T'; A ).,k) is equal to g(AT';}i,k).
The excellent fits obtained with the classifica-

tion model (Figures 0.6, #.7, and h.8) suggested that

 

prediction may proceed directly from the obtained dis- i
tribution of individual solvers, rather than going
through the fitted gamma distribution, g(AT';A_,k).

Let h(T') by the obtained cumulative relative frequency

 

distribution of individual solvers on a particular prob- l
lem. Using the obtained data, the theoretical probabi-
lities, WH(T'), may be obtained from
g

(26) WH‘T') = Z [Pr(A) h(AT‘)] .

A=0

An example of the computations involved in the

group predictions is given below. From Table 0.1,

2 = .735 of all subjects solved Problem 3 and .265 were

 

 

 

 

 

80.
non-solvers. Equation (21) becomes

PNA) =5 (:3 (.735)A (“265)”-A
A=0
Thus, for Problem 3 one has
(2?) WH(T') a .005 h (OT') + .052 h (T') +
.222 h (2T') + .021 h (3T') + .300 h
(hT').
For example, when T' a 70,
wH(70) = ).005) (o) + (.052) (.271) +
(.222) (.3h6) + (.h21) (.850) + (.300)
(.917) = .787 .
Here, h(0), h(70), h(1h0), h(210), and h(280) are obtained
from Figure n.6, with T' = TL50. The curve of predicted
group solutions under the Hierarchical Model was obtained
by applying equation (2?) at each time T'.
Similarly, for Problem 5, with 3 = .503, h obtained
from Figure 0.7, and T' a T200,
(28) wH(T') = .063 h (OT') + .250 h (T') +

.375 h (2T‘) + .250 h (BT') + .063 h

(“T') ;
and for Problem 8, with a a .h36, h from Figure h.8, and
T' = T590,
(29) . wH(T') = .099 h (OT') + .308 h (T') +

.366 h (2T') + .192 h (3T') + .037 h
(hT') .
The obtained cumulative relative frequency distri-

butions of group solutions appear in Figures b.12, h.13,

 

 

 

 

 

81.

and 0.10 for Problems 3, 5, and 8 respectively. The
obtained distributions are solid lines, the predictions
from the Hierarchical Model are dotted lines. Large
discrepancies between the observed and predicted pro-
babilities are evident. Kolmogorov-Smirnov one-sample,
goodness of fit tests are summarized on Table 4.8.

The prediction was statistically tenable for Problem
3, but could be rejected confidently for Problems 5 and 8.
These results indicate the Hierarchical Model to be
unsuitable--at least with regard to Problem 5 and Prob—
lem 8. A comparison of Figures 0.9, 4.10, b.11, with
b.12, b.13, 0.10 suggests that the predictions under
the Hierarchical Model are close to, and about as good

as, those attained with the Large-Solomon Model A (k-l).

The Equalitarian Model

The Equalitarian Model says that non-solvers
consume their share of the group's working time, though
they do not contribute to the solution. That is, A
solvers work on the problem; B non-solvers take their
share of the group's time. Consequently, if A solvers
are working, A/(A + B) is the rate at which they work.

The earlier statement of the Equalitarian Model

(equation 23) may be rewritten as

u
' = A A2 T'3A9k)
(30) WE(T ) Ago [PM ) géTB J

 

 

 

 

|
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85.

Table h.8 Kolmogorov-Smirnov one—sample, goodness-
of-fit tests for observed and predicted
group solutions under the Hierarchical

 

 

 

 

MOdel 0
Problem No. of D Sig. Values of D
Cases (Maximuml theoret.- for 25 df:
obs.l) .05 level .01 level
3 22 .ZHZ .270 .320
5 22 .371 .270 .320

 

 

 

 

 

 

 

86.

which is convenient for computations. Again, predictions

were made from the obtained distribution of individual
solvers, hi(T'), rather than the fitted gamma distribu-

tion, g(T';A~,k). The predictions are

4 2
(31) WE‘T" = Z} [?r(A) h( A T'fl
A=0

K_———
Sample computations for the group predictions under the
Equalitarian Model are given below. As in the case of
the Hierarchical Model, one has for Problem 3

#
mm = z (i) (.735)A 0250‘” .

A:
and thus
(32) wE(T') . .005 h (OT') + .052 h (.25T')
+ .222 h (T') + .021 h (2.25T') + .300
h (uT').
For example, when T' . 7o
WE(70) = (.005) ()) + (.052) (.038) +
(.222) (.271) + (.021) (.737) + (.300)
(.917) . .607.
The curve of predicted group solutions under the
Equalitarian Model was obtained by applying equation
(33) at each time, T'.
Similarly, for Problem 5 with h from Figure 0.7,
(33) WE(T') a .063 h (OT') + .250 h (.25T‘)
+ .375 h (T') + .250 h (2.250') + .063

h (uT'),

 

 

 

 

 

 

 

8?.
and for Problem 8, with h from Figure “.8,
(31+) wEW) - .099 h (01") + .308 h (.25T')
+ .366 h (T‘) + .192 h (2.251“ + .037
h (IIT') .

Predictions from the Equalitarian Model were com-
pared with the data from problem solving groups. The
predicted distributions of group solutions appear in
Figures b.12, b.13, and b.1h as broken lines. On all
three problems the corre5pondance between prediction
and observation is striking; the predicted distribu-
tions of group solution times are very close to the
observed distributions of group solution times.
Kolmogorov-Smirnov goodness-of—fit tests provide no
evidence against the hypothesis that the sample obser-
vations have been drawn from a theoretical population
having the predicted distribution. See Table h.9 for
a summary of these significance tests.

These results agree with the proposition that non-
solvers, those members of a group who are for some
reason on the wrong track, take a full part in group
discussion and consume their share of the group's time.
The successful Equalitarian Model has a variety of
implications about group structure. (a) All members
of the group contribute about equally to the discussion,
whether they are helpful or not. (b) The group does not

form a hierarchical structure with a particular member

 

 

 

 

III- . I h

 

 

Table h.9

88.

Kolmogorov-Smirnov one sample, goodness-
of-fit tests for observed and predicted
group solutions under the Equalitarian
Model.

 

Problem

D
No. of (maximum ltheoret.- Sig. Values of
Cases obs. I) D for 25 df:

.05 level .01 level

 

22 .159 .270 .320
22 .138 .270 .320
22 .093 .270 .320

 

 

 

 

 

 

 

89.
or members taking over, but is equalitarian. (c) The
superiority of groups which solve over groups which
fail to solve is explained merely on the basis of acci-
dents of individual successes, not on the basis of the
evolution of superior social organization.

The big question is whether non-solvers consume as
much time as solvers in the group discussion. Even
though the group process was observed in some detail,
this question remains unanswered because the observer
was unable to decide who the solvers and who the non-
solvers were. A variety of other observations were
made successfully, however, and can be used to test

some of the consequences of the mathematical model.
EMERGENT SOCIAL STRUCTURE

The Equalitarian Model implies that the group
members equally share the group's working time. In
structural terms, the implication is that an equali-
tarian organization should emerge during the social
interaction attending the attempt to solve the prob-
lem. The companion-choice data, gathered for each
group after each problem were cast in the form of
square matrices with binary entries—~a 1 represented a
choice and a 0 absence of choice. These h X b arrays
are called "sociomatrices" by Glanzer and Glaser (1959).

The three sets of post-problem choices generated three

 

 

 

 

 

 

-..—... _..._. .- ... ..-—..—

 

 

 

90.
sociomatrices for each group. A typical sociomatrix
is given in Figure b.15. Rows represent "choosers"
and columns represent those chosen.

Each column of the sociomatrix was summed to give
the total choices received by each group member. The
analysis proceeded on the column totals. An estimate
of the uncertainty, H, of each set of column totals
was computed. Uncertainty, in information theory, is
defined by the formula:

r
(35) H :3 --Z Pi logz P
1-1

1 i
where an event has r possible outcomes, the i-th of
which has a probability Pi’ and i = 1, ..., r. (Miller,
1955). The sample estimate, Q, may be written as
r
(36) Q = -.Z Pi 10g2 pi ,
i=1
where r is group size (0 in this case) and Pi is the
obtained relative frequency of the i-th column.
Uncertainty is at a maximum, 2.0 when the four cate-
gories are equiprobable (Pi = .250). Minimum uncer-
tainty is, of course, H = 0.0. Thus for the present
case 0.0<H<2.0.
Generally Speaking, the estimates H are biased.
For example, suppose that the true probabilities
P1 = P2 = P3 = Pu = .25, hence the true H = 2. If a

small sample of observations are recorded, the values

, and

3

of the sample relative frequencies, p1, p2, p

 

 

 

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92.
Ph make up a multinomial distribution. In general, the
sample relative frequencies will be unequal. These
random inequalities of the p's will always reduce a,
with the result that all values of a are less than or
equal to the true H = 2, and the mean E(§)< H. With
samples as small as those in the choice matrices
studied here, the bias is considerable.

The four categories of the sociomatrix were con-
sidered equally probable under the hypothesis of an
equalitarian choice structure. Given this hypothesis
it was possible to ascertain the expected value of the
estimated uncertainty for a particular grand total of
choices, M, in a sociomatrix and a particular number
of categories, r - h. Making use of tables prepared
by Rogers and Green (1955), theE(fi) was compared with
g computed for the sociomatrix of each group, where
E(E) is the first moment of sample information
(uncertainty) and g is the sample information as com-
puted from (36).

Like the more traditional indices of group
structure, the sampling distribution of Q is unknown.
Consequently, it was not possible to perform a signi-
ficance test in connection with the difference
(fi-E(%)), noted for a particular matrix. (Unlike
most indices of group structure, however, it was at

least possible to decide what value of uncertainty

 

 

 

 

T Iii.-- 4

 

 

93. :5‘
might be expected for a given number of choices and a t
given number of equiprobably categories). In the
absence of a suitable probabilistic statement, the
comparisons of a and E(fi) were scrutinized and the
general correspondence was deemed satisfactory.

However, interest was mainly in the central

tendency that characterized the structural properties
of the eXperimental groups. Consequently, the mean

A A
of the nvarious values of H was computed (i.e., H =

 

(1/n) i§;.1111)’ and the mean of the nvarious values of
E(H) as well (i.e., E(H) = (1/n) 2 E(H)i ). These
data are summarized in Table h.10:1 Inspection of this
table reveals rather striking similarities between the
mean uncertainty and the mean of the uncertainties ex-
pected for each group under the equalitarian hypo-
thesis. In fact, t-tests (solvers only) performed on
the difference (g -E(fi) ) for each of the three prob-
lems were not significant.

The inclusion of non—solving groups produced
almost no change in the mean uncertainties. In fact
the means of the non-solving groups behave in about the
same way as the solving groups. Although the mean un-
certainties of the non-solving groups are somewhat
lower than those of the solving groups (on both prob-
lems 5 and 8), the differences do not appear so great

as to lead one to speculate that non-solving groups

 

 

 

 

 

 

 

 

 

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9h.

developed a different structure.

Additional support for the hypothesis of equali—
tarian structure was developed by considering the
possible consequences should a non-equalitarian struc-
ture emerge from the problem-solving interaction. An
equalitarian structure might emerge when the problem
has a larger number of stages; more stages would, pre-
sumably, require more and diverse contributions from
the members. Assume further that these talkers or
contributors were recognized by others in the group.
Important talkers or contributors would be reflected
in the companion-choice structure; this structure
would become more equalitarian as the number of stages
increased.

To summarize, assume that successful problem
solvers 1) talk more and 2) are recognized by others
and become "leaders" or "stars" of the group. Both
assumptions are, of course, denied by the Equalitarian
Model. Inspection of the mean expectancies in Table
“.10, which are very similar for each of the three
problems, does not suggest the emergence of a non-
equalitarian structure as the number of stages in-
creases. In the analysis of variance, reported in
Table b.11, the mean square for problems was not Signi-
ficant. In this analysis (performed on groups which

solved all three problems), the criterion variable

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

95.
X
Table 4.10 Mean uncertainty, H, of member companion
choices--means and standard deviations
for problems 3, 5, and 8.
Solvers Solvers & Non-Solvers
Problem n H H E(H) n H H E(H)
3 (Rope) 21 1.609h .3505 1.5U82 22 1.6178 .thh 1.5h8?
5(Doub1e)17 1.55h2 .3991 1.5720 22 1.5b17 .h0h2 1.5719
8 (Gold 17 1.hu03 .h628 1.5h57 22 1.u161 .u633 1.5388
Dust)
Non-Solvers
3 (Rope) 1 --- --- -—-

5(Doub1e)

8 (Gold
Dust)

5 1.b992 .5211 1.5718
5 1.33U0 .5092 1.5152

 

 

-A

Tests of the hypothesis the population value is E(H) for
solving groups.

 

 

Problem t df Probability
3 (Rope) 0.800 20 <.50
5 (Double) -0.18h 16 <.50
8 (Gold Dust) -o.939 16 <1uo

 

 

 

 

 

 

 

 

 

 

 

96.
was sample uncertainty, computed from the choice
matrices, and the results are in accord with the equali-
tarian hypothesis.

Communication structure was analyzed in similar
fashion to provide support for the dispersion assump-
tion of the Equalitarian Model. As expected, a group
member addressed most of his communications to the
group at large rather than another member. It was
possible to assign a set of talking frequencies,
weighted for length of communication, to each group
member. A communication structure was obtained for
each group in this way. Each group's communication
structure on each problem was summarized by computing
the uncertainty, H, for each set of four frequencies.
Uncertainty was computed from equation (36), as before,
after the frequencies had been converted to relative
frequencies by dividing them by the total communica-
tions of all group members.

The equalitarian structure preposition may be
restated in terms of intermember communication; un-
certainty approaches 2.0 when members talked equally
often, equally long. Conversely, if one member
monopolizes the conversation, $.15 near 0.0.

As before, the first moment of the sampling dis-
tribution of H was approximated for each set of r a 4

equiprobable categories and the group total of M

 

 

 

 

 

 

 

 

97.

Table h.ll Analysis of variance where criterion
variable is sample uncertainty, com-
puted from the sociomatrices of choices.

 

 

 

 

Sum of dfi Mean Square F
Source Squares
Groups 2.u8u9 13 .1912 1.996 N.S.
Problems .1185 2 .0593 .-
Error 2.U902 26 .0958
Total 5.0936 #1

 

 

 

Analysis of variance where criterion variable is .
sample communication uncertainty. '

 

 

 

Sum of df Mean Square F
Source Squares .
Groups .b85h 13 .0373 1.719 N.S.
Problems .0515 2 .0258 1.189 N.S. ‘
Error .5653 26 .0217 i

 

Total 1.1022 #1 !

 

 

 

98.
communications. The Rogers and Green (1955) tables
A
for E(H) do not extend beyond the case where M = 10.

It was, therefore, necessary to make use of the Miller-

 

Madow approximation presented by Rogers and Green. The
formula was as follows:

A
(37) E(H) = log2 r - (10g. e) (r2 + 6M(r-1)-l), M>r .

2
12M

 

Again, ignorance of the sampling distribution of
H precluded testing the difference, H-E(H) for signi-
ficance in each of the experimental groups. The
difference of central interest, however, was g - H1H).
Table h.l summarizes these data for communication
structure. Significant differences (t-tests) were

found on each of the three problems between the

 

X
observed mean uncertainty of solving groups, H, and
the hypothesized value. Unfortunately, it was not
possible, experimentally, to distinguish between sol-

vers and non-solvers and one may only conclude that

 

some people talked more than others. This by no
means controverts the idea that non-solvers talk as
much as solvers; the result is neutral on that issue.

However, the (in fact) quite high estimates of H for

communication indicate that contributions were actually
widely spread. Tb make a more complete statement one

would have to know something about individual dif-

ferences. Concentrate for the moment on the absolute

 

 

 

 

 

99.

X
Uncertainty, H, of member communications--

 

 

 

 

Table n.12
means and standard deviations for Problems
3, 5, and 8.
Solvers Solvers & Non-SolverE—"_
Problem X A fl’A X A __A
N H sH E(H) N H sH E(H)
3 (Rope) 21 1.8085 .1601 1.9945 22 1.81#5 .1586 1.99h6
5 (Double) 17 1.7951 .1755 1.99h9 22 1.813h .177u 1.9952
8 (Gold 17 1.78%8 .1585 1.9956 22 1.771# .1685 1.9957
Dust)
Non-Solvers
3 (Rope) 1 ---- --- ---
5 (Double) 5 1.8755 .1888 1.9965
8 (Gold 5 1.7256 .2133 1.9962
Dust)

 

 

’A
Test of the hypothesis the population value is E(H) for
solving groups.

 

 

 

 

Problem t df Probability
3 (Rope) -5.296 20 .001
5 (Double) -u.69u 16 .001
8 (Gold Dust) -5 .usu 16 .001

 

 

 

 

 

100.

value of mean H; if all four subjects talk with equal
frequency, H - 2.0. Note that if three-and one-half
subjects talked with equal frequency, one would have
the uncertainty 1.80 which is about the g (communica-
tion) observed on each of the three problems. This
is 21232 to no organization, regardless of the de-
cision required by the statistical test.

As before, (p. 81, paragraphs 1 and 2) compari-
sons among the mean communication uncertainties,‘H,
on the three problems revealed them to be quite similar.
In the analysis of variance reported in the bottom half
of Table b.11, the mean square for problems was not
significant. In contrast to the just preceding, inde-
cisive analysis of communication structure, the
analysis of variance was interpreted as offering

further indirect support of the equalitarian structure

prOposition.

SUMMARY OF RESULTS

The major points of this chapter may be summarized

as follows:

1) It was found that word problems could be classi-
fied, without resorting to subjective decisions, by
means of rational curve-fitting in accord with the
mathematical model. Gamma distributions were success-

fully fitted to distributions of solution times.

 

 

 

 

101.
Estimates of the number of stages in each problem were
close to judgments of number of stages obtained in
subsidiary observations.

2) A mathematical model was developed for pre-
dicting the distributions of group solution times from
the distributions of individual solution-times; the
theoretical predictions (the Equalitarian Model) were
very close to the obtained data on all three of the
problems employed.

3) Observations of the groups' social inter-
action and emergent structure were found to be in
general accord with subsidiary assumptions of the
successful Equalitarian Model.

0) Group performance was inferior to individual
performance when the latter was pooled according to the

non-interactional model of Lorge and Solomon.

 

 

 

 

 

CHAPTER V

DISCUSSION AND CONCLUSIONS

Suppose that the mean times to completion of two

processes are equal at, say, 10 minutes. Imagine the

first process yields highly variable data with a
standard deviation of 10 minutes and a J-shaped distri-
bution. This would be interpreted as a one-stage pro-

cess in which the probability of the event is very low,

the rate parameter being 0.1 per minute. Suppose that

the other process has a mean of 10 minutes but much
smaller variance, having a standard deviation equal to
1 minute and a more nearly symmetrical, bell-shaped
distribution. One would interpret this as a loo-stage

process, each stage of which has a rate (probability

per unit time) 100 times greater than in the first pro-
cess. Thus, while solving a eureka problem and reading
an essay may take the same (average) time, the problem
solving process, having one or two lowhprobability
stages, would yield very variable data; and the reading,
being made up of a great many high-probability stages,
would yield much less variable data. These general
relationships seem intuitively compelling, and the

waiting-time theory used in this study is merely a

mathematical idealization of this line of argument.

102.

 

 

 

 

 

 

 

103.

CLASSIFICATION MODEL

The mathematical model takes advantage of a great
deal of information; the distribution of individual
solution times as well as the proportion of right and
wrong answers. The theoretical model begins with simple
assumptions; that the probability of solving a stage is
a constant over time until solution, and that the times
to solve the several stages of a problem are independent
and have equal means. From these assumptions, and the
concept of continuous time, precise logical argument
leads to the gamma distribution. The derivation is, in
fact, a fairly routine application of the methods of
probability theory.

This model had two empirical hurdles to surmount.
First, it claimed that times to solution would have a
gamma distribution. The obtained distributions were
close to the fitted gamma and did not differ in any
significant way. Second, it was possible to estimate
k, the number of equally-difficult stages, for each of
the three problems. The writer's intention and
intuition was that the three problems differ in number

of stages, and have approximately 1, 2 or 3, and 3 or
b stages, respectively. Miami University students
judged that the problems would have about 3, b, and 6
stages. The best estimates from the data are 1.3,

3.0, and 5.0, which progress in the direction intended

 

 

 

 

 

 

10h.
and are of about the magnitude expected. Thus, the
estimates of k from the data (depending on the squared
mean divided by the variance) have some apparent vali-
dity. Only further investigations of stages in prob-
lem-solving will determine whether this is an accident.
At present, except for an intuitive judgment of the
number of distinct "ideas" required to obtain the
solution, there is no adequate description of the
stages involved.

The number of stages appears to be the number of
distinct ideas which must be conjoined to obtain the
correct answer. The subject must have idea 1 Egg idea
2 and so forth, if he is to complete stage 1 and stage
2, etc. If there are several answers to the problem,
so that the subject may have idea 1 g; idea 2 23 ...
and so forth, this should make the problem easier,
increasing the rate A‘. This disjunction might very
well correspond to a single stage. Thus, to some
degree, the notion of stages is related to the logical
requirements of the problem, at least to the logical
interrelationships of the ideas involved. This does
not yield a complete description of the hypothetical
process, however. Given any set of ideas one could
specify the various compounds which would solve the
problem. However, we do not know for sure what ideas

the subjects entertained, so that we know the logical

 

 

 

 

 

 

 

 

 

 

105.
structure and not the elements embedded in that struc-
ture. Only further empirical separation of stages can
fulfill the final requirement.

While the concept of "number of stages," k, has
some value in beginning an analysis of problem—solving,
it should be emphasized that the number of stages esti-
mated by the present method is only an approximation.
One would expect that in most problems with several
stages, the stages differ in difficulty and have dif-
ferent values of)‘. Given any set of stages and their
)k-values one can construct the distribution of total
times. However, any empirically-determined distribu-
tion of total times can be closely approximated by
various combinations of stages and..A-values. The only
workable unigue answer to the question, How many stages
does this problem involve? is attained by first assum-
ing that all the component stages have equal values of

)\. The resulting estimate is something like the
information-theory idea that the information transmitted
through a given system, if it is 2 bits, is "equivalent"
to a system which can discriminate exactly four equi- .
probable categories. The system may actually be dis-
criminating nine unequally probable categories, and the
fact that information transmitted equals 2 bits does
not reveal the fine grain. In a similar way, the esti-

mate of 3 stages says that the subject's performance is

 

 

 

 

 

 

 

 

 

 

 

106.
"equivalent" to three equiprobable stages, though it
might actually consist of 5 or 6 stages which differ
widely in probability or rate,;‘.

Experimental separation of stages is needed before
definite conclusions can be drawn. A first step in
that direction was taken in the followbup study, in
which reading-times were estimated and subtracted from
problempsolving data. Intuitively, reading the problem
is a multi-stage process in which each stage has a high
value of A . Compared with problem-solving, reading
the problem consumes a relatively constant amount of
time. Similarly, it may be presumed that time to write
the solution is relatively constant, and since it can
easily be measured it should be separated from pure
problem-solving time. Possibly, by the use of proto-
cols, it is possible to determine when the subject
overcomes various stages of the problem, such as
"understanding" the problem, obtaining an hypothesis,
etc. Such logical or psychological analysis may lead
to an understanding of the stages of problempsolving,
but each such analysis should be adepted with caution.
The data of this study suggest that some problems in-
volve compounds of "part problems", and that each
part (even though it may involve formulation of the
problem, discovery of an hypothesis, devising a test

of the hypothesis, etc.) is mastered in a single, all-

 

 

—_.-..—.-. -w‘mmJ- _ ~ -

 

 

 

 

 

 

 

107.

or—nothing stage. This view is supported by the close
correlation of estimated number of stages with what
appear to be the number of parts into which the problem
can be divided; particularly, it is interesting to note
that the gold-dust problem has answers which involve
about six measurements, and is estimated to involve
about six stages.

The method of classifying problems according to
the number of stages involved can, in principle, be
applied to other than Eureka problems. For example,

a geometrical proof might involve roughly as many
stages as there are steps in the proof. A cross-word
puzzle might involve as many stages as there are diffi-
cult words to discover. A task which yields relatively
invariable times, such as manipulating a large number
of blocks in a routine way, would presumably have a
very large number of stages. One way of investigating
the present model is by studying tasks which, prima
facie, differ widely in number of stages.

This discussion has resolutely used the phrase
"number of stages" in place of the more natural ad-
jective, "complexity." In the model, the difficulty
of a problem (that is, for example, mean time to
solution) depends upon both k, the number of stages,
and A(, the probability of accomplishing a given stage

per unit time. The mean is, in fact, k(1/)\) . The

 

 

 

108.

term "complexity" refers to both of these factors, for
a problem may be complex because it involves many
stages to completion, or complex because, at the one
stage required, there are many alternative possibilities.
The number of wrong alternatives does not affect k, but
when there are many alternatives the probability of the
correct one is presumably low, hence4Kwould be low.

Though the present data give no such indication, it
is plausible to suppose that the emergent structure of a
permanent problem-solving group might be affected by the
kind of problems they work on. If each problem is main-
ly of a single stage, the group might develop a
hierarchical structure, attempting to let the best prob-
lem-solver remain in control. Multi-stage problems, on
the other hand, would tend to favor an equalitarian
structure so that various members can contribute to the
solution.

GROUPS VS. INDIVIDUALS

Although the present data are more extensive, and
analyzed in more detail, than previous results, the
general relationship between individual and group per-
formance agrees well with earlier findings. Groups
outperformed individuals, as has been almost universally
true in earlier studies. Furthermore, the groups were

less efficient than would be expected by pooling the

 

 

 

 

 

109.
abilities or accomplishments of individuals. The
observation that actual groups lie between individuals
alone and pooled individuals agrees with results of
Husband (19h0) and Taylor and Faust (1952), and the
observation of Watson (1928) though Faust (1959) found
a slight superiority of real groups over pooled indi-
viduals. The present data indicate that real groups,
while as £233 as pooled individuals, are comparatively
slow. In Figs. n.9, “.10, and b.11, in which group
data are compared with Large-Solomon predictions, the
largest discrepancy is at some intermediate time,
roughly at the mean time-to—solution. With small
numbers of groups it is possible that various results
would occur depending on the total time given the sub-
jects; the usual procedure gives only one point of the
cumulative distribution.

In the present study, groups failed to reach the
level predicted by the Lorge-Solomon pooling model, or
by the Hierarchical Model which is, in effect, the
continuous analog of the pooling model. The interpre-
tation offered, which is embedded in the assumptions
of the Equalitarian Model, is that group members who
are on the wrong track consume as much group time as do
members who are approaching solution. The pooling
models pool the abilities or accomplishments but do not

pool the disabilities and errors of the members in the

 

 

 

 

 

 

 

 

 

 

 

 

110.
group. That is, the Large-Solomon model combines the
wheat and leaves behind the chaff. The Equalitarian
model which fits the present data, pools both wheat and
chaff.

In a sense, the Large-Solomon model is impossible,
for until the problem is solved no member of the group
knows who is on the right and who on the wrong track.
Before solution there is no possible mechanism within
the group for singling out the wheat from the chaff.
The Large-Solomon model, and the Hierarchical Medel
formulated here, both set an unrealistic standard for
groups.

The data and the Equalitarian Model indicate that,
in terms of man-hours, it is cheaper to have individuals
solve problems separately than to use groups. Four indi-
viduals in separate rooms will, on the average, solve a
problem sooner than the same four individuals in a face
to face group. On this criterion, groups are clearly
inefficient problem-solvers. Yet this conclusion is
seriously misleading.

First, the Equalitarian Model differs from the
Hierarchical Model only when the probability of solu-
tion, a, is less than one. If all subjects would solve
the problem, there would be no difference between the
models. What is more, in that case a group of size

r would have a rate of r,&_, if individuals have a rate

 

 

 

 

 

 

 

 

111.
of A; whence groups would be precisely as efficient as
individuals. It is only the probability of an indivi-
dual failing to solve the problem, which slows down
the group average. In the case of individual failures,
the group has an advantage over individuals quite aside
from Speedp-for the group yields a single answer, and
the individuals working separately may offer several
different answers. It seems improbable that a group of
size h, for example, could attain agreement on a wrong

(answer in problems like those use in this study. Hence

 

it is very probable that the group will yield the
correct answer. Individuals, on the same problem, may

have a considerable probability of giving a wrong

 

 

answer.

In an experiment, one tends to ignore the indivi-
dual failures for the experimenter knows the right
answer to the problem. In any real life application,
there would be no exPerimenter with the correct answer
written down. If there were, there would be no need
for either the individuals in separate rooms or the
group.

Though the individuals will, on the average,
obtain a correct answer before a group, they may also
obtain some wrong answers. Working separately, four
individuals would not separate the correct from the

wrong answers, and as a result the first individual

 

 

 

112.
solution (which on the average will be before the
group solution) cannot be accepted with any confidence.
The group solution has a probability 1 - (l-a)r of
being correct, when 3 is the probability that an indi-
vidual is correct and the group is of size r. The
first individual solution has only probability g of
being correct. In this sense the group is slower but
surer.

It should be remarked that the present model makes
sense only if the problem is one of searching for a
unique correct answer. In a "human relations" problem
(Large, et a1.,l958) the task may be to assemble a

"answer 0 "

large number of ideas none of which is the
In such a task, mutual stimulation may be an important
advantage, whereas the drawback of groups, that non-
solvers take up as much time as solvers, may not be
found if virtually all ideas are accepted as valuable.

Groups may have advantages, in real life, which

are not reflected in the experimental conditions

 

studied here. For example, it is possible that a group
will persevere longer on a problem than the individuals
would separately, thus enhancing the probability of an

eventual solution. Work in a group may be conducive to

higher or more appropriate motivation than work alone.

 

The exigencies of group management may lead to such

benefits as regular work hours, orderly arrangement of

   

 

 

 

 

113.
materials, etc., which would give group workers an
advantage over individuals. By the same reasoning, of
course, members of groups may have the disadvantage of
working at other than their own best time, of strugg-
ling with an inappropriate organization of material,
etc. Such factors are probably more characteristic of
stable than of the ad hoc groups discussed in this
study, and there is no reason to believe that the simple
models employed here would apply to very different
situations with new properties.

The observations of groups in this study agreed
quite well with the supposition that they were
organized in an equalitarian structure. The success
of the Equalitarian Model suggests that, at any rate,
members who are on the wrong track contribute their
share to the discussions. The data on choice of
future partners failed to reveal any noticeable trace
of stable leadership within groups. An equalitarian
structure is, as was discussed above, relatively slow
and sure for it devotes time to members who contribute
only confusion, but at the same time it provides a
check against error. An hierarchical structure should,
according to the present analysis, increase Speed but
also increase the probability of accepting a wrong
solution. The conditions under which a hierarchy would

arise may be at least imagined; if the group were under

 

 

 

 
 
  
    

H . --..___-HH...H ,

.. .-.-—. ..

H...__
-—.....

 

 

 

 

 

llh.
extreme time pressure and could not afford to "suffer
fools gladly," it might be induced to adopt a hierarchi-
cal structure. The experimental conditions employed
here, giving enough time for almost all individuals to
solve the problems or arrive at a satisfying error, are
clearly not those which would most probably lead to a

hierarchy.
THE EFFECTS OF INDIVIDUAL DIFFERENCES

If there are large and stable individual dif-
ferences between subjects in their abilities to solve
these problems, then the number of stages in the prob-
lems is underestimated. Individual differences would
inflate the variances of time scores, and since k is
estimated by the squared.mean divided by the variance,
this would decrease the estimate of k below what it
should be.

Large and stable differences should, however, pro-
duce discernable hierarchical structures in groups, and
should lead groups to prefer to choose one bright
member over the others. The lack of any such hierarchy
suggests that any differences between subjects were
imperceptible to the subjects themselves.

Furthermore, an investigation was made of solvers
and non-solvers among the individuals working on the

three problems. While the form of the data.makes it

 

...- ,.k__~_‘_,=_‘___n_.__

 

 

 

115.
difficult to perform a suitable statistical test, the
indication was that there was little correlation
between success on the three problems. This is not
unexpected when it is noted that the subjects were all
drawn from an elementary psychology course and were
thus of a restricted range of talent, and when it is
recalled that the present three-problem experiment
would constitute a very short and unreliable intelli-

gence test.

CONCLUSIONS

 

It was found that the distributions of solution
times on three problems were adequately described by
gamma distributions. Gamma distributions arise from
a rational consideration of the stochastic theory of E
waiting times. Estimates of the number of equally-

I

I

difficult stages involved in the three problems, 1
l

 

agreed in substance with the intentions of the experi-
menter and with judgments by a group of students.
Furthermore, inspection of the logical structure of
the problems suggested that the number of stages is
close to the number of ideas which must be conjoined E'
to arrive at the solution. §
Groups of four subjects solved a significantly
higher proportion of problems than did individuals.

However, groups which solved took about as long as ‘

 

 

 

 

 

 

116.
individuals who solved. It was found that group
performance could‘ggt be predicted accurately from
individual performance on the assumption that
accomplishments were simply pooled. If, however, it
is also assumed that group members on the wrong track
consume their share of group time, so that both
accomplishments and failures of the individuals are
pooled, the entire distribution of group solution-times
could be predicted accurately.

The Equalitarian Model, which accurately predicted
group performance, led to the expectation that there
would be no hierarchical structures within groups.
Sociometric choices made after each problem supported
this hypothesis, since they were in agreement with an

hypothesis of random, equiprobable choice between

group members.

 

 

 

 

 

APPENDIX I

Use of Pearson's Tables of the IncompleteléTunction

In this study P(k) = (k-l) E (M an integer).

Let AT = x. If

(1)

(2)

(3)

(it)

(5)

(6)

Q
P(k) =/e'xxk'1dx

0

(see Mood, 1950), let

x
[fix(k) =Jére-xXk-ldx

The ratio of these two functions is the density

r8x(k) .
r (1:)

Pearson (1922) defines

x
(1(1) + 1) ./ e-xxpdx .
o
In this notation, equation (3) may be rewritten as

F'xho)
F (1))

Pearson defines
1 o
I(x.1:°)==)_nlt(p+ )

(8(p + 1)

Due to computational difficulties, Pearson tables

not I(x,p) but the auxillary function I(u,p) where

i 1
u = x/(P + 1)2 or x = u(p + l)2

117.

 

 

 

“...—H. -

 

 

118.
If p + l is taken equal to k, then by substitution
u a x/k%.
Furthermore p = k - 1.

Recall x = A.T. Hence

(7) u a x/k% = T

”he >4

Equation (7) takes T into u, and it is the u argument

which has been tabled extensively by Pearson (1922).
However, the transformation to u is facilitated

by noting that k = [E(T)] 2/(y2('r) and A: E(T) /g2(T) .

Substituting these expressions in (7)

 

 

 

' E(T)
(8) u H T O (T_ v) = '72:;-
_J.'.'.. T .
[Em] 2 2 U
L (72(T) 5)

 

 

Using the sample value 5, equation (8) may be written as
u=T/So
The tables are entered with appropriate u and

A
p a k-l, where k is the sample estimate of k.

 

 

APPENDIX 2

Proof that g(T;aA,k) u g((XT;)k,k)

 

The function. Wi(T) =AEE0 [Pr(A) g(T;CX)\,k)1
(generalized form) is not gonvenient for computational
purposes. In the second term of Wi(T) it is more
advantageous for the coefficient(1.to appear with the
variable T rather than the parameter 8.. Therefore, it

is necessary to prove:

(1) g(T:C(/1.k) = you» Am) .
Recall that
(2) g(T; A .k) = _)\_ e'AT(/\T)k-1.
(k-l) 2
Thus, AK
-0( T _
(3) emalm = 0:3: : e (dink 1
and
on you; 1,1.) = 31 ..‘QATm/mk'l,

In order to abbreviate notation, let G(T) = c(T;O(A,k)
be the cumulative distribution of g(T3CXal,k) and H(T)

be the cumulative distribution of g(o('r; )L ,k) e q(T).

It is sufficient to prove that G(T) a H(T).

It follows that

(5) d G(T) -O(. (T) d d HT = (T).
dT q an 'dIQ'T) q

Let CIT a z; then

119.

(t
)1
).
(‘2

   

 

 

 

120.

(6) a G(TL=O(fd Hum] .
dT [dz

Multiplying both sides of (6) by l/CK ,

(7) (1/00 [3T G(T) ]= d Hm

dz

Obviously G(T) = H(T) if d G(T) = d H(T), since
d d

G(O) a H(O) a O and G(ao) a H(°°) ... 1.

Now one may write

(8) d H(T) e pg, H(r)_ . dz ,
dT dz dT

but from (7)

d H(T) e <1/OL)[d G(T) ]

 

 

 

dz l_dT
Hence,
(9) d H(T) = (1/O()[d 6(1)] . 2.2.
dT l_dT dT
Since 2 =C(T, it follows that
(10) 2.2. = O( -
dT

Substituting (10) in (9). one has

 

dT dT

(11) d H(r) a (l/Cx) [ d G(T)]CX a d G(T) . Q.E.D.
dT

 

 

 

 

 

 

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l.
i
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. . .-
‘ . u . . .
. .
. . . . .
. n a s u q . v
s n
e . a. u I u u
n c I _. . x .
s . . . . n
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n u . 4 c w
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4 n
n .
u
v . .
.
. c n V n .
. n a
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c . . . a .
u .. . .
n
O s I . .
' ~
n . t
»
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