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Journal of Personality and Social Psychology
Copyright 2006 by the American Psychological Association
2006, Vol. 90, No. 4, 644 – 651
0022-3514/06/$12.00
DOI: 10.1037/0022-3514.90.4.644
Groups Perform Better Than the Best Individuals on Letters-to-Numbers
Problems: Effects of Group Size
Patrick R. Laughlin, Erin C. Hatch, Jonathan S. Silver, and Lee Boh
University of Illinois at Urbana–Champaign
Individuals and groups of 2, 3, 4, or 5 people solved 2 letters-to-numbers problems that required
participants, on each trial, to identify the coding of 10 letters to 10 numbers by proposing an equation in
letters, receiving the answer in letters, proposing a hypothesis, and receiving feedback on the correctness
of the hypothesis. Groups of 3, 4, and 5 people proposed more complex equations and had fewer trials
to solution than the best of an equivalent number of individuals. Groups of 3, 4, and 5 people had fewer
trials to solution than 2-person groups but did not differ from each other. These results suggest that
3-person groups are necessary and sufficient to perform better than the best individuals on highly
intellective problems.
Keywords: group problem solving, group size, intellective tasks
Cooperative groups perform better than independent individuals
al. (2002, 2003).We then review the surprisingly small amount of
on a wide range of problems (for representative reviews, see
previous research on the effects of group size in problem solving.
Hastie, 1986; Hill, 1982; Kerr & Tindale, 2004; Levine & More-
From these considerations, we predicted (a) better performance for
land, 1998). This research has traditionally compared an equal
groups of each of size two, three, four, and five than an equivalent
number of groups and individuals (e.g., 20 four-person groups with
number of individuals and (b) major improvement in performance
20 individuals), thus comparing groups and the average individual.
from group size two to three, with decreasing improvement from
A more stringent test of group versus individual performance is a
group sizes three to four to five.
comparison of n groups of size m with an equivalent number of
n
m individuals (e.g., 20 groups of size four with 80 individu-
Letters-to-Numbers Problems
als). This allows comparison of groups with the best, second best,
and so forth through the mth best of an equivalent number of
Previous to the experiment, we randomly assigned each of the
individuals rather than the usual comparison of groups and the
10 letters A, B, C, D, E, F, G, H, I, and J (without replacement) to
average individual. Extending traditional research to this more
1 of the 10 numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The objective
stringent comparison, two recent experiments reported that four-
for the problem solvers was to identify this mapping of the 10
person groups (Laughlin, Bonner, & Miner, 2002) and three-
letters to the 10 numbers in as few trials as possible. On each trial,
person groups (Laughlin, Zander, Knievel, & Tan, 2003) per-
the problem solvers proposed an equation in letters (e.g., A
D
formed better than the best of an equivalent number of individuals
?). The experimenter then gave the answer to the equation in letters
on a highly intellective task, letters-to-numbers problems.
(e.g., A
D
B). The problem solvers then proposed one specific
Comparisons of the performance of cooperative groups of a
mapping of a letter to a number (e.g., A
3) and were told whether
given size and individuals are a special case of the larger issue of
the hypothesis was true or false (e.g., true, A equals 3; or false, A
the relationship between group size and performance. The current
does not equal 3). The problem solvers then filled out the coding
experiment addressed this larger issue by a comparison of groups
of the 10 letters to the 10 numbers on their answer sheets. The full
of size two, three, four, and five people and the best of an
correct coding solved the problem, whereas an incorrect coding
equivalent number of individuals on letters-to-numbers problems.
required another trial. A maximum of 10 trials was allowed. The
In this article, we first explain letters-to numbers problems, con-
Appendix gives the instructions with four illustrative trials. How
sider possible strategies, and review the experiments of Laughlin et
might these problems be solved? We first consider a simple but
inefficient two-letter substitution strategy. We then consider more
efficient multiletter strategies.
Patrick R. Laughlin, Erin C. Hatch, Jonathan S. Silver, and Lee Boh,
Two-Letter Substitution Strategy
Department of Psychology, University of Illinois at Urbana–Champaign.
Erin C. Hatch is now at The Shepherd’s Table, Silver Spring, Maryland.
Table 1 illustrates a two-letter substitution strategy. Once any
Jonathan S. Silver is now at the College of Education, University of Iowa.
one of the letters is identified by logic (e.g., the two-letter answer
Lee Boh is now at the 3rd Divisional Air Defence Artillery Battalion,
to the equation A
J
ED means that the first letter, E, must be
Republic of Singapore Airforce, Singapore.
1) or experimenter feedback on the hypothesis (e.g., true, E is 1),
Correspondence concerning this article should be addressed to Patrick
the letter may be used in a series of two-letter equations to identify
R. Laughlin, Department of Psychology, University of Illinois at Urbana–
Champaign, 603 East Daniel Street, Champaign, IL 61820. E-mail:
the other letters. Nine of the letters have been identified by Trial 9,
plaughli@uiuc.edu
and the remaining letter follows by exclusion. Hence, the problem
644
GROUPS PERFORM BETTER THAN THE BEST INDIVIDUALS
645
Table 1
Groups Versus the Best Individuals on Letters-to-
Two-Letter Substitution Strategy
Numbers Problems
Trial
Equation
Hypothesis
Feedback
Laughlin et al. (2002) compared 82 four-person groups and an
equivalent number of 82
4
328 individuals on one letters-to-
1
A
J
ED
E
1
True
numbers problem. The groups proposed more complex equations
2
E
E
D
D
2
True
(letters per equation) and had fewer trials to solution than each of
3
E
D
A
A
3
True
4
E
A
G
G
4
True
the best, second-best, third-best, and fourth-best individuals. In a
5
E
G
B
B
5
True
subsequent experiment, Laughlin et al. (2003) compared 100
6
E
B
F
F
6
True
three-person groups and an equivalent number of 300 individuals
7
E
F
H
H
7
True
on two successive problems under five instruction conditions: (a)
8
E
H
C
C
8
True
9
E
C
J
J
9
True
standard or unconstrained, (b) letter coded as 1 known at outset, (c)
letter coded as 9 known at outset, (d) use at least three letters on
Note.
A
3; B
5; C
8; D
2; E
1; F
6; G
4; H
7; I
all equations, and (e) use at least four letters on all equations. The
0; J
9.
groups proposed more complex equations and had fewer trials to
solution than each of the best, second-best, and third-best individ-
uals. Performance for both groups and individuals was best with
is solved in nine trials. For simplicity, we have illustrated a
the letter coded as 9 known at the outset and instructions to use at
two-letter substitution strategy with the identified letter coded as 1,
least four letters on each equation, and the other three instruction
but a comparable strategy may be used once any of the letters is
conditions did not differ significantly from each other. Because
identified. For convenience of exposition, we subsequently refer to
with the random assignment to group or individual conditions, the
best of four group members would be equivalent to the best of four
the equation on Trial 1 of a given illustration as Equation 1, the
individuals, these results also indicate that the groups performed
equation on Trial 2 as Equation 2, and so on.
better than their best member would have performed alone.
Although a two-letter substitution strategy will generally solve
However, although the groups used more complex equations
the problem in the allowed maximum of 10 trials, it is clearly
and solved in fewer trials than the best individuals, neither exper-
suboptimal. The hypotheses on each trial redundantly test what is
iment reported the direct relation between letters per equation and
known, and there is no substitution of known letters in previous
trials to solution. Accordingly, we regressed trials to solution on
equations to identify further letters (e.g., after the letter E has been
letters per equation in each of these previous experiments. In
identified as 1 on Equation 1, the letter D has been identified as 2
Laughlin et al. (2002), there was an R of .41 ( p
.001) for the
on Equation 2, and the letter A has been identified as 3 on Equation
single problem. In Laughlin et al. (2003), there was an R of .39
3, substitution of these three letters in Equation 1 identifies the
( p
.001) for Problem 1 and an R of .44 ( p
.001) for Problem
letter J as 9).
2. This indicates that the use of more complex equations led to
better performance in both experiments.
The current experiment compared groups and an equivalent
Multiletter Strategies
number of individuals at each of group size two, three, four, and
five on two successive letters-to-numbers problems. From the
Table 2 illustrates a multiletter strategy. Rather than simply
results of Laughlin et al. (2002, 2003) and the previous consider-
adding the known E and D to identify A as 3, Equation 3 adds
ations of effective strategies on letters-to-numbers problems, we
combinations of E and D to identify A as 3, G as 4, and B as 5.
predicted that the groups would have fewer trials to solution and
Equation 4 then uses combinations of the known letters D and A to
propose more complex equations (letters per equation) than the
identify F as 6, H as 7, C as 8, and J as 9. The remaining letter I
best of an equivalent number of individuals at each of group size
is known by exclusion. The basic insight is that multiletter equa-
two, three, four, and five.
tions identify more than one letter and are therefore more infor-
mative than two-letter equations.
Group Size and Problem-Solving Performance
Table 3 illustrates a sophisticated multiletter strategy. The sum
of the integers from 1 to N is given by the formula N(N
1)/2.
Does problem-solving performance (a) improve linearly with
Because the letter coded as 0 (here I) has no effect, the answer to
increasing group size; (b) improve up to a point and then level off;
Equation 1 that adds all 10 letters is known in advance to be
9(10)/2
45, identifying G as 4 and B as 5. The letter G is then
Table 2
used in the multiletter Equation 2 to identify E as 1, A as 3, and D
Combined Two-Letter and Multiletter Strategy
as 2. The letters E, D, A, G, and B are then used in the multiletter
Equation 3 to identify F as 6, H as 7, C as 8, and J as 9. The
Trial
Equation
Hypothesis
Feedback
remaining letter, I, is known by exclusion, and the problem is
1
A
J
ED
B
0
False
solved in three trials. In summary, the illustrative strategies of
2
E
E
D
C
0
False
Tables 1, 2, and 3 demonstrate the basic principle that as the
3
EEE
EED
EDD
AGB
F
0
False
number of letters in an equation increases, the equation will
4
DDDA
DDAA
DAAA
FHCJ
I
0
True
identify progressively more letters, and hence the problem will be
Note.
A
3; B
5; C
8; D
2; E
1; F
6; G
4; H
7; I
solved in fewer trials.
0; J
9.
646
LAUGHLIN, HATCH, SILVER, AND BOH
Table 3
Sophisticated Multiletter Strategy
Trial
Equation
Hypothesis
Feedback
1
A
B
C
D
E
F
G
H
I
J
GB
A
0
False
2
GG
GG
GG
EAD
C
0
False
3
EDAG
BBBB
FHCJ
I
0
True
Note.
A
3; B
5; C
8; D
2; E
1; F
6; G
4; H
7; I
0; J
9.
or (c) improve up to a point, level off, and then decrease? Indeed,
In summary, previous research suggests that performance im-
is there any systematic relation between group size and problem-
proves with increasing group size for problems of moderate diffi-
solving performance? Despite the theoretical and practical impor-
culty that require understanding of verbal, quantitative, or logical
tance of the relation between group size and performance, there is
conceptual systems, but performance has not been shown to im-
a surprisingly small amount of previous research.
prove with increasing group size for so-called eureka problems.
Thorndike (1937) had students in intact classes from five uni-
Letters-to-numbers problems require knowledge of arithmetic, al-
versities respond as individuals and then as four-person, five-
gebra, and logic as well as information processing over a series of
person, or six-person groups to factual world knowledge items
trials. The previous studies of Laughlin et al. (2002, 2003) indicate
(e.g., geography, economics, politics) and items with correct an-
that letters-to-numbers problems are challenging but not exces-
swers defined by the judgment of experts (e.g., the better of two
sively difficult for the participants. Thus, previous research sug-
poems, the better of two paintings). Four-person groups were
gests linear improvement with increasing group size in the present
correct on 59% of the items, five-person groups on 61%, and
experiment.
six-person groups on 63%. Taylor and Faust (1952) had individ-
However, we have proposed that the crucial aspect of the
uals, two-person groups, and four-person groups play the parlor
superior performance of groups over individuals is the use of more
game Twenty Questions under the standard procedures (starting
complex, multiletter strategies rather than the simple, obvious, but
from a known category of animal, vegetable, or mineral and asking
less effective two-letter substitution strategy. The 328 individuals
up to a maximum of 20 yes–no questions to identify the object).
in Laughlin et al. (2002) had a probability of .71 of using two-letter
They played five games on each of 3 successive days. Four-person
equations on all trials for their single problem. The 180 individuals
groups solved more of the 15 problems in the allowed 20 questions
over the three instruction conditions—(a) unconstrained, (b) letter
than two-person groups, but there was a nonsignificant difference
coded as 1 given at outset, and (c) letter coded as 9 given at
for the number of questions (e.g., 17) on the problems that were
outset— of Laughlin et al. (2003) had a probability of .67 of using
correctly solved. Lorge and Solomon (1959, 1960) compared
two-letter equations on all trials on their first problem and .65 on
various group sizes from two to seven in intact classes in different
their second problem (the participants in the other two conditions
years on the Tartaglia (husbands and wives) river-crossing prob-
were instructed to use at least three or four letters on all equations).
lem. There was no consistent relationship between percentage of
This gives an overall individual probability of .68 of using two-
solvers and group size. For example, the solution rate was 15% for
letter equations on all trials. If we assume that multiletter equations
three-person groups and 13% for six-person groups in one class
are demonstrably preferable to two-letter equations if proposed by
and 66% for four-person groups and 46% for seven-person groups
at least one group member, the probability of the groups using a
in another class. Similarly, Thomas and Fink (1961) found no
multiletter strategy is 1
.68N (where N
group size). This
significant differences among groups of size two, three, four, and
predicts probabilities of .54, .69, .79, and .85 that groups of sizes
five on the Maier and Solem (1952) horse-trading problem.
two, three, four, and five will use a multiletter strategy. From these
Laughlin, Kerr, Davis, Halff, and Marciniak (1975) first gave
considerations, we predicted a major improvement in trials to
individual college students the 115 vocabulary items of the Ter-
solution from two-person to three-person groups but progressively
man (1956) Concept Mastery Test. After dichotomizing the indi-
decreasing improvement from three-person to four-person to five-
vidual scores at the median into high and low halves, they ran-
person groups.
domly assigned participants within each half to retake the same
items as individuals or in cooperative groups of size 2, 3, 4, or 5.
Method
There was significant linear improvement with increasing group
size for the high-ability groups but no effect of group size for the
Participants and Design
low-ability groups. Bray, Kerr, and Atkin (1978) compared male
and female groups of size 2, 3, 6, and 10 on “gold dust” (modified
The participants were 760 students at the University of Illinois at
Luchins, 1942, water jar) problems of low, medium, or high
Urbana–Champaign who received course credit for participation. Two
difficulty. For problems of low difficulty, there was no difference
hundred participants were randomly assigned to solve two successive
letters-to-numbers problems as individuals, 80 as 40 two-person groups,
among group sizes, probably because of a ceiling effect. For
120 as 40 three-person groups, 160 as 40 four-person groups, and 200 as
problems of moderate difficulty, male groups of size 10 had more
40 five-person groups. There were 20 random codings of the 10 letters to
correct answers than groups of sizes 3 and 6, but female groups did
the 10 numbers. In Replications 1–10 and 21–30, the first 10 of the random
not differ significantly from each other. For problems of high
codings were used for Problem 1 and the second 10 were used for Problem
difficulty, there was no effect of group size, probably because of a
2; in Replications 11–20 and 31– 40, the second 10 codings were used for
floor effect.
Problem 1 and the first 10 were used for Problem 2.
GROUPS PERFORM BETTER THAN THE BEST INDIVIDUALS
647
Instructions and Procedure
viduals, 15.00 for the second best, 16.90 for the third best, 18.58
for the fourth best, and 19.78 for the fifth best. Because the 40
The instructions are given in the Appendix. The experimenter gave these
codings of letters to numbers differed in difficulty, we conducted
instructions orally with the four illustrative trials on a blackboard, answer-
ing any questions on the procedure. In the group conditions, the members
a randomized blocks analysis of variance (ANOVA) on the num-
discussed to consensus on each proposed equation and hypothesis, and
ber of trials to solution, with the 40 replications as a blocking
each member wrote the proposed group equation in letters, answer in
variable. This analysis indicated a significant main effect of the
letters, group hypothesis, and feedback on the hypothesis on his or her
best, second-best, third-best, fourth-best, and fifth-best individuals,
response sheets on each trial. Individuals followed the same procedure
F(4, 156)
156.68, p
.001, MSE
0.9498. Tukey’s compar-
without discussion. This ensured that all proposed equations and answers
isons indicated that all 10 pairwise differences were significant
in letters, hypotheses, and feedback on hypotheses were available to each
( p
.01).
person throughout the problem, reducing demands on memory. All group
members and individuals had additional scratch paper for computations
The 40 five-person groups were compared with the 40 best,
and notes. After the two problems were completed, the experimenter
second-best, third-best, fourth-best, and fifth-best individuals.
explained the purpose of the research to the participants, answered any
Four of the 5 individuals in each replication were randomly se-
questions, gave them a written debriefing with a reference for further
lected, and the 40 four-person groups were compared with the 40
reading, asked them not to discuss the experiment with potential future
best, second best, third best, and fourth best of these 4 individuals.
participants, and thanked them for their participation.
Similarly, 3 of the 5 individuals in each replication were randomly
selected, and the 40 three-person groups were compared with the
Results
40 best, second best, and third best of these 3 individuals. In
We first determined the best, second best, third best, fourth best,
addition, 2 of the 5 individuals in each replication were randomly
and fifth best of the 5 individuals in each of the 40 replications by
selected, and the 40 two-person groups were compared with the 40
the number of trials over the two problems (if 2 individuals had the
best and second best of these 2 individuals.
same number of trials, we randomly assigned them to the appro-
Table 4 gives the means and standard deviations for Problems 1
priate conditions). Nonsolvers in the allotted 10 trials were con-
and 2 for number of trials to solution and letters per equation for
sidered to require 11 trials. Means were 12.98 for the best indi-
the best, second best, third best, fourth best, and fifth best of five
Table 4
Trials to Solution and Letters per Equation for Best, Second-Best, Third-Best, Fourth-Best, and Fifth-Best Individuals
Problem 1
Problem 2
Group Size
Individual
Variable
M
SD
M
SD
Five
Best
Trials
6.83
1.34
6.15
1.23
Letters per equation
2.33
0.67
2.37
0.93
Second
Trials
7.95
1.50
7.05
1.28
Letters per equation
2.15
0.29
2.29
0.73
Third
Trials
8.73
1.43
8.18
1.47
Letters per equation
2.09
0.23
2.16
0.66
Fourth
Trials
9.73
1.11
8.85
1.44
Letters per equation
2.11
0.35
2.15
0.40
Fifth
Trials
10.20
1.14
9.58
1.39
Letters per equation
2.01
0.19
2.11
0.41
Four
Best
Trials
7.15
1.48
6.40
1.36
Letters per equation
2.20
0.50
2.34
1.04
Second
Trials
8.23
1.56
7.53
1.41
Letters per equation
2.15
0.32
2.31
0.76
Third
Trials
9.33
1.21
8.70
1.49
Letters per equation
2.04
0.08
2.09
0.29
Fourth
Trials
10.18
1.20
9.35
1.35
Letters per equation
2.02
0.20
2.09
0.35
Three
Best
Trials
7.30
1.44
6.40
1.26
Letters per equation
2.32
0.67
2.36
0.94
Second
Trials
8.68
1.42
7.83
1.69
Letters per equation
2.17
0.43
2.30
0.80
Third
Trials
10.08
1.16
9.15
1.51
Letters per equation
2.01
0.20
2.11
0.41
Two
Best
Trials
8.00
1.89
7.33
1.64
Letters per equation
2.22
0.59
2.27
0.87
Second
Trials
9.60
1.45
8.83
1.60
Letters per equation
2.06
0.31
2.20
0.74
648
LAUGHLIN, HATCH, SILVER, AND BOH
individuals; best, second best, third best, and fourth best of four
Table 6
individuals; best, second best, and third best of three individuals;
Comparison of Groups and Equivalent Number of Individuals at
and best and second best of two individuals. Table 5 gives the
Each Group Size for Trials to Solution and Letters per Equation
means and standard deviations for five-person, four-person, three-
person, and two-person groups.
Trials
Letters
Group and
individual
t
SES
t
SES
Groups Versus Individuals
Five
Table 6 gives the results of one-tailed t tests and standard effect
Best
2.72
0.69
3.35
0.47
sizes for comparisons of the groups and the best, second best, and
Second
8.10
1.30
4.09
0.59
so forth of an equivalent number of individuals at each group size
Third
13.15
1.67
4.63
0.66
Fourth
17.60
1.81
4.60
0.66
for trials to solution and letters per equation. As indicated in Table
Fifth
20.79
1.87
4.99
0.73
6, the five-person groups had significantly fewer trials to solution
Four
and more letters per equation than each of the best, second-best,
Best
3.11
0.50
2.76
0.41
third-best, fourth-best, and fifth-best individuals. Similarly, the
Second
8.47
1.15
3.02
0.47
Third
14.02
1.55
4.10
0.64
four-person groups had significantly fewer trials to solution and
Fourth
17.67
1.69
4.05
0.65
more letters per equation than each of the four types of individuals,
Three
and three-person groups had significantly fewer trials to solution
Best
3.46
0.61
2.88
0.48
and more letters per equation than each of the three types of
Second
9.53
1.29
3.40
0.59
individuals. The two-person groups did not differ significantly
Third
15.43
1.65
4.22
0.75
Two
from the best individuals and had significantly fewer trials to
Best
1.13a
0.61a
solution than the second-best individuals.
Second
6.24
1.10
1.31a
Group Size
Note.
All ps
.001, except where marked with a superscript a.
a nonsignificant.
A 4 (group size: two, three, four, five)
2 (problems: one, two)
randomized blocks ANOVA with repeated measures on the second
variable for trials to solution indicated a significant main effect of
Regression of Trials to Solution on Letters per Equation
group size, F(3, 117)
11.04, p
.001, MSE
2.9565. Tukey’s
pairwise comparisons indicated fewer trials to solution for each of
Regression of the total trials to solution over the two problems
three-person, four-person, and five-person groups than two-person
on the total letters per equation over the two problems indicated an
groups, with nonsignificant differences among three-person, four-
R of .38 (R2
.14, p
.02) for two-person groups, R of .65 (R2
person, and five-person groups. The main effect of successive
.42, p
.001) for three-person groups,; R of .34 (R2
.11, p
problems was significant, F(1, 117)
63.38, p
.001, MSE
0.8721, with fewer trials to solution on Problem 2 (M
5.96) than
.03) for four-person groups, and R of .60 (R2
.36, p
.001) for
Problem 1 (M
6.79). The Size
Problems interaction was
five-person groups. Separate regressions for Problems 1 and 2
nonsignificant, F(1, 117)
2.04.
ranged from .18 for four-person groups on Problem 1 to .73 for
A similar ANOVA for letters per equation indicated a nonsig-
three-person groups on Problem 2. The Rs were higher on Problem
nificant main effect of group size, F(3, 117)
1.53. The main
2 than Problem 1 for all group sizes, indicating a fuller realization
effect of successive problems was significant, F(1, 117)
17.22,
of the effectiveness of multiletter equations on the second problem.
p
.001, MSE
0.4142, with more letters per equation on
Thus, as predicted, the number of trials to solution was influenced
Problem 2 (M
2.89) than Problem 1 (M
2.60). The Size
by the complexity of the proposed equations.
Problems interaction was nonsignificant, F(1, 117)
1.
Enjoyment
Table 5
After the second problem, the group members and individuals
Trials to Solution and Letters per Equation for Five-Person,
rated their enjoyment of the experiment on a 1–10 scale (10 being
Four-Person, Three-Person, and Two-Person Groups
the highest). Ratings were averaged over the members of a given
Problem 1
Problem 2
group. Both the individuals (M
7.96) and the group members
(M
7.24) rated their enjoyment quite highly, although individ-
Group size
Variable
M
SD
M
SD
uals rated their enjoyment more highly than the group members,
Five
Trials
6.25
1.13
5.70
1.18
F(1, 334)
19.06, p
.001. Enjoyment ratings did not differ for
Letters per equation
2.77
1.37
3.14
1.96
the members of different-sized groups, F(3, 145)
1. The main
Four
Trials
6.50
1.38
5.78
1.31
effect of individuals (best, etc.) was significant, F(4, 182)
4.42,
Letters per equation
2.63
1.38
2.85
1.50
p
.01. Each of the best (M
8.33), second-best (M
8.39), and
Three
Trials
6.45
1.52
5.65
1.41
Letters per equation
2.71
1.38
3.16
1.84
third-best (M
8.18) individuals enjoyed the experiment more
Two
Trials
7.95
1.62
6.70
1.77
than the fifth-best individuals (M
7.15). All other pairwise
Letters per equation
2.27
0.84
2.42
1.10
comparisons were nonsignificant.
GROUPS PERFORM BETTER THAN THE BEST INDIVIDUALS
649
Discussion
Group Size
Group Versus Individual Performance
Each of the three-person, four-person, and five-person groups
had fewer trials to solution and proposed more complex equations
Each of the three-person, four-person, and five-person groups
than the two-person groups. The three-person, four-person, and
had significantly fewer trials to solution and more letters per
five-person groups did not differ significantly from each other on
equation than the best of an equivalent number of individuals. In
either trials to solution or letters per equation. As there were 40
contrast, the two-person groups did not differ significantly from
replications of each group size, we may be confident that these
the best individuals, although they had significantly fewer trials to
nonsignificant differences between group sizes three, four, and
solution than the second-best individuals. These results replicate
five were not due to insufficient statistical power.
the previous superiority of three-person groups over the best of
Our review of previous research suggests that performance
three independent individuals (Laughlin et al., 2003) and four-
improves with increasing group size for problems of moderate
person groups over the best of four independent individuals
difficulty that require understanding of verbal, quantitative, or
(Laughlin et al., 2002) and extend them to the superiority of
logical conceptual systems, but performance has not been shown to
five-person groups over the best of five independent individuals.
improve with increasing group size for eureka problems. The
To our knowledge, these three experiments are the only reports
current finding that three-person groups performed better than
that groups of size three, four, and five perform better than the best
two-person groups is consistent with this research, as letters-to-
of an equivalent number of individuals. As the best group member
numbers problems require understanding of arithmetic, algebra,
is comparable to the best independent individual, these results also
and logic and systematic reasoning over a series of trials rather
indicate that the groups performed better than their best member
than a single insight, and they are of moderate but not excessive
would have performed alone.
difficulty for intelligent and motivated college students.
We attribute this superiority of three-person, four-person, and
Why was there no further improvement as group size increased
five-person groups over the best of an equivalent number of
from three to four to five? We have suggested that the crucial
individuals to the highly intellective nature of letters-to-numbers
aspect of effective performance on letters-to-numbers problems is
problems, which allow recognition and adoption of correct re-
the use of multiletter equations to identify two or more letters per
sponses, recognition and rejection of erroneous responses, and
equation rather than the use of a simple but inefficient two-letter
substitution strategy that identifies only one letter per trial. As-
effective collective information processing (Hinsz, Tindale, &
suming the individual probability of using two-letter equations on
Vollrath, 1997; Laughlin, VanderStoep, & Hollingshead, 1991).
all trials in previous research and assuming that multiletter equa-
Letters-to-numbers problems strongly fulfill Laughlin and Ellis’s
tions are demonstrably preferable to two-letter equations if pro-
(1986) four conditions of demonstrability. First, the group mem-
posed by at least one group member predicts progressively smaller
bers understand and agree on the underlying conceptual systems of
improvement with group size beyond three members. Coordination
arithmetic, algebra, and logic. Second, there is sufficient informa-
difficulties (Steiner, 1972) and production blocking (Diehl & Stro-
tion to demonstrate the superiority of strategies such as multiletter
ebe, 1987, 1991; Valacich, Dennis, & Connolly, 1994) might also
equations relative to obvious but less effective two-letter substitu-
have increased with increasing group size beyond three members.
tion strategies and to demonstrate and reject erroneous inferences.
Although these processes should be minimal in three-person
Virtually any sequence of equations of any degree of complexity
groups, they may be more detrimental in four-person and five-
contains some information, and the groups were able to process
person groups. In contrast, we do not believe that there was an
this information more effectively than the best individuals. Third,
appreciable motivation loss as a result of free riding (Kerr, 1983;
the members who had not considered effective strategies and
Olson, 1965) from three-person groups to four-person and five-
reasoning recognized the effectiveness when such strategies were
person groups, as the group members rated the experiment as quite
proposed by other members. Fourth, the members who proposed
enjoyable and there were nonsignificant differences for group size.
the effective strategies and reasoning had the ability, motivation,
Thus, three group members were necessary and sufficient for the
and time to demonstrate the effectiveness to the other members.
groups to perform better than the best of an equivalent number of
Thus, the group members combined their abilities and resources to
independent individuals. If groups of size three perform as well as
perform better than the best of an equivalent number of individuals
groups of larger size, it is obviously a more efficient use of human
on the highly intellective complementary group task (Steiner,
and logistic resources to use three-person groups. Further research
1966).
should be conducted to determine whether three persons are nec-
Tindale and Kameda (2000) and Kameda, Tindale, and Davis
essary and sufficient for groups to perform better than the best of
(2003) generalized the first of these conditions in their concept of
an equivalent number of individuals on other problem-solving
social sharedness, the degree to which preferences and cognitions
tasks, such as survival problems (e.g., Littlepage, Schmidt,
are shared among group members at the outset of group interac-
Whisler, & Frost, 1995). The research program on team perfor-
tion. Building on these shared preferences (the objective of solving
mance on realistic command and control tasks of Ilgen, Hollen-
in as few trials as possible and the norms of interpersonal influ-
beck, and colleagues has used four-person groups (e.g., Hedlund,
ence, e.g., accepting a demonstrably effective strategy) and cog-
Ilgen, & Hollenbeck, 1998; see Ilgen, Hollenbeck, Johnson, &
nitions (the conceptual systems and operations of arithmetic, al-
Jundt, 2005, for a review), and it is theoretically and practically
gebra, and logic), the groups combined the abilities, skills, and
important to determine whether group performance is comparable
insights of their members and thus performed better than the best
for smaller groups of size three and improves with groups of size
of an equivalent number of individuals.
five and larger.
650
LAUGHLIN, HATCH, SILVER, AND BOH
Future Research: Training, Transfer, and Expertise
groups: Toward the solution of a riddle. Journal of Personality and
Social Psychology, 53, 497–509.
Letters-to-numbers problems combine aspects of hypothesis
Diehl, M., & Stroebe, W. (1991). Productivity loss in idea-generating
testing (e.g., Klayman & Ha, 1987), mathematical and logical
groups: Tracking down the blocking effect. Journal of Personality and
reasoning (e.g., Laughlin & Ellis, 1986; Stasson, Kameda, Parks,
Social Psychology, 61, 392– 403.
Zimmerman, & Davis, 1991), cryptographic reasoning (e.g., New-
Gigone, D., & Hastie, R. (1993). The common knowledge effect: Infor-
ell & Simon, 1972; Singh, 1999), and collective induction (e.g.,
mation sharing and group judgment. Journal of Personality and Social
Psychology, 65, 959 –974.
Crott, Giesel, & Hoffman, 1998; Laughlin, 1999). These corre-
Hastie, R. (1986). Review essay: Experimental evidence on group accu-
spondences indicate that letters-to-numbers problems are a useful
racy. In G. Owen & B. Grofman (Eds.), Information pooling and group
and interesting domain for future research on the interrelated
accuracy (pp. 129 –157). Westport, CT: JAI Press.
issues of training, transfer, and expertise in small-group problem
Hedlund, J., Ilgen, D. R., & Hollenbeck, J. R. (1998). Decision accuracy in
solving (e.g., Bonner, 2004; Bonner, Baumann, & Dalal, 2003;
computer-mediated versus face-to-face decision-making teams. Organi-
Hollingshead, 1998; Liang, Moreland, & Argote, 1995; Littlepage
zational Behavior and Human Decision Processes, 76, 30 – 47.
et al., 1995). For example, individuals could be trained to use
Hill, G. W. (1982). Group versus individual performance: Are N
1 heads
effective multiletter substitution or known answer strategies (see
better than one? Psychological Bulletin, 91, 517–539.
Laughlin et al., 2003, for further discussion of effective strategies)
Hinsz, V. B., Tindale, R. S., &Vollrath, D. A. (1997). The emerging
before assembling for further group problem solving. Through
conceptualization of groups as information processors. Psychological
such member training, groups may be able to overcome the com-
Bulletin, 121, 43– 64.
mon knowledge (e.g., Gigone & Hastie, 1993) and hidden profile
Hollingshead, A. B. (1998). Group and individual training: The impact of
practice on performance. Small Group Research, 29, 254 –280.
(e.g., Stasser & Stewart, 1992) effects, whereby groups discuss
Ilgen, D. R., Hollenbeck, J. R., Johnson, M., & Jundt, D. (2005). Teams in
common initial preferences and shared information, fail to discuss
organizations: From input-output models to IMOI models. Annual Re-
unique unshared information, and hence make suboptimal deci-
view of Psychology, 56, 517–543.
sions. In contrast to this research, in which a single member with
Kameda, T., Tindale, R. S., & Davis, J. H. (2003). Cognitions, preferences,
critical information may not be able to convince other members of
and social sharedness: Past, present, and future directions in group
the validity of the information, letters-to-numbers problems enable
decision making. In S. L. Schneider & J. Shanteau (Eds.), Emerging
a member who has been trained on an effective strategy to dem-
perspectives on judgment and decision research (pp. 458 – 485). Cam-
onstrate the effectiveness of the strategy to the other group mem-
bridge, England: Cambridge University Press.
bers. Similarly, letters-to-numbers problems are a useful and in-
Kerr, N. L. (1983). Motivation losses in task-performing groups: A social
teresting task for future research on group-to-individual transfer:
dilemma analysis. Journal of Personality and Social Psychology, 45,
Does effective group problem solving transfer to subsequent indi-
819 – 828.
vidual problem solving?
Kerr, N. L., & Tindale, R. S. (2004). Group performance and decision
making. Annual Review of Psychology, 56, 623– 655.
Klayman, J., & Ha, Y.-M. (1987). Confirmation, disconfirmation, and
Conclusions
information in hypothesis testing. Psychological Review, 94, 211–228.
Laughlin, P. R. (1999). Collective induction: Twelve postulates. Organi-
Groups of size three, four, and five performed better than the
zational Behavior and Human Decision Processes, 80, 50 – 69.
best of an equivalent number of individuals, but groups of size two
Laughlin, P. R., Bonner, B. L., & Miner, A. G. (2002). Groups perform
performed at the level of the best of two individuals. Groups of
better than the best individuals on letters-to-numbers problems. Orga-
size three, four, and five performed better than groups of size two
nizational Behavior and Human Decision Processes, 88, 605– 620.
but did not differ from each other. These results suggest that
Laughlin, P. R., & Ellis, A. L. (1986). Demonstrability and social combi-
nation processes on mathematical intellective tasks. Journal of Experi-
groups of size three are necessary and sufficient to perform better
mental Social Psychology, 22, 177–189.
than the best of an equivalent number of individuals on intellective
Laughlin, P. R., Kerr, N. L., Davis, J. H., Halff, H. M., & Marciniak, K. A.
problems.
(1975). Group size, member ability, and social decision schemes on an
intellective task. Journal of Personality and Social Psychology, 31,
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Appendix
Instructions for Letters-to-Numbers Problems
This is an experiment in problem solving. The objective is to figure out
a code in as few trials as possible. The numbers 0 –9 have been coded as
Trial
Equation
Hypothesis
Feedback
the letters A–J in some random order. You will be trying to find out which
1
A
B
C
A
1
False
letter corresponds to which number. It is important to remember that all we
2
B
C
EA
A
8
False
are doing is changing the characters used to represent the numbers. We are
3
F
A
D
EE
E
1
True
not changing the way that the number system works. That is, we are still
4
H
J
D
I
0
True
using the same decimal number system you have been using all of your life.
Below is an example of a random code:
On the first trial the person chooses the equation A
B
?, and the
experimenter tells the person that the solution to this equation is C. This is
A
3, B
5, C
8, D
2, E
1, F
6, G
4, H
7, I
0,
because A
3 and B
5, which sums to 8, the letter represented by C. The
J
9
person then guesses that A represents the number 1, and the experimenter
First you will come up with addition or subtraction equations using the
indicates that this is not the case, or False. On the second trial the person
letters A–J that will be solved by the experimenter who will give you the
asks the solution to the equation B
C
? and is told that the answer is
answer in letter form. Then you will make a guess as to what one of the letters
EA. This is because B
5 and C
8, which sums to 13 or EA. Note that
represents, and the experimenter will tell you whether or not the guess is
on the third trial the person chooses to add three letters together. You may
correct (True) or incorrect (False). When you feel you know the full coding
use as many letters as you desire in your equations. Note that on the fourth
of letters to numbers, propose it in the space provided on your data sheet.
trial the person chooses to use a subtraction equation. You may use either
When you have correctly mapped all ten letters to all ten numbers, you will
addition or subtraction equations as you see fit.
have solved the problem. Remember, the objective is to solve the problem
in as few trials as possible.
Received June 8, 2004
Here are four example trials using the random code above. Note that
Revision received July 22, 2005
underlined letters represent experimenter feedback.
Accepted August 3, 2005