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Moves in the World Are Faster than Moves in the Head: Interactivity in the River Crossing Problem

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In solving a variety of problems people interact with their external environment, often using artefacts close at hand to supplement and augment their problem solving skills. The role of interactivity in problem solving was investigated using a river-crossing problem. All participants performed the task twice, once in a high interactivity condition and once in a low interactivity condition. Moves to completion were higher in the high interactivity condition but latency per move was much shorter with high than with low interactivity. Moves in the world were easier to implement than to simulate mentally and acted as epistemic actions to facilitate thinking. In addition, when participants experienced the low interactivity version of the task second, their performance reflected little learning. However, when the high interactivity version was completed second, latency to solution and latency per move were substantially reduced. These results underscore the importance of investigating problem solving behaviour from a distributed cognition perspective.
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Moves in the World Are Faster than Moves in the Head:
Interactivity in the River Crossing Problem
Frédéric Vallée-Tourangeau, Lisa G. Guthrie and Gaëlle Villejoubert
Department of Psychology, Kingston University
Kingston-upon-Thames UNITED KINGDOMKT1 2EE
f.vallee-tourangeau/l.guthrie/g.villejoubert@kingston.ac.uk
Abstract
In solving a variety of problems people interact with
their external environment, often using artefacts close at
hand to supplement and augment their problem solving
skills. The role of interactivity in problem solving was
investigated using a river-crossing problem. All
participants performed the task twice, once in a high
interactivity condition and once in a low interactivity
condition. Moves to completion were higher in the high
interactivity condition but latency per move was much
shorter with high than with low interactivity. Moves in
the world were easier to implement than to simulate
mentally and acted as epistemic actions to facilitate
thinking. In addition, when participants experienced the
low interactivity version of the task second, their
performance reflected little learning. However, when the
high interactivity version was completed second, latency
to solution and latency per move were substantially
reduced. These results underscore the importance of
investigating problem solving behaviour from a
distributed cognition perspective.
Keywords: Problem solving, interactivity, epistemic
actions, distributed cognition
Introduction
Scientists and lay people alike naturally create and build
artefacts or recruit existing ones to help them solve
problems. To be sure, artefacts such as calculators, data
management software, computers can facilitate complex
computations. But others, of more modest complexity,
such as pen and paper, can help articulate and structure
thinking. Space itself is a tool that can facilitate thinking,
that is it can be structured, designed (and redesigned)
such as to make thinking easier (Kirsh, 1995, 1996,
2010). Thus solving jigsaw puzzles involves physically
juxtaposing different pieces to gauge their fit; in Scrabble,
letter tiles are physically rearranged to facilitate word
production; in Tetris, tetrominoes are physically rotated to
determine their optimal place along a line. And beyond
puzzles and games, experts structure an external
environment to support thinking. Scientists use physical
objects and their arrangement in space to formulate and
test hypotheses: Watson (1968, pp. 123-125) describes
how he cleared his desk, cut out shapes corresponding to
the four nucleobases, and manipulated them until he saw
which ones could be paired to hold the double helix
together. Artefacts recruited in thinking are rich, varied
and modifiable. Their recruitment is at times strategic,
such that their users actively engage in their design and
engineer their function, and at others, opportunistic, that
is they are picked up from the environment in an ad hoc
fashion to help solve a problem, capitalizing on a
fortuitous interaction.
From a distributed cognition perspective, thinking is
the product of a cognitive system wherein internal and
external resources are coupled to create a dynamic, fluid,
and distributed problem representation (Villejoubert &
Vallée-Tourangeau, 2011; Weller, Villejoubert, & Vallée-
Tourangeau, 2011). The nature of the external resources
recruited in thinking and their functional role are guided
by principles of cognitive economy, effort and efficiency
(Clark, 1989; Kirsh, 2010). Actions complement and
augment thinking by providing new information,
unveiling new affordances, and can sometimes serve to
create a more cognitively congenial problem presentation
(Kirsh, 1996). Through the creation, recruitment and
manipulations of artefacts, new perspectives are gained,
encouraging the development or retrieval of problem
solving strategies, and improving the prospect of solving
the problem (Magnani, 2007). As the environment
shoulders some of the representational and computational
burden, valuable cognitive resources such as working
memory capacity and executive functions may be freed to
draw on stored knowledge or develop new solutions
(Magnani, 2007). For example, recent work on mental
arithmetic indicates that people are more accurate, more
efficient, and create more congenial interim totals when
they can manipulate number tokens that configure the
problem presentation, than when they perform the mental
arithmetic without (Vallée-Tourangeau, in press).
River Crossing
Transformation problems have been the focus of research
in cognitive psychology for the past 50 years. In these
problems, a well-defined space connects an initial and a
goal state. Legal moves are defined in terms of simple
rules and enacted with simple operators. Participants must
reach the goal state by transforming the initial state
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through a series of intermediate states. A well-studied
class of transformation problems are river-crossing
problems. In these problems, objects (people, animals, or
things) must be carried from one “riverbank to another
on a boat but with a set of constraints on moves that can
be selected to reach the goal. A common version involves
three missionaries and three cannibals (Reed, Enrst, &
Banerji, 1974; or three hobbits and three orcs, Thomas,
1974). In transporting all cannibals and missionaries from
one bank to the other, cannibals must not outnumber
missionaries or either bank. The boat can take at most two
passengers, and at least one. The problem space is
relatively narrow since illegal moves cannot produce
blind alleys of any depth (Reed et al., 1974) and can be
completed in 11 steps. In different versions, problem
difficulty is a function of the rules that constrain the
number of objects that can be moved at any one time,
which combinations of objects are allowed on the boat,
and which combinations can be left on either bank. The
number of objects and the rules that govern their transport
map out a problem space that links the initial state with all
objects on one side of the river to a goal state with all
objects on the other riverbank. Cognitive psychologists
have used this task as a window onto problem solving,
particularly planning (Greeno, 1978), search and move
selection (Reed et al., 1974; Simon & Reed, 1976). As
such river crossing problems have been used as a testing
platform for a number of process models of search and
move selection, strongly influenced by developments in
AI (Greeno, 1978; Simon & Reed, 1976).
The river-crossing task involves moving people or
things across a surface and as such foregrounds the
importance of interacting with an external task
representation. However, interactivity in river crossing
problem solving has never been the explicit focus of
investigation. The manner with which the river-crossing
task has been implemented varies a great deal across
studies. For example, Reed et al. (1974) used different
types of coins to represent missionaries and cannibals.
Jeffries et al. (1976) developed a basic computer interface
where participants typed in the objects they wanted to put
in the boat on a given crossing. The interface accepted
only legal moves and updated the simple representations
(often with letters and numbers, such as ‘3M’ for three
missionaries) on either side of the riverbank. Participants
kept on typing in their moves until they managed to
transport all objects from one bank to the other. Knowles
and Delaney (2005) designed a more realistic interface
with icons representing travellers against a backdrop of a
river with two banks and a boat. Participants selected
moves by clicking on the travellers, which then appeared
next to the boat on the screen. In all these instances
participants were never offered a three-dimensional work
surface on which objects transparently corresponding to
the scenario protagonists are manipulated and moved by
hand. In contrast, developmental psychologists who
worked with the river crossing task, being less sanguine
about ‘formal operations’ presumably, have taken care to
design rich interactive thinking environments with
physical materials representing the boat, the river, and
figurines corresponding to the cover story characters (e.g.,
Gholson, Dattel, Morgan, & Aymard, 1989).
A more explicit experimental focus on interactivity
may unveil interesting aspects of problem solving
performance. For example, there is evidence that in other
transformation problems interactivity substantially
transformed problem solving behaviour. Vallée-
Tourangeau, Euden and Hearn (2011) reported that
mental set is significantly reduced in Luchins’s volume
measurement problems when participants interact with an
actual physical presentation of the problem. The
manipulation of water jars created a dynamic problem
representation revealing solutions that were not simulated
mentally. The selection of moves was guided and
governed by three-dimensional perceptual feedback and
participants were less likely to persevere using a more
complicated solution for the test problems. In a river-
crossing task, interactivity may help participants work out
the quality of different moves not by simulating their
consequences mentally, but rather by simply completing
the move and observing the results. Such moves then are
epistemic actions’ (Kirsh & Maglio, 1994)moves that
may not, in themselves, necessarily help narrow the gap
with the goal state, but rather provide information as to
what to do next. As such, move selection can be
opportunistic, although not necessarily mindless; rather
the strategic consequences of a certain move can simply
be observed. Kirsh and Maglio (1994) demonstrated that
it is faster and easier to physically rotate the tetrominoes
in Tetris than to simulate their rotation mentally, leading
to better and more efficient problem solving behaviour. In
a similar vein, moves in the world, rather than moves in
the head, may help participants solve river-crossing
problems more efficiently as the reduced cognitive costs
of physical moves will enable them to select more moves
more quickly, than they would if they were completing
the task with a non-interactive problem presentation.
The Present Experiment
The present experiment examined performance in the
river crossing problem when presented with or without
artefacts as an aid to solution. This was measured in terms
of number of moves, latency to completion and latency
per move. In a high interactivity condition, the problem
was presented with a board, a raft and six figurines:
Participants had to move the raft and the figurines across
the board to register a move until they had moved all six
figurines from one bank to the other. In a low interactivity
version, the problem was described on a piece of paper
and participants were asked to verbalise the moves they
would make to reach the goal. They performed the
problem twice, once with the high interactivity version
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and once with the low interactivity version; the order was
counterbalanced across participants. This experiment
employed a mixed design with interactivity level as the
repeated measures factor and orderlow interactivity
first, high interactivity firstas the between subjects
factor. As moves can act as epistemic actions, we
predicted that participants would produce more moves,
would solve the problem more quickly and that hence
latency per move would be shorter in the high compared
to the low interactivity condition. We also predicted that
participants would complete the second presentation of
the task more quickly than the first since they would be
familiar with the procedure and may well exploit an
episodic record of their trajectory to help them select
better moves, and select them more quickly. A high
interactivity problem solving environment may more
clearly showcase evidence of learning because of the ease
and speed with which moves can be made in the world.
Method
Participants
Sixty-four university undergraduates participated in the
experiment in return for course credits. Due to testing
errors the data from three participants were incomplete,
therefore unsuitable for analysis. Of the remaining sixty-
one participants, nine did not complete the river crossing
problem and were excluded from further analyses. The
final sample was composed of 52 participants (45 females,
7 males, Mage = 21.4, SD = 5.1)
Procedure
Chickens and wolves were the protagonists in the river-
crossing scenario used for this experiment. The objective
was for the six animals to be transported from the left
riverbank to the right one. The selection of a move had to
comply with the constraints and rules of the problem. The
same instruction sheet explaining the objective of the task
and the rules of the problem was used for both conditions
and could be read by the participants throughout the
duration of the task. The sheet read:
Three wolves and three chickens on the left bank of a
river seek to cross the river to the right bank. They have
a boat which can carry only two animals at a time, but
there must always be an animal on the boat for it to
move.
However if at any time the wolves outnumber the
chickens on either bank the wolves will eat the chickens.
Thus you cannot move the animal(s) in a manner that
will result in the wolves outnumbering the chickens on
either bank.
The goal of the task is to move all the animals from
the left bank to the right bank.
In the low interactivity version of the task, the
researcher transcribed each move as verbalised by the
participant onto a record sheet. The record sheet was a
simple representation of the raft between the left and right
banks of the river, with slots to record the nature and
number of the animals on either side (which was denoted
with a ‘C’ for chickens and ‘W’ for wolves; see Fig. 1);
each page represented only one move. At any one time,
participants could only inspect their previous move as they
dictated their next move to the experimenter. As soon as
the next move was dictated, the sheet with the previous
move was turned over. Thus participants could not inspect
a historical record of previous moves. Illegal moves
proposed were noted, but the experimenter did not
transcribe the move on the recording sheet. Rather,
participants were invited to re-read the task instructions to
discover why such a move was not allowed.
Figure 1: Record sheet for the river crossing moves in the
low interactivity condition.
The high interactivity version of the task involved the
use of six plastic figurines, three wolves (9cm x 7cm x
2cm) and three chickens (4cm x 5cm x 1.5cm), one pop-
stick raft (9cm x 6cm) and a painted board (60cm x 45cm)
representing the river and banks (see Fig. 2). As the
participants interacted with the artefacts, the experimenter
recorded the moves, but this record was never shown to the
participants. An illegal move prompted the experimenter to
instruct participants to move the raft and the animals back
to the previous state and, as in the static condition, they
were invited to re-read the instruction sheet to determine
which moves were possible.
Figure 2: Board, raft and figurines in the high interactivity
condition.
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Figure 3: Mean latency to completion (left panel), mean number of legal and illegal moves (middle panel), mean latency
per move (right panel) as a function of order (completed first or second) in the low interactivity (light grey) and high
interactivity condition (dark grey). Error bars are standard errors of the mean.
The river crossing task was embedded in a testing
session during which participants completed a number of
other problem solving tasks unrelated to the present
experiment. In the low interactivity version, the boxes on
the first record sheet were completed with three C’s and
three W’s on the left bank. Prior to the selection of a move,
the researcher would draw an arrow above the raft to
represent the direction in which it was travelling. The
participants were discouraged from touching or pointing to
the record sheet; they could not sketch out a move using
pen and paper beforehand.
In the high interactivity condition, the board was placed
on a table in front of the participant with the researcher
placing all animals on the bank closest to the participant
and positioning the raft on the river. This ensured all
participants commenced the task with the same
presentation. A move was defined as completed when
whichever wolf (wolves) or chicken(s) being transported
for that particular move were removed from the raft onto
the other bank. Illegal moves were identified before they
were completed, with animals and raft returned to the
previous position on the board. In both conditions
participants were given 15 minutes to complete the river
crossing problem.
A 20-minute interval was designed between the two
presentations of the river crossing problem during which
participants completed a number of non-verbal puzzles,
including finding similarities and differences between
series of pictures, and identifying the odd picture in a
series of thematically related pictures. Finally, the river
crossing task was presented again in the alternate condition
(either low or high interactivity) to that which was
presented first; the order was counterbalanced across
participants. Thus, the independent variables manipulated
were condition (static, interactive) and order (static first,
interactive first) in a 22 mixed design. Performance in
both conditions was measured in terms of latency to
solution, number of legal and illegal moves, and latency
per move.
Results
Latency
Latencies to solution, displayed in the left panel of Figure
3, suggest that order had little effect on participants in the
low interactivity condition but the problem was completed
much quicker in the high interactivity condition when it
was experienced second. A 22 mixed analysis of variance
(ANOVA) revealed that the main effect of interactivity
condition was not significant, F(1, 49) = 2.14, p = .150,
while the main effect of order was significant F(1, 49) =
4.20, p =.046, as well as the condition by order interaction
F(1, 49) = 5.32, p = .025. Post hoc tests indicated that
latencies in the low interactivity condition did not decrease
significantly from the first to the second presentation, t(49)
= 0.090, p = .929. In turn, participants were quicker in the
second than in the first presentation of the problem in the
high interactivity condition, t(49) = 3.744, p < .001.
Moves
The mean number of legal and illegal moves are plotted in
the middle panel of Figure 3. The high interactivity
condition elicited a higher number of legal moves to solve
the river crossing problem compared to the low
interactivity condition and this was observed for both
orders. In a 22 ANOVA the main effect of condition was
significant F(1, 49) = 11.63, p =.001, while the main effect
of order was not significant, F< 1, nor was the condition
by order interaction, F(1, 49) = 1.26, p = .267.
In turn, the mean number of illegal moves was greater
in the high interactivity condition when it was experienced
5
10
15
20
Mean
10
15
20
25
30
FIRST SECOND
Mean Latency per Move (s)
Moves Latency Latency per Move
0
2
4
6
8
FIRST SECOND
Mean
Legal
Illegal
Presentation Order
200
250
300
350
400
450
500
Mean Latency (s)
Interactivity
FIRST SECOND
Low
High
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first, but the frequency of illegal moves was relatively
stable in the low interactivity condition across both
presentations. In a 2X2 ANOVA the main effect of
condition was significant, F(1, 49) = 7.16, p =.010, while
the main effect of order was not significant, F(1, 49) =
3.34, p = .074 nor was the condition by order interaction,
F(1, 49) = 2.69, p = .108.
Latency per Move
The latency per move data are shown in the right panel of
Figure 3. Latency per move in the low interactivity
condition was unaffected by order, however participants
appeared faster at enacting moves in the high interactivity
condition, especially the second time the participants
engaged with the task. In a 22 mixed ANOVA the main
effect of condition was significant, F(1, 49) = 20.0, p <
.001, but the main effect of order was not, F(1, 49) = 2.33,
p = .133; the condition by order interaction was significant
F(1, 49) = 11.4, p < .001. Post hoc tests revealed that the
mean latency per move in the low interactivity condition
did not decrease significantly from the first to the second
presentation, t(49) = 0.858, p = .395; in turn moves were
selected faster in the high interactivity condition when that
condition was experienced second, t(49) = 4.60, p < .001.
Discussion
This experiment investigated the impact of interactivity on
problem solving performance for a river crossing problem.
All participants were required to solve the problem twice,
once in a low interactivity context in which move selection
could only be simulated mentally and once in a high
interactivity context where moves could be implemented in
the world with a three-dimensional manipulable
presentation of the problem. The repeated measures design
eliminated random variance arising from between-subjects
differences: Any performance improvement emerging in
the high interactivity condition could not be attributed to a
different group of participants with a differing pool of
internal resources.
A high level of interactivity encouraged participants to
make more moves in reaching a solution than when they
completed the problem in the low interactivity condition;
however, the order in which participants completed the
task had no effect on the number of moves. In turn, the
order in which the conditions was experienced had an
effect on latency to solution. More important still, the main
effect of order was qualified by a significant interaction:
solution latencies in the low interactivity condition were
similar whether this was completed first or second, while
latencies dropped substantially when the high interactivity
condition was experienced second. The latency per move
data indicated that participants were always quicker to
select a move in the high interactivity condition, and were
generally quicker to select a move during the second
presentation of the problem. However, the more important
pattern in these data was the condition by order interaction:
Latency per move dropped precipitously when the second
presentation of the problem occurred in the interactive
condition.
As Kirsh (2010, p. 442) puts it: “Cognitive processes
flow to wherever it is cheaper to perform them”. Moves
were cheap in the high interactivity condition it is easier
to move the pieces in the world than to simulate their
movement in the head. More moves were made when the
participants were given the freedom to transport the
artefacts around the board to reach the solution than when
moves were simulated mentally.
Learning Manifest Through Interactivity
The second presentation of the problem offered the
opportunity to gauge the degree of learning and transfer.
There was much evidence of learning, when the second
opportunity to solve the problem took place in a context
that favoured a high level of interactivity: Participants
completed the problem in less time and selected moves at a
faster rate than when the second presentation of the
problem was in the low interactivity condition. In fact,
when the low interactivity condition was experienced
second, performance reflected little learning and transfer.
This pattern of results suggests two competing
explanations: (i) the process and quality of knowledge
acquisition is different as a function of the level of
interactivity or (ii) interactivity is a performance facilitator
and a high level of interactivity more clearly showcases
learning. Let’s take each in turn.
First exposure to the problem without much
interactivity might have fostered the acquisition of a
sounder and more actionable representation of the task and
appreciation of an efficient sequence of moves to solution.
In contrast, experiencing the problem in a context that
fosters a high level of interactivity might not be
accompanied by the same investment in cognitive effort,
proceeding primarily on the basis of procedural learning,
which in turn might interfere with the development of an
accessible and transferable conceptual representation of the
problem. As a result, when the problem is encountered for
the second time in a condition without much interactivity,
the procedural knowledge does not facilitate performance;
however, when the second presentation occurs in the high
interactivity condition, performance substantially benefits
from the knowledge acquired on the basis of the
experience in the low interactivity condition.
Alternatively, the substantial improvement in the high
interactivity condition when participants are presented the
problem a second time might not reflect differences in the
type and quality of experience but rather release from a
performance bottleneck. In other words, interactivity is a
performance facilitator. Cognitive efforts and task
demands are more exacting with low interactivityas
evidenced by the significantly longer latency per move.
When participants encounter the problem a second time
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but this time can indulge in cheaper move selection by
moving artefacts on the board, they experience a release
from the cognitive demands of the low interactivity
condition and are quicker at producing moves.
The moves data offer some support for the performance
facilitating interpretation. The number of legal moves to
completion increased from an average of 15.8 when the
high interactivity condition was experienced first to 17.2
when it was experienced second. And while this was not a
significant increase, the pattern suggests that participants
did not acquire an appreciation of a more efficient path to
solutionwhich would lead to the selection of fewer
moveson the basis of their experience with the low
interactivity condition. The release from the cognitively
demanding experience with the low interactivity condition
coupled with familiarity with the problem lead participants
to select more moves, and interactivity enabled them to do
so quickly. Moves provide information, and as participants
produced more moves, they were able to reach the goal
state faster.
A higher level of interactivity led to improved
performance in the river-crossing problem, when preceded
with the experience of solving the problem in a context
that did not afford the physical manipulation of the
problem presentation. Learning from previous experience
with the problem, coupled with the reduction in the mental
cost of making moves through interactivity provided the
solver with the freedom to experiment with more moves.
Through the interaction with artefacts, individuals were
provided with the opportunity to extend the process of
thinking beyond the mind and into the physical world.
These data underscore the importance of pursuing a
program of research that explicitly contrasts performance
when participants can manipulate a physical problem
presentation and when they cannot. In addition, we would
argue that such research efforts offer a more representative
window onto problem solving behavior observed outside
the psychologist’s laboratory.
Acknowledgments
We would like to thank Natalie Dorman and K’Dee
Bernard for their assistance with the recruitment and
running of the participants, and Chris Askew for helpful
suggestions.
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... This may reflect the presumption that it is the same thing to think in the head as it is to think with objects in the world. In contrast, developmental psychologists who worked with the river crossing task, being less sanguine about 'formal operations' presumably, have taken care to design rich interactive thinking environments with physical materials representing the boat, the river, and figurines corresponding to the cover story characters (e.g., Gholson et al. 1987; see also Vallée-Tourangeau et al. 2013). ...
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