Abstracts
Talks
Multiplication-format bias in algebraic modeling
Author: Miriam Bassok, Katja Borchert & Kristie Fisher
Affiliation: U. Washington
The hallmark of expertise is that domain knowledge is interconnected.
However, even in college students, algebraic knowledge appears to be
conceptually disconnected from arithmetic knowledge. In a series of studies,
we asked students to construct or select algebraic equations that represent
comparison statements such as "There are six times as many students (S) as
professors (P)." We found that most students prefer to represent such
statements with multiplicative equations (e.g., 6*P = S) and refrain from
using mathematically isomorphic division equations (e.g., S/6 = P). This
preference for multiplicative equations leads to a high proportion of
modeling errors, whereby students reverse the order of the variables (e.g.,
6*S = P). Such errors can be readily eliminated by explicitly asking
students to construct division equations. Nonetheless, students believe that
only multiplication equations are appropriate algebraic representations of
quantitative relations, whereas division equations denote arithmetic
problem-solving procedures. We discuss the instructional practices that lead
to this "multiplication-format" bias in algebraic modeling, which blocks
vertical transfer of knowledge from arithmetic to algebra.
Towards a hierarchical neighbor clustering model of human performance on the traveling salesperson problem
Author: Matt Dry
Affiliation: K.U. Leuven
The Traveling Salesperson Problem (TSP) has received much interest in
recent years as an example of an optimization task which human observers
are highly adept at solving despite being computationally difficult. A
number of models have been proposed in an attempt to describe the
heuristics employed by the human observers. In this paper we outline a new
model, the Hierarchical Neighbor Cluster (HNC) model, which is based upon
a growing body of evidence suggesting that the visual system is generating
a Voronoi tessellation-like representation at an early stage in visual
processing. We approach the problem of describing human performance on the
TSP from a bottom-up perspective, highlighting the similarity between the
TSP and other cluster- and structure-perception tasks involving random and
semi-random dot patterns.
The Law of Large Numbers in Sampling-based Choice:
Greater Sample Size Leads to Lower Decision Thresholds
Author: Laurel Evans and Marc. J. Buehner
Affiliation: Cardiff University
Bernoulli's law of large numbers dictates that confidence in sample
estimations increases with sample size. Here we assert that an optimal
strategy for sampling-based choice follows from this law. We consider the
value of a choice option to be the percentage of positive information for it
and use the contingency (the value of option A minus the value of option B)
to determine which of two options is superior. Our main claim is that as
people gain confidence in the two sample values, they gain confidence in the
difference between them, and therefore are willing to accept a smaller
contrast between them when deciding when to stop sampling and make a choice.
Thus any decision threshold, when measured by contingency, must lower as
sample size increases. We present simulation evidence that shows that a
model of choice utilizing such a dynamic threshold is superior to a
comparable model that uses a fixed-contingency threshold and report
experimental data that show that humans indeed employ this superior
strategy. When presented with fixed small (4, 8) or large (16, 32) samples
of mixed positive and negative information for each of two options,
participants make choices, on average, for higher contingency situations
when sample sizes are small, and endorse choices based on lower
contingencies only when sample sizes are large. This behavior is consistent
with theories of information foraging, and Charnov's (1976) marginal value
theorem.
Collective problem solving
Author: Robert Goldstone & Michael Roberts
Affiliation:
Indiana U.
Problem solving research tends to focus on the behavior of single
individuals. However, members of teams, businesses, committees, clubs, and
political groups often need to solve problems together. Even if all members
of a group have a common goal, they still may have difficulty achieving the
goal because of failures to coordinate their contributions. To study group
coordination in reaching a shared goal, we have devised the "Group Binary
Search Task" in which group members, without communication, submit numbers
in an attempt to collectively sum to a randomly selected target number.
After receiving group feedback, members adjust their submitted numbers until
the target number is reached. For all groups, performance improves with task
experience, and group reactivity decreases over rounds. Our empirical
results provide evidence for adaptive coordination in human groups, and as
the coordination costs increase with group size, large groups adapt through
spontaneous role differentiation across members and increasing
self-consistency within members. The empirical results are best fit by an
Agent Based Model featuring a flexible, adaptive agent strategy in which
agents decrease their reactions when the group feedback changes.
Causal reasoning in problem solving
Author: York Hagmayer & Bjoern Meder
Affiliation:
U. Goettingen
There has been considerable debate on the role of causal learning and
reasoning in problem solving. On the one hand, causal knowledge seems to
enhance performance in control tasks and facilitates transfer to novel
situations. On the other hand, people showing a good performance (e.g., in
controlling a complex dynamic system) do not necessarily acquire explicit
knowledge about the underlying causal structure. This finding contrasts with
results from research on causal learning demonstrating that people have the
capacity to infer causal structures from a number of different cues.
Although both lines of research seem to be highly relevant for each other,
there was little interaction between the two areas in the past. One reason
might be their different focus. While research on problem solving often
focuses on participants' task performance and transfer to novel settings,
research on causal learning mainly focuses on the acquisition and use of
causal knowledge. Another reason might result from using different causal
systems. While research on problem solving tends to examine how people learn
to control dynamic, non-linear systems, causal learning research has been
dominated by the use of rather simple and non-dynamic systems. In a recent
project we started to bridge the gap between causal learning, repeated
decision making, and problem solving research. To examine the interplay
between these areas, we developed a new experimental paradigm, which allows
examining and comparing performance, causal learning, and transfer to novel
situations. First results show that mere good performance may result from
quite different strategies, yielding causal or non-causal representations of
a given decision problem. The acquired representations, however, strongly
affect later performance and problem solving strategies in novel situations.
Should (insight) problem solving research go neuro?
Author: Guenther Knoblich
Affiliation:
U. Birmingham
During the last two decades the cognitive neuroscience wave has led to a
focus on basic perceptual, motor, and cognitive processes. Research on
thinking and problem solving has become somewhat unfashionable. Could it be
time for problem solving researchers to regroup in order to stop a trend
that increasingly neglects insights of the cognitive revolution? This could
take the form of criticizing neuroscience's inability to deal with cognitive
processes that take longer than a few hundred milliseconds. It could also
take the form of developing methods that allow us to link models of thinking
and problem solving to neuroscientific evidence. Of course, the latter would
mean relaxing the dear constraint that implementation does not matter for
cognitive processing. In my talk I will discuss opportunities and problems
with using neuroscience methods in problem solving research, taking the
example of insight problem solving.
What lurks beneath the surface: perceptual and conceptual prerequisites for
the learning of complex problem solving
Author: Ken Koedinger
Affiliation:
Carnegie-Mellon U.
The goal of the presentation is to emphasize how much of the development
of expert problem solving in academic domains (e.g., algebra, geometry,
chemistry, physics, etc.) may look like the acquisition of general or
higher-order strategies, but is more often about implicit learning of deep
concepts or percepts that provide the basis for complex reasoning and
inference.
How to find Good, Optimal and Robust TSP solutions?
Author: Walter Kropatsch, Yll Haxhimusa
Affiliation:
Vienna University of Technology
The traveling salesperson (TSP) problem consists in finding the shortest
tour through n cities. The locations of these cities determine the solution
and the difficulty of finding it. One tour consists of the order in which
the n cities are visited while the length of the tour should be as short as
possible. Finding the optimal tour is known to be NP-hard. However humans
find "good" solutions in close to linear time. In the real world human's
input exclusively comes from sensors which do not provide perfect data.
Hence their solutions need to be first of all robust against the noise and
the imperfection of the data. "Good" solutions are in most cases
satisfactory. Furthermore tours are invariant to translation, rotation and
uniform scaling. New tours with different cities but the same tour length
can be constructed by simple local edit operations: insertion and deletion
of cities, moving a city's location, rearranging the connectivity of small
local configurations. These operations relate different problems and allow
to collect them in classes of same or similar tour length. We will explore
some properties of the 'solution space' with respect to robustness and
optimality and compare potential solution strategies which might well be
useful for solving problems with similar properties.
A perceptual-motor account of formal notational reasoning
Author: David Landy
Affiliation: UIUC
The marks people make on paper are important parts of our cognitive
environment. When we look at drawings, what makes those images realistic is
that they engage properties of our perceptual system which usually respond
to real physical objects and situations -- we treat the drawing as a thing.
Treating depictions as things makes it possible to apply powerful cognitive
and perceptual processes to them, to infer facts about images as one would
about their referents. I will present several studies suggesting that users
of modern algebraic notations can and do interact with notational exemplars
in the same way, treating them as literal physical objects such that learned
interactions with those objects guarantee successful formal behavior. The
result is that difficult abstract tasks are translated into physical
manipulations--symbol pushing. On this view, the specific perceptual
properties of external notations are of central significance to symbolic
reasoning. Although such reasoning often conforms to abstract mathematical
principles, it is implemented by perceptual and perceptual-motor processes
operating over the notational expressions themselves. Although this
perspective is largely compatible with existing cognitive architectures used
in problem-solving, it suggests approaches to the allocation of tasks:
rather than treating the perceptual processes as a way of getting problem
information into a formal reasoning system, we can see formal reasoning as
emerging from the coordination of perceptual, motor, and cognitive,
physical, and cultural processes.
Heuristic models of human performance on bandit problems
Author: Michael D. Lee
Affiliation:
U. California, Irvine
In bandit problems, people must choose between a set of alternatives to
maximize the total reward over a series of trials. Each alternative has a
fixed but unknown rate of reward, drawn from a fixed but unknown
environmental distribution. In the finite horizon bandit problem we study,
there are only a small number of trials, and so people must balance between
exploring alternatives searching for the most rewarding one, versus
exploiting those alternatives already tried and known to be reasonably good.
Because it captures the exploration-exploitation trade-off, the bandit
problem is representative of many real-world learning and optimization
problems. In this study we develop various interpretable heuristic models of
people's decision-making in bandit problem. These models include standard
models from reinforcement learning and game theory, as well as
psychologically motivated models. Each model is fit to human data using
Bayesian graphical models, and their parameters are estimated using Markov
Chain Monte Carlo methods. These heuristic models are used to understand
human performance, to characterize the nature of optimal decision-making,
and to compare human to optimal decision-making.
Rebus, RAT and restructuring: Relationships among candidate insight problems
Author: James N. MacGregor & J. Barton Cunningham
Affiliation:
U. Victoria
Recently, new sources of candidate insight problems have been identified,
including matchstick arithmetic, Remote Associates, and rebus puzzles. These
new sources have the potential to improve the pool of problems for studying
insight problem solving. Traditionally, the available stimulus problems have
been limited to an ad hoc and heterogeneous collection of verbal riddles and
spatial puzzles, some of which are close to impossible to solve without
hints. To make matters worse, the insight status of some of these has been
questioned. The new sources represent a promising development, since each
consists of a large pool of similar problems that range in level of
difficulty. This paper explores relationships among some of the more recent
types of problem and a variety of "classic" insight tasks. The results
reinforce the insight status of some types of problem, while raising doubts
about others.
Structured Representation, Layered Processing, Memory Retrieval and
Affordances: All the Stuff You Need to Get to Heuristic Search
Author: Stellan Ohlsson
Affiliation:
U. Illinois, Chicago
The 1972 theory of problem solving was one of the major breakthroughs in
the study of higher cognition, because it presented a theory that was
precise, sufficient and responsible to data. There is no reason to doubt
that people engage in tentative action, mental lookahead and problem state
evaluation, the three central processes in the heuristic search theory. The
theory erred less in being false than in being incomplete: It said nothing
about how the thinker arrives at the point where heuristic search can
begin. The present talk will outline the major processing components that I
believe a thinking agent/system needs to have to be in a position to conduct
heuristic search. These include percepts with constituent structure,
constructed in layers of processing, which serve as memory retrieval probes,
especially retrieval of the possibilities for action inherent in the current
state of the world. Taken together, these components, plus the major
processes of heuristics search, provides a reasonably complete specification
of what it means to think.
Satisficing and optimizing in spatial problem-solving
Author: Tom Ormerod
Affiliation:
Lancaster U.
Since the seminal work of Newell & Simon, researchers have endeavoured to
identify the heuristics that humans use to tackle complex and unfamiliar
problems. Generally, such heuristics have provided sub-optimal but
sufficient methods. However, recent research from Gigerenzer and colleagues
points to a range of environmentally-determined heuristics that allow
near-optimal performance across a range of problem domains. In this paper,
we examine how different classes of heuristic can be brought to bear on
'vehicle routing problems', optimisation problems which contain perceptual
and computational task components, and which consequently might be variously
amenable to satisficing or optimising heuristic strategies.
Solving the Traveling Salesman Problem in two and three dimensions
Author: Zygmunt Pizlo, Joseph Catrambone, Edward Carpenter, Emil
Stefanov, David Foldes, and Yll Haxhimusa
Affiliation:
Purdue U.
Previous studies of how human subjects solve TSP involved problems on a
plane. There have been three main models of this ability: (i) convex hull
followed by cheapest insertion, (ii) crossing avoidance, and (iii)
coarse-to-fine pyramid approximations. We will present new results on how
humans solve TSP in a 3D volume, as well as on 3D surfaces. 3D volumetric
TSP will provide new tests for existing models. TSP on 3D surfaces will test
humans and models in natural cases of a non-Euclidean TSP.
Mathematical problem solving: Establishing a bridge between cognitive
science and education
Author: Bethany Rittle-Johnson
Affiliation:
Vanderbilt U.
Basic research in cognitive science has great potential for informing the
design and evaluation of interventions to improve learning in schools. The
problem solving literature should be particularly fruitful for identifying
learning processes and environments that enhance learning in mathematics and
science. However, such a multidisciplinary use of research on problem
solving poses many challenges. I will discuss efforts to bridge between
research on mathematical problem solving and mathematics teaching and
learning, focusing on two illustrative examples. One is the need to
establish a common meaning of problem solving and its relations to other
important learning outcomes in mathematics. For instance, procedural
knowledge is defined differently in the two literatures, and mathematics
education researchers focus more attention on procedural flexibility and
conceptual knowledge. The second example illustrates application of a basic
process, comparison, which supports problem solving in laboratory studies.
Indeed, supporting comparison in mathematics classrooms supported greater
student learning (Rittle-Johnson & Star, 2007; in press). At the same time,
this research revealed implicit assumptions and suggested new directions for
research in cognitive science on comparison. Opportunities for productive
interchanges between problem solving research and educational practice
exist, but they require open dialogue and multi-disciplinary collaboration
between the different fields.
Human problem solving; Influences of memory and conceptual organization
Author: Brian Ross
Affiliation:
U. Illinois, Urbana-Champaign
Research in human problem solving continues to be heavily affected
by Newell and Simon's view, but it has also been influenced by approaches
from other areas of cognition. I examine some of these influences from
memory and categorization on how people solve problems. I then discuss some
recent work on the learning of conceptual knowledge in a complex problem
solving domain. Finally, I speculate on some of the issues that might be
important for future work in the field.
Causality, decision making, and problem solving
Author: Steven Sloman
Affiliation:
Brown U.
Most problems occur within causal systems, systems that have temporal
dynamics and on which agents can intervene. I will discuss the possibility
that these are the kinds of problems that people are best at solving, and
that people impose causality to solve problems even where it does not
belong. Suggestive evidence is that people construe mathematical relations
as causal, preferring particular mathematical forms to others. Although this
constrains the symbolic representations people have available, it makes it
possible for us to use the powerful mechanisms we have for reasoning and
deciding in the real world to think abstractly.
Recursive problem solving strategies
Author: Ulrike Stege
Affiliation:
U. Victoria
The concept of recursion is usually introduced in the first year of a
Computer Science undergraduate university degree. Anecdotal evidence tells
us that students and instructors are often not comfortable with the concept
when learning or teaching it. We investigate the abilities of K-12 students
with respect to recursion. In particular, we investigate whether young
students use recursion as one of the possible problem solving methods. We
report on preliminary experiments when the task given is sorting items using
the comparison based sorting model.
Minimal control: problem solving in the real world
Author: Niels Taatgen
Affiliation:
Carnegie-Mellon U.
Problem solving is traditionally studied without acknowledging
interaction with the outside world. If perception has a role at all, it is
only to extract information from the world. More recently, the embedded
cognition movement has forwarded the idea that behavior is largely
controlled by the interaction with the world, leaving almost no role for
internal search, planning and problem solving processes. Nevertheless,
control cannot be completely relinquished to the environment, because the
way we act in certain circumstances depends on what our current goals and
intentions are, and in what particular state these goals are. To achieve
human-level flexibility and robustness in problem solving, though, it is
necessary to offload as much control as possible to the environment (Taatgen,
2005; 2007). This is especially necessary if problem solving is studied in
the context of multi-tasking, where multiple problems have to be interleaved
(Salvucci & Taatgen, 2008). In my talk, I will show examples from the domain
of airplane automation to highlight how these principles can improve
instruction (Taatgen et al., in press).
What makes a problem hard (or easy)? A computational perspective
Author: Iris van Rooij
Affiliation:
Radboud U. Nijmegen
There are many ways in which a problem can be hard or easy. In this talk
I will focus on one such meaning: a problem is hard if solving it requires
an excessive amount of time. NP-complete---or otherwise NP-hard---problems
are traditionally considered to be hard in this sense. This notion of
hardness has been playing an important role in debates in cognitive science
over the last decades, among them debates on the modularity of mind and the
heuristic nature of human rationality. In these debates often claims have
been made (explicitly or implicitly) about what it is that makes a given
problem hard. Reasons that are commonly listed include the following: (1)
optimization is hard, (2) solving a problem exactly is hard, (3) problems
with large search spaces are hard. On the other hand, there are also claims
about what characterizes easy problems, including: (4) satisficing is
relatively easy, (5) heuristics are relatively easy, and (6) approximation
is relatively easy. In this talk I discuss the misleading nature of these
claims. Drawing on insights from complexity theory, I propose an alternative
way of addressing the question "What makes a problem hard (or easy)?", one
that recognizes that the hardness or easiness of a problem often depends on
a complex interplay of a problem's parameters.
On the Computational Complexity of Analogy-Based Models of Problem Solving: Implications and Opportunities
Author: Todd Wareham
Affiliation: Memorial University of Newfoundland
In addition to being an object of study in its own right, the process of
deriving the best possible analogy between two situations or concepts is
also of use in general problem solving. For example, given a set of pairs of
problem-instances and problem-solving strategies that proved useful in in
the past in solving each of these instances, one way of selecting an
appropriate strategy for a new instance is to find the stored instance that
is most analogous (and hence relevantly similar) to the new instance and
then employ that stored situation's strategy. If this strategy in turn is
phrased in terms of roles or slots that are filled by particular aspects of
the stored instance, e.g., move(X, Y) where X and Y are entities in instance
I, the derived analogy is also useful in establishing the corresponding
aspects of the new instance. Though deriving analogies is known to be
computationally difficult (i.e., NP-hard) in general, it may yet be feasible
to derive analogies in the restricted situations encountered by human beings
in everyday life. In this talk, I will summarize known computational
complexity results (including parameterized results presented in van Rooij
et al. (2008)) for analogy derivation within the popular structure-mapping
framework proposed in Gentner (1983) and discuss the implications of these
results for analogy-based models of problem solving.
Posters
Use of Metacognitive Prompts and Manipulatives Promotes Efficient and Innovative Learning
Author: Belenky, D.M., & Nokes, T.J.
Affiliation: University of Pittsburgh
How does the type of learning material impact what is learned? The current research investigates
the nature of the benefits observed in students’ learning of math concepts when using manipulatives
(Uttal, Scudder & DeLoache, 1997). We examine how the type of manipulative (abstract or concrete)
and problem solving prompt (metacognitive or problem-focused) affects the efficient versus innovative
learning of probability concepts. Schwartz, Bransford, & Sears (2005) have hypothesized that adaptive
expertise consists of a balance between efficient learning, which results in quick, accurate
performance on a target task but does not transfer easily to new situations, and innovative learning,
which transfers more easily but does not show the same gains in skill acquisition in the original
context. Preliminary results suggest that pairing concrete manipulatives with meta-cognitive prompts
facilitates efficient and innovative learning as measured by problem solving transfer.
Causality and the perception of time
Author: Marc J Buehner
Affiliation: Cardiff University, UK
From working out questions such as "Will there still be a crop yield this year -- it seems awfully late in
the season?" to synchronizing perceptual input from multiple modalities (which process information at
different speeds, and the information itself arrives not necessarily at the same time, e.g. vision and
sound), people are constantly faced with solving timing problems. More specifically, having and maintaining
a sense of how much time has passed between one event and another is of fundamental importance to adaptive
cognition. Recent demonstrations of “intentional binding” (e.g. Haggard et al., 2002) suggest that people
experience a subjective shortening of time between actions and their consequences relative to unrelated
events. In this talk I will present data that suggests that intentional binding is a special form of
‘causal binding’ (Eagleman & Holcombe, 2002). In a reverse interpretation of Hume’s principles of causality,
according to which temporal contiguity is a key to forming causal associations, I shall argue that
experienced causality warps our perception of time in line with our expectations of natural timeframes.
TSP in 3D
Author: Joseph Catrambone, Emil Stefanov, Yll Haxhimusa & Zyg Pizlo
Affiliation: Purdue University
Lookahead and Feeling-of-Warmth in Insight Problem Solving
Author: Yun Chu & Edward P. Chronicle
Affiliation: SUNY Purchase Psychology Department
Sixty participants gave feeling-of-warmth (FOW) ratings to a computerized version of the cheap necklace problem
(CNP) in one of two conditions: a 6-move CNP sequence leading to the correct solution or a maximizing sequence
leading to no solution. Each move was presented on the screen for 15 seconds in an attempt to control for
participants’ lookahead. The participants were asked to give an FOW rating from 1-7. The results indicate that
contrary to the generally held belief that insight solutions appear all of a sudden, a somewhat gradual FOW
pattern emerged in both conditions. However, the specific rating changes for each move between the groups were
different. In addition, individual differences in lookahead were still observed even though exposure time to the
moves was limited. The role of lookahead in insight problem solving needs further investigation.
Is cost-benefit analysis possible for complex cognition? The case of investigative interviewing
Author: Coral J. Dando
Affiliation: Lancaster University, UK
Most theories of problem-solving and reasoning embody an implicit assumption that humans are intuitive cost-benefit
analysts. One domain in which the assumption is thought to hold is in police decision-making, specifically in the
context of investigative interviewing. The superiority of the Cognitive Interview (CI) method for optimising memorial
performance of witnesses has been demonstrated empirically. The CI is a homogenous procedure comprising several
components that contribute individually and incrementally to the CI superiority effect. However, the CI requires
considerable cognitive effort on the part of the interviewer, and investigators do not always apply the procedure
as a whole. Some components are more demanding than others, and it has been suggested that these are rejected in order
to simplify the task, particularly in time critical situations. We report findings that challenge this assumption.
Experienced investigators favour some of the demanding components over more straightforward techniques, despite often
perceiving the former to be less effective and more difficult to apply than the latter. One explanation for the absence
of cost-benefit analysis is that each component of CI gains its costs and benefits from conceptually different sources,
so monitoring relative costs and benefits itself becomes cognitively costly. Instead, we suggest that an investigator’s
context-bound goals may yield reward from the application of the most complex components in a way that determines and
then reinforces their ‘value’ in a manner that overrides cognitive economy.
Curing Recursion Aversion
Author: Katherine Gunion
Affiliation: University of Victoria, Canada
In Search of Transfer: Do Concrete Symbols Sometimes Make the Best Foundation?
Author: Percival Matthews
Affiliation: Vanderbilt University
When teaching abstract principles, we generally use symbols to represent the underlying ideas we wish to communicate.
This experiment examined how the nature of the symbols used for teaching affected learning and transfer in a novel
abstract mathematical domain. We found that the degree to which symbol use aligned with prior integer arithmetic
schemas affected both learning and transfer. Symbols used in a way misaligned or competing with prior arithmetic
schemas impeded learning, compared to aligned and abstract symbols. Although learning was equivalent with abstract
and with aligned symbols, abstract symbols supported transfer better than misaligned symbols. On the other hand,
patterns in the data suggest that misaligned concrete symbols may have supported transfer better than abstract
symbols. This suggests that concrete symbols can sometimes facilitate transfer better than abstract ones,
contrary to some other recent findings (Sloutsky, Kaminski & Heckler, 2005).
When three heads are better than one: Effects of collaboration and mixed expertise on effective problem solving
Author: Jennifer Wiley, Patrick Cushen & Andrew Jarosz
Affiliation: University of Illinois at Chicago
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