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Purdue Winer Memorial Lectures: New Perspectives on Human Problem Solving



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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.


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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