Modeling Mathematical Reasoning as Trained Perception-Action Procedures

Goldstone, R. L., Weitnauer, E., Ottmar, E., Marghetis, T., & Landy, D. H. (2016).   Modeling Mathematical Reasoning as Trained Perception-Action ProceduresIn R. Sottilare, A. Graesser, X. Hu, A. Olney, B. Nye, and A. Sinatra (Eds.) Design Recommendations for Intelligent Tutoring Systems: Volume 4 – Domain Modeling. Orlando, FL: U.S. Army Research Laboratory. (pp. 213-223).

We have observed that when people engage in algebraic reasoning, they often perceptually and spatially transform algebraic notations directly rather than first converting the notation to an internal, non spatial representation. We describe empirical evidence for spatial transformations, such as spatially compact grouping, transposition, spatially overlaid intermediate results, cancelling out, swapping, and splitting. This research has led us to understand domain models in mathematics as the deployment of trained and strategically crafted perceptual-motor processes working on grounded and strategically crafted notations. This approach to domain modeling has also motivated us to develop and assess an algebra tutoring system focused on helping students train their perception and action systems to coordinate with each other and formal mathematics. Overall, our laboratory and classroom investigations emphasize the interplay between explicit mathematical understandings and implicit perception action training as having a high potential payoff for making learning more efficient, robust, and broadly applicable.

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Comparison versus reminding

Tullis, J. G. & Goldstone, R. L. (2016).  Comparison versus reminding.  Cognitive Research: Principles and Implications, 1, 1-20, DOI 10.1186/s41235-016-0028-1.

Comparison and reminding have both been shown to support learning and transfer. Comparison is thought to support transfer because it allows learners to disregard non-matching features of superficially different episodes in order to abstract the essential structure of concepts. Remindings promote memory for the individual episodes and generalization because they prompt learners to retrieve earlier episodes during the encoding of later related episodes and to compare across episodes. Across three experiments, we compared the consequences of comparison and reminding on memory and transfer. Participants studied a sequence of related, but superficially different, proverb pairs. In the comparison condition, participants saw proverb pairs presented together and compared their meaning. In the reminding condition, participants viewed proverbs one at a time and retrieved any prior studied proverb that shared the same deep meaning as the current proverb. Experiment 1 revealed that participants in the reminding condition recalled more proverbs than those in the comparison condition. Experiment 2 showed that the mnemonic benefits of reminding persisted over a one-week retention interval. Finally, in Experiment 3, we examined the ability of participants to generalize their remembered information to new items in a task that required participants to identify unstudied proverbs that shared the samemeaning as studied proverbs. Comparison led to worse discrimination between proverbs related to studied proverbs and proverbs unrelated to studied proverbs than reminding. Reminding supported better memory for individual instances and transfer to new situations than comparison.

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Mastering algebra retrains the visual system to perceive hierarchical structure in equations

Marghetis, T., Landy, D., & Goldstone, R. L. (2016).  Mastering algebra retrains the visual system to perceive hierarchical structure in equations.  Cognitive Research: Principles and Implications, 1(25), 1-10, DOI 10.1186/s41235-016-0020-9.

Formal mathematics is a paragon of abstractness. It thus seems natural to assume that the mathematical expert should rely more on symbolic or conceptual processes, and less on perception and action. We argue instead that mathematical proficiency relies on perceptual systems that have been retrained to implement mathematical skills. Specifically, we investigated whether the visual systemin particular, object-based attentionis retrained so that parsing algebraic expressions and evaluating algebraic validity are accomplished by visual processing. Object-based attention occurs when the visual system organizes the world into discrete objects, which then guide the deployment of attention. One classic signature of object-based attention is better perceptual discrimination within, rather than between, visual objects. The current study reports that object-based attention occurs not only for simple shapes but also for symbolic mathematical elements within algebraic expressionsbut only among individuals who have mastered the hierarchical syntax of algebra. Moreover, among these individuals, increased object-based attention within algebraic expressions is associated with a better ability to evaluate algebraic validity. These results suggest that, in mastering the rules of algebra, people retrain their visual system to represent and evaluate abstract mathematical structure. We thus argue that algebraic expertise involves the regimentation and reuse of evolutionarily ancient perceptual processes. Our findings implicate the visual system as central to learning and reasoning in mathematics, leading us to favor educational approaches to mathematics and related STEM fields that encourage students to adapt, not abandon, their use of perception.

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Discovering Psychological Principles by Mining Naturally Occurring Data Sets

Goldstone, R. L., & Lupyan, G. (2016).  Harvesting naturally occurring data to reveal principles of cognitionTopics in Cognitive Science, 8, 548-568.

The very expertise with which psychologists wield their tools for achieving laboratory control may have had the unwelcome effect of blinding psychologists to the possibilities of discovering principles of behavior without conducting experiments. When creatively interrogated, a diverse range of large, real-world data sets provides powerful diagnostic tools for revealing principles of human judgment, perception, categorization, decision-making, language use, inference, problem solving, and representation. Examples of these data sets include patterns of website links, dictionaries, logs of group interactions, collections of images and image tags, text corpora, history of financial transactions, trends in twitter tag usage and propagation, patents, consumer product sales, performance in high-stakes sporting events, dialect maps, and scientific citations. The goal of this issue is to present some exemplary case studies of mining naturally existing data sets to reveal important principles and phenomena in cognitive science, and to discuss some of the underlying issues involved with conducting traditional experiments, analyses of naturally occurring data, computational modeling, and the synthesis of all three methods.This article serves as the introduction to a TopiCS topic with the same name.  The rest of the downloadable papers in this Topic are:

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Berger, J. (2016). Does presentation order impact choice after delay? Topics in Cognitive Science.

Christiansen, M. H., & Monaghan, P. (2016). Division of labor in vocabulary structure: Insights from corpus analyses. Topics in Cognitive Science, 8, 670684. doi: 10.1111/tops.12205.

Griffiths, T. L., Abbott, J. T., & Hsu, A. S. (2016). Exploring human cognition using large image databases. Topics in Cognitive Science, 8, 569588. doi: 10.1111/tops.12209.

Heit, E., & Nicholson, S. P. (2016). Missing the party: Political categorization and reasoning in the absence of party label cues. Topics in Cognitive Science, 8, 697714. doi: 10.1111/tops.12206.

Koedinger, K. R., Yudelson, M. V., & Pavlik Jr., P. I. (2016). Testing theories of transfer using error rate learning curves. Topics in Cognitive Science, 8, 589609. doi: 10.1111/tops.12208.

Moat, H. S., Olivola, C. Y., Chater, N., & Preis, T. (2016). Searching choices: Quantifying decision making processes using search engine data. Topics in Cognitive Science, 8, 685696. doi: 10.1111/tops.12207.

Pope, D. G. (2016). Exploring psychology in the field: Steps and examples from the used-car market. Topics in Cognitive Science, 8, 660669. doi: 10.1111/tops.12210.

Vincent-Lamarre, P., Blondin Masse, A., Lopes, M., Lord, M., Marcotte, O., & Harnad, S. (2016). The latent structure of dictionaries. Topics in Cognitive Science, 8, 625659. doi: 10.1111/tops.12211.

 

An in vivo study of self-regulated study sequencing in introductory psychology courses

Carvalho, P.F., Braithwaite, D.W., de Leeuw, J.R., Motz, B.A., & Goldstone, R.L. (2016). An in vivo study of self-regulated study sequencing in introductory psychology courses. PLoS ONE 11(3): e0152115.

Study sequence can have a profound influence on learning. In this study we investigated how students decide to sequence their study in a naturalistic context and whether their choices result in improved learning. In the study reported here, 2061 undergraduate students enrolled in an Introductory Psychology course completed an online homework tutorial on measures of central tendency, a topic relevant to an exam that counted towards their grades. One group of students was enabled to choose their own study sequence during the tutorial (Self-Regulated group), while the other group of students studied the same materials in sequences chosen by other students (Yoked group). Students who chose their sequence of study showed a clear tendency to block their study by concept, and this tendency was positively associated with subsequent exam performance. In the Yoked group, study sequence had no effect on exam performance. These results suggest that despite findings that blocked study is maladaptive when assigned by an experimenter, it may actually be adaptive when chosen by the learner in a naturalistic context.

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Index of Supplemental Videos for “An Integrated Computational Model of Perception and Scientific Discovery in a Very Simple World, Aiming at Psychological Realism”

Below is an index of supplemental videos for the manuscript:

Lara-Dammer, F., Hofstadter, D. R., & Goldstone, R. L. (under review).An Integrated Computational Model of Perception and Scientific Discovery in a Very Simple World, Aiming at Psychological Realism

Overview V1

Plausible Approach V1

Plausible Approach V2

Object Identification V1

Object Identification V2

Object Identification V3

Object Identification V4

Object Identification V5

Same Angle V1

Same Angle V2

Ambiguous Event V1

Ambiguous Event V2

Direction Parameter V1

Direction Parameter V2

Direction Parameter V3

Direction Parameter V4

Direction Parameter V5

Stability V1

Stability V2

First Discovery 1

First Discovery 2

First Discovery 3

First Discovery 4

Pressure 1

Pressure 2

Pressure 3

Pressure 4

Kinetic Energy 1

Kinetic Energy 2

Kinetic Energy 3

Kinetic Energy 4

Free Space 1

Free Space 2

Free Space in a Circle (Tricycle)

Boyle’s Law Sophisticated A (Tricycle)

Boyle’s Law Sophisticated B (Tricycle)

Boyle’s Law Sophisticated C (Tricycle)

Non-ideal Gas A (non-success, Tricycle)

Non-ideal Gas B (Tricycle)

Non-ideal Gas C (Tricycle)

Understanding Noise (Tricycle)

Failed Discovery (Tricycle)

Thinking in Groups A (Tricycle)

Thinking in Groups B (Tricycle)

Informal mechanisms in mathematical cognitive development: The case of arithmetic

Braithwaite, D. W., Goldstone, R. L., van der Maas, H. L . J., & Landy, D. H. (2016).  Informal mechanisms in mathematical cognitive development: The case of arithmetic.  Cognition, 149, 40-55.

The idea that cognitive development involves a shift towards abstraction has a long history in psychology. One incarnation of this idea holds that development in the domain of mathematics involves a shift from non-formal mechanisms to formal rules and axioms. Contrary to this view, the present study provides evidence that reliance on non-formal mechanisms may actually increase with age. Participants – Dutch primary school children – evaluated three-term arithmetic expressions in which violation of formally correct order of evaluation led to errors, termed foil errors. Participants solved the problems as part of their regular mathematics practice through an online study platform, and data were collected from over 50,000 children representing approximately 10% of all primary schools in the Netherlands, suggesting that the results have high external validity. Foil errors were more common for problems in which formally lower-priority sub-expressions were spaced close together, and also for problems in which such sub-expressions were relatively easy to calculate. We interpret these effects as resulting from reliance on two non-formal mechanisms, perceptual grouping and opportunistic selection, to determine order of evaluation. Critically, these effects reliably increased with participants’ grade level, suggesting that these mechanisms are not phased out but actually become more important over development, even when they cause systematic violations of formal rules. This conclusion presents a challenge for the shift towards abstraction view as a description of cognitive development in arithmetic. Implications of this result for educational practice are discussed.

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Integration of social information by human groups

Granovskiy, B., Gold, J. M., Sumpter, D., & Goldstone, R. L. (2015). Integration of social information by human groups. Topics in Cognitive Science, 7, 469-493.

We consider a situation in which individuals search for accurate decisions without direct feedback on their accuracy, but with information about the decisions made by peers in their group. The “wisdom of crowds” hypothesis states that the average judgment of many individuals can give a good estimate of, for example, the outcomes of sporting events and the answers to trivia questions. Two conditions for the application of wisdom of crowds are that estimates should be independent and unbiased. Here, we study how individuals integrate social information when answering trivia questions with answers that range between 0% and 100% (e.g., “What percentage of Americans are left-handed?”). We find that, consistent with the wisdom of crowds hypothesis, average performance improves with group size. However, individuals show a consistent bias to produce estimates that are insufficiently extreme. We find that social information provides significant, albeit small, improvement to group performance. Outliers with answers far from the correct answer move toward the position of the group mean. Given that these outliers also tend to be nearer to 50% than do the answers of other group members, this move creates group polarization away from 50%. By looking at individual performance over different questions we find that some people are more likely to be affected by social influence than others. There is also evidence that people differ in their competence in answering questions, but lack of competence is not significantly correlated with willingness to change guesses. We develop a mathematical model based on these results that postulates a cognitive process in which people first decide whether to take into account peer guesses, and if so, to move in the direction of these guesses. The size of the move is proportional to the distance between their own guess and the average guess of the group. This model closely approximates the distribution of guess movements and shows how outlying incorrect opinions can be systematically removed from a group resulting, in some situations, in improved group performance. However, improvement is only predicted for cases in which the initial guesses of individuals in the group are biased.

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Effectiveness of learner-regulated study sequence: An in-vivo study in introductory psychology courses

Cavalho, P. F., Braithwaite, D. W., de Leeuw, J. R., Motz, B. A., & Goldstone, R. L. (2015).  Effectiveness of learner-regulated study sequence: An in-vivo study in introductory psychology courses.  Proceedings of the Thirty-Seventh Annual Conference of the Cognitive Science Society.  (pp. 309-314).  Pasadena, CA: Cognitive Science Society.

Study sequence can have a profound impact on learning. Previous research has often shown advantages for interleaved over blocked study, though the reverse has also been found. Learners typically prefer blocking even in situations for which interleaving is superior. The present study investigated learner regulation of study sequence, and its effects on learning in an ecologically valid context – university students using an online tutorial relevant to an exam that counted toward their course grades. The majority of participants blocked study by problem category, and this tendency was positively associated with subsequent exam performance. The results suggest that preference for blocked study may be adaptive under some circumstances, and highlight the importance of identifying task environments under which different study sequences are most effective.

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Brain self-portraits: the cognitive science of visualizing neural structure and function

Goldstone, R. L., Börner, K., & Pestilli, F. (2015).  Brain self-portraits: the cognitive science of visualizing neural structure and function.  Trends in Cognitive Sciences, 19, 462-474.

With several large-scale human brain projects currently underway and a range of neuroimaging techniques growing in availability to researchers, the amount and diversity of data relevant for understanding the human brain is increasing rapidly. A complete understanding of the brain must incorporate information about 3D neural location, activity, timing, and task. Data mining, highperformance computing, and visualization can serve as tools that augment human intellect; however, the resulting visualizations must take into account human abilities and limitations to be effective tools for exploration and communication. In this feature review, we discuss key challenges and opportunities that arise when leveraging the sophisticated perceptual and conceptual processing of the human brain to help researchers understand brain structure, function, and behavior.

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Homo Socialis and Homo Sapiens

Goldstone, R. L. (2015).  Homo Socialis and Homo Sapiens.  Review of Behavioral Economics, 2, 77-87.

[This paper is a commentary on the following article: Gintis, H., & Helbing, D. (2015). Homo Socialis: An Analytical Core for Sociological Theory.  Review of Behavioral Economics.]

Explaining how patterns of collective behavior emerge from interactions among individuals with diverse, sometimes opposing, goals is a societally crucial and particularly timely pursuit. It is timely because humans are more tightly connected to one another now than ever before. From 1984 to 2014 there has been more than a million-fold increase in the number of devices that can reach the global digital network. Although web technology is new and transformative, from a broader perspective, it is also just a recent manifestation of humanity’s perpetual drive to become more intermeshed. Earlier manifestations of this drive include the printing press, global transportation networks, telecommunication systems, and the academy. These social networks have catalyzed the formation of otherwise unattainable social patterns. Understanding the origins and possible destinations of these social patterns is both scientifically and pragmatically consequential.

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Benefits of variation increase with preparation

Braithwaite, D. W., & Goldstone, R. L. (2014).  Benefits of variation increase with preparation.  Proceedings of the Thirty-Sixth Annual Conference of the Cognitive Science Society.  (pp. 1940-1945).  Quebec City, Canada: Cognitive Science Society.

Abstract concepts are characterized by their underlying structure rather than superficial features. Variation in the examples used to teach abstract concepts can draw attention towards shared structure and away from superficial detail, but too much variation can inhibit learning. The present study tested the possibility that increasing attention to underlying structural relations could alleviate the latter difficulty and thereby increase the benefits of varied examples. Participants were trained with either varied or similar examples of a mathematical concept, and were then tested on their ability to apply the concept to new cases. Before training, some participants received pre training aimed at increasing attention to the structural relations underlying the concept. The relative advantage of varied over similar examples was increased among participants who received the pretraining. Thus, preparation that promotes attention to the relations underlying abstract concepts can increase the benefits of learning from varied examples.

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Cycles and predictability in human collective behavior: An experimental generalization of Rock-Scissors-Paper

Here are some reports of our PLoS One paper on human collective behavior studying cyclic patterns in a generalization of the familiar rock-scissors-paper game.  We find situations in which groups of people grow increasingly predictable as they cycle around a set of choice options in a game similar to rock-scissors-paper but with 24 rather than 3 choices.

Social Learning Strategies in a Networked Group

Wisdom, T. N., Song, X., & Goldstone, R. L. (2013). Social Learning Strategies in a Networked Group.  Cognitive Science, 37, 1383-1425

When making decisions, humans can observe many kinds of information about others’ activities, but their effects on performance are not well understood. We investigated social learning strategies using a simple problem-solving task in which participants search a complex space, and each can view and imitate others’ solutions. Results showed that participants combined multiple sources of information to guide learning, including payoffs of peers’ solutions, popularity of solution elements among peers, similarity of peers’ solutions to their own, and relative payoffs from individual exploration. Furthermore, performance was positively associated with imitation rates at both the individual and group levels. When peers’ payoffs were hidden, popularity and similarity biases reversed, participants searched more broadly and randomly, and both quality and equity of exploration suffered. We conclude that when peers’ solutions can be effectively compared, imitation does not simply permit scrounging, but it can also facilitate propagation of good solutions for further cumulative exploration.

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See a movie of Creature League experiment in action

Integrating formal and grounded representations in combinatorics learning

Braithwaite, D. W., & Goldstone, R. L. (2013).  Integrating formal and grounded representations in combinatorics learning.  Journal of Educational Psychology, 105, 666-682.

The terms concreteness fading and progressive formalization have been used to describe instructional approaches to science and mathematics that use grounded representations to introduce concepts and later transition to more formal representations of the same concepts. There are both theoretical and empirical reasons to believe that such an approach may improve learning outcomes relative to instruction employing only grounded or only formal representations (Freudenthal, 1991; Goldstone & Son, 2005; McNeil & Fyfe, 2012; but see Kaminski, Sloutsky, & Heckler, 2008). Two experiments tested the effectiveness of this approach to instruction in the mathematical domain of combinatorics, using outcome listing and numerical calculation as examples of grounded and formal representations, respectively. The study employed a pretest-training, posttest design. Transfer performance, that is, participants’ improvement from pretest to posttest, was used to assess the effectiveness of instruction received during training. In Experiment 1, transfer performance was compared for 4 types of instruction, which differed only in the types of representation they employed: pure listing (i.e., listing only), pure formalism (i.e., numerical calculation only), list fading (i.e., listing followed by numerical calculation), and formalism first (i.e., listing introduced after numerical calculation). List fading instruction led to transfer performance on par with pure formalism instruction and higher than formalism first and pure listing instruction. In Experiment 2, an enhanced version of list fading training was again compared to pure formalism. However, no difference in transfer performance due to training was found. The results suggest that combining grounded and formal representations can be an effective approach to combinatorics instruction but is not necessarily preferable to using formal representations alone. If both grounded and formal representations are employed, the former should precede rather than follow the latter in the instructional sequence.

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Cyclic game dynamics driven by iterated reasoning

Frey, S., & Goldstone, R. L. (2013). Cyclic game dynamics driven by iterated reasoning. PLoS One, 8(2): e56416. doi:10.1371/journal.pone.0056416

Recent theories from complexity science argue that complex dynamics are ubiquitous in social and economic systems. These claims emerge from the analysis of individually simple agents whose collective behavior is surprisingly complicated. However, economists have argued that iterated reasoning–what you think I think you think–will suppress complex dynamics by stabilizing or accelerating convergence to Nash equilibrium. We report stable and efficient periodic behavior in human groups playing the Mod Game, a multi-player game similar to Rock-Paper-Scissors. The game rewards subjects for thinking exactly one step ahead of others in their group. Groups that play this game exhibit cycles that are inconsistent with any fixed-point solution concept. These cycles are driven by a ‘‘hopping’’ behavior that is consistent with other accounts of iterated reasoning: agents are constrained to about two steps of iterated reasoning and learn an additional one-half step with each session. If higher-order reasoning can be complicit in complex emergent dynamics, then cyclic and chaotic patterns may be endogenous features of real-world social and economic systems.
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See a movie of actual humans (shown as Xs) playing the Mod Game.  Notice the clumping of their moves and their regular progression around the circle of choices.

Benefits of Graphical and Symbolic Representations for Learning and Transfer of Statistical Concepts

Braithwaite, D. W, & Goldstone, R. L. (2013).  Benefits of Graphical and Symbolic Representations for Learning and Transfer of Statistical Concepts.  Proceedings of the Thirty-Fifth Annual Conference of the Cognitive Science Society.  (pp. 1928-1933).  Berlin, Germany: Cognitive Science Society.

Past research suggests that spatial configurations play an important role in graph comprehension. The present study investigates consequences of this fact for the relative utility of graphs and tables for interpreting data. Participants judged presence or absence of various statistical effects in simulated datasets presented in various formats. For the statistical effects introduced earlier in the study, performance was better with graphs than with tables, while for the effect introduced last in the study, this trend reversed. Additionally, in the later sections of the study, responses with graphs, but not tables, reflected increasing influence from the presence of stimulus features which had been relevant earlier in the study, but were no longer relevant. The findings suggest that graphs, relative to tables, may better facilitate perception of complex relationships among data points, but may also bias readers more strongly to favor some perspectives over others when interpreting data.

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An experiment on the cognitive complexity of code

Hansen, M. E., Lumsdaine, A., & Goldstone, R. L. (2013).  An experiment on the cognitive complexity of code.  Proceedings of the Thirty-Fifth Annual Conference of the Cognitive Science Society. Berlin, Germany: Cognitive Science Society.

What simple factors impact the cognitive complexity of code? We present an experiment in which participants predict the output of ten small Python programs. Even with such simple programs, we find a complex relationship between code, expertise, and correctness. We use subtle differences between program versions to demonstrate that small notational changes can have profound effects on comprehension. We catalog common errors for each program, and perform an in-depth data analysis to uncover effects on response correctness and speed.

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Cognitive Architectures: A Way Forward for the Psychology of Programming

Hansen, M. E., Lumsdaine, A., & Goldstone, R. L. (2012).  Cognitive Architectures: A Way Forward for the Psychology of Programming.  Onward! Workshop at the Third Annual SPLASH Conference 2012.

Programming language and library designers often debate the usability of particular design choices. These choices may impact many developers, yet scientific evidence for them is rarely provided. Cognitive models of program comprehension have existed for over thirty years, but lack quantitative definitions of their internal components and processes. To ease the burden of quantifying existing models, we recommend using the ACT-R cognitive architecture: a simulation framework for psychological models. In this paper, we provide a high-level overview of modern cognitive architectures while concentrating on the details of ACT-R. We review an existing quantitative program comprehension model, and consider how it could be simplified and implemented within the ACT-R framework. Lastly, we discuss the challenges and potential benefits associated with building a comprehensive cognitive model on top of a cognitive architecture.

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Inducing mathematical concepts from specific examples: The role of schema-level variation

Braithwaite, D. W., & Goldstone, R. L. (2012).  Inducing mathematical concepts from specific examples: The role of schema-level variation.  Proceedings of the Thirty-Fourth Annual Conference of the Cognitive Science Society.  (pp. 138-143).  Sapporo, Japan: Cognitive Science Society.

Previous research suggests that comparing multiple specific examples of a general concept can promote knowledge transfer. The present study investigated whether this approach could be made more effective by systematic variation in the semantic content of the specific examples. Participants received instruction in a mathematical concept in the context of several examples, which instantiated either a single semantic schema (non-varied condition) or two different schemas (varied condition). Schema-level variation during instruction led to better knowledge transfer, as predicted. However, this advantage was limited to participants with relatively high performance before instruction. Variation also improved participants’ ability to describe the target concept in abstract terms. Surprisingly, however, this ability was not associated with successful knowledge transfer.
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Teaching the Perceptual Structure of Algebraic Expressions: Preliminary Findings from the Pushing Symbols Intervention

Ottmar, E., Landy, D., & Goldstone, R. L. (2012).  Teaching the Perceptual Structure of Algebraic Expressions: Preliminary Findings from the Pushing Symbols Intervention.  Proceedings of the Thirty- Fourth Annual Conference of the Cognitive Science Society.  (pp. 2156-2161).  Sapporo, Japan: Cognitive Science Society.

We describe an intervention being developed by our research team, Pushing Symbols (PS). This intervention is designed to encourage learners to treat symbol systems as physical objects that move and change over time according to dynamic principles. We provide students with the opportunities to explore algebraic structure by physically manipulating and interacting with concrete and virtual symbolic systems that enforce rules through constraints on physical transformations. Here we present an instantiation of this approach aimed at helping students learn the structure of algebraic notation in general, and in particular learn to simplify like terms. This instantiation combines colored symbol tiles with a new touchscreen software technology adapted from the commercial Algebra Touch software. We present preliminary findings from a study with 70 middle-school students who participated in the PS intervention over a three-hour period.
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Improving Perception to Make Distant Connections Closer

Goldstone, R. L., Landy, D., & Brunel, L. (2011). Improving Perception to Make Distant Connections Closer. Frontiers in Perception Science, 2.
doi:10.3389/fpsyg.2011.00385

One of the challenges for perceptually grounded accounts of high-level cognition is to explain how people make connections and draw inferences between situations that superficially have little in common.  Evidence suggests that people draw these connections even without having explicit, verbalizable knowledge of their bases.  Instead, the connections are based on sub-symbolic representations that are grounded in perception, action, and space.  One reason why people are able to spontaneously see relations between situations that initially appear to be unrelated is that their eventual perceptions are not restricted to initial appearances.  Training and strategic deployment allow our perceptual processes to deliver outputs that would have otherwise required abstract or formal reasoning.  Even without people having any privileged access to the internal operations of perceptual modules, these modules can be systematically altered so as to better subserve our high-level reasoning needs.  Moreover, perceptually-based processes can be altered in a number of ways to closely approximate formally sanctioned computations.

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Innovation, imitation, and problem-solving in a networked group

Wisdom, T. N., & Goldstone, R. L. (2011).  Innovation, imitation, and problem-solving in a networked group.  Nonlinear Dynamics, Psychology, and Life Sciences, 15, 229-252.

We implemented a problem-solving task in which groups of participants simultaneously played a simple innovation game in a complex problem space, with score feedback provided after each of a number of rounds. Each participant in a group was allowed to view and imitate the guesses of others during the game. The results showed the use of social learning strategies previously studied in other species, and demonstrated benefits of social learning and nonlinear effects of group size on strategy and performance. Rather than simply encouraging conformity, groups provided information to each individual about the distribution of useful innovations in the problem space. Imitation facilitated innovation rather than displacing it, because the former allowed good solutions to be propagated and preserved for further cumulative innovations in the group. Participants generally improved their solutions through the use of fairly conservative strategies, such as changing only a small portion of one’s solution at a time, and tending to imitate solutions similar to one’s own. Changes in these strategies over time had the effect of making solutions increasingly entrenched, both at individual and group levels. These results showed evidence of nonlinear dynamics in the decentralization of innovation, the emergence of group phenomena from complex interactions of individual efforts, stigmergy in the use of social information, and dynamic tradeoffs between exploration and exploitation of solutions. These results also support the idea that innovation and creativity can be recognized at the group level even when group members are generally cautious and imitative.

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Going with the group in a competitive game of iterated reasoning

Frey, S., & Goldstone, R. L. (2011).  Going with the group in a competitive game of iterated reasoning.  Proceedings of the Thirty-Third Annual Conference of the Cognitive Science Society.  (pp. 1912-1917).  Boston, Massachusetts: Cognitive Science Society.

In some strategic games, thinking ahead about other players’ reasoning can lead to better predictions about what they will do. In other games, infinitely iterated reasoning ultimately prescribes random play. In an online experiment of strategic thinking in groups, we tested participants in a game with the formal structure of a random game, but the superficial struc- ture of a game that rewards iterated reasoning. We found that participants conformed to the superficial structure of the game, and earned more than they would have by playing randomly. We estimated how many steps participants thought ahead in the game and discovered implicit coordination at the group level. Participants unexpectedly “matched” their degree of iterated thinking to each other.

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Adaptive group coordination and role differentiation

Roberts, M. E., & Goldstone, R. L. (2011).  Adaptive Group Coordination and Role Differentiation.  PLoS One, 6, 1-8.

Many real world situations (potluck dinners, academic departments, sports teams, corporate divisions, committees, seminar classes, etc.) involve actors adjusting their contributions in order to achieve a mutually satisfactory group goal, a win-win result. However, the majority of human group research has involved situations where groups perform poorly because task constraints promote either individual maximization behavior or diffusion of responsibility, and even successful tasks generally involve the propagation of one correct solution through a group. Here we introduce a group task that requires complementary actions among participants in order to reach a shared goal. Without communication, group members 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 and self-consistency among members. We suggest several agent-based models with different rules for agent reactions, and we show that the empirical results are best fit by a flexible, adaptive agent strategy in which agents decrease their reactions when the group feedback changes. The task offers a simple experimental platform for studying the general problem of group coordination while maximizing group returns, and we distinguish the task from several games in behavioral game theory.

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Learning near-optimal search in a minimal explore/exploit task

Sang, K., Todd, P. M., & Goldstone, R. L. (2011). Learning near-optimal search in a minimal explore/exploit task. Proceedings of the Thirty-Third Annual Conference of the Cognitive Science Society. (pp. 2800-2805). Boston, Massachusetts: Cognitive Science Society.

How well do people search an environment for non-depleting resources of different quality, where it is necessary to switch between exploring for new resources and exploiting those already found? Employing a simple card selection task to study exploitation and exploration, we find that the total resources accrued, the number of switches between exploring and exploiting, and the number of trials until stable exploitation becomes more similar to those of the optimal strategy as experience increases across searches. Subjects learned to adjust their effective (implicit) thresholds for exploitation toward the optimal threshold over 30 searches. Those implicit thresholds decrease over turns within each search, just as the optimal threshold does, but subjects’ explicitly stated exploitation threshold increases over turns. Nonetheless, both the explicit and learned implicit thresholds produced performance close to optimal.

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Connecting instances to promote children’s relational reasoning

Son, J. Y., Smith, L. B., & Goldstone, R. L. (2011). Connecting instances to promote children’s relational reasoning.  Journal of Experimental Child Psychology, 108, 260-277.

The practice of learning from multiple instances seems to allow children to learn about relational structure. The experiments reported here have focused on two issues regarding relational learning from multiple instances: (1) what kind of perceptual situations foster such learning and (2) how particular object properties, such as complexity or similarity, interact with relational learning. Two kinds of perceptual situations were of interest here: simultaneous view, where instances are viewed at once, and sequential view, instances are viewed one at a time, one right after the other. We examine the influence of particular perceptual situations and object properties using two tests of relational reasoning: a common match-to-sample task (where new instances are compared to a common sample) and a variable match-to-sample task (where new instances are compared to a sample that varies on each trial). Experiments 1 and 2 indicate that simultaneous presentation of even highly dissimilar instances, one simple and one complex, effectively connects them together and improves relational generalization in both match-to-sample tasks. Experiment 3 showed simple samples are more effective than complex ones in the common match-to-sample task. However, when one instance is not used a common sample and various pairs of instances are simply compared (Experiment 4), simple and rich instances are equally effective at promoting relational learning. These results bear on our understanding of how children connect instances and how those initial connections affect learning and generalization.

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Analogical Transfer from a Simulated Physical System

Day, S., & Goldstone, R. L. (2011).  Analogical transfer from a simulated physical system.  Journal of Experimental Psychology: Learning, Memory, and Cognition37,551-567.

Previous research has consistently found that spontaneous analogical transfer is strongly tied to concrete and contextual similarities between the cases. However, that work has largely failed to acknowledge that the relevant factor in transfer is the similarity between individuals’ mental representations of the situations, rather than the overt similarities between the cases themselves. Across several studies, we find that participants are able to transfer strategies learned from a perceptually concrete simulation of a physical system to a task with very dissimilar content and appearance. This transfer is reflected in better performance on the transfer task when its underlying dynamics are consistent rather than inconsistent with the preceding training task. Our data indicate that transfer in these tasks relies on the perceptual and spatial nature of the training task, but does not depend on direct interaction with the system, with participants performing equally well after simply observing the concrete simulation. We argue that participants in these studies are using the concrete, spatial, dynamic information presented in the training simulation as the basis for a concretely similar mental model of the dissimilar transfer task. Unexpectedly, our data consistently showed that transfer was independent of reported recognition of the analogy between tasks: while such recognition was associated with better overall performance, it was not associated with better transfer (in terms of applying an appropriate strategy). Together, these findings suggest that analogical transfer between overtly dissimilar cases may be much more common—and much more relevant to our cognitive processing—than is generally assumed.

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Distinguishing Levels of Grounding that Underlie Transfer of Learning

Byrge, L. A., & Goldstone, R. L. (2011).  Distinguishing levels of grounding that underlie transfer of learning.  Proceedings of the Thirty-Third Annual Conference of the Cognitive Science Society.  (pp. 2818-2823).  Boston, Massachusetts: Cognitive Science Society.

We find that transfer of learning from a perceptually concrete simulation to an isomorphic but superficially dissimilar text- based problem is sensitive to the congruence between the force dynamics common to both systems and the kinesthetic schema induced via action in the first, perceptually concrete, simulation. Counterintuitively, incompatibility between the force dynamics and the kinesthetic schema has a beneficial effect on transfer, relative to compatibility as well as an unrelated control. We suggest that this incompatibility between action and system dynamics may make the system’s relational structure more salient, leading to a more flexible conceptualization that ultimately benefits transfer. In addition, we suggest that too much “action concreteness” in hands-on learning may actually limit transfer, by fostering an understanding that is tied to that action and therefore less available for transfer in situations where that action is no longer relevant.

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Functional structure and coordination failure in real-time group behavior

Frey, S., & Goldstone, R. L. (2010). Functional Structure and Coordination Failure in Real-Time Group Behavior. Proceedings of the Thirty-Second Annual Conference of the Cognitive Science Society.  (pp. 2093-2098).  Portland, Oregon: Cognitive Science Society.

We reveal spontaneous group formation and differentiation in an online dynamic coordination experiment. We observe increased group stratification and attribute it to increases in pairwise cooperative behavior, rather than uncooperative behavior. Our network analyses document the fine scale structure of coordination failure in the face of many established determinants of coordination success. We explore previous work in coordination failure to frame our own findings. Factors that have been previously shown to improve coordination in discretetime, forced-decision experimental games do not prevent decisive coordination failure in our real-time, asynchronous group decision-making environment.

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Priming a central executive search process: exploration and exploitation in generalized cognitive search processes

Hills, T. T., Todd, P. M., & Goldstone, R. L. (2010). Priming a Central Executive Search Process: Exploration and Exploitation in Generalized Cognitive Search Processes.  Journal of Experimental Psychology: General, 139, 560-609.

The trade-off between exploration and exploitation is common to a wide variety of problems involving search in space and mind. The prevalence of this trade-off and its neurological underpinnings led us to propose domain-general cognitive search processes (Hills, Todd, & Goldstone, 2008). We propose further that these are consistent with the idea of a central executive search process that combines goal-handling across subgoal hierarchies. In the present study, we investigate 3 aspects of this proposal. First, the existence of a unitary central executive search process should allow priming from 1 search task to another and at multiple hierarchical levels. We confirm this by showing cross-domain priming from a spatial search task to 2 different cognitive levels within a lexical search task. Second, given the neural basis of the proposed generalized cognitive search process and the evidence that the central executive is primarily engaged during complex tasks, we hypothesize that priming should require search in the sense of a self-regulated making and testing of sequential predictions about the world. This was confirmed by showing that when participants were allowed to collect spatial resources without searching for them, no priming occurred. Finally, we provide a mechanism for the underlying search process and investigate 3 alternative hypotheses for subgoal hierarchies using the central executive as a search process model (CESP). CESP envisions the central executive as having both emergent and unitary processes, with one of its roles being a generalized cognitive search process that navigates goal hierarchies by mediating persistence on and switching between subgoals.

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Proximity and precedence in arithmetic

Landy, D. H., & Goldstone, R. L. (2010).  Proximity and precedence in arithmetic. The Quarterly Journal of Experimental Psychology63, 1953-1968.

How does the physical structure of an arithmetic expression affect the computational processes engaged in by reasoners? In handwritten arithmetic expressions containing both multiplications and additions, terms that are multiplied are often placed physically closer together than terms that are added. Three experiments evaluate the role such physical factors play in how reasoners construct solutions to simple compound arithmetic expressions (such as “2 + 3 × 4”). Two kinds of influence are found: First, reasoners incorporate the physical size of the expression into numerical responses, tending to give larger responses to more widely spaced problems. Second, reasoners use spatial information as a cue to hierarchical expression structure: More narrowly spaced subproblems within an expression tend to be solved first and tend to be multiplied. Although spatial relationships besides order are entirely formally irrelevant to expression semantics, reasoners systematically use these relationships to support their success with various formal properties.

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When do words promote analogical transfer?

Son, J. Y., Doumas, L. A., & Goldstone, R. L. (2010).  When do words promote analogical transfer?  The Journal of Problem Solving3, 52-92.

The purpose of this paper is to explore how and when verbal labels facilitate relational reasoning and transfer. We review the research and theory behind two ways words might direct attention to relational information: (1) words generically invite people to compare and thus highlight relations (the Generic Tokens [GT] hypothesis), and/or (2) words carry semantic cues to common structure (the Cues to Specific Meaning [CSM] hypothesis). Four experiments examined whether learning Signal Detection Theory (SDT) with relational words fostered better transfer than learning without relational words in easily alignable and less alignable situations (testing the GT hypothesis) as well as when the relational words matched and mismatched the semantics of the learning situation (testing the CSM hypothesis). The results of the experiments found support for the GT hypothesis because the presence of relational labels produced better transfer when two situations were alignable. Although the CSM hypothesis does not explain how words facilitate transfer, we found that mismatches between words and their labeled referents can produce a situation where words hinder relational learning.

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Recognizing group cognition

Theiner, G, Allen, C., & Goldstone, R. L. (2010). Recognizing group cognition. Cognitive Systems Research11, 378-395.

In this paper, we approach the idea of group cognition from the perspective of the ”extended mind” thesis, as a special case of the more general claim that systems larger than the individual human, but containing that human, are capable of cognition (Clark, 2008; Clark & Chalmers, 1998). Instead of deliberating about ”the mark of the cognitive” (Adams & Aizawa, 2008), our discussion of group cognition is tied to particular cognitive capacities. We review recent studies of group problem solving and group memory which reveal that specific cognitive capacities that are commonly ascribed to individuals are also aptly ascribed at the level of groups. These case studies show how dense interactions among people within a group lead to both similarity-inducing and differentiating dynamics that affect the group’s ability to solve problems. This supports our claim that groups have organization-dependent cognitive capacities that go beyond the simple aggregation of the cognitive capacities of individuals. Group cognition is thus an emergent phenomenon in the sense of Wimsatt (1986). We further argue that anybody who rejects our strategy for showing that cognitive properties can be instantiated at multiple levels in the organizational hierarchy on a priori grounds is a ”demergentist,” and thus incurs the burden of proof for explaining why cognitive properties are ”stuck” at a certain level of organizational structure. Finally, we show that our analysis of group cognition escapes the ”coupling-constitution” charge that has been leveled against the extended mind thesis (Adams & Aizawa, 2008).

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Social Learning and Cumulative Mutual Improvement in a Networked Group

Wisdom, T. N., & Goldstone, R. L. (2010). Social Learning and Cumulative Mutual Improvement in a Networked Group.  Proceedings of the Thirty-Second Annual Conference of the Cognitive Science Society.  (pp. 1405-1410).  Portland, Oregon: Cognitive Science Society.

We used a simple problem-solving game task to study imitation and innovation in groups of participants. Guesses were composed of multiple elements with linear and interactive effects on score, and score feedback was provided after each of a number of rounds. Participants were allowed to view and imitate the guesses of others during each round, and the score information accompanying others’ guesses was either shown or hidden in two conditions. When scores were not visible, social learning was impeded; participants were less efficient in their searching of the problem space and achieved lower performance overall. When scores were visible, higher performance was observed, and results indicated a more equitable sharing of productive exploration among participants within groups as a result of selective imitation and cross-participant cumulative mutual innovations.

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How you named your child: Understanding the relationship between individual decision-making and collective outcomes

Gureckis, T. M., & Goldstone, R. L. (2009). How you named your child: Understanding the relationship between individual decision-making and collective outcomes. Topics in Cognitive Science1, 651-674.

We examine the interdependence between individual and group behavior surrounding a somewhat arbitrary, real world decision: selecting a name for
one’s child. Using a historical database of the names given to children over the last century in the United States, we nd that naming choices are influenced by both the frequency of a name in the general population, and by its “momentum” in the recent past in the sense that names which are growing in popularity are preferentially chosen. This bias toward rising names is a recent phenomena: in the early part of the 20th century, increasing popularity of a name from one time period to the next was correlated with a decrease in future popularity. However, more recently this trend has reversed. We evaluate a number of formal models that detail how individual decision-making strategies, played out in a large population of interacting agents, can explain these empirical observations. We argue that cognitive capacities for change detection, the encoding of frequency in memory, and biases towards novel or incongruous stimuli may interact with the behavior of other decision makers to determine the distribution and dynamics of cultural tokens such as names.

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Analogical transfer from interaction with a simulated physical system

Day, S. B., & Goldstone, R. L. (2009).  Analogical transfer from interaction with a simulated physical system. Proceedings of the Thirty-First Annual Conference of the Cognitive Science Society, 1406-1411. Amsterdam, Netherlands: Cognitive Science Society.

[See also  Day & Goldstone, (2011) Journal of Experimental Psychology: Learning Memory, and Cognition]

In two studies, we find that participants are able to transfer strategies learned while interacting with a simulated physical system to a dissimilar and less perceptually-concrete domain. Interestingly, performance on the transfer task was completely unrelated to explicit knowledge of the structural correspondences between the systems. We suggest that direct interaction with a concrete system may lead to a kind of procedural knowledge that provides a good basis for analogical transfer.

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Collective behavior

Goldstone, R. L. & Gureckis, T. M. (2009).  Collective behavior. Topics in Cognitive Science1, 412-438.

The resurgence of interest in collective behavior is in large part due to tools recently made available for conducting laboratory experiments on groups, statistical methods for analyzing large data sets reflecting social interactions, the rapid growth of a diverse variety of online self-organized collectives, and computational modeling methods for understanding both universal and scenario-specific social patterns. We consider case studies of collective behavior along four attributes: the primary motivation of individuals within the group, kinds of interactions among individuals, typical dynamics that result from these interactions, and characteristic outcomes at the group level. With this framework, we compare the collective patterns of noninteracting decision makers, bee swarms, groups forming paths in physical and abstract spaces, sports teams, cooperation and competition for resource usage, and the spread and extension of innovations in an online community. Some critical issues surrounding collective behavior are then reviewed, including the questions of ‘‘Does group behavior always reduce to individual behavior?’’ ‘‘Is ‘group cognition’ possible?’’ and ‘‘What is the value of formal modeling for understanding group behavior?’’

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How much of symbolic manipulation is just symbol pushing?

Landy, D. H., & Goldstone, R. L. (2009).  How much of symbolic manipulation is just symbol pushing? Proceedings of the Thirty-First Annual Conference of the Cognitive Science Society, 1072-1077. Amsterdam, Netherlands: Cognitive Science Society.

This paper explores the hypothesis that schematic abstraction—rule following—is partially implemented through processes and knowledge used to understand motion. Two experiments explore the mechanisms used by reasoners solving simple linear equations with one variable. Participants solved problems displayed against a background that moved rightward or leftward. Solving was facilitated when the background motion moved in the direction of the numeric transposition required to solve for the unknown variable. Previous theorizing has usually assumed that such formal problems are solved through the repeated application of abstract transformation patterns (rules) to equations, replicating the steps produced in typical worked solutions. However, the current results suggest that in addition to such strategies, advanced reasoners often employ a mental motion strategy when manipulating algebraic forms: elements of the problem are “picked up” and “moved” across the equation line. This demonstration supports the suggestion that genuinely schematic reasoning could be implemented in perceptual-motor systems through the simulated transformation of referential (but physical) symbol systems.

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Adaptive group coordination

Roberts, M. E., & Goldstone, R. L. (2009).  Adaptive group coordination, Proceedings of the Thirty-First Annual Conference of the Cognitive Science Society. 2698-2704. Amsterdam, Netherlands: Cognitive Science Society.

Human groups exhibit poor performance in many social situations because task constraints promote either individual maximization behavior or diffusion of responsibility. We introduce a group task that tests human coordination when only a shared group goal exists. Without communication, group members submit numbers in an attempt to collectively sum to a randomly selected number. After receiving group feedback, members adjust their submitted numbers in the next round. Small groups generally outperform large groups, and for all groups, performance improves with task experience, and reactivity to feedback decreases over rounds. Our empirical results and computational modeling provide evidence for adaptive coordination in human groups despite minimal shared history and only indirect communication, and perhaps most interestingly, as the coordination costs increase with group size, large groups adapt through spontaneous role differentiation and self-consistency among members.

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