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 Procedures.  In R. Sottilare, A. Graesser, X. Hu, A. Olney, B. Nye, and A. Sinatra (Eds.) Design Recommendations for Intelligent Tutoring Systems: Volume 4 РDomain 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 system‚ÄĒin particular, object-based attention‚ÄĒis 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 expressions‚ÄĒbut 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 cognition.  Topics 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, 670‚Äď684. 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, 569‚Äď588. 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, 697‚Äď714. 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, 589‚Äď609. 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, 685‚Äď696. 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, 660‚Äď669. 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, 625‚Äď659. doi: 10.1111/tops.12211.


Peer review and competition in the art exhibition game

Balietti, S., Goldstone, R., & Helbing, D. (2016).  Peer review and competition in the art exhibition game.  Proceedings of the National Academy of Sciences, 113, 8414-8419.

To investigate the effect of competitive incentives under peer review, we designed a novel experimental setup called the Art Exhibition Game. We present experimental evidence of how competition introduces both positive and negative effects when creative artifacts are evaluated and selected by peer review. Competition proved to be a double-edged sword: on the one hand, it fosters innovation and product diversity, but on the other hand, it also leads to more unfair reviews and to a lower level of agreement between reviewers. Moreover, an external validation of the quality of peer reviews during the laboratory experiment, based on 23,627 online evaluations on Amazon Mechanical Turk, shows that competition does not significantly increase the level of creativity. Furthermore, the higher rejection rate under competitive conditions does not improve the average quality of published contributions, because more high-quality work is also rejected. Overall, our results could explain why many ground-breaking studies in science end up in lower-tier journals. Differences and similarities between the Art Exhibition Game and scholarly peer review are discussed and the implications for the design of new incentive systems for scientists are explained.

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Robert Goldstone Elected to the American Academy of Arts and Science

On the 20th of April, 2016, Robert Goldstone was elected to the American Academy of Arts and Sciences.  He joins several others at Indiana University, including Richard Shiffrin, Linda Smith, and Robert Nosofsky from the Department of Psychological and Brain Sciences, and Douglas Hofstadter and Mike Dunn from the Cognitive Science Program, who are also members of AAAS.  Read about it at:

Drs. Paulo Carvalho and Joshua de Leeuw Graduate

Congratulations to the pair o’docs, Dr. Paulo Carvalho and Dr. Joshua de Leeuw, who commenced on the 6th of May, 2016. ¬†Speaking of paradox,¬†¬†If two graduate students organize all of the activities in a laboratory for those people, and only those people, who do not organize activities for themselves, then how will the laboratory continue to operate after they have departed?

Dr. Joshua de Leeuw will start in the Fall of 2016 as Assistant Professor in Cognitive Science at Vassar College.  Meanwhile, Dr. Paulo Carvalho will start a position as postdoctoral research scientist at Carnegie-Mellon University, working with Dr. Ken Koedinger.  Hearty congratulations to the both of them!


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)

Disagreeing to Agree


Without ever explicitly discussing it, groups often times establish norms.  A family or committee might develop a norm about when it is acceptable or not for members to interrupt each other.  People greeting each other in different countries have very different norms for whether to shake hands or kiss, and if to kiss, how many times and in what cheek order.  In some countries, tipping is not the norm, but if it is, violating the tipping norm could make you a persona non grata at a restaurant.  We (Hawkins & Goldstone, 2016) were interested in how social norms emerge in a group without its members explicitly deciding on them, and the factors that promote effective norms.

To help explore these questions, we started by considering a simple scenario we call ‚ÄúBattle of the Exes.‚Ä̬† You and your romantic partner live in a small town and both love coffee.¬† Your shared loved of coffee was not, alas, enough to keep you together, and you have now broken up.¬† There are only two coffee shops in your town, one with much better coffee than the other.¬† Both you and your ex want to go every day for coffee during your simultaneously occurring coffee breaks, but if you pick the same place and run into one another, neither of you will enjoy your break at all.

Neither you nor your ex want to sit down to negotiate a schedule, but can you nonetheless develop a satisfactory routine?  One of you could always go to the better coffee shop, but that would not be fair.  Each of you could choose randomly, but that would end up with you and your ex often seeing each other, which would not maximize your duo’s happiness, and would not provide a stable solution in the long run.

These three features — fairness, happiness maximization, and stability are generally useful ways to assess the quality of a group‚Äôs behavior.¬† To study scenarios like ‚ÄúBattle of the Exes‚ÄĚ in the laboratory, we developed an interactive, real-time, online game.¬† On each of the 60 rounds of the game, two players are given the choice of moving their avatar to one of two circles — one that they can visibly see will give them a small monetary prize and one that will give them a large payoff.¬† The only catch is that if both players move to the same circle, then neither player gets anything for that round.¬† For half of the groups, there was a small discrepancy between the prizes (1 cent vs 2 cents), and for the other half, there was a large discrepancy (1 cent versus 4 cents).¬† Also, for half of the groups, each of the players could see the other player‚Äôs moment-to-moment position as they moved to the circles (Dynamic movement), while for the other half of the groups, the players only see the final choice that the other player made (Ballistic movement).

568 players were matched together to create 284 two-player groups.  Some groups developed behaviors that were fair and stable, and led to both players earning a lot of money.  These groups tended to develop social norms even without explicit communication.  For example, the players A and B would alternate over rounds who got the large payoff, first A then B then A…., leading to a pattern like ABABABABAB.

In terms of maximizing happiness, the dynamic condition led to better earnings for the players than the ballistic condition.  When the players can see each others’ moment-to-moment inclinations, that helps them coordinate.  The dynamic condition also led to fairer solutions than the ballistic condition, with players earning similar amounts of money.  An implication of these results is that giving the members in a group more information about what each person in the group is currently thinking about doing can help the group achieve well-coordinated, fair and happy solutions.  This is something for politicians, social network providers, and amusement parks to consider when they are trying to design social spaces for their groups.  Mutual visibility of group members is often an effective way to promote coordination.

In terms of developing stable strategies, there was a striking interaction between payoffs and movement type.  When there was not a large difference in payoffs, choices in the ballistic condition were more stable than in the dynamic condition.  When the stakes were low, players in the dynamic condition simply relied on moment-to-moment visual information to figure out who should get the larger payoff on any given round.  They did not feel a strong pressure to develop a norm because they could use their continuous information as a crutch to help them coordinate.  However, when the stakes were high, with one circle earning four times what the other circle earned, then the dynamic condition developed significantly more stable solutions than the ballistic condition.  For these particularly contentious, high stakes situations, it is useful for the players to develop strong norms to help them coordinate, and the moment-to-moment information about player positions helps to create these norms.

One clear measure of how much contention there is in a group is how long both players move toward the same high payoff option before one ‚Äúpeels off‚ÄĚ and lets the other player have the high payoff prize.¬† Using this objective measure, groups have more contention at the beginning of the experiment session than the end.¬† The higher stakes condition has more contention early on than the lower stakes condition, but by the end of the experiment, that ordering is flipped.¬† Groups that have more contention at the beginning of the experiment tend to have less contention by the end of experiment, and are more likely to develop clever strategies like alternating who gets the high payoff option from round to round.¬† A take-home message from this result is that contention in groups is not something to be avoided.¬† For the groups in our ‚ÄúBattle of the Exes‚ÄĚ game, early contention gives rise to well-coordinated, fair, efficient, and happiness maximizing solutions by the end of the experiment.¬† It may be tempting to try to pave over contention and disagreement in a group, but letting the group work through these contentions is often key to giving them the motivation and insight that they need to develop creative, well-coordinated norms like alternating who gets the better payoff over rounds.¬† So, although it may have been contention that broke you and your ex up in the first place, there is hope that this kind of early contention may allow you to enjoy your superior cup of coffee in peace.¬† At least on Mondays, Wednesdays, and Fridays.

Hawkins, R. X. D., & Goldstone, R. L. (2016). The formation of social conventions in real-time environments.  PLoS One, 11(3): e0151670



The formation of social conventions in real-time environments

Hawkins, R. X. D., & Goldstone, R. L. (2016). The formation of social conventions in real-time environments.  PLoS One, 11(3): e0151670.

Why are some behaviors governed by strong social conventions while others are not? We¬†experimentally investigate two factors contributing to the formation of conventions in a¬†game of impure coordination: the continuity of interaction within each round of play¬†(simultaneous vs. real-time) and the stakes of the interaction (high vs. low differences¬†between payoffs). To maximize efficiency and fairness in this game, players must¬†coordinate on one of two equally advantageous equilibria. In agreement with other¬†studies manipulating continuity of interaction, we find that players who were allowed to¬†interact continuously within rounds achieved outcomes with greater efficiency and¬†fairness than players who were forced to make simultaneous decisions. However, the¬†stability of equilibria in the real-time condition varied systematically and dramatically¬†with stakes: players converged on more stable patterns of behavior when stakes are high.¬†To account for this result, we present a novel analysis of the dynamics of continuous¬†interaction and signaling within rounds. We discuss this previously unconsidered¬†interaction between within-trial and across-trial dynamics as a form of social¬†canalization. When stakes are low in a real-time environment, players can satisfactorily¬†coordinate `on the fly,’ but when stakes are high there is increased pressure to establish¬†and adhere to shared expectations that persist across rounds.

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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 ‚ÄstDutch primary school children ‚Äď 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‚Äô 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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Learning as Coordination

Schwartz, D. L, & Goldstone, R. L. (2016).  Learning as coordination: Cognitive psychology and education.  In L. Corno & E. M. Anderman (Eds.) Handbook of Educational Psychology, 3rd edition. New York: Routledge (pp. 61-75).

The chapter follows a central thesis: A major task of teaching and instruction is to help learners coordinate categories of cognitive processes, capabilities, and representations. While nature confers basic abilities, education synthesizes them to suit the demands of contemporary culture. So, rather than treating categories of learning and instruction as an either‚Äďor problem, the problem is how to coordinate learning processes so they can do more together than they can alone. This thesis, which proposes a systems level analysis, is not the norm when thinking about teaching and learning. More common is the belief that learning involves strengthening select cognitive processes rather than coordination across processes. Our chapter, therefore, needs to develop the argument for learning as coordination. To do so, we introduce findings from the field of cognitive psychology.

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Avatars and behavioral experiments: methods for controlled quantitative social behavioral research in virtual worlds

Hmeljak, D., & Goldstone, R. L. (2016).  Avatars and behavioral experiments: methods for controlled quantitative social behavioral research in virtual worlds.  In Y. Silvan (Ed.) Handbook on 3D3C Virtual Worlds.  Zurich, Switzerland: Springer International Publishing.

Three-dimensional, Community, Creation, and Commerce (3D3C) worlds can support real-time, quantitatively controlled experiments for studying human group behavior. This chapter provides a review of social behavioral research in virtual worlds, their methodologies and goals, such as studies of socio-economical trends, interpersonal communications between virtual world residents, automated survey studies, etc. The chapter contrasts existing research tools in virtual worlds with the goals of studying human group behavior as a complex system‚ÄĒhow interacting groups of people create emergent organizations at a higher level than the individuals comprising such groups. Finally, the chapter presents features of virtual world-based group behavior experiments that allow the recreation of controlled quantitative experiments previously conducted in supervised lab sessions or web-based games.

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