Young Students’ Functional Thinking Modes: The Relation Between Recursive Patterning, Covariational Thinking, and Correspondence Relations

A theoretical model describing young students’ (Grades 1–3) functional-thinking modes was formulated and validated empirically (n = 345), hypothesizing that young students’ functional-thinking modes consist of recursive patterning, covariational thinking, correspondence-particular, and correspondence-general factors. Data analysis suggested that functional-thinking tasks can be categorized on the basis of the proposed model. Analysis traced three categories of students that represent different functional-thinking profiles. Category 1 students exhibited a recursive-thinking profile. Category 2 students utilized a combination of recursive and contextual strategies and exhibited an emergent covariational and correspondence-particular thinking. Category 3 students approached functional-thinking situations flexibly, using a combination of covariational and correspondence strategies. A structural model showed two parallel paths from recursive patterning to correspondence-general through correspondence-particular or covariational.

The Pósa method with ATD lenses: Praxeological analysis on math problems in Hungarian talent care education with ‘recursion’ in their logos blocksLa méthode Pósa avec des lentilles TAD: analyse praxéologique des problèmes mathématiques dans l’enseignement hongrois de soins aux talents avec «récursion» dans leurs blocs de logos

The praxeological analysis of selected questions used in the Hungarian Pósa method is presented, focusing on a common element in their logos blocks, called recursive thinking. As part of a broader research with reverse didactic engineering methodology, aiming at theorizing the ‘intuitively’ developed Pósa method, the present findings are also compared to previous results and re-interpret the concepts of kernel and web of problem thread. Based on these results gained by using tools of the Anthropological Theory of the Didactic, the paper offers a partial description of the didactic strategy of the Pósa method for inquiry-based learning mathematics and raises questions for further research.RésuméNous présentons l'analyse praxéologique de certaines questions utilisées dans la méthode hongroise Pósa, en nous concentrant sur un élément commun à leurs logos blocs, appelé pensée récursive. Dans le cadre d’une recherche plus large qui met en place une méthodologie d’ingénierie didactique inverse visant à théoriser la méthode de Pósa développée "intuitivement", les résultats actuels réinterprètent les concepts de noyau et de réseau de fils de problèmes. Sur la base des résultats obtenus en utilisant les outils de la théorie anthropologique du didactique, l'article offre une description partielle de la stratégie didactique de la méthode Pósa pour l'apprentissage des mathématiques basé sur l'enquête, et soulève des questions pour des recherches ultérieures.

The computation of strategic learning in repeated social competitive interactions: Learning sophistication, reward attractor points and strategic asymmetry

ABSTRACTSocial interactions rely on our ability to learn and adjust our behavior to the behavior of others. Strategic games provide a useful framework to study the cognitive processes involved in the formation of beliefs about the others’ intentions and behavior, what we may call strategic theory of mind. Through the years, the growing field of behavioral economics provided evidence of a systematic departure of human’s behavior from the optimal game theoretical prescriptions. One hypothesis posits that human’s ability to accurately process the other’s behavior is somehow bounded. The question of what constraints the formation of sufficiently high order beliefs remained unanswered. We hypothesize that maximizing final earnings in a competitive repeated game setting, requires moving away from reward-based learning to engage in sophisticated belief-based learning. Overcoming the attraction of the immediate rewards by displaying a computationally costly type of learning might not be a strategy shared among all individuals. In this work, we manipulated the reward structure of the interaction so that the action displayed by the two types of learning becomes (respectively not) discriminable, giving a relative strategic (resp. dis) advantage to the participant given the role endorsed during the interaction. We employed a computational modeling approach to characterize the individual level of belief learning sophistication in three types of interactions (agent-agent, human-human and human-agent). The analysis of the participants’ choice behavior revealed that the strategic learning level drives the formation of more accurate beliefs and eventually leads to convergence towards game optimality (equilibrium). More specifically we show that the game structure interacts with the level of engagement in strategically sophisticated learning to explain the outcome of the interaction. This study provides the first evidence of a key implication of strategic learning heterogeneity in equilibrium departure and provides insight to explain the emergence of a leader-follower dynamics of choice.AUTHOR SUMMARYDynamic interaction between individuals appears to be a cornerstone for understanding how humans grasp other minds. During a strategic interaction, in which the outcome of one’s action depends directly on what the other individual decides, it appears crucial to anticipate the other’s actions in order to adjust our own behavior. In theory, choosing optimally in a strategic setting requires that both players hold correct beliefs over their opponent’s behavior and best-respond to it. However, in practice humans systematically deviate from the game-theoretical (equilibrium), suggesting that our ability to form accurate beliefs is cognitively and/or contextually constrained. Previous studies using computational modelling suggested that during a repeated game interaction humans vary in the sophistication of their learning process leading to the formation of beliefs over their opponent’s behavior of different orders of complexity (level of recursive thinking such as “I think that you think that …”). In this work we show that the individual engagement in sophisticated (belief-based) learning drives the convergence towards equilibrium and ultimately performance. Moreover, we show that this effect is influenced by both the game environment and the cognitive capacity of the participants, shaping the very dynamic of the social interaction.DATA AVAILABILITYThe authors confirm that upon publication the raw behavioral data and Matlab code for reconstruction of all figures, computational models and statistical analyses will be made available for download at the following URL: https://zenodo.org/

Estimating the Use of Higher-Order Theory of Mind Using Computational Agents

When people make decisions in a social context, they often make use of theory of mind, by reasoning about unobservable mental content of others. For example, the behavior of a pedestrian who wants to cross the street depends on whether or not he believes that the driver of an oncoming car has seen him or not. People can also reason about the theory of mind abilities of others, leading to recursive thinking of the sort ‘I think that you think that I think…’. Previous research suggests that this ability may be especially effective in simple competitive settings. In this paper, we use a combination of computational agents and Bayesian model selection to determine to what extent people make use of higher-order theory of mind reasoning in a particular competitive game known as matching pennies. We find that while many children and adults appear to make use of theory of mind, participants are also often classified as using a simpler reactive strategy based only on the actions of the directly preceding round. This may indicate that human reasoners do not primarily use their theory of mind abilities to compete with others.

Theory of Mind Development in Adolescence and Early Adulthood: The Growing Complexity of Recursive Thinking Ability

This study explores the development of theory of mind, operationalized as recursive thinking ability, from adolescence to early adulthood (N= 110; young adolescents = 47; adolescents = 43; young adults = 20). The construct of theory of mind has been operationalized in two different ways: as the ability to recognize the correct mental state of a character, and as the ability to attribute the correct mental state in order to predict the character’s behaviour. The Imposing Memory Task, with five recursive thinking levels, and a third-order false-belief task with three recursive thinking levels (devised for this study) have been used. The relationship among working memory, executive functions, and linguistic skills are also analysed. Results show that subjects exhibit less understanding of elevated recursive thinking levels (third, fourth, and fifth) compared to the first and second levels. Working memory is correlated with total recursive thinking, whereas performance on the linguistic comprehension task is related to third level recursive thinking in both theory of mind tasks. An effect of age on third-order false-belief task performance was also found. A key finding of the present study is that the third-order false-belief task shows significant age differences in the application of recursive thinking that involves the prediction of others’ behaviour. In contrast, such an age effect is not observed in the Imposing Memory Task. These results may support the extension of the investigation of the third order false belief after childhood.