Recursive Thinking Research Articles

Fibonacci Numbers between History, Semiotics, and Storytelling: The Birth of Recursive Thinking

The aim of this paper is to discuss the emergence of recursive thinking through the famous problem posed by Fibonacci regarding the growth of the rabbit population. This paper qualitatively analyzes and discusses the semiotic aspects raised by the students working with this historical source in the form of a story. From this perspective, the value of the historical problems as socio-cultural references (voices) and of the narrations as mediating factors to enhance students’ learning of new mathematical concepts, such as recursion, is explored in depth. The focus lies on the pivotal role played by the students’ construction of personal senses during in-group mathematical activities, in dialectics with the normative and mathematical meanings. It is highlighted that fostering environments conducive to dialogue among peers, as well as linking various shapes and contexts of knowledge, is necessary. Here, storytelling and history are regarded as fruitful resources aiding students in the gradual construction of a personal sense of mathematical concepts, including recursion.

A Narrative Approach to Foster the Construction of Recursive Thinking in High School Students

In this paper, we will explain the development of a mathematical activity involving narrative and short stories in particular, with the aim of investigating whether it is possible to use the narrative approach to promote the construction of recursive thinking in high school students from a four-year scientific high school (Grades 11 and 12). We present qualitative research based on the networking of two theoretical frameworks used to analyze students’ protocols and the issues surfacing during class discussion: Abstraction in Context (AiC) and Documenting Collective Activity (DCA). In our research, the students, divided into small groups, dealt with a highly immersive “story problem” with the Sierpinski Triangle as its central element. The task was designed to ensure consistency with the story and involved the construction, with GeoGebra 6.0 software, of a fractal city, Fractlandia, with squares and sinkholes. The preliminary results show that the story proposed functioned as a motivation to solve the problem, and the last questions of the task proved the most engaging for the students, mainly because of the connection with the story, and also because they involved some reflection about the behavior to the infinity of the perimeter and the area of the Sierpinski Triangle.

Interdisciplinary insights into navigating the maze of landscape multifunctionality

Abstract Increasing demands for land to deliver multiple and sometimes conflicting services to people and nature have led to the development of an extensive body of research focussed on multifunctional landscapes. However, this has created both insight and confusion, as authors from a variety of disciplines have independently tackled the question of how to manage the trade‐offs and synergies inherent in landscapes that are required to produce multiple functions and services. We employed an interdisciplinary perspective to formulate some key questions that researchers of multifunctional landscapes can use to identify blind spots. Our process resulted in a question‐based analysis support scheme that supports reflection and recursive thinking about multifunctional landscapes, beginning with objective setting and visions for addressing it, grounded in baseline mapping, then assessing landscape functions and their single and multiple interactions; as well as the analysis of sensitivity to spatial and temporal dimensions. Other key points identified are the need for clarity and examination of unstated assumptions, from aims to definitions; accounting for scale; incorporating stakeholder needs throughout the process and applying suitable methods of measurement and aggregation. The focus on asking guided questions derives from the insight that there is no universal correct approach to multifunctional landscapes; the aim should instead be to find the most appropriate methods for the given circumstances and goals. Policy implications. Tackling current and future socio‐ecological challenges is an interdisciplinary undertaking, necessitating collaborative efforts between research fields that each bring valuable and distinct insights. To effectively combat these challenges, multifunctional landscapes require a clear process and focused objective in their implementation. Multifunctionality should be seen as a means to an end, rather than an end in itself. Read the free Plain Language Summary for this article on the Journal blog.

Self-Awareness of Goals Task (SAGT) and Planning Skills: The Neuroscience of Decision Making.

A goal's self-awareness and the planning to achieve it drive decision makers. Through a neuroscientific approach, this study explores the self-awareness of goals by analyzing the explicit and implicit processes linked to the ability to self-represent goals and sort them via an implicit dominant key. Thirty-five professionals performed a novel and ecological decision-making task, the Self-Awareness of Goals Task (SAGT), aimed at exploring the (i) self-representation of the decision-making goals of a typical working day; (ii) self-representation of how these goals were performed in order of priority; (iii) temporal sequence; and (iv) in terms of their efficacy. Electrophysiological (i.e., alpha, beta, and gamma band), autonomic, behavioral, and self-report data (General Decision Making Style and Big Five Inventory) are collected. Higher self-awareness of goals by time as well as efficacy and the greater activation of alpha, beta, and gamma bands in the temporoparietal brain area were found. Correlations reported positive associations between the self-awareness of goals via a time and dependent decision-making style and a conscientious personality, but also between the self-awareness of goals via an efficacy and rational decision-making style. The results obtained in this study suggest that the SAGT could activate recursive thinking in the examinee and grasp individual differences in self-representation and aware identification of decision-making goals.

Recursive expectations approach in policymaking

The primary contribution of modern economics to policy analysis may be the recognition of the crucial role of expectations in policy intervention. The essence of expectations is to think about others’ thinking. We argue that policymakers need recursive thinking, that is, the ability to think about thinking and review three policy episodes related to the lack of recursive thinking. We see that the disciplinary divide or the limited scope of recursive thinking on the side of policymakers can cause huge damage to social welfare in times of crisis. Finally, we consider two examples of future agendas in the recursive approach.

Is recursive "mindreading" really an exception to limitations on recursive thinking?

The ability to mindread recursively-for example, by thinking what person 1 thinks person 2 thinks person 3 thinks-is a prime example of recursive thinking in which one process, representation, or idea becomes embedded within a similar one. It has also been suggested that mindreading is an exceptional example, with five recursive steps commonly observed for mindreading, in comparison with just one or two in other domains. However, conceptual analysis of existing recursive mindreading tasks suggests that conclusions about exceptional mindreading are insecure. Revised tasks were devised to provide a more rigorous test of recursive mindreading capacity. Study 1 (N = 76) found significantly worse performance at level-5 recursive mindreading on the revised tasks (17% correct) compared with the original tasks (80% correct), and no effect of moderate financial bonuses for good performance. Study 2 (N = 74) replicated poor performance at level-5 recursive mindreading on the revised tasks (15% correct) in the absence of bonuses, but found better performance (45% correct) when participants were offered large bonuses for accuracy, encouraged to take as much time as needed, and assisted with a strategy for recursive reasoning. These findings suggest that, like recursive thinking in other domains, recursive mindreading is effortful and limited. We discuss how the proposed role for high levels of recursive mindreading in communication, culture, and literature might be reconciled with these limitations. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Supplemental Material for Is Recursive “Mindreading” Really an Exception to Limitations on Recursive Thinking?

Supplemental Material for Is Recursive “Mindreading” Really an Exception to Limitations on Recursive Thinking?

Darwin and the golden rule: how to distinguish differences of degree from differences of kind using mechanisms

Darwin claimed that human and animal minds differ in degree but not in kind, and that ethical principles such as the Golden Rule are just an extension of thinking found in animals. Both claims are false. The best way to distinguish differences in degree from differences in kind is by identifying mechanisms that have emergent properties. Recursive thinking is an emergent capability found in humans but not in other animals. The Golden Rule and some other ethical principles such as Kant’s categorical imperative require recursion, so they constitute ethical thinking that is restricted to humans. Changes in kind have tipping points resulting from mechanisms with emergent properties.

Task Development to Measure Functional Thinking: Focusing on Third Graders’ Understanding

The purpose of this study was to develop the tasks to measure functional thinking and to verify their suitability by applying them to third-grade students. For this purpose, various tasks related to functional thinking were analyzed and developed according to the different modes of functional thinking and functional relationships. To verify the tasks, the Cronbach’s α value for each task type was calculated with the results of how the third graders solved the tasks. Exploratory factor analysis (EFA) was performed as needed. As a result, tasks were developed to measure four types of thinking: recursive, covariation, correspondence-particular, and correspondence-general. In particular, recursive patterning tasks were divided into repeated geometric patterns, growing geometric patterns, repeated numeric patterns, and growing numeric patterns through EFA. Only growing geometric patterning and growing numeric patterning tasks were found to be suitable for measuring third-grade students’ recursive thinking. Growing numeric patterning tasks were divided by arithmetic sequences and other sequences through EFA. The covariation, correspondence-particular, and correspondence-general tasks developed in this study were also verified. Contrary to the consistently high performance regarding the additive relationships, students’ functional thinking varied depending on the different types of functional relationships. This study is expected to inspire researchers to pay more attention to task development and verification to measure students’ functional thinking.

Encouraging research on recursive thinking through the lens of a model of the spread of contagious diseases

Recursive reasoning is a powerful tool used extensively in problem solving. For us, recursive reasoning includes iteration, sequences, difference equations, discrete dynamical systems, pattern identification, and mathematical induction; all of these can represent how things change, but in discrete jumps. Given the school mathematics curriculum’s later emphasis on calculus—the mathematics of change in continuous contexts—it is surprising that the curriculum seems to neglect recursive thinking after the early grades. Research shows that recursion supports the learning of algebra among younger students, but the lack of similar research with older students is concerning. In this paper we suggest possible affordances from teaching recursive modeling, including a basic model of the spread of contagious diseases. We also discuss different ways to present these models at various points in the curriculum that might develop connections between mathematics and the real world, and support students’ learning of mathematics. This leads to what we, as mathematicians, think would be interesting research questions for mathematical educators.

How do children overcome their pragmatic performance problems in the true belief task? The role of advanced pragmatics and higher-order theory of mind.

The true belief (TB) control condition of the classical location-change task asks children to ascribe a veridical belief to an agent to predict her action (analog to the false belief (FB) condition to test Theory of Mind (ToM) abilities). Studies that administered TB tasks to a broad age range of children yielded surprising findings of a U-shaped performance curve in this seemingly trivial task. Children before age four perform competently in the TB condition. Children who begin to solve the FB condition at age four, however, fail the TB condition and only from around age 10, children succeed again. New evidence suggests that the decline in performance around age four reflects pragmatic confusions caused by the triviality of the task rather than real competence deficits in ToM. Based on these results, it can be hypothesized that the recovery of performance at the end of the U-shaped curve reflects underlying developments in children's growing pragmatic awareness. The aim of the current set of studies, therefore, was to test whether the developmental change at the end of the U-shaped performance curve can be explained by changes in children's pragmatic understanding and by more general underlying developmental changes in recursive ToM or recursive thinking in general. Results from Study 1 (N = 81, 6-10 years) suggest that children's recursive ToM, but not their advanced pragmatic understanding or general recursive thinking abilities predict their TB performance. However, this relationship could not be replicated in Study 2 (N = 87, 6-10 years) and Study 3 (N = 64, 6-10 years) in which neither recursive ToM nor advanced pragmatic understanding or recursive thinking explained children's performance in the TB task. The studies therefore remain inconclusive regarding explanations for the end of the U-shaped performance curve. Future research needs to investigate potential pragmatic and general cognitive foundations of this developmental change more thoroughly.

Anxiety sensitivity, uncertainty and recursive thinking: A continuum on Cyberchondria conditions during the Covid Outbreak

Background. Cyberchondria is a term used to refer to excessive surfing the web looking for health care information, excessive checking behavior being related to health-related anxiety. This period of quarantine for the Covid-19 pandemic is increasing the pathological use of the internet, and the excessive surfing the web looking for health care information. Another dimension related to the Covid-19 outbreak refers to uncertainty intolerance, for this reason being necessary for the healthcare professionals to provide clear and linear information. Aim. The aim of this review is to identify the psychological correlations connected to cyberchondria in the quarantine period. Methods. Following the PRISMA guidelines, we carried out a systematic review of the literature on PubMed. The terms used for the search were “Cyberchondria” OR “Anxiety” AND “Quarantine”. Results. As resulting from the reviewed literature, there is a relationship between anxiety for one’s own state of health and cyberchondria, with negative psychological effects of quarantine, including post-traumatic stress symptoms, depression, anxiety, low mood, irritability, insomnia, uncertainty, emotional exhaustion, this condition being associated with hypervigilance, and catastrophic misinterpretation of bodily signs. Conclusion. In the light of this and according to the literature, it would be desirable that research can further explore the factors influencing the increase in cyberchondria in the future.

Supporting Children’s Second-order Recursive Thinking and Advanced ToM Abilities: A Training Study

ABSTRACT The present study evaluated: i) the effects of two training programs designed for promoting Theory of Mind (ToM) skills in children aged 7/8, and ii) the relations between second-order recursive thinking (II-order-RT), advanced-ToM (Adv_ToM) and metacognition. Ninety-one 7- to 8-year-old children were assigned to one of three training conditions: a II-order-RT, an Adv_ToM, and a control condition. Groups were equivalent at pretest for SES, executive functions and reading comprehension. The main effect of time was significant on II-order-RT, but not on Adv_ToM and metacognition. Children of the II-order-RT group improved their performance on over the training period II-order-RT while children of the control group did not) and they also outperformed children of the control condition at posttest. This positive effect generalized to Adv_ToM. Children of the Adv_ToM group improved over the training period their performance on both Adv_ToM and II-order-RT (while children of the control group did not); moreover, they outperformed children of the control condition at posttest on Adv_ToM, but not on II-order-RT. The benefits of both ToM training programs extended to metacognitive knowledge. Our findings throw light upon generative mechanisms of acquisition that explain the relation between II-order-RT and Adv_ToM, and between these mature ToM skills and metacognition.

Neural correlates of recursive thinking during interpersonal strategic interactions.

To navigate the complex social world, individuals need to represent others' mental states to think strategically and predict their next move. Strategic mentalizing can be classified into different levels of theory of mind according to its order of mental state attribution of other people's beliefs, desires, intentions, and so forth. For example, reasoning people's beliefs about simple world facts is the first‐order attribution while going further to reason people's beliefs about the minds of others is the second‐order attribution. The neural substrates that support such high‐order recursive reasoning in strategic interpersonal interactions are still unclear. Here, using a sequential‐move interactional game together with functional magnetic resonance imaging (fMRI), we showed that recursive reasoning engaged the frontal‐subcortical regions. At the stimulus stage, the ventral striatum was more activated in high‐order reasoning as compared with low‐order reasoning. At the decision stage, high‐order reasoning activated the medial prefrontal cortex (mPFC) and other mentalizing regions. Moreover, functional connectivity between the dorsomedial prefrontal cortex (dmPFC) and the insula/hippocampus was positively correlated with individual differences in high‐order social reasoning. This work delineates the neural correlates of high‐order recursive thinking in strategic games and highlights the key role of the interplay between mPFC and subcortical regions in advanced social decision‐making.

Elementary students’ functional thinking: From recursive to correspondence

This study aims to identify elementary students’ functional thinking processes in solving pattern problems. Previous studies showed that elementary students' functional thinking still often experience errors in solving pattern problems. The study of the functional thinking process in solving pattern problems is a fundamental key as a solution to find out the strengths and weaknesses of elementary school students, so that they are better prepared in generalizing relationships, representing and analyzing function behavior in advanced algebra classes. This study used a descriptive qualitative approach with a case study method. Participants of study was sixty-five elementary students who had not yet received generalization patterns material. The instruments were tasks and interview guidelines. Based on the task results, students who had correct answers were chosen using purposive sampling to be given an in-depth interview. The finding indicated that elementary students are able to think functionally in different ways. Students’ functional thinking begins with recursive thinking in the pre-finding formula in the entry stage. Students find the formula by corresponding thinking in the attack stage. Finally, students use the formula to get inverse in the review stage.

Children’s Capacity for Algebraic Thinking in the Early Grades

Patterns pervade our everyday life and are natural and familiar to children. Exciting new research in the field of Early Algebra describes the development of young children’s algebraic thinking through the meaningful study of patterns. This paper contributes to this research by reporting on a case study of two primary schools in South Africa and looking at the capacity of learners for algebraic thinking in the early grades. Data was gathered using Grade 3 learner tests and focus group interviews based on pattern tasks. The data analysis process examined learners’ pattern matching and building explanations related to each data collection instrument using Blanton, Brizuela, Gardiner, Sawrey and Newman-Owens’s (2015. A learning trajectory in 6-year-old’s thinking about generalising functional relationships. Journal for Research in Mathematics Education, 46, 511–558) levels of sophistication in learner’s thinking about functional relationships. The results of the study suggest that young learners have the potential to think algebraically when offered opportunities to do so, particularly in terms of recursive and functional thinking when solving pattern problems. Traditionally instructional practice emphasises lengthy exposure to recursive thinking as a first route to algebraic thinking: this recent research suggests that young children also have the capacity to engage in functional thinking. Thus, we may need to rethink our approach to algebraic thinking through the development of a pattern-based approach to algebra.

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

Abstract 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.

Simulation in children's conscious recursive reasoning.

When do children acquire the ability to understand recursion-that is, repeated loops of actions, as in cookery recipes or computer programs? Hitherto, studies have focused either on unconscious recursions in language and vision or on the difficulty of conscious recursions-even for adults-when learning to program. In contrast, we examined 10- to 11-year-old fifth-graders' ability to deduce the consequences of loops of actions in informal algorithms and to create such algorithms for themselves. In our experiments, the children tackled problems requiring the rearrangement of cars on a toy railway with a single track and a siding-an environment that in principle allows for the execution of any algorithm-that is, it has the power of a universal Turing machine. The children were not allowed to move the cars, so each problem's solution called for them to envision the movements of cars on the track. We describe a theory of recursive thinking, which is based on kinematic simulations and which we have implemented in a computer program embodying mental models of the cars and track. Experiment 1 tested children's ability to deduce rearrangements of the cars in a train from descriptions of algorithms containing a single loop of actions. Experiment 2 assessed children's spontaneous creation of similar sorts of algorithms. The results showed that fifth-grade children with no training in computer programming have systematic abilities to deduce from and to create informal recursive algorithms.

The role of the insula in intuitive expert bug detection in computer code: an fMRI study

Software programming is a complex and relatively recent human activity, involving the integration of mathematical, recursive thinking and language processing. The neural correlates of this recent human activity are still poorly understood. Error monitoring during this type of task, requiring the integration of language, logical symbol manipulation and other mathematical skills, is particularly challenging. We therefore aimed to investigate the neural correlates of decision-making during source code understanding and mental manipulation in professional participants with high expertise. The present fMRI study directly addressed error monitoring during source code comprehension, expert bug detection and decision-making. We used C code, which triggers the same sort of processing irrespective of the native language of the programmer. We discovered a distinct role for the insula in bug monitoring and detection and a novel connectivity pattern that goes beyond the expected activation pattern evoked by source code understanding in semantic language and mathematical processing regions. Importantly, insula activity levels were critically related to the quality of error detection, involving intuition, as signalled by reported initial bug suspicion, prior to final decision and bug detection. Activity in this salience network (SN) region evoked by bug suspicion was predictive of bug detection precision, suggesting that it encodes the quality of the behavioral evidence. Connectivity analysis provided evidence for top-down circuit “reutilization” stemming from anterior cingulate cortex (BA32), a core region in the SN that evolved for complex error monitoring such as required for this type of recent human activity. Cingulate (BA32) and anterolateral (BA10) frontal regions causally modulated decision processes in the insula, which in turn was related to activity of math processing regions in early parietal cortex. In other words, earlier brain regions used during evolution for other functions seem to be reutilized in a top-down manner for a new complex function, in an analogous manner as described for other cultural creations such as reading and literacy.

Effects of adolescent sociocognitive development on the cortisol response to social evaluation.

Adolescents become increasingly sensitive to social evaluation. Some previous studies have related this change to pubertal development. The present longitudinal study examined the role of sociocognitive development. We investigated whether or not the transition to recursive thinking, the ability to think about (others') thoughts, would be associated with changes in the magnitude and timing of the cortisol response to social evaluation. Salivary cortisol was obtained during the Leiden Public Speaking Task. The task was administered twice with a 2-year interval to 221 participants, aged 9-17 years at Time 1. The area under the curve was computed to assess the magnitude of the overall cortisol response. Two difference scores, reflecting speech anticipation and speech delivery, were computed to assess the timing of the cortisol response. Recursive thinking was measured with a cartoon description task. Regression analyses with clustered bootstrap controlling for pubertal development, age, and general cognitive functioning showed that the transition to recursive thinking predicted an increase in the cortisol response to speech anticipation, but was unrelated to the magnitude of the overall cortisol response. This is in line with the view that increasing sensitivity to social evaluation in adolescence is mainly due to the effects of pubertal hormones on affective regions of the brain. Sociocognitive development affected the timing rather than the magnitude of the cortisol response. The results suggest that recursive thinking enables earlier realization of social-evaluative threat. (PsycINFO Database Record