Marr’s seminal distinction between computational, algorithmic, and implementational levels of analysis has inspired research in cognitive science for more than 30 years. According to a widely-used paradigm, the modelling of cognitive processes should mainly operate on the computational level and be targeted at the idealised competence, rather than the actual performance of cognisers in a specific domain. In this paper, we explore how this paradigm can be adopted and revised to understand mathematical problem solving. The computational-level approach applies methods from (...) computational complexity theory and focuses on optimal strategies for completing cognitive tasks. However, human cognitive capacities in mathematical problem solving are essentially characterised by processes that are computationally sub-optimal, because they initially add to the computational complexity of the solutions. Yet, these solutions can be optimal for human cognisers given the acquisition and enactment of mathematical practices. Here we present diagrams and the spatial manipulation of symbols as two examples of problem solving strategies that can be computationally sub-optimal but humanly optimal. These aspects need to be taken into account when analysing competence in mathematical problem solving. Empirically informed considerations on enculturation can help identify, explore, and model the cognitive processes involved in problem solving tasks. The enculturation account of mathematical problem solving strongly suggests that computational-level analyses need to be complemented by considerations on the algorithmic and implementational levels. The emerging research strategy can help develop algorithms that model what we call enculturated cognitive optimality in an empirically plausible and ecologically valid way. (shrink)

Capacities for abstract thinking and problem‐solving are central to human cognition. Processes of abstraction allow the transfer of experiences and knowledge between contexts helping us make informed decisions in new or changing contexts. While we are often inclined to relate such reasoning capacities to individual minds and brains, they may in fact be contingent on human‐specific modes of collaboration, dialogue, and shared attention. In an experimental study, we test the hypothesis that social interaction enhances cognitive processes of rule‐induction, which in (...) turn improves problem‐solving performance. Through three sessions of increasing complexity, individuals and groups were presented with a problem‐solving task requiring them to categorize a set of visual stimuli. To assess the character of participants’ problem representations, after each training session, they were presented with a transfer task involving stimuli that differed in appearance, but shared relations among features with the training set. Besides, we compared participants’ categorization behaviors to simulated agents relying on exemplar learning. We found that groups performed superior to individuals and agents in the training sessions and were more likely to correctly generalize their observations in the transfer phase, especially in the high complexity session, suggesting that groups more effectively induced underlying categorization rules from the stimuli than individuals and agents. Crucially, variation in performance among groups was predicted by semantic diversity in members’ dialogical contributions, suggesting a link between social interaction, cognitive diversity, and abstraction. (shrink)

In this paper, we focus on the development of geometric cognition. We argue that to understand how geometric cognition has been constituted, one must appreciate not only individual cognitive factors, such as phylogenetically ancient and ontogenetically early core cognitive systems, but also the social history of the spread and use of cognitive artifacts. In particular, we show that the development of Greek mathematics, enshrined in Euclid’s Elements, was driven by the use of two tightly intertwined cognitive artifacts: the use of (...) lettered diagrams; and the creation of linguistic formulae. Together, these artifacts formed the professional language of geometry. In this respect, the case of Greek geometry clearly shows that explanations of geometric reasoning have to go beyond the confines of methodological individualism to account for how the distributed practice of artifact use has stabilized over time. This practice, as we suggest, has also contributed heavily to the understanding of what mathematical proof is; classically, it has been assumed that proofs are not merely deductively correct but also remain invariant over various individuals sharing the same cognitive practice. Cognitive artifacts in Greek geometry constrained the repertoire of admissible inferential operations, which made these proofs inter-subjectively testable and compelling. By focusing on the cognitive operations on artifacts, we also stress that mental mechanisms that contribute to these operations are still poorly understood, in contrast to those mechanisms which drive symbolic logical inference. (shrink)

Collaborative interaction pervades many everyday practices: work meetings, innovation and product design, education and arts. Previous studies have pointed to the central role of acknowledgement and acceptance for the success of joint action, by creating affiliation and signaling understanding. We argue that various forms of explicit miscommunication are just as critical to challenge, negotiate and integrate individual contributions in collaborative creative activities. Through qualitative microanalysis of spontaneous coordination in collective creative LEGO constructions, we individuate three interactional styles: inclusive, characterized by (...) acknowledgment and praise; instructional, characterized by self-repair; and integrative, characterized by widespread self- and other-repair. We then investigate how different interaction styles leave distinct material traces in the resulting LEGO models. The inclusive interaction style generally results in concatenations of individual contributions with little coherence and core narrative. The instructional style produces coherent, but largely individually driven models. Finally, the integrative style generates more innovative models, synthesizing individual contributions in shared narratives or schemas. (shrink)

Many types of everyday and specialized reasoning depend on diagrams: we use maps to fnd our way, we draw graphs and sketches to communicate concepts and prove geometrical theorems, and we manipulate diagrams to explore new creative solutions to problems. While the linear and symbolic character of verbal language has long served as the predominant model of human thought, it is remarkable how — through a range of contexts — thinking and communication critically depend on manipulations of external, ofen non-linear, (...) and manipulable iconic-diagrammatic vehicles. In this issue of Pragmatics and Cognition we explore such diagrammatic reasoning, that is, of individual and intersubjectively distributed cognitive processes relying on external representations (i.e. diagrams in a broad sense) as tools for thinking and communicating. (shrink)

This paper outlines a method to assess the effectiveness of diagrams, from semiotic foundations. In doing so, we explore the Peircian notion of signification, as applied to diagrammatic representations. We review a history of diagrams, with particular emphasis on schematics used for representing systems, and uncover the neglect of semiotic analysis of diagrammatic representations. Through application of category theory to the Peircian triadic model, we propose a set of quantitative quality measures for diagrams, and a framework for their assessment, based (...) on the properties of their encoding, pragmatic and perceptual morphisms. These measures include diagram complexity, utility, aesthetics and expert assessment of semiotic content, together with qualitative feedback. We consider the diagrams as an aid to cognitive processes, rather than a purely communication media. This utility-focused perspective on diagram quality dimensions allows for fresh insights into the creation of effective diagrams. (shrink)