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      How children perceive fractals: Hierarchical self-similarity and cognitive development

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          • Visual recursion becomes available around the same age as iteration (9–10 years-old).

          • Acquisition of visual recursion is facilitated by previous experience with iteration.

          • Both visual recursion and iteration correlate with grammar comprehension.

          • Representing hierarchies recursively improves detection of nested mistakes.

          • Recursion in vision follows learning constraints similar to language.


          The ability to understand and generate hierarchical structures is a crucial component of human cognition, available in language, music, mathematics and problem solving. Recursion is a particularly useful mechanism for generating complex hierarchies by means of self-embedding rules. In the visual domain, fractals are recursive structures in which simple transformation rules generate hierarchies of infinite depth. Research on how children acquire these rules can provide valuable insight into the cognitive requirements and learning constraints of recursion.

          Here, we used fractals to investigate the acquisition of recursion in the visual domain, and probed for correlations with grammar comprehension and general intelligence. We compared second ( n = 26) and fourth graders ( n = 26) in their ability to represent two types of rules for generating hierarchical structures: Recursive rules, on the one hand, which generate new hierarchical levels; and iterative rules, on the other hand, which merely insert items within hierarchies without generating new levels. We found that the majority of fourth graders, but not second graders, were able to represent both recursive and iterative rules. This difference was partially accounted by second graders’ impairment in detecting hierarchical mistakes, and correlated with between-grade differences in grammar comprehension tasks. Empirically, recursion and iteration also differed in at least one crucial aspect: While the ability to learn recursive rules seemed to depend on the previous acquisition of simple iterative representations, the opposite was not true, i.e., children were able to acquire iterative rules before they acquired recursive representations. These results suggest that the acquisition of recursion in vision follows learning constraints similar to the acquisition of recursion in language, and that both domains share cognitive resources involved in hierarchical processing.

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          Most cited references 37

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          Cognitive control, hierarchy, and the rostro-caudal organization of the frontal lobes.

           David Badre (2008)
          Cognitive control supports flexible behavior by selecting actions that are consistent with our goals and appropriate for our environment. The prefrontal cortex (PFC) has an established role in cognitive control, and research on the functional organization of PFC promises to contribute to our understanding of the architecture of control. A recently popular hypothesis is that the rostro-caudal axis of PFC supports a control hierarchy whereby posterior-to-anterior PFC mediates progressively abstract, higher-order control. This review discusses evidence for a rostro-caudal gradient of function in PFC and the theories proposed to account for these results, including domain generality in working memory, relational complexity, the temporal organization of behavior and abstract representational hierarchy. Distinctions among these frameworks are considered as a basis for future research.
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            The development of human functional brain networks.

            Recent advances in MRI technology have enabled precise measurements of correlated activity throughout the brain, leading to the first comprehensive descriptions of functional brain networks in humans. This article reviews the growing literature on the development of functional networks, from infancy through adolescence, as measured by resting-state functional connectivity MRI. We note several limitations of traditional approaches to describing brain networks and describe a powerful framework for analyzing networks, called graph theory. We argue that characterization of the development of brain systems (e.g., the default mode network) should be comprehensive, considering not only relationships within a given system, but also how these relationships are situated within wider network contexts. We note that, despite substantial reorganization of functional connectivity, several large-scale network properties appear to be preserved across development, suggesting that functional brain networks, even in children, are organized in manners similar to other complex systems. 2010 Elsevier Inc. All rights reserved.
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              Pathways to language: fiber tracts in the human brain.

              The human language function is not only based on the grey matter of circumscribed brain regions in the frontal and the temporal cortex but moreover on the white matter fiber tracts connecting these regions. Different pathways connecting frontal and temporal cortex have been identified. The dorsal pathway projecting from the posterior portion of Broca's area to the superior temporal region seems to be of particular importance for higher-order language functions. This pathway is particularly weak in non-human compared to human primates and in children compared to adults. It is therefore considered to be crucial for the evolution of human language, which is characterized by the ability to process syntactically complex sentences.

                Author and article information

                1 October 2014
                October 2014
                : 133
                : 1
                : 10-24
                [a ]Department of Cognitive Biology, University of Vienna, Althanstrasse 14, A-1090 Vienna, Austria
                [b ]Language Research Laboratory, Lisbon Faculty of Medicine, Av. Professor Egas Moniz, 1649-028 Lisboa, Portugal
                [c ]Department of Linguistics, University of Vienna, Porzellangasse 4, A-1090 Vienna, Austria
                [d ]Department of Linguistics & Asian/Middle Eastern Languages, San Diego State University, 5500 Campanile Drive, San Diego, CA 92182-7727, USA
                Author notes
                [* ]Corresponding author at: Department of Cognitive Biology, Althanstrasse 14, Wien A-1090 Vienna, Austria. mauricio.martins@
                © 2014 The Authors

                This is an open access article under the CC BY-NC-ND license (



                visuo-spatial, recursion, iteration, hierarchy, language evolution, development


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