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

Abstract

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.

Keywords: Development; Hierarchy; Iteration; Language evolution; Recursion; Visuo-spatial.

Fig. 1

Examples of rules used to generate hierarchical relationships.

Fig. 2

Recursive process generating a visual fractal.

Fig. 3

Schematic depiction of the different representations required to perform our non-recursive iterative task (A), and our recursive task (B) (adapted from Martins et al. (2014)).

Fig. 4

Example of a typical Visual Recursion Task trial. Initially, the first three iterations of a fractal generation are depicted, sequentially, from left to right, with an interval of 2 s between iteration (top). On the presentation of each new image, the previous iterations remain visible on the screen, in the positions depicted in the figure. Then, while the first three iterations remain visible, two additional images are presented simultaneously in the bottom part of the screen, corresponding to the ‘correct’ fourth iteration (bottom right) and a foil (bottom left), and the participant chooses between them. In this example, the foil is a ‘positional foil’ (see Fig.5 for details on foils).

Fig. 5

Examples of fractals used in the Visual Recursion Task. The first four iterations of a fractal generation, as well as one foil (‘incorrect’ fourth iteration), were produced. There were different categories of ‘visual complexity’ – 3, 4 and 5 – and different categories of foils. In ‘Odd constituent’ foils two elements within the whole hierarchy were misplaced; in ‘positional error’ foils all elements within new hierarchical levels were internally consistent, but inconsistent with the previous iterations; in ‘Repetition’ foils no additional iterative step was performed after the third iteration.

Fig. 6

Example of a typical embedded iteration task trial. In this example, the correct answer is on the bottom left. The foil on the bottom right is a “repetition foil” (see Fig.5 for details on foils).

Fig. 7

Performance in Visual Recursion Task (VRT) and Embedded Iteration Task (EIT), across grades. Fourth graders had higher scores than second graders, in both VRT and EIT. Within each grade, the difference between tasks was not significant.

Fig. 8

Performance across different task-sequence conditions. In the sequence condition ‘VRT-EIT’ (right) participants performed the Visual Recursion Task (VRT) first; in the condition ‘EIT-VRT’ (left) participants performed the Embedded Iteration Task (EIT) first. Children who performed the iterative task first scored globally better than those who started with recursion. Crucially, starting the procedure with VRT decreased EIT accuracy. This suggests that children transferred knowledge from simple iteration to recursion, but not the other way around.