MIND, BRAIN, AND EDUCATION In Search of Structures: How Does the Mind Explore Infinity? Florence Mihaela Singer 1 and Cristian Voica 2 ABSTRACT—When reasoning about infinite sets, children seem to activate four categories of conceptual structures: geometric (g-structures), arithmetic (a-structures), fractal- type (f -structures), and density-type (d-structures). Students select different problem-solving strategies depending on the structure they recognize within the problem domain. They naturally search for structures in challenging learning contexts. This tendency to search for structure might be a characteristic of human cognition and a necessary condition for human knowledge development. For example, specific fractal structures are intrinsic to concepts such as the numerical system that have been developed by the human race over a long period of time. When these structures are emphasized within teaching, they can facilitate the deep understanding of several basic concepts, in mathematics and beyond. Infinity is a topic of great interest for theorists of various domains, as well as for the larger public. The concept of infinity has always had a transcendental dimension that makes it difficult to approach. However, mathematics offers some tools to deal with infiniteness. Thus, the infinite mathematical entities, such as infinite sets of numbers and infinite sets of points, might be seen as tools for entering into the world of infinities. For this world, mathematics gives a language, descriptions, and ways to operate. By putting students in specific operational contexts within mathematics, we can learn more about how they relate concrete, tangible objects with intangible infinite representations. Because infinity is not a topic as such in the school curriculum, students mobilize their intuition rather than their more grounded knowledge in order to answer questions. By asking students about infinity in a mathematical context 1 University of Ploiesti 2 University of Bucharest Address correspondence to Florence Mihaela Singer, Fac- ulty of Letters and Science, University of Ploiesti, G-ral Berthelot 38, 010169 Bucharest, Romania; e-mail: mikisinger@ gmail.com. we can use a mathematical language to better describe the results of this exploration, and we obtain a snapshot of students’ depth of mathematical understanding. In turn, this exploration casts light on how the mind works when it operates with transcendental entities such as infinity and infiniteness. Moreover, such a study might lead to possible strategies for better teaching and learning. Infinity is inexorably related to numbers; consequently we begin by exploring studies on number perception in humans. A large body of cognitive science and neuroscience research has revealed that children have stronger native predispositions for processing numbers than assumed by the Piagetian perspective (Carey, 1999, 2001; Dehaene, 2001, 2007; Hartnett & Gelman, 1998; Karmiloff-Smith, 1992; Mix, Levine, & Huttenlocher, 1999; Singer, 2007a; Wynn, 1992). For example, in a study with 8- to 12-year-olds, Smith, Solomon, and Carey (2005) identified a high level of coherence between children’s thinking about the infinite divisibility of weight, on the one hand, and the infinite divisibility of number, on the other, concluding that there are multiple, two-way interactions between the domains of number and matter, as children construct an understanding of the continuity of matter, space, and weight and an understanding of rational number. A recent study (Anderson et al., 2008) made a distinction between spatial imagery and object imagery, showing that only spatial imagery is related to success in mathematical problem solving. This distinction is important for our perspective because it demonstrates the existence of separate cognitive mechanisms for processing objects (which are based on shape) and relations among objects (which are based on spatial relationships). Because our study included children from a wide age range, we also examined our results in light of recent research in cognitive developmental psychology. The research of Fischer and Bidell (2006), Dawson-Tunik, Commons, Wilson, and Fischer (2005), and Commons, Trudeau, Stein, Richards, and Krause (1998) discuss the shape of development, emphasizing the idea of hierarchical complexity. Does the increase in conceptual complexity through cognitive development stimulate or hinder the perception of infinity? Although the present study cast some light on this issue, it is still a question open to debate.  2010 the Authors Volume 4—Number 2 Journal Compilation  2010 International Mind, Brain, and Education Society and Blackwell Publishing, Inc. 81