Final Report Summary - MATHCONSTRUCTION (Constructing Mathematical Knowledge beyond Core Intuitions) Research in psychophysics, neuroscience, developmental and comparative psychology converge on the idea that humans and other animals possess dedicated systems of “core knowledge” representing numeric and geometric information. Such intuitive knowledge is thought to guide the acquisition of elaborate concepts of numbers and geometry. However, core systems of representations for numbers and geometry fall short of providing the representational power to support even the most fundamental mathematical concepts: Integers, and Euclidean geometry. Within this framework, the MathConstruction project sought to understand the origins of numeric and geometric concepts, from birth to adulthood. We focused mostly on two basic concepts: exact numbers, and plane angles.A first line of research aimed at characterizing infants’ numerical competences at the very start of life (0-3 days of age). Beyond newborn infants’ capacity to perceive abstract numerosity across the visual and auditory modalities, we found that they possess a generic sense of quantity applying equally to numerosity, to spatial extent, and to duration. Second, we found a discontinuity in newborns’ reactions to small and to large numerosities, much like the discontinuity observed in older infants – just like older infants, newborns thus seem to possess a separate system for representing small sets. Lastly, we showed that newborns detect amodal correspondences between numerically structured small sets (AB vs. ABB), again showing sensitivity to abstract information.In a second line of research, we investigated children’s acquisition of exact knowledge concepts. In contrast to both leading theories of numerical development, we found that young children (2 years ½) are not bound to approximate quantities, nor do they possess a full-fledged concept of exact number. More specifically, they do not understand that one-to-one correspondence indexes number, but rather interpret it as indexing set extension. In follow-up work, we found that this limitation is resolved in children one year older, but this development does not relate to learning number words.We also tested whether children grasp some essential structural properties of the integers: the fact that integers form a non-circular structure (they do not loop back, unlike for example the days of the week), and the fact that all integers can be generated by iterating a ‘successor function’ (+1). We designed new tasks to approach these questions, with school-aged children. We found an understanding of the non-circular structure already in 1st grade; however, children do not show any awareness of the successor function structure until grade 2.Our third line of research focused on geometry. We used angle as a test case to probe children’s knowledge of Euclidean geometry, the geometry that has been historically called “natural”. To our surprise, our results converge to show little understanding of angles in the first years of life. First, contra claims published in the literature, we found that infants are totally insensitive to angle. Second, we found that preschool children gradually develop an ability to extract angle, yet initially their perception is highly biased towards cues of size and distance. Third, we found that when asked to reason about angle, preschool children use distance between sides as an (inaccurate) proxy, this tendency slowly decaying as children attend primary school. Lastly, although 5th graders showed limitations in their apprehension of angle (sometimes stating that angle increases when a figure increases in size, for example), adults were no better. We thus propose that Euclidean geometry, the “natural” geometry that has angle as one of its core concepts, is actually the result of a mental construction – and that a crucial step in this mental construction involves the creation of two separate concepts for angle and distance.
