Beyond Core Knowledge: Natural Geometry

Abstract

For many centuries, philosophers and scientists have pondered the origins and nature of human intuitions about the properties of points, lines, and figures on the Euclidean plane, with most hypothesizing that a system of Euclidean concepts either is innate or is assembled by general learning processes. Recent research from cognitive and developmental psychology, cognitive anthropology, animal cognition, and cognitive neuroscience suggests a different view. Knowledge of geometry may be founded on at least two distinct, evolutionarily ancient, core cognitive systems for representing the shapes of large-scale, navigable surface layouts and of small-scale, movable forms and objects. Each of these systems applies to some but not all perceptible arrays and captures some but not all of the three fundamental Euclidean relationships of distance (or length), angle, and direction (or sense). Like natural number (Carey, 2009), Euclidean geometry may be constructed through the productive combination of representations from these core systems, through the use of uniquely human symbolic systems.

Nonsymbolic number and cumulative area representations contribute shared and unique variance to symbolic math competence

“…Such intuitions, it is believed, rest on perceptual properties of the visual system and universal spatial experiences. That geometry is highly visual and perhaps grounded in navigation abilities (59) and that arithmetic of small sets of objects recruits parallel individuation processes (60) may provide some basis for protomathematical intuitions. The present findings suggest a further basis for such intuitions by showing that even formally taught mathematics performed by college students may be rooted in analog representations of number and cumulative area, with nonsymbolic magnitude precision predicting individual differences in advanced arithmetic and school-relevant geometry.…”

Intuitive geometry and visuospatial working memory in children showing symptoms of nonverbal learning disabilities

“…According to Spelke et al (2010), natural geometry is founded on at least two evolutionarily ancient and cross-culturally universal cognitive systems that capture abstract information about the shape of the surrounding world: two core systems of geometry. The first represents the shapes of large-scale navigable surface layouts, while the second represents small-scale movable forms and objects.…”

“…The concept of intuitive geometry has been recently introduced by Dehaene, Izard, Pica, and Spelke (2006). These authors investigated whether some principles of geometry can be considered as core culture-free concepts (see also Spelke, Lee, & Izard, 2010) by examining the spontaneous geometrical knowledge of an Amazonian native group that was not exposed to a geometrical instruction. Dehaene and colleagues hypothesized that people might possess primitive principles of geometry, similar to the case for numerical knowledge.…”

The relationship among geometry, working memory, and intelligence in children

“…The former includes abilities that are believed to be independent of a person's cultural background and education; the latter depends on learning and education. In fact, it has been suggested (Spelke et al, 2010; see also Dehaene, Izard, Pica, & Spelke, 2006) that geometry includes a core intuitive knowledge. Numerous intuitions seem to develop during childhood and spontaneously accord with the principles of Euclidean geometry, even in the absence of training in mathematics (Izard, Pica, Spelke, & Dehaene, 2011).…”

“…Moreover, some researchers actually hypothesize the existence of more than two systems (Carey, 2009) or propose that geometry systems are distinct from number systems (Spelke, Lee, & Izard, 2010), thus further articulating the mathematical domain.…”