When thought overwhelms the mind, the mind uses the body and the world. Several studies reveal ways that people alone or together use gesture and marks on paper to structure and augment their thought for comprehension, inference, and discovery. The studies show that the mapping of thought to gesture or the page is more direct than the arbitrary mapping to language and suggest that these forms of visual/spatial/action representation are used to “translate” language into mental representations. It is argued that (...) actions in space create patterns in the world that reflect abstractions, that the actions are incorporated into gestures and the patterns into diagrams, a network that integrates gesture, action, the designed world, and abstraction dubbed spraction. (shrink)

When thought overwhelms the mind, the mind uses the body and the world. Several studies reveal ways that people alone or together use gesture and marks on paper to structure and augment their thought for comprehension, inference, and discovery. The studies show that the mapping of thought to gesture or the page is more direct than the arbitrary mapping to language and suggest that these forms of visual/spatial/action representation are used to “translate” language into mental representations. It is argued that (...) actions in space create patterns in the world that reflect abstractions, that the actions are incorporated into gestures and the patterns into diagrams, a network that integrates gesture, action, the designed world, and abstraction dubbed spraction. (shrink)

Depictive expressions of thought predate written language by thousands of years. They have evolved in communities through a kind of informal user testing that has refined them. Analyzing common visual communications reveals consistencies that illuminate how people think as well as guide design; the process can be brought into the laboratory and accelerated. Like language, visual communications abstract and schematize; unlike language, they use properties of the page (e.g., proximity and place: center, horizontal/up–down, vertical/left–right) and the marks on it (e.g., (...) dots, lines, arrows, boxes, blobs, likenesses, symbols) to convey meanings. The visual expressions of these meanings (e.g., individual, category, order, relation, correspondence, continuum, hierarchy) have analogs in language, gesture, and especially in the patterns that are created when people design the world around them, arranging things into piles and rows and hierarchies and arrays, spatial-abstraction-action interconnections termed spractions. The designed world is a diagram. (shrink)

In this paper, we focus on the development of geometric cognition. We argue that to understand how geometric cognition has been constituted, one must appreciate not only individual cognitive factors, such as phylogenetically ancient and ontogenetically early core cognitive systems, but also the social history of the spread and use of cognitive artifacts. In particular, we show that the development of Greek mathematics, enshrined in Euclid’s Elements, was driven by the use of two tightly intertwined cognitive artifacts: the use of (...) lettered diagrams; and the creation of linguistic formulae. Together, these artifacts formed the professional language of geometry. In this respect, the case of Greek geometry clearly shows that explanations of geometric reasoning have to go beyond the confines of methodological individualism to account for how the distributed practice of artifact use has stabilized over time. This practice, as we suggest, has also contributed heavily to the understanding of what mathematical proof is; classically, it has been assumed that proofs are not merely deductively correct but also remain invariant over various individuals sharing the same cognitive practice. Cognitive artifacts in Greek geometry constrained the repertoire of admissible inferential operations, which made these proofs inter-subjectively testable and compelling. By focusing on the cognitive operations on artifacts, we also stress that mental mechanisms that contribute to these operations are still poorly understood, in contrast to those mechanisms which drive symbolic logical inference. (shrink)

Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols are not only used to (...) express mathematical concepts—they are constitutive of the mathematical concepts themselves. Mathematical symbols are epistemic actions, because they enable us to represent concepts that are literally unthinkable with our bare brains. Using case-studies from the history of mathematics and from educational psychology, we argue for an intimate relationship between mathematical symbols and mathematical cognition. (shrink)