How do people develop well-structured representations for non-perceptual, quantitative concepts? For example, when mathematicians reason about five-dimensions, do they depend on internal spatial representations, do they apply a series of symbolic rules, or do they use a combination of both? The current research addresses this question in the context of children learning about zero and the negative numbers. By the time most children begin learning the integers, they have an internal spatial representation that supports their abilities to reason about natural number magnitude, even when presented symbolically as digits. Recent evidence, however, indicates that children?s reasoning with zero and negative numbers relies on the application of syntactic rules. This contrasts with most adults who have developed a spatial representation of negative numbers in their own right. A combination of instructional, behavioral, and fMRI methods are examining the relative influences of spatial and symbolic experiences on brain reorganization and children?s development of integer concepts.
Outcome and broad impact: Researchers at Stanford University brought together systems neuroscience, the science of learning, and mathematics education to improve children’s learning of an abstract mathematical concept – negative numbers. The researchers found that brain systems associated with detecting visual symmetry activate when adults solve problems involving integers. The researchers created a novel hands-on and computerized instructional sequence to emphasize the symmetry of the integers. They found that this instruction improved children’s abilities to solve novel problems involving the integers compared to prevailing instructional techniques. Experiments: Through a series of experiments involving response-time recordings and neuroimaging, the research team found that adults implicitly use symmetry perception to solve integer arithmetic problems. The researchers tested the educational implications by teaching 9-10-year old children about integer properties, with or without a symmetry emphasis. Children in all conditions showed equal gains in integer addition and number comparisons, but students who received the symmetry framing were better able to apply their integer knowledge to unfamiliar problems and new topics like fractions. Intellectual merit: For the natural numbers, perceptual-motor systems are widely understood to contribute to learning and problem solving: Children combine touching, visual grouping, and an innate sense of perceptual magnitude to develop an understanding of operations on numerical symbols. Beyond natural numbers, it is unclear whether and how perception is involved, as numbers beyond the naturals often are imperceptible and due to cultural invention. The present research shows that perceptual computations support humans’ understanding of the integers, a number system that is more complex and taught later in school than the natural numbers.
