Does the human capacity for mathematical intuition depend on linguistic competence or on visuo-spatial representations? A series of behavioral and brain-imaging experiments provides evidence for both sources. Exact arithmetic is acquired in a language-specific format, transfers poorly to a different language or to novel facts, and recruits networks involved in word-association processes. In contrast, approximate arithmetic shows language independence, relies on a sense of numerical magnitudes, and recruits bilateral areas of the parietal lobes involved in visuo-spatial processing. Mathematical intuition may emerge from the interplay of these brain systems.

http://science.sciencemag.org/content/sci/284/5416/970.full.pdf

Sources of Mathematical Thinking: Behavioral and Brain-Imaging Evidence

in micro-sensor and radio technology will enable small but smart sensors to be deployed for a wide range of environmental monitoring applications. In order to constrain communication overhead, dense sensor networks call for new and highly efficient methods for distributing queries to nodes that have observed interesting events in the network. A highly efficient data-centric routing mechanism will offer significant power cost reductions (17), and improve network longevity. Moreover, because of the large amount of system and data redundancy possible, data becomes disassociated from specific node and resides in regions of the network (10)(7)(8). This paper describes and evaluates through simulation a scheme we call Rumor Routing, which allows for queries to be delivered to events in the network. Rumor Routing is tunable, and allows for tradeoffs between setup overhead and delivery reliability. It's intended for contexts in which geographic routing criteria are not applicable because a coordinate system is not available or the phenomenon of interest is not geographically correlated.

/pdf/rumor-routing-algorithm-for-sensor-networks-5fhjbo7v78.pdf

Rumor Routing Algorithm For Sensor Networks

Two experiments examined kindergartners', first graders', and second graders' numerical estimation, the internal representations that gave rise to the estimates, and the general hypothesis that developmental sequences within a domain tend to repeat themselves in new contexts. Development of estimation in this age range on 0-to-100 number lines followed the pattern observed previously with older children on 0-to-1,000 lines. Between kindergarten and second grade (6 and 8 years), patterns of estimates progressed from consistently logarithmic to a mixture of logarithmic and linear to a primarily linear pattern. Individual differences in number-line estimation correlated strongly with math achievement test scores, improved estimation accuracy proved attributable to increased linearity of estimates, and exposure to relevant experience tended to improve estimation accuracy.

Development of Numerical Estimation in Young Children

http://www.unicog.org/publications/DehaeneCohen_TwoAcalculias_cortex1997.pdf

Cerebral Pathways for Calculation: Double Dissociation between Rote Verbal and Quantitative Knowledge of Arithmetic

This article highlights key findings from the small body of research on mathematics difficulties (MD) relevant to early identification and early intervention. The research demonstrates that (a) for many children, mathematics difficulties are not stable over time; (b) the presence of reading difficulties seems related to slower progress in many aspects of mathematics; (c) almost all students with MD demonstrate problems with accurate and automatic retrieval of basic arithmetic combinations, such as 6 + 3. The following measures appear to be valid and reliable indicators of potential MD in kindergartners: (a) magnitude comparison (i.e., knowing which digit in a pair is larger), (b) sophistication of counting strategies, (c) fluent identification of numbers, and (d) working memory (as evidenced by reverse digit span). These are discussed in terms of the components of number sense. Implications for early intervention strategies are explored.

Early Identification and Interventions for Students With Mathematics Difficulties

Researchers have assumed that adults solve simple arithmetic problems by retrieving answers from a network of stored facts. In 2 studies, undergraduates described their solutions of single-digit multiplication problems. They reported direct retrieval on approximately 80% of trials, but also reported rules (e.g., anything times 0 is 0), repeated addition (e.g., 2 × 4 = 4 + 4), number series (e.g., 3 × 5 = 5, 10, 15), and derived facts (e.g., 6 × 7 = [6 × 6] + 6). Participants were slower to retrieve problems that were most likely to be solved by nonretrieval procedures and faster to retrieve problems that were usually solved by retrieval. These results indicate that direct-retrieval models are incomplete accounts of adults' performance and support a continuing influence of learning and experience on the mental representation of simple multiplication problems.

Lisa Buffone

Papers

Multiple routes to solution of single-digit multiplication problems.