Math proficiency in early school age is an important predictor of later academic achievement. Thus, an important goal for society should be to improve math readiness in pre-school age children, especially in low-income children who typically arrive in kindergarten with less mathematical competency than their higher-income peers. The majority of existing research-based math intervention programs target symbolic, verbal number concepts in young children. However, very little attention has been paid to the preverbal, intuitive ability to approximately represent numerical quantity, which is hypothesized to be an important foundation for full-fledged mathematical thinking. Here, we test the hypothesis that repeated engagement of non-symbolic approximate addition and subtraction of large array of items results in improved math skills in very young children, an idea that stems from our previous studies in adults. Three to five year-old children showed selective improvements in math skills after multiple days of playing a tablet-based non-symbolic approximate arithmetic game compared to children who played a memory game. These findings, collectively with our previous reports, suggest that mental manipulation of approximate numerosities provides an important tool for improving math readiness, even in preschoolers who have yet to master the meaning of number words.

Given a fibred, compact, orientable 3‐manifold with single boundary component, we show that a fibration with fiber surface of negative Euler characteristic can be perturbed to yield taut foliations realizing an open interval of boundary slopes about the boundary slope of the fibration. These taut foliations extend to taut foliations in the corresponding surgery manifolds. 2000 Mathematics Subject Classification: primary 57M25; secondary 57R30.

We investigate group actions on simply-connected (second countable but not necessarily Hausdorff) 1-manifolds and describe an infinite family of closed hyperbolic 3-manifolds whose fundamental groups do not act nontrivially on such 1-manifolds. As a corollary we conclude that these 3-manifolds contain no Reebless foliation. In fact, these arguments extend to actions on oriented R \mathbb R -order trees and hence these 3-manifolds contain no transversely oriented essential lamination; in particular, they are non-Haken.

We extend the Eliashberg-Thurston theorem on approximations of taut oriented C 2 -foliations of 3-manifolds by both positive and negative contact structures to a large class of taut oriented C 1;0 -foliations, where by C 1;0 foliation we mean a foliation with continuous tangent plane field. These C 1;0 -foliations can therefore be approximated by weakly symplectically fillable, universally tight, contact structures. This allows applications of C 2 -foliation theory to contact topology and Floer theory to be generalized and extended to constructions of C 1;0 -foliations. 57M50; 53D10

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