Symbolic arithmetic knowledge without instruction.

Symbolic arithmetic is fundamental to science, technology and economics, but its acquisition by children typically requires years of effort, instruction and drill. When adults perform mental arithmetic, they activate nonsymbolic, approximate number representations, and their performance suffers if this nonsymbolic system is impaired. Nonsymbolic number representations also allow adults, children, and even infants to add or subtract pairs of dot arrays and to compare the resulting sum or difference to a third array, provided that only approximate accuracy is required. Here we report that young children, who have mastered verbal counting and are on the threshold of arithmetic instruction, can build on their nonsymbolic number system to perform symbolic addition and subtraction. Children across a broad socio-economic spectrum solved symbolic problems involving approximate addition or subtraction of large numbers, both in a laboratory test and in a school setting. Aspects of symbolic arithmetic therefore lie within the reach of children who have learned no algorithms for manipulating numerical symbols. Our findings help to delimit the sources of children's difficulties learning symbolic arithmetic, and they suggest ways to enhance children's engagement with formal mathematics.

The Hippo pathway effector YAP is an essential regulator of ductal progenitor patterning in the mouse submandibular gland.

Salivary glands, such as submandibular glands (SMGs), are composed of branched epithelial ductal networks that terminate in acini that together produce, transport and secrete saliva. Here, we show that the transcriptional regulator Yap, a key effector of the Hippo pathway, is required for the proper patterning and morphogenesis of SMG epithelium. Epithelial deletion of Yap in developing SMGs results in the loss of ductal structures, arising from reduced expression of the EGF family member Epiregulin, which we show is required for the expansion of Krt5/Krt14-positive ductal progenitors. We further show that epithelial deletion of the Lats1 and Lats2 genes, which encode kinases that restrict nuclear Yap localization, results in morphogenesis defects accompanied by an expansion of Krt5/Krt14-positive cells. Collectively, our data indicate that Yap-induced Epiregulin signaling promotes the identity of SMG ductal progenitors and that removal of nuclear Yap by Lats1/2-mediated signaling is critical for proper ductal maturation.

Non-symbolic arithmetic abilities and mathematics achievement in the first year of formal schooling.

Children take years to learn symbolic arithmetic. Nevertheless, non-human animals, human adults with no formal education, and human infants represent approximate number in arrays of objects and sequences of events, and they use these capacities to perform approximate addition and subtraction. Do children harness these abilities when they begin to learn school mathematics? In two experiments in different schools, kindergarten children from diverse backgrounds were tested on their non-symbolic arithmetic abilities during the school year, as well as on their mastery of number words and symbols. Performance of non-symbolic arithmetic predicted children's mathematics achievement at the end of the school year, independent of achievement in reading or general intelligence. Non-symbolic arithmetic performance was also related to children's mastery of number words and symbols, which figured prominently in the assessments of mathematics achievement in both schools. Thus, non-symbolic and symbolic numerical abilities are specifically related, in children of diverse socio-economic backgrounds, near the start of mathematics instruction.