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Students exhibiting mature number sense make sense of numbers and operations, use reasoning to notice patterns, and flexibly select the most effective and efficient problem-solving strategies (McIntosh et al., 1997; Reys et al., 1999; Yang, 2005). Despite being highlighted in national standards and policy documents (CCSS, 2010; NCTM, 2000, 2014), students’ mature number sense and its nomological network are not yet well specified. For example, how does students’ mature number sense relate to their knowledge of fractions and their grade-level mathematics achievement? We analyzed 129 middle school students’ scores on measures of mature number sense, fraction and decimal computation, and grade-level mathematics achievement. We found mature number sense to be measurably distinct from their fraction and decimal knowledge and uniquely associated with students’ grade-level mathematics achievement.

Background: Dinoflagellates are taxonomically diverse and ecologically important phytoplankton that are ubiquitously present in marine and freshwater environments. Mostly photosynthetic, dinoflagellates provide the basis of aquatic primary production; most taxa are free-living, while some can form symbiotic and parasitic associations with other organisms. However, knowledge of the molecular mechanisms that underpin the adaptation of these organisms to diverse ecological niches is limited by the scarce availability of genomic data, partly due to their large genome sizes estimated up to 250 Gbp. Currently available dinoflagellate genome data are restricted to Symbiodiniaceae (particularly symbionts of reef-building corals) and parasitic lineages, from taxa that have smaller genome size ranges, while genomic information from more diverse free living species is still lacking. Results: Here, we present two draft diploid genome assemblies of the free-living dinoflagellate Polarella glacialis, isolated from the Arctic and Antarctica. We found that about 68% of the genomes are composed of repetitive sequence, with long terminal repeats likely contributing to intra-species structural divergence and distinct genome sizes (3.0 and 2.7 Gbp). For each genome, guided using full-length transcriptome data, we predicted > 50,000 high-quality protein-coding genes, of which ~40% are in unidirectional gene clusters and ~25% comprise single exons. Multi-genome comparisonmore »unveiled genes specific to P. glacialis and a common, putatively bacterial origin of ice-binding domains in cold-adapted dinoflagellates. Conclusions: Our results elucidate how selection acts within the context of a complex genome structure to facilitate local adaptation. Because most dinoflagellate genes are constitutively expressed, Polarella glacialis has enhanced transcriptional responses via unidirectional, tandem duplication of single-exon genes that encode functions critical to survival in cold, low-light polar environments. These genomes provide a foundational reference for future research on dinoflagellate evolution.« less