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This book consists of conclusions drawn from the expertise shared at the Conference on Standards for Prekindergarten and Kindergarten Mathematics Education. It offers substantive detail regarding young students' understandings of mathematical ideas.

Proving mathematical theorems at the olympiad level represents a notable milestone in human-level automated reasoning 1 – 4 , owing to their reputed difficulty among the world’s best talents in pre-university mathematics. Current machine-learning approaches, however, are not applicable to most mathematical domains owing to the high cost of translating human proofs into machine-verifiable format. The problem is even worse for geometry because of its unique translation challenges 1 , 5 , resulting in severe scarcity of training data. We propose AlphaGeometry, a theorem prover for Euclidean plane geometry that sidesteps the need for human demonstrations by synthesizing millions of theorems and proofs across different levels of complexity. AlphaGeometry is a neuro-symbolic system that uses a neural language model, trained from scratch on our large-scale synthetic data, to guide a symbolic deduction engine through infinite branching points in challenging problems. On a test set of 30 latest olympiad-level problems, AlphaGeometry solves 25, outperforming the previous best method that only solves ten problems and approaching the performance of an average International Mathematical Olympiad (IMO) gold medallist. Notably, AlphaGeometry produces human-readable proofs, solves all geometry problems in the IMO 2000 and 2015 under human expert evaluation and discovers a generalized version of a translated IMO theorem in 2004. A new neuro-symbolic theorem prover for Euclidean plane geometry trained from scratch on millions of synthesized theorems and proofs outperforms the previous best method and reaches the performance of an olympiad gold medallist.

Multiple-choice examinations play a critical role in university admissions across the world. A key question is whether imposing penalties for wrong answers on these examinations deters guessing from women more than men, disadvantaging female testtakers. We consider data from a large-scale, high-stakes policy change that removed penalties for wrong answers on the national college entry examination in Chile. The policy change reduced a large gender gap in questions skipped. It also narrowed gender gaps in performance, primarily among high-performing test-takers, and in the fields of math, social science, and chemistry.

Each year, 15 million babies worldwide are born preterm. Preterm birth is associated with adverse neurodevelopmental outcomes across the life span. Recent registry-based studies suggest that preterm birth is associated with decreased wealth in adulthood, but the mediating mechanisms are unknown. This study investigated whether the relationship between preterm birth and low adult wealth is mediated by poor academic abilities and educational qualifications. Participants were members of two British population-based birth cohorts born in 1958 and 1970, respectively. Results showed that preterm birth was associated with decreased wealth at 42 years of age. This association was mediated by decreased intelligence, reading, and, in particular, mathematics attainment in middle childhood, as well as decreased educational qualifications in young adulthood. Findings were similar in both cohorts, which suggests that these mechanisms may be time invariant. Special educational support in childhood may prevent preterm children from becoming less wealthy as adults.

Recently, there has been a growing emphasis on basic number processing competencies (such as the ability to judge which of two numbers is larger) and their role in predicting individual differences in school-relevant math achievement. Children's ability to compare both symbolic (e.g. Arabic numerals) and nonsymbolic (e.g. dot arrays) magnitudes has been found to correlate with their math achievement. The available evidence, however, has focused on computerized paradigms, which may not always be suitable for universal, quick application in the classroom. Furthermore, it is currently unclear whether both symbolic and nonsymbolic magnitude comparison are related to children's performance on tests of arithmetic competence and whether either of these factors relate to arithmetic achievement over and above other factors such as working memory and reading ability. In order to address these outstanding issues, we designed a quick (2 minute) paper-and-pencil tool to assess children's ability to compare symbolic and nonsymbolic numerical magnitudes and assessed the degree to which performance on this measure explains individual differences in achievement. Children were required to cross out the larger of two, single-digit numerical magnitudes under time constraints. Results from a group of 160 children from grades 1-3 revealed that both symbolic and nonsymbolic number comparison accuracy were related to individual differences in arithmetic achievement. However, only symbolic number comparison performance accounted for unique variance in arithmetic achievement. The theoretical and practical implications of these findings are discussed which include the use of this measure as a possible tool for identifying students at risk for future difficulties in mathematics.

Start young children off with Common Core math using these innovative activities Teaching the Common Core Math Standards with Hands-On Activities, Grades K-2 provides teachers with the help they need to begin teaching to the new standards right away. The book outlines the Common Core math standards from kindergarten to second grade, providing one classroom-ready activity for each standard, plus suggestions for variations and extensions for students of different learning styles and abilities. Along with teaching the required mathematical concepts and skills, many of the activities encourage collaboration, technology utilization, written and oral communication, and an appreciation of the significance of mathematics in modern life. As the Common Core is adopted across the nation, teachers are scrambling to find information on CCSS-aligned lesson planning and classroom activities. This comprehensive guide answers that need, providing both the background information and practical, applicable guidance that can bring the Common Core into the classroom today. The activities include: * Abstract and critical thinking using mathematical reasoning * Problem-solving strategies and calculation proficiency * Math fluency, and an understanding of mathematical concepts and skills * Applying mathematical understanding to real life problems Early confidence and success in math is critical to a student's future performance. Math anxiety and a shaky foundation can hinder a student's potential far into the future, giving elementary math teachers a huge role in shaping their students' academic lives. The Common Core has set the bar, and Teaching the Common Core Math Standards with Hands-On Activities, Grades K-2 brings the standards to life.

Eight middle school mathematics teachers’ perceptions and uses of curriculum materials and the Common Core State Standards for Mathematics (CCSSM) were investigated. Adapting a noticing framework and models of dialogic instruction and direct instruction, teachers’ noticing practices with curriculum materials and the CCSSM when planning, enacting, and reflecting on lessons were examined. Teachers who were committed to implementing the CCSSM and who were using one of two substantively different curriculum programs were purposefully selected. Data sources included multiple forms of interviews and classroom observations. The teaching evidenced three distinct noticing patterns. These patterns indicated that teachers’ curriculum materials were associated with how teachers perceived and enacted the CCSSM. Teaching with a curriculum program that was designed as a thinking device prioritized the Standards for Mathematical Practice of CCSSM evidenced noticing that was consistent with dialogic instruction. Teaching with a curriculum program that was designed as a delivery mechanism prioritized the Content Standards of CCSSM and evidenced noticing consistent with direct instruction. Findings indicated that the designated curriculum and contributed to differing interpretations of CCSSM and served as a lens for noticing. However, a dialogic curriculum program was not sufficient to support dialogic approaches in practice. One pattern showed teachers planning dialogic lessons, but the lesson enactments were not consistent with teachers’ plans, with evidence that the teachers were not aware that their practices differed from dialogic approaches. Implications for research and practice are discussed.

ABSTRACT BACKGROUND Low health literacy is considered a potential barrier to improving health outcomes in people with diabetes and other chronic conditions, although the evidence has not been previously systematically reviewed. OBJECTIVE To identify, appraise, and synthesize research evidence on the relationships between health literacy (functional, interactive, and critical) or numeracy and health outcomes (i.e., knowledge, behavioral and clinical) in people with diabetes. METHODS English-language articles that addressed the relationship between health literacy or numeracy and at least one health outcome in people with diabetes were identified by two reviewers through searching six scientific databases, and hand-searching journals and reference lists. FINDINGS Seven hundred twenty-three citations were identified and screened, 196 were considered, and 34 publications reporting data from 24 studies met the inclusion criteria and were included in this review. Consistent and sufficient evidence showed a positive association between health literacy and diabetes knowledge (eight studies). There was a lack of consistent evidence on the relationship between health literacy or numeracy and clinical outcomes, e.g., A1C (13 studies), self-reported complications (two studies), and achievement of clinical goals (one study); behavioral outcomes, e.g., self-monitoring of blood glucose (one study), self-efficacy (five studies); or patient-provider interactions (i.e., patient-physician communication, information exchange, decision-making, and trust), and other outcomes. The majority of the studies were from US primary care setting (87.5 %), and there were no randomized or other trials to improve health literacy. CONCLUSIONS Low health literacy is consistently associated with poorer diabetes knowledge. However, there is little sufficient or consistent evidence suggesting that it is independently associated with processes or outcomes of diabetes-related care. Based on these findings, it may be premature to routinely screen for low health literacy as a means for improving diabetes-related health-related outcomes.

The purpose of this paper is to explain at the simplest possible level why finite mathematics based on a finite ring of characteristic p is more general (fundamental) than standard mathematics. The belief of most mathematicians and physicists that standard mathematics is the most fundamental arose for historical reasons. However, simple mathematical arguments show that standard mathematics (involving the concept of infinities) is a degenerate case of finite mathematics in the formal limit p→∞; standard mathematics arises from finite mathematics in the degenerate case when operations modulo a number are discarded. Quantum theory based on a finite ring of characteristic p is more general than standard quantum theory because the latter is a degenerate case of the former in the formal limit p→∞.