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Understanding of numerical development is growing rapidly, but the volume and diversity of findings can make it difficult to perceive any coherence in the process. The integrative theory of numerical development posits that a coherent theme does exist-progressive broadening of the set of numbers whose magnitudes can be accurately represented-and that this theme unifies numerical development from infancy to adulthood. From this perspective, development of numerical representations involves four major acquisitions: (a) representing magnitudes of nonsymbolic numbers increasingly precisely, (b) linking nonsymbolic to symbolic numerical representations, (c) extending understanding to increasingly large whole numbers, and (d) extending understanding to all rational numbers. Thus, the mental number line expands rightward to encompass larger whole numbers, leftward to encompass negatives, and interstitially to include fractions and decimals. KEYWORDS-numerical development; numerical magnitudes; mathematical development; fractions; negative numbers; number line Research on numerical development is expanding rapidly, with large literatures emerging on numerical development in infancy, childhood, and adolescence; development of subitizing, counting, estimation, and arithmetic; knowledge of whole numbers, fractions, decimals, and negatives; and nonsymbolic and symbolic representations. Researchers have also examined conceptual and procedural knowledge; underpinnings of numerical development in evolutionary processes, neural processes, cognitive processes, and emotional processes; longitudinal stability of individual differences; and numerical competence in normal and special populations. The list does not end there: Researchers have also looked at relations to numerical knowledge of variations in economic status, culture, language, and instruction; relations among numerical, spatial, and temporal knowledge; relations of numerical knowledge to more advanced mathematics; and relations of interventions that improve numerical knowledge to subsequent learning, to name a subset of areas within the field (see Table S1 in the online Supporting Information for references for each area). Discoveries in these areas attest to the health of the field of numerical development. However, the sheer number of discoveries and areas can make it difficult to perceive any coherence in the developmental process. Is there such coherence or is numerical development just one thing after another? THE INTEGRATED THEORY OF NUMERICAL DEVELOPMENTThe integrated theory of numerical development proposes that the continuing growth of understanding of numerical magnitudes provides a unifying theme for numerical development. Within this perspective, numerical development is a process of broadening the set of numbers whose magnitudes, individually or in arithmetic combination, can be accurately represented. The theory identifies four main trends in numerical development: (a) representing increasingly precisely the magnitudes of numbers expressed nonsymbolically,...

Understanding an arithmetic operation implies, at minimum, knowing the direction of effects that the operation produces. However, many children and adults, even those who execute arithmetic procedures correctly, may lack this knowledge on some operations and types of numbers. To test this hypothesis, we presented preservice teachers (Study 1), middle school students (Study 2), and math and science majors at a selective university (Study 3) with a novel direction of effects task with fractions. On this task, participants were asked to predict without calculating whether the answer to an inequality would be larger or smaller than the larger fraction in the problem (e.g., "True or false: 31/56 ‫ء‬ 17/42 Ͼ 31/56"). Both preservice teachers and middle school students correctly answered less often than chance on problems involving multiplication and division of fractions below 1, though they were consistently correct on all other types of problems. In contrast, the math and science students from the selective university were consistently correct on all items. Interestingly, the weak understanding of multiplication and division of fractions below 1 was present even among middle school students and preservice teachers who correctly executed the fraction arithmetic procedures and had highly accurate knowledge of the magnitudes of individual fractions, which ruled out several otherwise plausible interpretations of the findings. Theoretical and educational implications of the findings are discussed.

There are a growing number of large-scale educational Randomized Controlled Trials (RCTs). Considering their expense, it is important to reflect on the effectiveness of this approach. We assessed the magnitude and precision of effects found in those large-scale RCTs commissioned by the EEF (UK) and the NCEE (US) which evaluated interventions aimed at improving academic achievement in K-12 (141 RCTs; 1,222,024 students). The mean effect size was 0.06 standard deviations (SDs). These sat within relatively large confidence intervals (mean width 0.30 SDs) which meant that the results were often uninformative (the median Bayes factor was 0.56). We argue that our field needs, as a priority, to understand why educational RCTs often find small and uninformative effects. Rigorous Large-Scale Educational RCTs are Often Uninformative: Should We Be Concerned? Large-scale Randomized Controlled Trials (RCTs) are now regularly used to evaluate educational interventions. For example, the US-based National Center for Educational Evaluation and Regional Assistance (NCEE) started funding large-scale RCTs in 2002, and the UK-based Education Endowment Foundation (EEF) has funded more than 160 since 2012. This trend is not limited to these two countries: in recent years funding organizations in the European Union (e.g., European Schoolnet), Japan (e.g., Nippon Foundation), Australia (e.g., Social Ventures), Switzerland (e.g., Jacob's Foundation), Brazil (e.g., Lemann Foundation) and Bangladesh (e.g., BRAC) have also prioritized RCTs in education. Evaluating the efficacy of educational programs before implementation is important to avoid wasting resources. In medicine, there are many instances where RCTs have shown that promising treatments were ineffective or harmful (Sibbald & Roland, 1998). However, conducting large-scale RCTs is expensive. For example, the EEF spends around £500,000 per trial (EEF, 2015a). Given the growing number of large-scale RCTs in education, and their expense, it is important to reflect on how informative this new research focus has been. To our knowledge, no study has systematically evaluated this recent trend. In this paper we use empirical data from two prominent educational funding bodies to evaluate the typical effects produced by large-scale educational RCTs. Our aim is to provide an empirical basis for discussions of the field's efforts to build rigorous scientific evidence. Randomized Control Trials RCTs are widely regarded as the 'gold standard' for measuring the efficacy of interventions (Pocock, 1983). In their simplest form, participants are randomly assigned to an experimental group which receives the intervention, or a control group that receives an alternative treatment or possibly no treatment. The effectiveness of the intervention is then determined by comparing the outcomes between groups. RCTs are highly regarded because, compared with other types of studies (e.g., case studies), they ensure that the groups are probabilistically identical at the outset and that any difference in outcome are...

Fraction and decimal arithmetic pose large difficulties for many children and adults. This is a serious problem, because proficiency with these skills is crucial for learning more advanced mathematics and science and for success in many occupations. This review identifies two main classes of difficulties that underlie poor understanding of rational number arithmetic: inherent and culturally contingent. Inherent sources of difficulty are ones that are imposed by the task of learning rational number arithmetic, such as complex relations among fraction arithmetic operations, and that are present for all learners. Culturally contingent sources of difficulty are ones that vary among cultures, such as teacher understanding of rational numbers, and that lead to poorer learning among students in some places than others. We conclude by discussing interventions that can improve learning of rational number arithmetic.

reasoning is critical for science and mathematics, but is very difficult. In 3 studies, the hypothesis that alternatives generation required for conditional reasoning with false premises facilitates abstract reasoning is examined. Study 1 (n = 372) found that reasoning with false premises improved abstract reasoning in 12- to 15-year-olds. Study 2 (n = 366) found a positive effect of simply generating alternatives, but only in 19-year-olds. Study 3 (n = 92) found that 9- to 11-year-olds were able to respond logically with false premises, whereas no such ability was observed in 6- to 7-year-olds. Reasoning with false premises was found to improve reasoning with semiabstract premises in the older children. These results support the idea that alternatives generation with false premises facilitates abstract reasoning.

In two studies (N's = 55 and 54), we examined a basic form of conceptual understanding of rational number arithmetic, the direction of effect of decimal arithmetic operations, at a level of detail useful for informing instruction. Middle school students were presented tasks examining knowledge of the direction of effects (e.g., "True or false: 0.77 * 0.63 > 0.77), number line estimation of decimal magnitudes, and knowledge of decimal arithmetic procedures. Their confidence in their direction of effect judgments was also assessed. We found (1) most students incorrectly predicted the direction of effect of multiplication and division with decimals below one; (2) this pattern held for students who accurately estimated the magnitudes of individual decimals and correctly executed decimal arithmetic operations; (3) explanations of direction of effect judgments that cited both the arithmetic operation and the numbers' magnitudes were strongly associated with accurate judgments; and (4) judgments were more accurate when multiplication problems involved a whole number and a decimal below one than with two decimals below one. Implications of the findings for instruction are discussed.

Research findings regarding the effects of educational interventions—typically reported in units of standard deviations (e.g., Cohen’s d)—are often translated into more intuitive metrics before being communicated to teachers. However, there is no consensus about the most suitable metric, and no study has systematically examined how teachers respond to the different options. We conducted two preregistered studies addressing this issue. We found that teachers have strong preferences concerning effect size metrics in terms of informativeness, understandability, and helpfulness. These preferences challenge current research reporting recommendations. Most importantly, we found that different metrics induce different perceptions of an intervention’s effectiveness—a situation that could cause teachers to have unrealistic expectations about what a given intervention may achieve. Implications for how educational effects should be communicated are discussed.