Young Children Intuitively Divide Before They Recognize the Division Symbol.

Children bring intuitive arithmetic knowledge to the classroom before formal instruction in mathematics begins. For example, children can use their number sense to add, subtract, compare ratios, and even perform scaling operations that increase or decrease a set of dots by a factor of 2 or 4. However, it is currently unknown whether children can engage in a true division operation before formal mathematical instruction. Here we examined the ability of 6- to 9-year-old children and college students to perform symbolic and non-symbolic approximate division. Subjects were presented with non-symbolic (dot array) or symbolic (Arabic numeral) dividends ranging from 32 to 185, and non-symbolic divisors ranging from 2 to 8. Subjects compared their imagined quotient to a visible target quantity. Both children (Experiment 1 N = 89, Experiment 2 N = 42) and adults (Experiment 3 N = 87) were successful at the approximate division tasks in both dots and numeral formats. This was true even among the subset of children that could not recognize the division symbol or solve simple division equations, suggesting intuitive division ability precedes formal division instruction. For both children and adults, the ability to divide non-symbolically mediated the relation between Approximate Number System (ANS) acuity and symbolic math performance, suggesting that the ability to calculate non-symbolically may be a mechanism of the relation between ANS acuity and symbolic math. Our findings highlight the intuitive arithmetic abilities children possess before formal math instruction.

Finite Mixture Modeling for Program Evaluation: Resampling and Pre-processing Approaches.

BACKGROUND: Finite mixture models cluster individuals into latent subgroups based on observed traits. However, inaccurate enumeration of clusters can have lasting implications on policy decisions and allocations of resources. Applied and methodological researchers accept no obvious best model fit statistic, and different measures could suggest different numbers of latent clusters. OBJECTIVES: The purpose of this article is to evaluate and compare different cluster enumeration techniques. RESEARCH DESIGN: Study I demonstrates how recently proposed resampling methods result in no precise number of clusters on which all fit statistics agree. We recommend the pre-processing method in Study II as an alternative. Both studies used nationally representative data on working memory, cognitive flexibility, and inhibitory control. CONCLUSIONS: The data plus priors method shows promise to address inconsistencies among fit measures and help applied researchers using finite mixture models in the future.