Prior research substantially overstates the cost of retention under test-based promotion policies to both taxpayers and students who delay labor market entry because it omits two important factors. First, there is a delay between the intervention and the taxpayer's expenditure. Second, on average, the treatment leads to less than a full year of…

In this article, I analyze episodes from two third-grade classrooms drawn from a larger classroom teaching experiment to explore how these students began to incorporate nonnumerical symbols in their mathematical expressions when asked to represent indeterminate quantities. The article addresses two research questions: What understandings did these…

One crucial issue in mathematics development is how children come to spontaneously apply arithmetical principles (e.g. commutativity). According to expertise research, well-integrated conceptual and procedural knowledge is required. Here, we report a method composed of two independent tasks that assessed in an unobtrusive manner the spontaneous…

Four hundred and three 3rd- and 5th-grade Chinese students took the Multiplication Estimation Test or participated in the interview on it, designed to assess their computational estimation performance on whole-number multiplication. Students perform better when tasks are presented visually than orally. Third graders tend to use rounding based…

When asked to calculate the area of a particular shape, one student responded by asking, "Is that the outside or the inside?" while another student replied, "I think it's the one where you put a little 2 next to it." Both of these responses indicate a lack of conceptual understanding of area and reinforce the research findings…