How many apples make a quarter? The challenge of discrete proportional formats.

Proportional judgments are easier for children in continuous formats rather than discretized ones (e.g., liquid in a beaker vs. in a beaker with unit markings). Continuous formats tap a basic sense of approximation magnitude, whereas discretized formats evoke erroneous counting strategies. On this account, truly discrete formats with separated objects should be even harder. This study (N = 565 7- to 12-year-old children) investigated that prediction. It also examined whether the format effects vary with children's fraction knowledge (FK; part-whole relations, computation, and fraction number line estimation). As found previously, discretized formats were more challenging than continuous ones; as predicted, discrete formats were yet harder. The format effect interacted with FK. Low-FK children were above chance only with continuous formats, medium-FK children struggled with discrete formats only, and high-FK children did well with all three formats.

Stereotype Threat Effects on Learning From a Cognitively Demanding Mathematics Lesson.

Stereotype threat-a situational context in which individuals are concerned about confirming a negative stereotype-is often shown to impact test performance, with one hypothesized mechanism being that cognitive resources are temporarily co-opted by intrusive thoughts and worries, leading individuals to underperform despite high content knowledge and ability (see Schmader & Beilock, ). We test here whether stereotype threat may also impact initial student learning and knowledge formation when experienced prior to instruction. Predominantly African American fifth-grade students provided either their race or the date before a videotaped, conceptually demanding mathematics lesson. Students who gave their race retained less learning over time, enjoyed the lesson less, reported a diminished desire to learn more, and were less likely to choose to engage in an optional math activity. The detrimental impact was greatest among students with high baseline cognitive resources. While stereotype threat has been well documented to harm test performance, the finding that effects extend to initial learning suggests that stereotype threat's contribution to achievement gaps may be greatly underestimated.