In this paper we summarize recent findings from large-scale international surveys (PISA and TIMSS) of relevance to the predictability gradient hypothesis (Stankov, 2013). Non-cognitive measures are divided into two groups on the basis of their correlation with mathematics achievement. Many have low (r < 0.20) correlations. The best predictors of cognitive performance are a cluster of self-beliefs consisting of confidence, self-efficacy, anxiety, and self-concept. These appear to be the most potent influences underpinning Cattell's (1987) investment theory of fluid and crystallized intelligence. Self-beliefs affect cognitive performance either as impediments (anxiety) or facilitators (good calibration of self-efficacy and confidence). This information about the role of self-beliefs can inform future efforts at intervention.