Much time and effort, as well as funding, is being devoted to Genome Wide Association Studies (GWAS) for identifying genetic causes of variation (single nucleotide polymorphisms or SNPs) in human cognitive abilities and educational attainments (CA and EA). After years of finding only very weak associations, usually failing to replicate, attention has turned to aggregates of otherwise non-significant SNPs (called polygenic scores, or PGS) and some associations with traits are now being reported. Here we show how, in the context of CA and EA as approximation measures, spurious correlations in GWAS/PGS can arise in a number of ways, particularly from genetic population structure. We review recent studies suggesting that attempts to control for such confounds have been quite inadequate, and also criticize the underlying statistical assumptions and genetic model.