According to this theory, formative assessment of mathematics learning may involve a range of methods, but summative assessement requires external examinations. These should be based on a hierarchical view of mathematical subject matter, and at a number of levels, corresponding to mathematical ‘ability’. However difficult, the excellence of the mathematically talented will shine trough, and any move to make examinations more accessible or less trying for student, must represent a dilution of standards. Competition in examinations provides a means of identifying the best mathematicians.