This study seeks to determine the nature of the intellectual demands of the Higher Grade course in Mathematics with a view to early, more accurate prediction of individual pupil success in this course. The need for such early prediction is clearly indicated by the interest shown on the part of parents and pupils alike during the Standard Seven year where the realisation exists that Matriculation Mathematics is a subject sometimes found to be "overwhelmingly difficult". The "drop out" figure from the Higher Grade course to the Standard Grade course in most schools further demonstrates the need for more careful selection at the Standard Seven level. Both old (1973) and new (1984) syllabuses are analysed to determine the nature of the content and the intellectual level at which this should be taught. In addition, a series of past Cape Senior Certificate examination papers are investigated to reveal information about the nature and level of examining. Mental processes involved in the examination items are classified and the general composition of the examination papers is discussed. A test device suitable for Standard Seven pupils is developed on the basis of the composition of the Higher Grade Matriculation examination papers analysed. The object of this test is to provide that early indication to pupils of their ability to cope with the level of mental process required by the Higher Grade course in Mathematics. The investigation describes the construction, administration and further development of the test device and, furthermore, seeks to show its predictive validity for the Matriculation examination in Mathematics by comparing test results with successive school examination results over a three year period. The possibility of sex differences in Mathematics achievement and prediction are also investigated on the basis of the results obtained during the course of this experiment. General conclusions are drawn, the difficulties encountered are discussed and some suggestions for further research are offered.