Findings about the nature of adult motivation and the rapid expansion of knowledge about mathematics and its pedagogy, suggested that teacher training in mathematics should be planned and executed under two major guidelines. Further, a school mathematics programme should be concerned, in the first place, with children learning mathematics usefully and meaningfully. A second concern, which derived its importance mainly from the first, should be the preparation of teachers to teach mathematics so that the pupils could learn it in a 'non-arbitrary substantive fashion.