This thesis is a record of experiments in the education of mathematics teachers for Irish Secondary schools conducted at Thomond College of Education, Limerick during the years 1975–77 inclusive. But it is more than a mere record of successes and failures. In its analyses and syntheses, based on experiments and programmes conducted under actual conditions, it endeavours in a true spirit of research in mathematical education to provide new insights. The research culminates in the redefinition of an old problem in mathematical education, and a first step towards a viable solution to the redefined problem is presented.