What are the special and particular problems that lie in the way of effective teaching and learning in pure mathematics? If we restrict our attention to a context of higher education in the UK, there are a number of signals and signs that can be read. Among these are current issues in mathematical education, reported problems in contemporary literature (a good recent example is Brake, 2001) and-to a lesser extent-the results of the Assessment of the Quality of Education, which has recently been initiated by Quality Assurance Agency acting on the behalf of the funding bodies.
At the heart of the matter is the nature of the discipline itself. Everyone should have a working definition of their subject and there are a number of famous descriptions to hand. For example, if we consult the first edition of Encyclopædia Britannica, we find a relatively short but interesting entry under 'Mathematics':
Mathematics are commonly distinguished into pure and speculative, which consider quantity abstractly; and mixed which treat of magnitude as subsisting in material bodies.
Pure mathematics have one peculiar advantage…it is easy to put an end to controversies, by shewing either that our adversary has not stuck to his definitions, or has not laid down true premises, or else that he has drawn false conclusions from true principles.
These disciplines…instruct by profitable rules…ensure and corroborate the mind to a constant diligence of study…perfectly subject us to the