The confusion on these questions, among teachers too, is quite understandable. Even University departments and educational administrators are sometimes at war about the nature, place and purpose of Mathematics! I was asked, a few years ago, to act as consultant in such a dispute at a University in Southern Africa, where the Science Faculty academics and administrators were, according to the outraged `pure' party in the Mathematics Department, plotting to eliminate all the pure mathematics courses from the BSc degree and force the Maths Department to teach only what was required and useful for the other Science departments (Chemistry, Physics, Biology, Geology, Computer Science, Statistics,...). At another University in Africa, the lone pure mathematician in the Department of Mathematics and Statistics finds herself launching a one-woman rescue operation, striving to restore the core mathematics courses and the advanced Pure courses which have been gradually `killed'. She has a strong case: it is impossible to go on to do significant work and research in applied mathematics, computer science, probability or physics, without mastering a fair amount of pure mathematics.
As some mathematicians see it, such moves to eliminate `pure' mathematics are nothing less than an assassination attempt - striking at the real heart of mathematical life; or (to change the metaphor) an attempt to dethrone the rightful Queen of the Sciences (whose adoring subjects include many who are not scientists) and relegate her to a menial role as Servant of the Sciences.
What was my response as arbitrator in the above-mentioned dispute? It went something like this:
It is the norm in most Universities for the Mathematics Department to teach `service courses' to the students from other Science Departments, and to Engineering students, alongside their own mathematics courses - and to be given the extra resources required to perform this service. But specialist mathematics must be preserved as the heritage of any reputable university. A mathematics course has a certain essential integrity, a logical structure and a concern for proof, while many service courses must perforce be merely a set of recipes for the consumption of user departments.
If current tight resources demand that some courses be BOTH integral mathematics courses worthy of the name AND service courses, then both sides will be required to show consideration and the spirit of healthy (and hopefully temporary) compromise, and major efforts must be directed urgently towards a healthier solution which will ensure the survival of an authentic mathematics degree - perhaps through the structure of an Honours degree. Meanwhile there are positive gains from having to teach mathematics with its applications in mind: a good example of this is the influential ``COMAP'' series of mathematics texts prepared in the USA with a philosophy of expounding mathematics as a subject in its own right, with its own natural logic of development (thus resisting pressures to be only a service subject), yet providing motivation through vividly concrete and practical topics like discrete mathematics and the calculus of differences.
Mathematics needs special time-tabling considerations. This is understood and accepted in most university science faculties. This means more lecture time in the earlier years than subjects which require practicals, more time for personal study and practice and problem-solving-especially in later years, and extra time and staff if servicing other department's needs. It would normally be expected of other departments, on grounds of academic professionalism and courtesy, that they be sensitive to the mathematics department's own perception of its unique needs.
On helping students discover their talent for pure/applied/etc. Generally the first year at a university is the great decider of aptitude. Failure rates tend to be quite high all over the world in first year mathematics, and mathematics majors are selected out at the end of that year. At that point it is often necessary to choose whether to concentrate on pure, applied or statistics options, and lecturers and advisors will usually take great trouble to help students find their own best direction.
When user departments make naive and unrealistic demands as to what they want taught and when (e.g. the Physics Department wanting Hilbert spaces taught in second year!), the only solution is to have distinct customized user-techniques courses taught by the Mathematics Department (if given extra resources to do so), or taught by user departments themselves.
On convincing the non-mathematicians that removing the core elements of mathematics (real analysis, complex analysis, linear algebra, group theory, topology,..) is a very short-sighted expedient -even suicidal for a fledgling university: see the article following this for my defence case for Pure Mathematics.