Recently, there has been some build-up of momentum in the MOOC (Massive Open Online Courses) movement, and its impact is felt by many academic institutions. The introduction of these courses undermines traditional university style teaching, especially low level courses where each class consists of up to several hundred students with one lecturer, leaving little to no opportunity to ask questions or interact with the instructor. In some sense the rigidity and lack of human interaction is similar to an online setting anyway, except it’s not as flexible scheduling wise.
This has many ‘pure’ or ‘hands-off’ fields of study worried. It is no surprise that some departments constantly have to justify their worth to people who provide funding, and others take it for granted. For instance, the biology department probably never has to worry about its inherent worth because all medical professionals have to do some sort of biology, and people will always get sick and injured. Same with engineering; there will never be a time when they are not in demand. This is not so with departments like English and Classical Studies; and even mathematics.
In most universities what saves mathematics from becoming a small niche department is the high demand for first year calculus. I don’t have on-hand some global statistics, but in my experience almost all students in science, engineering, and commerce must take first year calculus, and many arts majors do as well (such as economics). It is likely a reasonable estimate that at least half of first year students must take first year calculus; which creates an enormous demand. Hence math departments get a lot of funding for instructors.
However, this is not a happy relationship. While calculus is a required course for many programs, not nearly as many are happy with the way it’s taught. While most universities have different versions of calculus for different specialities, the differences are relatively hollow: almost all calculus courses are taught by mathematicians in a manner befitting of mathematicians. Every calculus course includes theoretical treatments of limits, continuity, differentiability, etc. which were developed by pure mathematicians (such as Cauchy and Weierstrass) over a century after Newton. It is said that Newton himself would not recognize a modern calculus course. It also warned, in Steven Krantz’s book, that some universities are abolishing their math department’s monopoly on calculus courses, and these courses taught by faculty from other departments are actually viewed as superior. This is a harrowing situation for mathematicians.
Firstly, I don’t feel that MOOCs by themselves will force math departments out of business, or even cause them to shrink. In history more efficient systems almost never led to shrinkage in an economy. Yes the demand for in-house instruction may decrease, but the demand for instructors need not decrease. From a purely economical perspective, the most logical conclusion of an expanded ability to teach is to teach more people, not to cut the number of instructors (since the number of people wanting to learn is not constant. Most universities have to turn down students who apply). But it does force departments and mathematicians to seriously think about what low level mathematical courses offer to non-specialists.
As a number theorist, I think the most important principle to pass along is that exact quantities or notions are rare and unrealistic. Thus in turn means techniques to approximate quantities, such as zeroes, critical points, integrals, etc., are much more valuable than exact techniques. For instance every low level calculus course I have encountered has an extremely unhealthy obsession with finding anti-derivatives. We KNOW that most elementary functions don’t have an anti-derivative which is also an elementary function, and even in cases where a solution exists they are normally impractical.
Next, an important thing to note is to teach students to think in discrete terms, not continuous terms. The idea that ‘continuous’ is the ‘truth’ and we ‘approximate continuity’ by discrete objects is (while technically correct) faulty. Our universe is fundamentally more discrete than continuous. At the very basic level, the universe is made of discrete parts, and the meta behaviour we observe is a continuous approximation of the complicated discrete interactions that underly everything. This I believe is a critically important idea that should be communicated to all students which will serve them much better than knowing how to find the anti-derivative for