Why are mathematics departments so large?

Why are mathematics departments so large?   Isn't mathematics basically done?  Is there really any research left to do?

A university the size of U.Victoria probably would only have 5 or 6 mathematicians, broadly construed, if it only cared about mathematics research functionality. Mathematics research is alive and well in the world, and it has both short-term and long-term real-world implications.  But our societal interest in mathematics research does not warrant the number of faculty in our research institutions. 

The reason U.Vic has 30+ mathematics and statistics faculty members comes down to service courses: these are the large-enrollment first and second year mathematics and statistics courses, like introductory calculus and statistics courses.  Here at U.Vic we have somewhere over 1000 students completing elements of our service calculus courses, every year.   The desire to have research faculty teaching those courses, and the follow-on courses needed by many faculties, notably engineering, is what drives our faculty size.  Universities have tried various ways to get those math faculty numbers down, but without much success.  

I don't know of any university that has a wildly successful service calculus stream.  Generally these are courses that students are required to take for their majors, and enthusiasm levels have a fairly low cap.   Measuring success of service courses isn't done in the absolute ways one might hope for (does it inspire students to do useful things? do they get involved in research?) but with the more relative (how does this way of offering it compare to what we've done for the past decade?).    At present, the best-known models for teaching calculus for the typical student is the small class of 16-20 students in a lecture format with plenty of time for one-on-one interaction.  At U.Vic we can't afford that kind of luxurious teaching model, so we put 200+ students in a big lecture hall, and have smaller tutorials where students can get a little more attention.  Needless to say, that still requires a good amount of lecturers. 

This begs the question: why are students required to take these courses?  Ultimately universities are supposed to be places of research, where students can learn useful things from inspired researchers at the cutting-edge of knowledge.  Service courses don't fit that model precisely -- service courses are offered in more like a high-school atmosphere. 

A century ago, calculus was offered less like a service course.  And students in universities were required to learn latin.  The latin requirement had a few motivations: medicine and biology used elements of the language.  But it also served as a measure of the incoming class, a gauge of their proficiency with logic and grammar.   In the Sputnik era, enrollment at universities surged.  An emphasis on engineering and the hard sciences brought with it a heavier demand for proficiency with basic mathematics.  The  latin requirement was swapped-out for a calculus requirement, and service calculus courses became mainstream. Mathematics departments swelled: not from a societal demand for more mathematics research, but from the requirement for more service courses, together with the expectation that instructors at universities should be researchers. 

Service courses have become more refined over the past century, but modern textbooks continue to look an awful-lot like textbooks from 1950.  In my opinion, the most dramatic change to service course landscape occurred in just the past couple of years.  Software like Photomath and Socratic allow you to take a photograph of typical service course material, even grade-school material, with your cell-phone.  If the software can parse the text well-enough, it will often comprehend it and give you a fully-worked-out solution, suitable for submitting for grading. 

This introduces a new debate.  If university material is so mundane that a common pocket-device can do it better than a typical student, what's the point of teaching the topic? 

An analogous debate occured over the years: it was once common to demand children remember multiplication tables up to 20x20 or 12x12.  When I was a child, it was down to 10x10.  As a student I believe I remembered only up to 6x6. I would work out the rest using arithmetic, eg: 8x3 = (10-2)x3 = 10x3 - 2x3 = 30 - 6 = 24, although I had a simplified way of thinking about this, internally.  I never liked having to remember things that I could derive. When pocket calculators came on the scene, people's perspective began to shift.  The memorization of multiplication tables was increasingly seen as a mundane machine task, not something a person should know.  So the requirement was loosened.   We are at the cusp where service-calculus problems are becoming machine-doable, so this same debate will open-up on calculus courses. 

A skeptical mathematician might say "but wait! We are teaching these students not how to simply get an answer. We are teaching them the logical infrastructure of calculus and how to make a coherenet logical argument!"    That is of course a valid point, but the counter-point is far more compelling: soon we will have students turning to their phones to get answers to pretty much any question we can throw at them in a service course.   You have likely already observed this phenomena: we forget phone numbers because they are on our cellular devices.  We forget material we know we can quickly find on the internet.  The ease with which we can acquire knowledge impacts how we value it, and how well we remember it.  And calculus knowledge is now available with a few clicks on your cell phone.