Your son will have great options in the end. Wishing him and you lots of luck @listener76!
I am a bit late to this discussion, but thought folks might want to hear from a faculty member at a small school. I am a tenured professor at a top tier liberal arts college. I started visiting CC recently, and find it (mostly this particular forum) interesting and informative, but don’t have time to post regularly. However, I can stick around for a few days if anyone would like to discuss this topic further.
Just a word on the choice of schools, copied from above.
St. Olaf, Oberlin, Kenyon, Amherst, Grinnell, Davidson, Furman, Vanderbilt, William and Mary
All are excellent choices – I know people at all of them except (I think) Furman, and all those I know are excellent mathematicians. In some departments (certainly St. Olaf, Davidson, Vanderbilt, and W&M) you’ll find some at the world-class level, but that doesn’t matter much. You should at this point be focused primarily on teaching quality, and the student’s learning experience. Given the interest in music, I think this makes St. Olaf an excellent choice.
I haven’t reviewed all of the posts on this thread, but looking at the original post:
I think your son would be in a good position to start a math major at my institution, or any of its peers. He would start in multivariable calculus in his first semester, as do the vast majority of our majors. He certainly wouldn’t lack a peer group who are very excited about mathematics, and wouldn’t run out of courses to take.
Our program (like that of many of our peers) is designed around learning experiences, rather than courses, that can be tailored to what an individual wants out of the major (folks at St. Olaf, for example, will tell you they are a “big tent” department, and take this point very seriously). We have students who go on to graduate school and get PhDs; others who work on wall street, as actuaries, as consultants; other students go to law or medical school, or into K-12 education.
On this point, I think an advantage in undergraduate education that a small college may have over a larger university is its flexibility. Since students are our primary focus, and we are encouraged to engage them in all sorts of learning experiences, it is easier for a strong student to get involved in close work with faculty than it may be at a large research institution. Students can sit in on graduate courses at a state flagship, but they can learn the same material from us one-on-one (or in small groups of a few students) if they are so inclined. We routinely offer these sorts of experiences, and have time for them built into our workload.
We are also able to use mathematics to help students develop their basic skills like writing and public speaking (my colleagues and I are discouraged by the fact that, notwithstanding their top test scores, many of our students do not begin with these basic skills). These are important skills even for a research mathematician like myself.
Before concluding, I should mention that the division between liberal arts colleges and universities is somewhat arbitrary from an undergrad’s point of view. I think the advantages that a small college has can also be found to some degree at a smaller university (e.g. Vanderbilt, Yale, Rice, Dartmouth) where undergraduate teaching is valued. I suspect that my department and peers would see more commonalities than differences with those departments in terms of the student experience.
Sorry for the lengthy post (I didn’t have time to compose a shorter one!), but I hope some of that was helpful. Congratulations on making excellent choices for your son, and I wish you both the best of luck!
What a helpful post, @mathprofdad! Thank you for chiming in with your insightful, first hand knowledge.
Having a kid at a SLAC who is majoring in math/something, I am very happy to hear your opinion. Big SLAC fans in my family, as you can probably see from my avatar.
I disagree with the math prof at a small school. Just as much, if not more, flexibility. People, not the system rule (what you see “in print” is not ironclad). Large U’s also have small classes for honors students plus small upper level class sizes. Plus they offer so many more diverse outside one’s major courses to choose from. I know students can only take a small percentage of available courses but the breadth is amazing at large schools. Foreign language- not just the ones offered in any HS. Linguistics, history of science and all sorts of cultures far from the mainstream. Likewise in literature. 100, not just 10, math majors in any given graduation year. Plus grad students. There are professors well known nationally/world wide do care about undergrad education in the large U’s. Big U’s are composed of many smaller subunits.
“Foreign language- not just the ones offered in any HS.”
If you think the top LACs are offering only Spanish, French, and Chinese you’d be very wrong. 
I include German and others in the HS mix. Urdu, Portuguese and so many others not always offered. I discovered Linguistics is not a major at some elite colleges as well. Yes, one can take only so many credits but it is fun to be exposed to a much wider world even if it is only friends who dabble in them.
There is also the case of having limited numbers of math professors to encounter. A professor may be a great teacher but everyone has a slightly different slant to things. Different approaches…
Solid LAC math programs tend to support 10+ faculty. Levels beyond this range would seem to be superfluous for even advanced mathematics students.
Let’s take St. Olaf, a school I first mentioned here back on page 2. Currently 17 mathematics professors to choose from.
Many of the better LACs offer 8 to 9 languages. Portuguese is offered at one of my kids’ small LACs, the other has it available in their consortium. Since 99.9% of kids will never have interest in Urdu, that would be an odd rationale for choosing a bigger school. If the desire is strong, there are many ways to go about language learning outside of the traditional 4 year college setting.
One can’t always assume the LAC will be lacking in academic offerings compared with their university counterparts.
For instance, one reason why I turned down some university admission offers was because my LAC(Oberlin) had far greater offerings in the academic areas I was interested in (Chinese history/politics) than those universities.
Same with foreign languages as my LAC’s offerings in East Asian languages was greater than those offered by many universities as shown by the fact a college classmate was deterred from taking a summer Chinese language class at his in-state U over the summer in an attempt to leapfrog into 3rd year Chinese after taking the LAC’s first-year course because that U taught the course at half the rate* of our LAC.
In short, taking that second-year Chinese language course at his in-state U would have meant he’d have a gap of one full semester to fill to be ready to take our LAC’s Chinese language placement exam for 3rd year Chinese. .
Also, there aren’t many universities, especially just a decade ago which offered 2+ years of classical Chinese on top of 3+ years of vernacular Chinese…much less LACs like the one I attended.
- One year's worth of foreign language at our LAC would be the equivalent of 2 years worth of foreign language at that in-state U.
If folks are still interested in talking about math departments, you may still find it useful to hear from someone who has spent time in a few.
It’s hard to compare math departments with modern languages or literature. One reason is that the relationship between math research and teaching is different from that in other fields. In most other fields, faculty members teach primarily in their own research area, but academic mathematicians are qualified to teach across the undergraduate curriculum, and generally do so. The forefront of math research is not accessible to most undergrads, so even directing undergrad research is primarily a teaching activity. When it comes to teaching undergraduates, one’s specific research area matters less than teaching skill, intellectual curiosity and vigor, and breadth of interests.
We hire with these principles in mind, and (at least after I was hired) have been successful in finding first-rate mathematicians who are also outstanding teachers. Balancing research and teaching (and making the two interact) is hard, but it makes the job very rewarding. I don’t know exactly how many faculty would be superfluous, but there are many exceedingly successful departments (St. Olaf among them) with around 10-20.
It’s true that there is a big world out there with lots to learn, but it’s mistaken to equate number of course sections with a department’s vitality. We have a somewhat smaller, but richer and more flexible undergraduate curriculum at my current place than at previous state flagship, primarily due to the close work we do with students. Even so, I had to look quite hard through the curricula of the large schools posted up thread (with the exception of Berkeley) to find any course we haven’t offered in the last few semesters. Maybe a few of our students would be better off at Berkeley, but the vast majority wouldn’t.
In the end, one may prefer liberal arts colleges or large u’s as a matter of personality or taste, but I encourage parents to look beyond courses in evaluating programs. Look instead for evidence that students are developing intellectual independence: learning to read and do math on their own, giving presentations outside the college/university, writing a significant piece of mathematics, seeking out their own resources and organizing their own activities, etc.
(Of course, personal happiness is really important too!)
I hope folks find something in there that is helpful or clarifying. Sorry I’m not really able to stick around to discuss things more, life is very busy these days!
I imagine the caliber of the departments also counts. Being lucky enough to have a flagship U with many top rated departments is different than being in what most are- average (remembering that one half need to be below the median- this isn’t Lake Wobegone). One half are in that 25-75% range…
Another thing for your son to consider is having summer experiences that can push him further. http://www.ams.org/programs/students/emp-reu I am hoping @mathprofdad can chime in on how he perceives these. And is the Budapest study abroad program actually worthwhile? https://www.budapestsemesters.com St Olaf has this on their department website.
I am a big believer in using summer programs to advance a kid’s self knowledge about how they want to shape their lives.
Summer REU are great for students who love math and want to learn more. They get their feet wet in research in a fun and encouraging environment, which is usually collaborative as opposed to competitive, and even get paid for it. Most REU these days even include a social program to help students connect. I would recommend any of the REU listed on the AMS website, and have sent students to some of them.
We’ve also sent students to Budapest, which has a similar ethos to an REU but is during the year and more course-oriented. Also there’s a wonderful cultural aspect to it. Both REUs and Budapest are worth exploring for a strong student and especially if they want to go to graduate school. Our best students usually do some kind of summer thing or semester away every year.
As for the question on caliber of institutions, it does matter, provided you have the right measure of a college’s caliber. I find it impossible to rank colleges linearly but if you must you probably shouldn’t take US News too seriously. Rankings of graduate programs are also not a good guide, as undergraduate and graduate education are very different things. IMHO some of my suggestions in the posts above could be one way to begin calibrating in a way that’s more relevant to you and your student.
Williams College has 25 Math faculty members for a school of ~2000 students. The math program there is exemplary and has consistently punched well above its weight class.
It may depend on how well distributed the math faculty are across the various subspecialties (which can affect what kind of elective math courses are offered).
It may also depend on whether the math faculty are also expected to cover statistics and computer science, or if there are separate departments to cover those subjects.
REU’s are competitive. Good experience and some of the best ones are at surprising institutions (colleges one may not consider for their academic caliber).