Are Honors Math Programs Worth Pursuing in College?

[max111]

I have looked into and am very interested in the Michigan Honors Math Program (specifically the 295-296-395-396 sequence). I am planning to apply to college right now, and I need help picking schools.

1) Do other schools have a similar honors math program?
2) Disregarding admissions requirements (I would be a competitive applicant virtually anywhere), would you recommend a specific school for research opportunities in math and physics? (I do not know what fields I am interested in; however, I would like to go to a school with enough resources to provide research opportunities in almost any field.)

 

[Topher925]

UofM is a big school and therefore does a lot of research on many topics. It's a good enough place to go as any. Personally, I have never exactly been impressed by any of their graduates but the school is very highly rated. Are you looking to go into math or physics?

Every school has an honors program in one form or another. If you want my opinion, don't put to much weight in honors programs or fancy statistics when making your decision. I believe a lot of that stuff is pretty overrated.

 

[max111]

I'm most likely going to double major.

By the time I graduate I will have done Calc I-III, Differential Eqns, Linear Algebra, Real Variables, and Complex Variables.

I would love to re-take these courses in a very rigorous, highly-theoretical program (like Michigan's 295-296-395-396 sequence). However, I don't want to take these courses again if I wouldn't get anything extra out of them.

Which schools have the option of taking these courses with a more proof-oriented focus (i.e. Spivak's book instead of Stewart's--not that Stewart's text is bad or anything; I've already taken a math sequence using his book).

A lot of the classes I have taken (with the exception of Complex Variables) were watered-down for high school students or pre-engineering students at the local community college.

 

[Vid]

Michigan's honors sequence does use Spivak's book for the first two semesters. Followed by Munkres' Analysis on Manifolds for the next two. Definitely a great course sequence for freshman math majors.

 

[trinitron]

Every school has an honors program in one form or another. If you want my opinion, don't put to much weight in honors programs or fancy statistics when making your decision. I believe a lot of that stuff is pretty overrated.

I have to put in my two cents here and say that I don't think this could be further from the truth. As a Michigan student, here is the comparison I will make: Our honors math program is quite good, and our physics one is really lacking. Looking back on my first two years here I can say that my math education was immensely beneficial, and my physics courses were essentially worthless.

I will first talk about the specifics of the program, before saying why I think it's a mistake to dismiss the idea of these honors tracts. Most years the math program is actually more advanced than Spivak/Munkres. Usually the first year will cover Spivak in addition to linear algebra at the level of Hoffman, and perhaps do some abstract algebra as well. Second year will typically cover topology and differential forms the first semester and then go deeper into geometry the second semester. Brian Conrad use to teach one of the harder versions of the 395-396 sequence (though he moved to Stanford this year), and you can find his notes for the course on his webpage- http://math.stanford.edu/~conrad/diffgeomPage/handouts.html.

Most students who take the entire sequence will be taking graduate courses by their Junior year (many even start in their sophomore year), and typically do quite well. Thus people who enter the program (and the people who do vary greatly, many with no experience with proof based math) exit the program with so called 'mathematical maturity'. Unless you know these subjects at a very high level I think you will certainly get a lot out of the sequence. As far as other schools to look into: Chicago has a similar tract. More elite schools such as Stanford and Harvard typically have a 395 type class as the intro (e.g. Math 55 at Harvard), rather than offering a rigorous course on single variable real analysis to freshman. If you can get into schools like this I would definitely recommend them.

But my main point in writing this was to point out that I think these types of honors sequences are quite effective. Michigan doesn't have an honors physics track like the one at Chicago, MIT or Harvard (typically K&K + Purcell), so I simply skipped the intro courses and took the junior level E+M and mechanics. It was not nearly as beneficial to me, first of all because although 'harder' material was supposedly being covered, it didn't assume the same kind of intellectual maturity of its students that an honors course can afford to assume. (My opinion on this pretty much reflects here: http://www.artofproblemsolving.com/Resources/AoPS_R_A_Calculus.php ) Thus looking back I actually learned very little physics, and didn't learn in a coherent way how to think physically the same way I did mathematically. The second thing that this was supposed to illustrate was that all honors programs are not created equal. Doing research on them (as you clearly are) is a very wise choice. I paid out of state tuition to come study at Michigan solely for its undergraduate math program; I had a very good picture of what it was before hand, and the ways I thought that it would benefit me have certainly panned out.

In any case, after reading this rambling post perhaps you will think my time would have been better spent in the English department. My final advice is this: continue to do a lot of research and put in a lot of work towards choosing the right school for you. It will certainly pay off.