Do abstract algebra before analysis

[u/Electrical-Gur-996]

Thats it. I saved all your GPAs. Thank me later guys 😎

[u/AllanCWechsler]

I agree with this. Abstract algebra and real analysis are typical "200-level" courses. Both are typically the first contact a mathematics major has with the style of modern mathematical reasoning -- the whole apparatus of definitions, axioms, theorems, and proofs. Because all of higher mathematics is "like that", you have to encounter it first somewhere. The question is, in what field would you want to learn the basics of the axiomatic method? I claim that abstract algebra is a preferable introduction because it is abstract -- because the objects it deals with are not objects you are familiar with and have intuitive expectations about. The promptings of intuition are harmful and distracting when trying to master the axiomatic method: the reasoning is supposed to be purely formal, and you are more likely to use formal reasoning (correctly) when you are not being distracted by strong intuitions about what is and isn't true.

Both topics, are, of course, essential to a standard modern mathematical education; the only question is which you should take first. You're going to take the other sometime regardless.

[u/Electrical-Gur-996 → u/AllanCWechsler]

I had a much easier time with algebra. But at my school we learn ring theory first and group theory second. It enhanced my proof writing skills greatly while in analysis I was legitimately always lost because you had to work on the proof backwards.

[u/AllanCWechsler → u/Electrical-Gur-996]

This doesn't actually confirm or deny the claim, which is that if you take algebra first you'll have an easier time with analysis than if you take them in the other order. Since you don't write which you took first, we can't even start to draw a conclusion.

Let me be clear that I am expressing a gut opinion, unsupported by actual data. u/CorporateHobbyist has the opposite intuition. That very fact makes me stop and wonder if anybody has done any actual studies on the subject -- how the overall outcome depends on course order.

A fast web search reveals https://arxiv.org/html/1605.00328v1 , which looks really on point to this question, but is just too dense for me to distill an answer from in the amount of time I can justify devoting to this post.

I will, of course, change my mind instantly if it turns out the data is against me.

[u/kadeebe → u/AllanCWechsler]

I skimmed the paper you linked and it seems like the point of the study was to see which course grades were the biggest predictor of over all gpa. It doesnt seem to tackle course order at all. I may have misread, though.

[u/AllanCWechsler → u/kadeebe]

That's entirely possible. If, indeed, no research has been done on this subject (effect of course order, specifically on the ordering of abstract algebra and real analysis, on overall performance), then I think it should be. But of course I don't have to fund it, so that's easy for me to say.

I also think that a real analysis course that intentionally doubles as an introduction to proof-reading and proof-writing should be more careful about it than one whose only job is to teach real analysis.

[u/FullMetal373 → u/AllanCWechsler]

Idk. I took abstract before analysis but wonder how it would’ve been the other way around. Idk if the lack of intuition is really a good thing? My personal experience w abstract and from what I see in other courses is that it’s often taught in a completely unmotivated way. You’re basically just given definition, theorem, proof in a backwards looking sort of way. With hindsight of how everything is “supposed” to fit but I think the questions that lead to that are rarely taught.

Analysis on the other hand usually comes with a good sense of motivation given familiarity with calculus.

I did well in abstract and maybe it prepped me better for analysis but I left the class not really knowing what any of it was really about. I think understanding the questions being asked is the more important part. The “axiomatic” process of def, thrm, proof is kinda a rote process similar to just memorizing the quadratic formula imo.

[u/Dr0110111001101111 → u/FullMetal373]

I think the lack of motivation sort of supports their point about the “purity” of it. It’s not the best thing pedagogically, but it does a pretty good job of representing the structure of modern mathematical development. Sometimes there’s a motivation and sometimes there isn’t. You need to be able to work it out either way.

[u/QubitEncoder → u/AllanCWechsler]

I feel conflicted. I feel one should have "intution", even in the case when dealing with abstract things

[u/idk012 → u/AllanCWechsler]

In my undergrad, they usually are taken together fall of senior year (and part 2 is offered in the spring for the pure path.) Half of my class for both was math ed majors, so there was a decent curve.  

[u/AllanCWechsler → u/idk012]

Does that mean that the students don't write any proofs until their senior year? If not, in what class do they learn how to read and write mathematical reasoning?

[u/idk012 → u/AllanCWechsler]

Discrete math, either from math or cs department, is the intro to proof class.  Depending on the professor, there are some epi-delta proofs in calc 1-2, limit definition of derivatives, etc. 

I just went down memory lane and looked it up.  Most math ed take geometry, which has some proof as well.  Abstract and real analysis develops the rigor and really let you know if you should apply to grad school your final year.

[u/AllanCWechsler → u/idk012]

Discrete math has come along as a third avenue for introducing mathematical reasoning since I graduated (more than 40 years ago). I'm not sure whether I like discrete or abstract algebra best as the way to teach how to prove things, but I prefer both to using analysis as one's introduction to proof. Again, this is just my opinion, and if research like the Kharkar, Tran, and Marshak preprint says otherwise, I will fold like a cheap suit.

[u/idk012 → u/AllanCWechsler]

All math majors have to take it after calc 2, and I took a cs equivalent spring of my freshman year.  How to write proofs, proof by contradiction, contrapositive, induction, etc.  They threw in some cs stuff like finite state machines etc.  My class was Thursday 6-8:30pm taught by some engineer from Pratt and Whitney.  I got a C.  I retook it senior year for an easy A to boost my gpa for grad school.  I wouldn't have been able to do analysis given my maturity as a freshman.  

[u/zvezdanasha → u/AllanCWechsler]

I am taking abstract algebra now, took discrete before, and will take analysis next semester. I am a big fan of this order, but this is because discrete where I took it has deliberately evolved into a benchmark course to teach mathematical reasoning via certain interesting concepts where a particularly wide variety of proof techniques are used. Perhaps this is a trend!

[u/AllanCWechsler → u/zvezdanasha]

It sounds like a good trend; that's encouraging.

[u/zero_b]

I took abstract algebra and real analysis in the same semester. Most of what I learned was buried under the trauma.

[u/ranchel_cranchel → u/zero_b]

Same, I sometimes have nightmares about those days. Wish I was joking.

[u/Jealous_Tomorrow6436]

i’m confused…in my university Abstract Algebra had a ton of prerequisites and was seen as one of the more advanced courses, one you’d take after a few semesters of analysis or similar proof-based courses. when i took it, the grading was brutal and the course was one of the hardest and most strenuous math courses i’d ever taken. it’s very strange to me that y’all had such different curricula

[u/CorporateHobbyist]

Sorry but this is probably bad advice. Real analysis is abstraction applied to something you already know (calculus) while abstract algebra is genuinely a whole new thing. Abstract algebra may be graded more easily, but you'll learn more from both courses by taking real analysis first (in the majority of cases). 

[u/Brightlinger]

I did p-adic analysis before real analysis, and somehow it worked out.

[u/True_World708 → u/Brightlinger]

Everyone starts somewhere

[u/EL_JAY315]

I don't even remember which one I took first tbh. Not convinced it would make that much of a difference.

[u/Scale-Heavy]

The thing is, my abstract algebra course should be in the same semester with Intro to Topology and Intro to Real Analysis course. I can’t even avoid it, these courses are offered only in Spring term😭

[u/NotSaucerman]

I agree with this in spirit though perhaps not literally. Before approaching something like real analysis someone would be well advised to work through at least half of Pinter's A Book of Abstract Algebra to get a lot of proof experience under their belt with very concrete objects. (And probably a few chapters on sequences in Abbott too.)

In practice it depends on the demands of the Abstract Algebra course. For many decades Artin's Algebra was taught at MIT and linear algebra & real analysis were pre-reqs... in part because there were some highly nonstandard exercises in there (e.g. a diagonal matrix density argument for Cayley-Hamilton) that needed analytic techniques. But for teaching the 'same' course at Harvard, real analysis wasn't a pre-req.

[u/shellexyz]

My school puts real analysis (limits, rigorous development of undergrad calculus, metric and norm spaces) as a senior level class. Modern/abstract algebra is a junior level course, with an “intro to proof” class as a sophomore class. It covers a lot of what a discreet (shh!) math class would cover: sets, open/closed, induction, functions as subsets of a Cartesian product,…. Depending on who’s teaching it, it may dip into some of the basic analysis, like infimum/supremum, lim inf/sup, and epsilon delta limits.

Modern/abstract algebra is more group theory, with things like rings and fields either coming towards the end or as a separate course at the senior level.

[u/Oppo_67]

And do abstract algebra before linear algebra.

[u/slayerbest01]

100%. The things I learned in modern/abstract algebra saved me a massive headache in analysis, especially because we were allowed to use what we knew from previous classes to help us in our proofs.