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    • CommentAuthorroberto
    • CommentTimeFeb 15th 2012
    I'm looking for series of exercises in mathematics, from the undergraduate level and up, which have a clearly defined bigger-goal, such as proving a theorem, or introducing a concept or theory.

    Some exaples (to be added later): has several "developed as a series of exercises" writings. "A principal ideal domain that is not Euclidean, developed as a series of exercises." "Quadratic reciprocity, developed from the theory of finite fields as a series of exercises."
    (quotations taken from the webpage)
    (Thanks to Arturo's comment at:

    Should each of these be in a separate answer, even if they are at the same webpage? or should people post the first URL, which points to them (as a "resource")? I think the first choice is better.

    I also know about some wonderful exercises (in Spanish), which guide the reader through some of the equivalences of AC, while introducing concepts such as towers and the principle of recursive definition. (Based, I think, on Munkre's "Topology")

    At the beginning of times, MO was open to this kind of CW, big-list questions. Is it still appropriate?

    Is "Guided exercises" the right wording in English? Some searching seems to suggest "Guided exercises" is too general and includes those meant to review a recently learned topic, which is not what I mean. In Spanish, "Ejercicios guiados" would be the fine.
    • CommentAuthorMariano
    • CommentTimeFeb 15th 2012

    As long as the question does not become an «add links to Bergman's web pages» list... :D

    • CommentAuthorroberto
    • CommentTimeFeb 15th 2012
    I'm really afraid of it, because I only know those two examples of the "big-list".
    • CommentAuthorroberto
    • CommentTimeFeb 15th 2012 edited
    But is Bergman's web page the only one for such valuable resources in all of the English internet?
    I'm not a big fan of this question -- the things you posted are really at the advanced undergraduate / first year graduate level, which I think is too low for MO (and my guess is that the answers you get will be at this level or lower). Perhaps it'd be better on But I don't spend much time there, so I'm not certain.
    You should write why you are interested in such a list. Do you plan on teaching a course based on this approach?

    I don't really think this is a good question without elaboration (e.g., as suggested by Dan Petersen), because giving a proof or introducing a concept as a series of exercises is basically a style of exposition. Practically any proof or topic can be written up in this fashion, although it is not always desirable to do so. Here are a few examples from textbooks:

    --Silverman and Tate, "Rational points on elliptic curves": the proof of Bezout's theorem (in one of the appendices) is given as a series of exercises with hints.

    --Munkres, "Topology": there's a section on nets that is almost entirely exercises.

    --Hartshorne, "Algebraic Geometry": A large number of results are given as either a series of exercises, or a "single" exercise with many "parts."

    And the list goes on. And on.

    Linguistic note: I don't think there is any two-word phrase in English that captures what you mean. "Proof via exercises" or "Exploration via exercises" or "Exposition via exercises" is probably the best you're going to get. Although you should probably look up "Moore method," if you are not familiar with this term already.


    Charles has this right, google "Moore method" and find notes from classes taught in that style.

    Not quite on target, but you might want to look through

    I wouldn't care enough to close this question, but it seems like a strange thing to ask. Is there anyone out there who is thinking "I'd really like to work through a set of guided exercises, but I don't care whether the end goal is the Prime Number theorem or the classification of semisimple Lie algebras"?

    Actually, you make me wonder now: When I was at MOP (the US Math Team training camp), there was a tradition that one MOPper would volunteer to write a set of exercises of this form, and the rest of us would work in small groups for the next 24 hours trying to solve as many as we could. These were called the "MOP marathons". If I remember correctly, the topics for the years I was there were Urysohn's metrization theorem, something in hyperbolic geometry (I don't recall the final result), and factorization into prime ideals in rings of algebraic integers. People put a lot of work into these; I wonder if they are still being done, and if they are online anywhere.
    • CommentAuthorroberto
    • CommentTimeFeb 16th 2012
    First (dan's times):
    I'm undergraduate, but thought the question might be of interest to the community, and even that it shouldn't matter if I understand the exposition pointed out in the answers.

    Some time later (Charles's times):
    But I see how it can turn into a "list of books" times "number of expositions in it" kind of question.

    And later:
    Thanks for all the feedback and the right wordings, specifically, as Noah points out, searching for class notes along with "Moore method" makes my question highly irrelevant.
    • CommentAuthorroberto
    • CommentTimeFeb 16th 2012
    Thanks! I didn't know about that question.

    I think I answered your concerns about the "end goal" in my last post (but without having read your post then), and you are right.
    • CommentAuthorabmiller
    • CommentTimeFeb 16th 2012
    @David: MOP marathons haven't been done for a while; I believe the last one was in 2002 (I do however have some hope that they may make a comeback in the future).

    To the best of my knowledge, none of the past ones are online.

    (If you or anyone else wants to know more, contact me off-thread.)