selfstudy

5 Problems with Students Self Studying Mathematics

For several years already I have been trying to help students who have been self-studying math. I did this completely free with no compensation for me. I saw this as a very enriching experience. Most of the people I helped came from physics forums. I usually contacted them and told them I might be able to help them. If they decided this was ok, then we would go further. My help usually consisted of providing resources, providing extra problems, checking problems, giving extra explanations, etc. As a rule, I did not want to help people with formal coursework, but only with self-studies.

Most of the self-study projects I engaged in ended up a failure, however. This happened for various very different factors. I will try to go over the most common ones. Whether you are self-studying completely on your own, or have somebody like me to help you, I think the following discussions will make you aware of how difficult self-studying is, and what the likely traps are.

1) The helper didn’t provide the right help

This needs to be said. I helped a lot of students, but a lot of times, I was not the correct person to help. I enjoy pure mathematics a lot. I like math for its own sake, and this is very clear from the help I give. Furthermore, by my very nature, I find myself not very capable of helping with most high school mathematics. It has been a shock that knowing high school mathematics well does not mean you can explain it well, or that you can explain it in an interesting way to somebody else. So in these cases, I could not help much.

What you should get out of this is that somebody helping you needs to be at the right level for you. If you struggle with basic reordering of equations, getting somebody too advanced may not help you very well, it might confuse you even further. On the other hand, if you want to do very pure math, then getting somebody to help who only cares about applications is probably not a good thing either.

There needs to be a very good bond between student and helper. If that bond is missing, then both student and helper could be interested, but they get nowhere together.

2) Students who go too quickly

This happened most frequently with people who are new to abstract math. I’m thinking of high school students who try their first steps in the territory of pure math. It happened quite often that they told me after two days that they finished the first chapter and understood everything. What’s more, they did all the problems as well. When I ask them to show me their solutions, they were all wrong. The student THOUGHT he understood everything, but in reality, he didn’t.

In studying mathematics, it is very important to take it slow. Reading a fiction book of 1000 pages will be much quicker than reading an abstract math book of 50 pages. Case in point: I once finished 1000 pages of a neat fantasy book in one day, but I struggle to do more than 5 pages of mathematics in that same time frame!

In studying math, every statement needs to be chewed on very carefully. It needs to be spit out and chewed on several times. This takes time. A lot of time. This is normal. Don’t think you can understand an analysis book in 2 months, there just isn’t enough time for that. Especially if you self-study.

3) People get discouraged

Naturally, if your progress is 5 pages a day (at most!), then you’ll start calculating. You might calculate that it will take you more than half a year to finish one single math book! This is discouraging. Very discouraging. This is why many people stop.

Also, math is hard. Very hard. You will regularly hit walls. You will regularly see exercises which you are completely unable to solve. And when this happens 10 times in a row, you get discouraged. And you quit.

If your hobby is collecting stamps, then you can see your collection of stamps growing every day (hopefully). But with math, your improvements will be minimal. You won’t see any improvement from day to day. And after a while, you just give up since things are too difficult. You DO improve, however, but this is not easily apparent. But if you keep detailed records of your math adventures, then looking back for 2 or 3 months will reveal that you have grown a lot since then. Math never becomes easy, however. The only easy math is the math you already know.

4) Life gets in the way

Studying math is something very intensive. You will need at least one hour every day. This is sometimes possible at first, but soon after that you end up in the hospital, you change your job, you find a new lover, etc., and what is possible first is not possible after. Life got in the way of mathematics. There’s little you can do about this. If you have more important priorities than math, then you can’t do the math. There are only 24 hours in a day after all, most of which we sleep.

5) You get bored

So you study math, and very quickly you see that it is not what you thought it was. You get bored of it. This happens very frequently since many people don’t really know what science is about before they try it. Once they do try it, it’s not what they think it is so they drop it. No need to shame yourself for this if this happened to you. You found out something new: that math is not for you. You tried and it didn’t work, but at least you tried and you know you’d rather do something different.

Conclusion

As you see, self-studying math is difficult. Studying it outside a formal classroom setting is even more difficult. Many people try it, but many stop, mostly for the reasons you see above. But please don’t be discouraged from studying math. Who knows? Maybe things will work out for you. You don’t know until you try.

 

 

49 replies
  1. mathwonk says:

    @mathemasochist: how long to spend on a book depends on how densely it is written and that reflects often how expert is the author. I am a professional algebraic geometer and have owned a copy of David Mumford's (fields medalist) red book Introduction to algebraic geometry for 50 years, perusing it from time to time over that period. Now I have been reading it carefully, in retirement, for about a year, and am less than half way through, in trying to actually understand everything I read. So even though I already "know" much of it I am spending about 2 years on this one book, and gladly so. of course i also do the yardwork. oh now i see you are long gone.

  2. mathemasochist says:

    Hi. Inexperienced self-studying student here. I know it's very commonly repeated that you can't force yourself to absorb some topic all at once, but I was wondering if you could feasibly get through a book on a topic like analysis with 2 months of time "on average." And by that, I mean something like working through four different subjects concurrently over the course of 8 months or so. Is this any different from working through each subject one at a time, spending two months on each? I'm thinking extending the study allows you to digest it better even if you spend the same amount of time at the desk, but I don't see much information about it. Can anyone compare the two different setups?

  3. ibkev says:

    [QUOTE="fluidistic, post: 5551139, member: 122352"]I'm a bit late to the party. My question is "Do you have any success story Micromass?". I mean by that student(s) who really learned a lot and had a good academic start or so.[/QUOTE]I have a success story, but it's not the one you're expecting. :)Success in this context means a lot of different things to different people. Folks of a variety of ages are here self studying and for a lot of different reasons. I've read posts from retirees, all the way down to teens and twenty somethings.Myself, I'm in my 40s, my kids are all in high school/university and my career trajectory is basically ballistic now. I started self-study late last year and have been slowly plugging away ever since. What got me going was how helping my kids with their math and physics homework reminded me just how much I enjoyed all this stuff as an engineering student, many years ago. I remember the insane pace of my degree program, having ~= no life and how I used to joke, half seriously, that true understanding and intuition of any given course came only mid-way through the follow-on course!!Lacking background in physics to even know what questions to ask, initially my goals were quite vague. I started out wanting to understand special and general relativity and to make sense of what the Higgs boson really was. This time I wanted to take my time and understand what I'm learning, deep in my bones, every step of the way.micromass (what a great guy!!!) in combination from a lot of reading on these forums, was kind enough to help me get started. I thought I did quite a lot of math during my degree but one of the first questions micromass asked was if the math I had done was computational only. I had to stop for a minute to even understand his question! You mean there's more than one kind?? :) So, it's been almost a year now, on micromass's recommendation, I've been working through a text on math proofs and started reading about real analysis (advanced, axiomatic calculus). Also, from a less rigorous perspective, I've brushed up on enough calculus 1 and 2 (which came back fairly quickly, kind of like riding a bicycle actually) to get about 1/4 of the way through the problems in Morin's classical mechanics book. Also, I'm about 1/3 through introductory linear algebra. (A beauty of self study is that when I start getting bored of one subject, I can just switch to something else. I don't have to worry about an exam in 2 days. :D I highly recommend having several subjects on the go at the same time to other self studiers.)But wow – it's slow going and it's easy to feel like my goals are infinitely far away. As is common for people who don't know much about a subject, I grossly underestimated how vast these subjects were and how long they would take to learn. It reminds of being back in high school, excited for aerospace engineering and imagining myself single handedly designing the next moon lander, not realizing the less glamorous and specialized reality!The time I'm able to spend on this has been wildly inconsistent. Some weeks I manage only an hour or two. A lot of days I come home pretty brain dead from my day job and the best I can do is surf physicsforums for an hour. So I can completely relate to the things micromass' has posted about being things you have to stay strong in the face of.That said, the flip side of this coin is that this hobby doesn't have an "end" – there'll always be something new to learn. And I do love the little eureka moments, and those moments where I make a connection between two topics I've been learning about.Anyways, I'm still happily chugging along. For my personal self study effort, this is what success looks like.p.s. I loved the goals that andrewkirk set for himself earlier in this thread. Paraphrasing:

    • understand the derivation of Newton's gravitational law as an approximation of Einstein's general relativity equations.
    • understand the derivation of the equations that describe a hydrogen atom and its electron orbitals.
    • understand the proof of the Jordan Curve Theorem.
    • understand the proof of Godel's Incompleteness Theorem.

    Seeing these has made me realize that it's time to take my initial vaguely formulated goals and be more specific!

  4. IGU says:

    [QUOTE="fluidistic, post: 5551139, member: 122352"]I'm a bit late to the party. My question is "Do you have any success story Micromass?". I mean by that student(s) who really learned a lot and had a good academic start or so.[/QUOTE]I'll offer my son as an example.  He self-studied starting in 8th grade when I pulled him out of public school in California.  After five years of studying only math (much self-study, plus auditing (mostly graduate) classes at local universities) he went to Cambridge.  He is now about to start his third and final year.  His knowledge and understanding of mathematics was and is well beyond his fellow undergraduates.Of course this early in his life it is hard to say whether he is going to be a real success story of any sort.  We shall see whether he stays interested in mathematics and where his interests lead him.  But his foundation of self-study is great for being independent and self-motivated, and I am very pleased that he seems to care about the math way more than the grades he gets.  I am guessing at this point that he will choose to go to grad school.By the way, I think his two (or four) favorite books for self-study were Tom Apostol's Calculus (Volumes I & II), and Nathan Jacobson's Basic Algebra (I & II).  His approach has always been to read everything and do every problem (although he would skip many exercises).  He likes his math dry and rigorous.

  5. jasonRF says:

    Great insight.  4 and 5 have been the most prevalent for me, especially when I try to work through an entire book.   I have only been successful working through the majority or more of a book a handful of times – the things they tend to have in common are1.  Book is at the right level 2. Book is concise – it covers most of what I want and not a lot of extra material I don't care about (Axler's Linear Algebra Done Right is good example of a concise book).  3.  I was fundamentally interested in the subject – I wasn't learning a topic to simply prepare me for something I was more interested in4. book was adequately engaging.  Finding the right book has been key for me.  One killer has been setting artificial goals, such as doing half or more of the problems in each chapter; when I do this I find that I don't spend my time very wisely. Most of my self-studying is more focused.  Sometimes it is just a chapter or two in a book to get a better understanding of a topic of current interest; or more often the standard research problem of seeing a result quoted in a paper that I don't understand or want to understand more fully, so I look up the references and/or books (or just try to work out on my own) until I am satisfied.  jason

  6. Rio Larsen says:

    That was a good article that I thought was mostly relevant to me. I do read textbooks on my own and some of the points are relevant to me; points one and five weren't as relevant to me.I understand the feeling of it being easier to read a fiction book than a math book. I don't read 1 thousand pages of a fiction book before I read five pages of a math book, however.

  7. fluidistic says:

    I'm a bit late to the party. My question is "Do you have any success story Micromass?". I mean by that student(s) who really learned a lot and had a good academic start or so.

  8. ProfuselyQuarky says:

    [QUOTE=”Isaac0427, post: 5489101, member: 552304″]I’d love to, but I have no contacts.[/QUOTE]
    Couldn’t your parents help you find somebody then?

  9. Isaac0427 says:

    [QUOTE=”micromass, post: 5488713, member: 205308″]Why don’t you find somebody to help/tutor you?[/QUOTE]
    I’d love to, but I have no contacts.

  10. Maxila says:

    [QUOTE=”micromass, post: 5484945, member: 205308″]micromass submitted a new PF Insights post

    [URL=’https://www.physicsforums.com/insights/problems-self-studying/’]Problems with Self-Studying[/URL]

    [IMG]https://www.physicsforums.com/insights/wp-content/uploads/2016/05/selfstudy.png[/IMG]

    [URL=’https://www.physicsforums.com/insights/problems-self-studying/’]Continue reading the Original PF Insights Post.[/URL][/QUOTE]

    Great article with many good insights; I wish more people here were as patient and generous helping others as you.

  11. micromass says:

    [QUOTE=”Isaac0427, post: 5488701, member: 552304″]but I do wish I could get questions answered without going through 20 articles or going on here [/QUOTE]

    Why don’t you find somebody to help/tutor you?

  12. ProfuselyQuarky says:

    [QUOTE=”Isaac0427, post: 5488701, member: 552304″](asking too much on here makes me feel a little stupid)[/QUOTE]
    A teacher one day said (when half the class missed an assignment because they didn’t understand):
    [QUOTE]Only stupid people are afraid to ask questions. What idiots![/QUOTE]
    He was pretty upset, or perhaps disappointed in the class. Of course I’m calling you neither, but just something to think about :smile:

  13. Isaac0427 says:

    [QUOTE=”ProfuselyQuarky, post: 5488301, member: 586985″]For me, it’s predominantly the second one: I go too quickly thinking that I understand only to realize that I [I]don’t[/I] really understand a couple of pages later when new material is added. Aaah, so frustrating sometimes…. you then have to go back and unwind all the progress that you *think* you made. But that’s learning for you.[/QUOTE]
    2 is a big one for me too. 3 and 4 also get me a lot, especially when seeing how long lectures are. 1 is a little less but I do wish I could get questions answered without going through 20 articles or going on here (asking too much on here makes me feel a little stupid).

  14. IGU says:

    [QUOTE=”A. Neumaier, post: 5487568, member: 293806″]You can do this only with one of the extremely rare books without any errors in it, or you’ll stall indefinitely trying to prove wrong things.[/QUOTE]

    I think that one of the hallmarks of the effective self-studier is the self-confidence to decide that the book is wrong. It’s actually pretty common that there are mistakes, so this is an essential skill. Then you figure out what the problem was supposed to be and what the real solution is. Or not. But yes, getting stuck is to be avoided.

  15. ProfuselyQuarky says:

    [QUOTE=”Isaac0427, post: 5487434, member: 552304″]I have ran into all problems listed (except number 5, I could truly never seem to get bored).[/QUOTE]
    For me, it’s predominantly the second one: I go too quickly thinking that I understand only to realize that I [I]don’t[/I] really understand a couple of pages later when new material is added. Aaah, so frustrating sometimes…. you then have to go back and unwind all the progress that you *think* you made. But that’s learning for you.

  16. boneh3ad says:

    The book [url=http://www.amazon.com/Make-Stick-Science-Successful-Learning/dp/0674729013][i]Make it Stick[/i] by Brown, Roediger, and McDaniel[/url] may be of interest to those of you who are students, self-studying something, or tutors/instructors. It’s all about the research into the cognitive psychology of learning and how to do it (or direct it) more effectively. I found it to be pretty interesting.

  17. Jimster41 says:

    I enjoyed this insight article – and found a lot of encouragement in it. I just started “Category Theory for The Sciences” by Spivak. The first chapter almost stopped me in my tracks because it was so freakin dry dry dry… I just can’t memorize abstract rules of maximal precision without connection to something… but now that he is introducing types, aspects and ologs it is easier going and I feel more hopeful it is going to be an exciting book.

  18. blue_leaf77 says:

    Completely agree with point 2), in self-studying abstract math, one occasionally needs to stop and muse what he/she just learned all mean. Abstract math as the name implies can sometimes be so abstract that no everyday events can be taken as an analogy. Definitions needs to be memorized (as it is all that can be done with definitions) in order to aid learner in doing proofs.

  19. A. Neumaier says:

    [QUOTE=”MathematicalPhysicist, post: 5487407, member: 72″]I don’t think it would be a waste to read and try solving all the problems, it would be time consuming but I want to gain a good grasp of the theory.[/QUOTE]
    You can do this only with one of the extremely rare books without any errors in it, or you’ll stall indefinitely trying to prove wrong things.

    The right method is o be a hybrid of ant and grasshopper, and to know when and where to switch from one mode to the other.

  20. Isaac0427 says:

    I must say, as a self-studier, I have ran into all problems listed (except number 5, I could truly never seem to get bored). Great article, however I wonder if there could be a set of ways to avoid all these problems.

  21. MathematicalPhysicist says:

    Yes, I read your post a year ago.

    As for reading the books for the parts relevant for me it seems reasonable but then I assume I would be redirected to previous chapters, so I prefer to be the ant here from Zeidler’s volume 1 quote:”
    We may distinguish between the ants, who read page n before reading page
    (n + 1), and the grashoppers who skim and skip until something of interest
    appears and only then attempt to trace its logical ancestry. For the sake of the
    grashoppers, herewith is a listing of certain basics.

    Never understood the Grasshoppers, what a headache is to trace the logical ancestry of something and you’re never sure if you haven’t missed something important,
    not that I say you couldn’t try reading a book from where you want if you know the other results that are covered in the previous pages.

    I don’t think it would be a waste to read and try solving all the problems, it would be time consuming but I want to gain a good grasp of the theory.

    That said I have returned to Munkres’ Topology book, and it would seem I’ll need to use the Grasshopper method, I believe it takes more time than the ants plain method.

    Anyway, haven’t we all returned to something we learned a few years ago just to notice that we forgot how to prove it, for this case it’s good to reread the proofs and do the exercises.

    Anyway this is a habbit I have since my first two logic books i learned from in high school years ago, to read them cover to cover and solving every exercise.

  22. A. Neumaier says:

    [QUOTE=”MathematicalPhysicist, post: 5487301, member: 72″]if you want to solve the claymath problem or at least understand it[/QUOTE]
    Did you see [URL=’http://www.physicsoverflow.org/21786/energy-mass-spectrum-yang-mills-bosons-infinite-and-discrete?show=21846#a21846′]my post on this[/URL]?

  23. A. Neumaier says:

    [QUOTE=”MathematicalPhysicist, post: 5487347, member: 72″]but got stuck on one exercise[/QUOTE]
    These books are not meant to be read from cover to cover until one gets stuck. Just read te sections you are interested in (e.g., there is one in Volume 2 I believe on Osterwalder-Schrader stuff, I believe), and see what of the earlier stuff is needed to understand these parts.

  24. A. Neumaier says:

    [QUOTE=”MathematicalPhysicist, post: 5487347, member: 72″]We really should have a 30 hours day, it would be more acceptable.[/QUOTE]
    I’d need a 60 hours day to do all that I’d like to do. But probably my appetite would grow proportional to the length of the day, so that nothing short of eternity would satisfy me….

  25. MathematicalPhysicist says:

    Thanks on the recommendations, I have Reed’s and Simon’s books (I started reading the first volume but got stuck on one exercise which I posted on MSE, here it is:
    [URL]http://math.stackexchange.com/questions/1394210/sa-b-is-dense-in-pca-b-in-the-cdot-infty);[/URL] Perhaps I’ll return to it this year, but I don’t think I’ll have time for it now.

    We really should have a 30 hours day, it would be more acceptable.

  26. A. Neumaier says:

    [QUOTE=”MathematicalPhysicist, post: 5487301, member: 72″]Yes I know, but if you want to solve the claymath problem or at least understand it, you should understand the previous work that the wiki entry describes here:
    [URL=’https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_existence_and_mass_gap’]https://en.wikipedia.org/wiki/Yang–Mills_existence_and_mass_gap[/URL]

    It’s indeed a problem that the people who might know the most about this area are old or unfortunately dead.
    Are there any other experts who are responsive via email whom I can ask questions in this field?[/QUOTE]
    You can try Buchholz, Schroer, or Yngvason. Also, the book by Glimm and Jaffe contains all about Osterwalder-Schrader theory. The 4 volumes of Reed and Simon’s functional analysis are also a good complement.

  27. MathematicalPhysicist says:

    [QUOTE=”A. Neumaier, post: 5487281, member: 293806″]Remember that the paper you studied is very old. [URL=’https://en.wikipedia.org/wiki/Konrad_Osterwalder’]Osterwalder[/URL] was 73 last year, and [URL=’https://de.wikipedia.org/wiki/Robert_Schrader’]Schrader[/URL] 76; he died last November.[/QUOTE]
    Yes I know, but if you want to solve the claymath problem or at least understand it, you should understand the previous work that the wiki entry describes here:
    [URL]https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_existence_and_mass_gap[/URL]

    It’s indeed a problem that the people who might know the most about this area are old or unfortuanetly dead.
    Are there any other experts who are responsive via email whom I can ask questions in this field?

  28. A. Neumaier says:

    [QUOTE=”MathematicalPhysicist, post: 5487089, member: 72″]not even from the authors of the paper whom I asked via the email.[/QUOTE]
    Remember that the paper you studied is very old. [URL=’https://en.wikipedia.org/wiki/Konrad_Osterwalder’]Osterwalder[/URL] was 73 last year, and [URL=’https://de.wikipedia.org/wiki/Robert_Schrader’]Schrader[/URL] 76; he died last November.

  29. MathematicalPhysicist says:

    [QUOTE=”A. Neumaier, post: 5487193, member: 293806″]Where?[/QUOTE]
    there:
    [URL]http://www.physicsoverflow.org/32749[/URL]

    It’s almost a year since I asked, since then I went to read other stuff in maths, I also read Rudolf Haag’s Local Quantum Physics but got stuck on something in some page over there, if I’ll have a question on this book when I’ll return to it I’ll post it in overflow once more.

  30. A. Neumaier says:

    [QUOTE=”MathematicalPhysicist, post: 5487089, member: 72″]I asked in Physicsoverflow a question on a paper of Schrader on constructive qft[/QUOTE]
    Where?

  31. MathematicalPhysicist says:

    [QUOTE=”jtbell, post: 5485465, member: 20524″]This is my experience also, both as a student (undergraduate and graduate levels) and in teaching. In physics at least, most textbooks contain more material than can be covered in a standard classroom-based course. Every instructor has his/her own preferences for secondary topics (applications, etc.). Textbook authors and publishers want to cater to as many potential customers as possible. An instructor has to choose a set of topics that fits within the time allotted for his/her course.

    When you self-study, you have no time restrictions, so you can cover an entire book if you want. However, it’s going to take longer than a typical classroom-based course that uses the same book, even if you can work through the material at the same pace.[/QUOTE]

    Yes I agree it can take even more than a year or two; but if you want to be a researcher you must be eager to learn by yourself since there will not always be the suitable teacher to teach you, some of them might be dead already.

    P.S
    I asked in Physicsoverflow a question on a paper of Schrader on constructive qft and still didn’t get an answer, not even from the authors of the paper whom I asked via the email.

    It seems I’ll need to solve my problems by myself…

    P.P.S
    Don’t forget a book is by itself written by a lecturer, so a good book is a good teacher.
    :-)

  32. SVN says:

    I believe people who are self-studying are in particular need of helping hand in two cases, which are in some sense are opposite of each other. Being so the cases may require different approaches.

    I guess everyone who is self-studying sooner or later run into the situation when he can’t understand a particular statement or derivation of a formula or unable to see a relation between ideas, concepts, etc. It is especially true in the case of mathematics. You mostly (and hopefully) understand everything before, but at this point things come to halt. When I was a pupil a was unable to understand the technique of long division, for many days my parents and relatives around me tried to explain it, but failed, until one of my fathers buddies managed to find proper words, after that everything continued to run smoothly as it had been doing before. It is exactly the moment you are in need of some hand.

    I believe a good indicator for such situation is your ability to correctly formulate the question. If you are capable of doing so, there is a good chance some wise guy would be willing to explain it to you.

    The second situation you need help originates from the fact that starting studying a new discipline you do not have a whole picture of it. Things which are intimately related, appear to you as something not related at all. Grasping this whole picture is often quite challenging (and, from my viewpoint, the most exciting part until you self-study for some practical reasons, learning a particular technique for example).

    Characteristic feature of this kind of challenge is inability to pose correst questions. You may understand lots of details but not what this all about. Since it is difficult for a potential answerer to get what you do not understand it may be not very effective helpful to ask questions on forums. But it can be helped by using one more textbook, since the whole picture may become clearer when seen from yet another perspective.

    As one of great american science fiction writers once said «In order to ask a question you must already know most of the answer».

  33. Stephen Tashi says:

    What things characterize situations when self-study projects are successful ?

    I think self-study is most often successful when the student is attempting to understand problems instead of seeking to understand legends and rumors.

    A student enrolled in a formal course of education had the advantage of being forced to study various things. For example, based on a legend or rumor that group theory is useful for understanding physics, a student may enroll in a course on group theory without having any idea what he is getting into. Then the pressure of exams and keeping up with his peers is motivation to keep studying the material, even if the connection to physics never becomes obvious.

    By contrast, a student pursuing the legend that group theory is useful in physics on his own is likely to drop-out of that line of study unless he can find study material that is a custom-fit to his current intellectual development. It would need to relate group theory to physics and deal with physics on the level the student is prepared to understand.

    Pursuing legends and rumors approaches all self-study topics at the level of vocabulary questions. For example, we can say “What is this “theory of relativity” that I hear so much about?” and “What are these “tensors” that one needs know in order to understand “relativity”. People who formulate self-study as a vocabulary exercise are the ones who want roadmaps like : “Name the topics I should study in order to understand what (the name) relativity means”.

    A student who is curious about a problem has a more focused goal. For example, a person who wants to know “What is the best time for me to invest in stock X (US Steel) during the next 3 months” hasn’t formulated a specific mathematical problem, but he at least has a general idea of his objective. Very abstract problems can be good motivators. For example, a person might have a burning curiosity about “How is it possible that we keep finding prime numbers that are close to each other”?

    Investigating problems involves sharpening vague questions into specific questions. Mathematics is an excellent tool for doing that and utility is a good motivation for studying it.

  34. QuantumQuest says:

    This is a great insight by[I] micromass[/I]. Being now about fifty years old and having done a lot of self-study on math and physics for about fifteen years, I agree to all points he makes, applying them to people of various ages. In my case, the formal education I have is in CS, with a certification on web programming and many years of application programming. What pushed me – boosted me I can say, to self study math, was a real love for this field, that I maintain intact. Along the process, something popped up inside me about physics – especially the field of macro-world is of great interest to me, so I found what specific areas of math were of direct relevance and most important in this regard. Had I keep to learn abstract math just for its own sake – although fascinating, wouldn’t get me far. So, I totally agree here to the notion of [I]goal [/I]as stated in above posts. Without a specific goal, nobody gets anywhere. The goal may mean different things to different people but it’s still a goal.

    [QUOTE=”jbunniii, post: 5485593, member: 81553″]I personally have never had much success studying new mathematical topics (other than superficially) without a first exposure/introduction in a classroom setting.[/QUOTE]

    Very well said in my opinion. In my case – given especially the #4 point [I]micromass[/I] makes in his insight, this translated into taking some online courses from (mainly) US universities and many of them at an advanced undergraduate/intro graduate level. This definitely – besides “whetting my appetite”, gave me most importantly the [I]motivation[/I] and [I]a good schedule[/I] to go on, regarding [I]what [/I]to pursue further and [I]what[/I] the relevant good resources are. Needless to say here that PF is [I]the [/I]great resource.

    [QUOTE=”A. Neumaier, post: 5485450, member: 293806″]Some people study math by themselves and succeed. I was one of them – at a time when the internet didn’t yet exist. From time to time I asked my math teacher at school something, or showed him what I had done – it was usually wrong, but he patiently corrected me until I saw where I needed to be more careful. Later I properly studied math as a student, doing all the required stuff (much of which I knew already but unsystematically) and in addition a lot of more advanced stuff that really interested me. Much later I learnt physics the same way – without taking any classes.

    Thus I am not as pessimistic as micromass. it takes a lot of time and determination for years, yes . But if you are interested enough to invest the effort there is nothing wrong with self study of math or physics[/QUOTE]

    [QUOTE=”A. Neumaier, post: 5485557, member: 293806″]The important point is that one needs to have enough motivation to keep going when the problems appear, and find enough strength and determination to overcome them. That makes the difference between failing and succeeding.[/QUOTE]

    Definitely agree to [I]A. Neumaier. If [/I]you [I]really [/I]want to learn something even at a very advanced level, you [I]can[/I] do it. No discouragement or boring or anything for that matter can take charge. This defines the difference between failure and success and is really particularly difficult at times, but nevertheless you [I]can [/I]succeed. I don’t think though that [I]micromass [/I]is pessimistic in his insight. Most people get discouraged or bored and give up, sooner or later. But this can be very well addressed if someone has enough patience and is strongly determined to succeed.

    I think it takes much time and a lot of efforts to [I]really learn – [/I]not just thinking so, math, physics or anything at the scientific level – i.e quantitatively and at a sufficient depth. However, this should not be a source of discouragement, because in life and in general, bad things happen quickly and effortlessly. Good things are developed through time, spending a lot of efforts, but this is why it is fascinating and rewarding to pursue them.

  35. vela says:

    [quote]In studying math, every statement needs to be chewed on very carefully. It needs to be spit out and chewed on several times. This takes time. A [b]loot[/b] of time. This is normal.[/quote]
    I didn’t know you were Canadian. :wink:

  36. jbunniii says:

    [QUOTE=”micromass, post: 5485525, member: 205308″]This is very true. Often people don’t really know what it means to really deeply study something. In a classroom based setting, the teacher will give the necessary explanations, but also will tell you what problems to solve. The teacher will then correct the problems. This means you are forced to do the problems and you’re forced to do them correctly. If you don’t do them correctly, you are very quickly set straight. In self-study, you are your own judge, and that may be misleading. Thus it happens that people overestimate their grasp of the material.[/quote]
    Additionally, a good teacher will provide motivation which may be missing from the textbook, and will also guide you through the most important topics and problems. This is very important because most textbooks contain far too much material to read from cover to cover, but a beginner has no idea what material is the most important and what should be skipped. Also, a teacher/formal course will enforce a strict schedule, which is surely more time-efficient than almost any self-studier would be able to maintain.

    I personally have never had much success studying new mathematical topics (other than superficially) without a first exposure/introduction in a classroom setting. When done well, this provides the motivation I need to self-study more in-depth, advanced material on the same topic. So, I am comfortable attempting quite advanced material in analysis, since that is where I had the most formal coursework, but when faced with something like algebraic topology, I don’t get very far because I never had an instructor (or anyone else) “sell” me on the topic, so I don’t see why it is interesting. Conversely, group theory, which seems quite pointless in many ways, is a subject I love because I had several great instructors who helped me to see its beauty. I have never seen an introductory textbook on the subject which would have conveyed this to me before I gave up in boredom.

  37. A. Neumaier says:

    [QUOTE=”micromass, post: 5485525, member: 205308″]if you choose to self-study you will run into some problems sooner or later.[/QUOTE]
    Like wiith every worthwhile endeavor in life. The important point is that one needs to have enough motivation to keep going when the problems appear, and find enough strength and determination to overcome them. That makes the difference between failing and succeeding.

  38. micromass says:

    [QUOTE=”Square1, post: 5485039, member: 365401″]I recognize your username from years ago when I first started using this website and you’ve helped many at least here I am sure.

    Interesting though the point that self studiers will want to rush through more so than the regular student. In fact I would say that that is a consequence of the school system. If you don’t understand something, often people just try and put it to memory instead, move on, do assignments, grab the marks, and that will probably be the extent of the “studying” on a particular topic. [/QUOTE]

    This is very true. Often people don’t really know what it means to really deeply study something. In a classroom based setting, the teacher will give the necessary explanations, but also will tell you what problems to solve. The teacher will then correct the problems. This means you are forced to do the problems and you’re forced to do them correctly. If you don’t do them correctly, you are very quickly set straight. In self-study, you are your own judge, and that may be misleading. Thus it happens that people overestimate their grasp of the material.

    [QUOTE=”andrewkirk, post: 5485122, member: 265790″]I really identify with section (3) about getting discouraged. Most Maths and Physics books seem so long, and the rate of progress is so slow that it truly is daunting sometimes. I think there are hardly any of the text books I own that I have read from cover to cover. The ones that were used as part of courses were not required to be finished because the course tended to pick out certain chapters, while omitting others.

    I have found it easier to maintain determination in self-studying a subject if one focuses on a major theorem, rather than on finishing a book.[/QUOTE]

    This is absolutely correct. If you have some goal in mind, things will go much easier on you. It is the same when conducting research. If you have a problem to solve, you will read the relevant literature and keep the problem in mind. This makes a lot of the material easier to digest and it really does keep you going.

    [QUOTE=”otto9K9otto, post: 5485449, member: 235832″]Another problem with math self-studying is that the student does not see the big picture that a teacher should. So the student may skip over proofs (why do I need this stuff, anyway!) and just learn to solve problems. This is OK to a point. You miss some intellectual maturation by skipping proofs.[/QUOTE]

    You’re right. Whenever I read a book, I always tell myself “I can skip this little proof”. But one little proof becomes many different proofs. And after a while, you’re just lost and you need to start over again. The mind can only handle so many concepts at once. And proofs really make it easier to digest the concepts.

    [QUOTE=”A. Neumaier, post: 5485450, member: 293806″]Some people study math by themselves and succeed.[/QUOTE]

    You’re right. But I do think more people try it and fail. Some people their will and personality is so strong that they succeed. But math self-study is by no means easy. A lot of people don’t like it or can’t handle it. I’m not meaning to discourage anybody from self-studying, but if you choose to self-study you will run into some problems sooner or later.

    [QUOTE=”jtbell, post: 5485465, member: 20524″]
    When you self-study, you have no time restrictions, so you can cover an entire book if you want. However, it’s going to take longer than a typical classroom-based course that uses the same book, even if you can work through the material at the same pace.[/QUOTE]

    That is certainly my experience. Things will be harder and take longer than a typical classroom-setting. I do still think it’s worth it.

  39. jtbell says:

    [QUOTE=”andrewkirk, post: 5485122, member: 265790″]I think there are hardly any of the text books I own that I have read from cover to cover.[/QUOTE]
    This is my experience also, both as a student (undergraduate and graduate levels) and in teaching. In physics at least, most textbooks contain more material than can be covered in a standard classroom-based course. Every instructor has his/her own preferences for secondary topics (applications, etc.). Textbook authors and publishers want to cater to as many potential customers as possible. An instructor has to choose a set of topics that fits within the time allotted for his/her course.

  40. A. Neumaier says:

    Some people study math by themselves and succeed. I was one of them – at a time when the internet didn’t yet exist. From time to time I asked my math teacher at school something, or showed him what I had done – it was usually wrong, but he patiently corrected me until I saw where I needed to be more careful. Later I properly studied math as a student, doing all the required stuff (much of which I knew already but unsystematically) and in addition a lot of more advanced stuff that really interested me. Much later I learnt physics the same way – without taking any classes.

    Thus I am not as pessimistic as micromass. it takes a lot of time and determination for years, yes . But if you are interested enough to invest the effort there is nothing wrong with self study math or physics. Those interested are recommended to read Chapter C4: How to learn theoretical physics of my[URL=’http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html’] theoretical physics FAQ[/URL].

  41. otto9K9otto says:

    Another problem with math self-studying is that the student does not see the big picture that a teacher should. So the student may skip over proofs (why do I need this stuff, anyway!) and just learn to solve problems. This is OK to a point. You miss some intellectual maturation by skipping proofs.

  42. andrewkirk says:

    I really identify with section (3) about getting discouraged. Most Maths and Physics books seem so long, and the rate of progress is so slow that it truly is daunting sometimes. I think there are hardly any of the text books I own that I have read from cover to cover. The ones that were used as part of courses were not required to be finished because the course tended to pick out certain chapters, while omitting others.

    I have found it easier to maintain determination in self-studying a subject if one focuses on a major theorem, rather than on finishing a book. For example:
    [LIST]
    [*]learning GR from Schutz’s text, my focus was on getting to the stage where I could understand the derivation of Newton’s gravitational law as an approximation, from Einstein’s equation.
    [*]learning QM from Shankar’s text, my goal was to be able to understand the derivation of the equations that describe a hydrogen atom and its electron orbitals.
    [*]furthering my study of algebraic topology from Greenberg and Harper (I had studied this topic before at uni, but only as far as the Excision Theorem), my goal was to be able to understand the proof of the Jordan Curve Theorem.
    [*]learning logic, my goal was to be able to understand the proof of Godel’s Incompleteness Theorem.
    [/LIST]
    In each case, the power and majesty of the goal – a really amazing theorem – was enough to keep me going through the tough times Micromass describes, so I did finally attain all those goals, to my great delight in each case. A wonderful theorem is much more inspiring than merely finishing a book. I never completely finished any of those books, because other interests came along to distract me. But the buzz I got from understanding the proofs of those theorems injected more excitement and momentum into the process so that in each case I went a long way past the original goal – eg for GR to learning the mathematics of gravitational collapse and cosmology.

    As well as being more inspirational, focusing on a major theorem is more achievable because proofs of such often seem to occur well before the end of a text, sometimes even only at halfway or earlier. Many of the texts I own seem to spend the last few chapters delving into esoteric niche areas that are of particular interest to the author, but not essential to a sound understanding of the subject.

  43. Square1 says:

    I recognize your username from years ago when I first started using this website and you’ve helped many at least here I am sure.

    Interesting though the point that self studiers will want to rush through more so than the regular student. In fact I would say that that is a consequence of the school system. If you don’t understand something, often people just try and put it to memory instead, move on, do assignments, grab the marks, and that will probably be the extent of the “studying” on a particular topic. And this is easily favoured when you are in a class with a schedule and a grade that is often most cared about. I am actually going back right now on my own to redo my intro physics textbook over the summer and I am most thankful that I can approach win flexibility and time so that I can have a better understanding of what I read. My mind can go on tangents, jump forwards and backward throughout the book as I please. This is very very helpful when something is confusing, but you see a parallel with another topic in later chapters. You can go and red those subjects as you need, so that you can immediately reconcile the two topics, which will help you learn the initial topic in a much more intuitive fashion.

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