I am a sophomore in high school. When I approach a math problem or concept, I normally try to determine a formula or systematic method of solving all similar problems. However, since I entered precalculus, I have found that this approach does not work. I really love math and find it interesting and exciting, but I find that I am unable to solve the kinds of conceptual problems that I have been doing more recently. For example, when I try to do a counting problem, it is often unclear how to approach it. How can I improve my problem-solving and critical thinking skills? Also, how can I approach problems logically to solve ones that are not clear at first?
122984
I'm a sophomore in university and seriously feel that I'm bad at solving mathematical and algorithmic problems (be it discrete math, calculus or just puzzles). I noticed that I'm only good at solving questions that are similar to the ones that have been taught to us.  Here's how I generally approach it:   What is the problem? What do I need to do here? Does it look like I've encountered this before?  Can I think of a smaller problem to solve instead?   If the answer is no to all the above then I sort of blank out. I stare at it and force my brain to run through a wide variety of stuff, almost like a brute force attempt of solving it. Obviously that leads me to nowhere everytime. I simply can't think "outside the box."   What can I do to improve my situation?
45254
I am looking to study mathematical logic, however, I find that introductory books are very daunting, which kind of disheartens me. You see, slowly but surely, I started to realize that the maths which I have learned did not just pop out of thin air, but is a collection of systems, which must of been developed via some other system, i.e, maths did not develop itself.  So I began to look into the origins of mathematics, and read that it was developed via a type of logic, which exists sort of by 'default', via a set of axioms, and then of course I looked up the definition of axioms.   So given that I'd be studying a type of logic whose origins are self evident axioms, naturally I believed there would be no prerequisites. However, in looking up mathematical logic, I have come across things such as Boolean algebra, sets, first order logic, some other type of logic, called 'traditional logic', as well as references to a sort of calculus, though not in a mathematical sense, I think.  So all in all, I am trying to develop a type of mental spider web, and I am trying to find out the strands which lye at the absolute bounds so that I may learn this mystique logic. Though I have no idea where to start.  Side note: This is the book I have started reading:   Credit goes to Wolfgang Rautenberg.
116074
I'm looking for mathematics books, courses + exercises for high school level. My daughter was high school certificated but she has stopped her studies for 5 years. She would like to be involved in new educational program but she needs to restore her mathematic capabilities.  Could you help me to identify which mathematics books can help her?  Thanks a lot for your reply,  FR.
28699
I just remembered about a problem/paradox I read years ago in the fun section of the newspaper, which has had me wondering often times. The problem is as follows:  A maths teacher says to the class that during the year he'll give a surprise exam, so the students need be prepared the entire year. One student starts thinking though:   The teacher can't wait until the last day of school, because then the exam won't be unexpected. So it can't be the last day. Since we've removed the last day from the list of possible days, the same logic applies to the day before the last day. By applying 1) and 2) we remove all the days from the list of possible days. So, it turns out that the teacher can't give a surprise exam at all.   Following this logic, our student doesn't prepare for this test and is promptly flunked when the teacher does give it somewhere during the middle of the year (but that's my own creative addition to the problem).  This problem reminds me about the  for finite number of turns - you have to betray at the last turn because tit-for-tat retaliation is no longer relevant (no next turn), but then that means that you have to betray at the turn before that, and so on, until you reach the conclusion that you can't cooperate at all.  So is the student's reasoning correct or not? Mathematically it looks like it should be, but that would imply that surprise exams are not possible (and they are).
40053
I've recently been reading about the Millenium Prize problems, specifically the Riemann Hypothesis. I'm not near qualified to even grasp the entire problem, but seeing the hypothesis and the other problems I can't help wonder: what is the practical use of solving it?  Many researchers have spent a lot of time on it, trying to prove it, but why is it so important to solve the problem?  I've tried relating the situation to problems in my field. For instance, solving the $P \ vs. NP$ problem has important implications should it be shown to be either $P = NP$ or $P \neq NP$: for instance, cryptographic algorithms, but it's hard to say WHY the Riemann Hypothesis, or other problems, are so important.  Perhaps a partial answer could be made by seeing which solutions proof of the Poincaré Conjecture has lead to.
5050
I am a third year (starting fourth year in the fall) PhD student in mathematics. I've passed all qualifying exams and am currently doing research. As far as I can tell, I am not doing poorly. I have the good fortune of having a great advisor, being in a very supportive department, and having friends and family who genuinely care about my success.   The fact is research is hard. It appears to consist primarily of staring at a problem for days and days and days without getting anywhere. Sometimes, rarely, I do figure something out and that feels wonderful, but the overwhelming majority of my time appears to be spent banging my head against a mostly figurative wall. I am not complaining about the material being hard, and I am not averse to putting in hard work, but I get frequently discouraged when I realize the vast volumes of mathematics that I yet know nothing about (and probably never will). It's very hard to quantify progress - in particular, there are too few tangible returns after too many hours worked. I find myself thinking along the lines of "Oh, if only someone actually smart were thinking about this problem they would have solved it in moments" and so on.   I've talked about this to some number of people; here is some advice I have received:   Take a day off. Putting in hours upon hours of trying things doesn't magically lead to a solution, particularly if the brain is tired and just wants to sleep. Have a hobby. Since math research doesn't exactly provide instant gratification, a hobby might provide some instead.  Talk to other graduate students. Realize that many graduate students go through this.   I'm interested to know how other folks have dealt with being discouraged as a graduate student. Does it get better with time and experience? Is this a sign that research is not for me and that I should seriously consider a life outside academia?
50126
I have started solving problems from the American Mathematical Monthly. While I will definitely continue this, I am not sure how much this will help for math graduate school admissions (assuming I submit my solutions and they are recognized by the Monthly). How significant would multiple solutions to Monthly problems be in admissions? What about having the published solution to a problem? One issue is that I don't know how difficult the problems are presumed to be: all I know is that I personally have solved at least one of them!
95205
[Context (Can be skipped)]  Hello, I am a grade 11 high school student in Canada. For most of my time as a student I have been pretty bad at math. It wasn't that I didn't like math or didn't have a natural aptitude for it, but rather had to do with a really bad education in elementary school that didn't get repaired until relatively recently.  Anyway, I decided to buckle down and fix the gaps in my knowledge during grade 10 which brought me from a 61% in grade 9 to a 75% in grade 10. I decided to go even further this year and pushed myself to a 92% in grade 11. Once again, I am looking to challenge myself.  [Question]  I want to have an advantage going into grade 12 by learning math from now until next fall. What subject should I learn and what resources do I need to learn it? I have gotten very good at self study, so that isn't an issue.  I have heard abstract algebra might be a good place to start. Is this true?  P.S I am in grade 11 and have not started any kind of calculus yet. My next course will be advanced functions and then calculus and vectors.
87063
I'm a computer science student who is a maths hobbyist. I'm convinced that I've proven a major conjecture. The problem lies in that I've never published anything before and am not a mathematician by profession. Knowing full well that my proof may be fallacious, erroneous, or simply lacking mathematical formality, what advice would you give me?
25294
I am a student in calculus II. I'm now failing tests solely because I cannot factor; I understand everything else. This is compounded by the fact it seems to exceedingly hard to find anything comprehensive online. This is a long-running problem I've been facing; I can't be the only one.  I know some methods, but I always run into more problems I can't factor. I know about synthetic/long division, binomial/perfect square trinomial factoring, difference and sum of cubes and squares, the grouping method, the quadratic formula (unsure how to use for factoring), guess and check, and some trinomial methods. I only know the difference/sum of squares/cubes well. For example, I can't factor $4x^2 - 4x - 3$.  It would be dandy if I could get a comprehensive list or a universal solution to all these factoring questions, but that's unlikely to happen (and would be too broad). I'll narrow this down to something more tangible: What are the methods I should know to factor trinomials?
105161
How is it that you read a mathematics book? Do you keep a notebook of definitions? What about theorems? Do you do all the exercises? Focus on or ignore the proofs?  I have been reading Munkres, Artin, Halmos, etc. but I get a bit lost usually around the middle. Also, about how fast should you be reading it? Any advice is wanted, I just reached the upper division level.
95985
I am hoping you can give me some suggestions. I am teaching in a very diverse (made of minority groups) college and the students are mostly Psychology majors. Most students are fresh from high school but some of them are older returning students above 40. Most of the students have motivational problems and aversion to math. But I am still looking for a book that covers the basic curriculum: from descriptive to sampling and testing all the way to ANOVA, and all in the context of experimental methods. The department requires me to use SPSS in class, but I like the idea of building the analysis in a spreadsheet such as excel.  p.s. the other teachers use a book that I don't like because of the extensive reliance on computational formulae. I find using these computational formulas - rather than the more intuitive and computationally intensive formula that is consistent with the rational and basic algorithm- unintuitive, unnecessary and confusing. This is the book I refer to Essentials of Statistics for the Behavioral Sciences, 7th Edition Frederick J Gravetter State University of New York, Brockport Larry B. Wallnau State University of New York, Brockport ISBN-10: 049581220X Thank you for reading!
122739
This is a question about studying for the Putnam examination (and, secondarily, other high-difficulty proof-based math competitions like the IMO). It is not about the history of the competition, the advisability of participating, the career trajectories of former participants, or other such things. This is a question about how one can most effectively prepare to do well.   Many such questions have been asked before on this site. The answers, while helpful, all seem to boil down to recommendations of the same standard canon of books with the encouragement to do more problems. This is very good advice, but I want to approach the topic from a different angle.   We have a wealth of talent on this website, and in particular many users who have done quite well at these competitions. I would like to hear their personal stories. In particular, how did you prepare, and what kind of time commitment did you put forth? And advice, especially practical study tips, is always welcome. :)  edit: Also, it seems to me that the majority of people who do very well at the Putnam had developed the majority of their skill in high school, and focused mainly on their classes in college (while attending whatever Putnam seminar their college offered). I would appreciate comments on this matter too.   (Motivation: It seems to me like the role of talent is vastly overestimated in mathematics, and in mathematics competitions in particular, to the point where the Putnam exam gets used as a sort of pseudo-IQ test. Of course the people who do well have gifts, but it also seems that, without fail, they all have a history of doing many hours of mathematics a day for years on end. For example, I recall reading an interview with Tao where he admitted his childhood consisted of nothing but math and computer games. I am trying to gather some evidence on this matter [and advice for Putnam preparation!]. Please keep the answers focused on the actual question, though.)
3142
Note: I will expand this question with more specific points when I have my own internet connection and more time (we're moving in, so I'm at a friend's house).  This question is broad, involved, and to some degree subjective.  (I started out as a physics-only student, but eventually decided to add a mathematics major. I am greatly interested in mathematics; the typical curriculum required for physics students is not deep or thorough enough; mathematics is more general (that means work!); and it only requires a few more classes. Naturally, I enjoy  immensely.)  This question asks mainly of undergraduate-level study, but feel free to discuss graduate-level study if you like.  Please do not rush your answer or try to be comprehensive. I realize the StackOverflow model rewards quick answers, but I would rather wait for a thoughtful, thorough (on a point) answer than get a fast, cluttered one. (As you probably know,  produces clear, useful writing; and a properly-done comprehensive answer would take more than a reasonable amount of time and effort.) If you think an overview is necessary, that is fine.  For a question this large, I think the best thing to do is focus on a specific area in each answer.    Update: To Sklivvz, Cedric, Noldorin and everyone else: I had to run off before I could finish, but I wanted to say I knew I would regret this; I was cranky and not thinking clearly, mainly from not eating enough during the day. I am sorry for my sharp responses and for not waiting for my reaction to pass. I apologize.  Re: Curricula:   Please note that I am not asking about choosing your own curriculum in college or university. I did not explicitly say that, but several people believed that was my meaning. I will ask more specific questions later, but the main idea is how a physics student should study mathematics (on his or her own, but also by choosing courses if available) to be a competent mathematician with a view to studying physics.  I merely mentioned adding a mathematics major to illustrate my conclusion that physics student need a deeper mathematical grounding than they typically receive.   And now I have to run off again.
35517
Also, I want to make it clear: Beginner.  I'm getting really frustrated trying to study for math competitions:   On the one hand, there are books teaching the high school curriculum, but that's it. I don't need these, I know (almost) everything on your regular high school curriculum. On the other hand, there are books for the IMO. But they assume you have a really strong olympiad mathematics background, and they are only there to prepare you for the IMO. They're basically composed of one or two lines of theory per chapter, and then a huge list of hard exercises and problems. I don't need these either: I'm not even near the IMO calliber, and the solutions to the problem seems to rely on theorems and techniques that aren't taught in high school.   What I am really looking is for a bridge between these two extrema: A book that makes a connection between high school basic math and IMO advanced math.  A book that covers  is what I am trying to find. I've never heard about any of these topics, though they seem to be essential for math competitions. Also, I've never found any book that covers them.   I'm not sure whether a single book will contain such a broad range of topics, so I believe that I'll need a book for Algebra, a book for Number Theory, a book for Combinatorics, etc.   But if that's the case, I also want to make it clear that I'm not looking for a book that teaches you college Algebra, college Number Theory, etc. I want something that covers topics that appear on a math competition, but doesn't assume anything other than high school maths.  Thank you in advance.
91631
I am working to get a PhD, maybe next year this time around. Currently, this is my second year. I have published 3 papers. All of them are not about one problem but consider different problems. My strategy to find a research idea/problem is given below.  1 - Read papers whose titles are either interesting or I have some apriori understanding of those titles.  2 - Understand those papers as best as I can. That is, I try to understand each detail presented in the paper except the simulation results (this is because I think simulation results are hard to reproduce).  3 - I analyze the paper and try to find a very small problem which can be addressed (i.e. I can solve it) by the methodology of the paper.  4 - If I am successful at step 3 I start working on that problem otherwise I repeat the process from step 1.  By applying this strategy I have been able to write 3 papers and I have submitted a fourth (I am not satisfied with those papers because at very best they can be considered average papers). But my concern is that the problems addressed in those papers do not belong to one area. As far as I have seen, for the profiles of some of the renowned researchers in my field their PhD work is focused on one area. One main thing I console myself with is that almost all of those researchers had good supervisors. My supervisor does not even read the draft/reviews of my papers and the suggestions that he provides during the meetings are so broad that the problem becomes intractable. Under these circumstances what are your opinions: Is there any better strategy to do research?   I will be very thankful to you for providing me suggestions or commenting on my dilemma.
37580
It's a question of a some time ago test, I've found a way to solve the problem using calculus, but always I've thought that exist a solution with algebra and geometry.  Thank you for your time.