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So after failing another math test today I have realized that I can't do math on my own. For whatever reason when I get to a test I just can't do math without extra resources. I think this is because when I do homework there are many problems that I just can't do on my own so I need help or I have to look at the answer. Once I get the answer I try to work it out again, and then when I go to review I make sure to go over those problems again. Unfortunately I think I am just learning how to do those problems and not learning how to do that kind of problem.

How do I stop myself from learning like this? I am not meaning to just learn processes to do math homework, I am trying to learn the math but it seems like I am not actually learning anything when I study.

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Perhaps you should be a bit more specific. Is it this specific class or is it an overall thing? (calculus? algebra?) Is there a specific aspect that's causing this trouble: "I cannot do algebraic manipulations without mistakes", "I do not know which trigonometric identity to apply", ...? –  Srivatsan Sep 26 '11 at 20:49
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Actually, we've already pointed out your algebra handicap a lot of times before. You seem to have the calculus down (more or less), but are always tripped up by the algebra. You really need to go back to your roots, so to speak... –  J. M. Sep 26 '11 at 20:55
    
Looking at the answers is always tempting. But it really does keep you from working through and understanding a problem. Resist the urge and move onto another problem in a set, and then another problem. Don't check your work until you've attempted several problems. Resist the temptation my friend! –  SamtheBrand Sep 27 '11 at 14:04
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Jordan, I know this is an old thread, but I don't think a comment bumps it, and it made me think of a piece of advice a math professor gave me which made a huge difference in my success on tests: Don't study with your book open. Take practice exams, problems copied from the book, whatever it is you will have to do on your actual exam, and go to the library with them and leave your book at home. Try to do them without ANY outside help whatsoever. Very often when you study sitting next to book or computer you use them as a crutch. Crutches are not available during your test. –  user23784 May 4 '12 at 3:22
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I know finals are likely approaching, so hopefully this makes it to your inbox, or someday someone sees it and benefits. I really, really, REALLY recommend following it though. When you sit down with no outside aids you will realize how much more difficult it is to do the problems... and of course, that is the situation you are in when being tested. –  user23784 May 4 '12 at 3:23
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7 Answers

up vote 31 down vote accepted

Jordan

Maybe what you need is a coach or a mentor. If you don't believe you can do this yourself, you never will do it. That happens to most of us in Maths at some stage. But if you have someone telling you the answers or "how to do it", you'll never actually learn how to do it for yourself. I used to sit in front of problems until I could solve them (I was bloody-minded in that kind of way). My dad could have helped me if I'd asked, but he knew I had to solve it for myself.

A couple of years ago I helped someone who hadn't even got their basic qualification in maths and had failed two or three times and was on her last chance. Basically I showed her how, if she'd put all her knowledge together, she would have passed every time - how she did know what she needed - nothing but confidence, actually. She flew through well ahead of any marginal grade (and someone who hoped she'd fail, and didn't believe I could help her in the time available, had a problem to deal with).

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I am still failing everything I attempt. When do I know when to quit? Eventually I will run out of money taking this class over so many times. –  user138246 Oct 5 '11 at 21:28
    
Hi Jordan: as the message said you need a mentor or a coach. I can coach you if you like message me here i think khanacademy.org may be a good solution for you. –  Ahmed Masud Jan 2 '12 at 0:54
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There are perhaps 4 distinct stages in Mathematical understanding, and a lot of people never get past the first (and indeed may never need to).

The stages are:

  1. Applying specific recipes to solve specific problems.
  2. Analysing problems to work out which are the right recipes to apply.
  3. Analysing recipes to understand why they give the right answer, and proving that they will always work (or alternatively, understanding their limitations).
  4. Exploring interesting (often artificial) scenarios to see whether it leads to anything interesting. Sometimes this might lead to the creation of new recipes, or even whole new fields of mathematics.

The point is that if you're limiting yourself to solving specific problems, then you are missing the stuff that really makes maths interesting – and it is no wonder you are having trouble retaining information. Why these things work is far more interesting than what steps you need to follow to reach the solution, and I found it much easier to remember.

Make sure you read around the topic in your text books, and understand the reason for stuff. Work out reasons and connections – make up stuff for the time being if you need to – make up crazy poems or rude acronyms or whatever you need to do to make connections.

A good teacher can help – but so can a site like this...

I remember the relationship between Sine, Cosine and Tangent because of a rude poem about one of my teachers, that the teacher himself taught me. It isn't really something I can post on the internet under my own name, though! I learned it 30 years ago, and I still remember it.

I know other people who simply recite S-O-H C-A-H T-O-A.

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Heh, reminds me: I was stuck with that "sohcahtoa" business up until I learned coordinate geometry... –  J. M. Sep 26 '11 at 21:51
    
One time after a trigonometry test in high school I saw that someone was attempting to cheat by writing the mnemonic on their desk in secret. Unfortunately, they wrote "SOH COH TOA." Oops. –  fluffy Sep 27 '11 at 0:16
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"Some Old Hippie Caught Another Hippy Tripping On Acid" –  Austin Mohr Sep 27 '11 at 0:45
    
Yeah - my poem began "Silly Old..." and went downhill rapidly from there! –  Bill Michell Sep 27 '11 at 11:18
    
I can never remember things like that, I just learn things by practicing them and then I forget them as soon as I stop practicing. –  user138246 Oct 7 '11 at 3:20
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I believe that all the above answers are quite splendiferous. (I do not use that adjective lightly. :)) I'd like to provide my own input, though it might be slightly Socratic:

What do you care about? Ask yourself this and answer honestly. Does mathematics happen to be one of the things you care about? Or do you find mathematics to be a torturous beast that has no soul? Hyperbole aside, this matters.

It is very easy to be good at math even if you don't care about it. That applies to all subjects, really. Nearly anyone can sit down, go over a text, and learn the associated processes and patterns of it. But, here is my point: It is so much easier if you care about it.

I can't convince you to do this; it is your own task. But let me clarify that caring about an exam grade or your math grades does not necessarily mean you care about math itself. To care about math involves doing things that most people think are crazy or odd. For example, I spend my spare time doing mathematics and felt literally sad when I learned that $\int x^x dx$ cannot be expressed in terms of elementary functions. Do you have this same amount of interest in mathematics? Note that I'm not saying you have to be like this, I'm just saying that it helps immensely to actually care about what you're learning.

Now, I would not say I am near the caliber of the majority of the users here, but this is what I have learned about learning math so far:

  • Be patient. You can't learn everything at once. Even Goethe, one of the smartest humans to live, was aware of this fact. (He made a play, Faust, that expresses this fact.) I like to take pointers from people like him.
  • Don't lose hope or give up early. I've realized that it's not my intelligence that causes me to be decent (I don't really think I can call myself "good") at mathematics, but rather my perseverance. I literally spent two weeks trying to solve one problem and I failed. But the point isn't that I failed: It's that I tried until I could do no more. I exhausted every thing I could think of and then acknowledged that this problem was simply beyond my current understanding. I like Edison's quote along these lines: "I have not failed. I've just found 10,000 ways that won't work."
  • Learn at your own pace. I can't emphasize this enough. I had a really interesting experience a few months ago with a friend. I was showing him how to solve quadratic equations. I tried to show him how to use a general approach and see the inherent process behind why a particular method works. This failed miserably. But, I did learn something: This was beyond his pace. So, instead of using this method, I illustrated it with concrete numbers. Then, I showed him how the general method is underlying these numbers. He had a lot of trouble with the general idea, but he did eventually solve it. Let me tell you, though: This wasn't a stupid person, by any means. He just hadn't learned this yet.
  • Don't take your grades too seriously. One of my favorite people in the world and one of my greatest teachers told me this, "It's just a number. You can't reduce a person to a number." This is one of the wisest pieces of advice I have ever heard. But be careful: My teacher here was not saying that grades are irrelevant, he was saying that your grades should not be the Gospel. They shouldn't be definitive concrete that roots you to a particular level. Too many people fall victim to this. I see it everyday, sadly.

Now, let me try my best to give you some specific and exact advice:

  • Pinpoint precisely why you fail. I'm not saying you are a failure, or anything to that extent. I'm saying that you, as a human (like us all), fail to some extent. Everyone fails somewhere. What's important is figuring out where, exactly, that you are failing. You cannot fix a problem without knowing what it is. For example, I struggle with English and particularly grammar. It pains me. But, once I realized where I was failing, I did a lot better. I realized, "Wow... My understanding is so elementary and inaccurate that it's no wonder I can't understand what a prepositional phrase is." I realized that, the very base of my understanding was horrible and inaccurate. I also noticed that I was failing because I was trying to put out a fire with an ice cube. My method of learning grammar was inherently lacking: I tried to understand the concepts of grammar based solely on their definitions and how they function, rather than trying to understand them intuitively and inherently by analyzing them in specific sentences. I learned that, to my surprise, a word's function and classification depends entirely on its use in a sentence.
  • I looked over a few of your questions and I think your issue is that you've skipped into waters too deep without any way of floating. That is to say, I think your class(es?) have went beyond your mathematical understanding. They are not at a comfortable level. Don't be embarrassed or upset over this: You can fix this. It has already been suggested, but I'd really recommend khanacademy. I recommend taking a week or as long as you need to work past the elementary exercises involving addition, subtraction, division, etc. It is best to do an exercise until you get a streak, which usually tells you to move on. I personally would not focus on the word problems, but that is a choice of preference. If you need help with any of the exercises, there are up to three videos explaining the concept. I find Sal to be impressively great at illustrating complex ideas. (It may help to learn about the concepts you're covering in class by using khanacademy while you work your way up to those concepts as a side project. That may complicate things, but it should help.)
  • Consult Wikipedia profusely. This is something that has helped me greatly in some of the most bizarre ways: I learned about Louville's proof involving the thing about integrals like $\int x^x dx$. Now, I in no way am at that level of mathematics to fully understand what Louville was doing, but I get the point he made. (There's a subtle difference between those two: If you fully understand, you can reproduce the result yourself. If you get the point, you can use the result to your advantage literally or as a supplement to how you view other concepts.) P.S. Don't be surprised if Wikipedia makes you feel stupid. It does this to me all the time. There is a profuse learning curve to the mathematics on that site, but you can get bits here and there that make a significant impact on how you understand things. Even one sentence can enlighten you a lot sometimes.

I have wrote you a book that could be even longer (I apologize if I am in any way verbose, but I have written this all in my best attempt to help), but I'd like to end on one final point:

  • Participate in the lovely community here. I've been using this site for a few days now and I find that it's one of the most interesting ways to learn about mathematics. Helping other users here is also a great way of enhancing your own understanding. Be careful, however. Do not try to answer a question unless you feel comfortable with what you are submitting. That is, you would put this confidently on an exam if the question came up in school. At the same time, try not to strain the denizens here with too many questions. Try to give as much as you take. Now, I have a feeling this is difficult at your current stage, but try to follow this guideline as much as you can. Hopefully the users here will be understanding about this fact and help you. They seem to be doing that, judging from this question and others. :)

Again, sorry for the book.

EDIT: I now see that I am quite late to answering this question. (I have a horribly bad sense of time. . .) I hope that is not too heavily frowned upon and Jordan actually has a chance to benefit from this.

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Thanks for the comment. I am still not sure why I am not learning though, I try but I retain nothing. I took 6 weeks off and went over college algebra, khan academy and such and I still struggle with algebra. I might not even be able to do school anymore, I am not sure, my grades are pretty bad and might not be recoverable. I was just trying to learn math but colleges don't care about that, I ruined my grades in the process and will likely have no choices of schools to transfer to. –  user138246 Jan 15 '12 at 6:33
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Don't take this dramatically, but is there any way that you could take a break from school entirely? I don't know of your age, so I am potentially offering horrible advice. It's an option to consider: Find a way of producing your own income (if you don't have this way already) and take a year or so off. Use this time to study math casually and for fun. Try to study what you need, but also what you find interesting. Also, there is a lot of unknown factors here, so I can't say much. I don't know how you do in other subjects or why you are in college (i.e. what you're majoring in and plan to do). –  000 Jan 15 '12 at 6:48
    
I am old enough where I do not want to take time off, I feel like I would just be wasting time. Everyone says to take a break but I have breaks between semesters and I try and study math but I really get no where. –  user138246 Jan 15 '12 at 6:55
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Try looking at this from an almost opposite angle: Since you are having so much trouble finding what works, ask yourself what has worked? What have you done, even if very insignificant, that has helped you? Try to do those things again. Try to do things like them. I'm sorry my tone here is very general, but sometimes it helps to ask yourself questions that may seem useless or obvious. –  000 Jan 15 '12 at 7:21
    
Honestly I feel like nothing has helped me learn math, my whole math knowledge is just non-existent and I constantly forget everything that I learn. I have taken college algebra several times and I doubt I need classes more basic than that. –  user138246 Jan 15 '12 at 7:28
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There is a book which title is «How to solve it» by G. Polya. It tries to understand the process of solving problems and its typical mental operations. You can download the entire book as pdf.

From that book:

First. You have to understand the problem

Second. Find the connection between the data and the unknown. You may bo obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution

Third. Carry out that plan.

Fourth. Examine the solution obtained.

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I've not read that book, but informally this is the same process I always use. Works wonders for developing a deep understanding of whatever system you're working within. –  Jordaan Mylonas Sep 26 '11 at 22:20
    
I have a step where I can't make any progress, keep trying and fail to make any progress than get frusrated and have to give up. For example I spent all day (yes, all day) working out maybe 6 simple math problems. –  user138246 Oct 7 '11 at 3:21
    
I gave this book to a highschool student I was tutoring and it did wonders for him. –  Nick Alger Jan 15 '12 at 6:39
    
What is the correct link? This serl.iiit.ac.in/cs6600/book.pdf returns a 404. –  Jeff Feb 15 '12 at 1:34
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About your question: I think there is nothing wrong with looking at the answer and asking for help /after/ you've attempted the problem yourself and thought about it extensively. I find that when I do this, and finally see the correct answer (be it calculation or proof), the solution reaally sticks. Unless you're extremely skilled (and clearly skill can only be achieved through a lot of practice), nobody's expecting you to be able to solve all the problems in your textbook on the first try.

As a fellow student, to remedy my weaknesses in a subject, there are usually four things I do.

  • I simply solve a lot of problems. If I can't solve a problem, I try my best to understand the solution and the general solution pattern of the problem type. I believe practice makes perfect in mathematics.
  • I try my best to ask a lot of questions. Of the lecturer, of my fellow students, basically anyone available. I can't overemphasize the importance of this. If there is something I don't understand, I note it down, and usually approach my lecturer when he has time, so he could explain things to me thoroughly. I prefer private sessions where his attention is focused on me, because I find that when asking questions in class I often get "half an answer" (and understandably so, we need to get on with the material).
  • I carefully think what exactly are my difficulties in the subject. Why am I having a hard time understanding, say, multivariable calculus? Is it because the explanations in class were unclear? Maybe my foundation in single-variable calculus is not strong enough? Maybe I just haven't had enough practice?
  • I look for supplemental material. Sometimes the way things were explained in class simply aren't enough for me. There are a lot of excellent, free resources online that can enrich my understanding.
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Regarding the second point, I will just mention that Math.SE is a great place to ask questions too! –  Srivatsan Sep 26 '11 at 21:36
    
Absolutely! ;-) –  josh Sep 26 '11 at 21:41
    
Most of the time I am so far behind in class it feels like that it would be embarassing to ask a question. For example today the instructor wrote on the board $x^5$ and said that it would be easy to find what would give us that derivative, he asked if anybody knew and someone said something close to $5/x^5$ i think and that just blew my mind, that said it almost instantly. For about 10 minutes after I tried to figure it out and couldn't do it in my head at all. –  user138246 Sep 26 '11 at 23:14
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I think you'll find there are a lot of people in your class that are in the same situation you are at, all of them too embarrassed to ask. I often ask "stupid" questions, it's nothing to be ashamed of! Your instructor will be glad to clarify things for you in class and not on your exam paper ;). That being said, you can always approach your instructor privately after class, or even ask here, if you don't feel comfortable about asking in class. –  josh Sep 27 '11 at 0:25
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By the way, the expression that will give us a derivative of r^5 is (1/6)*r^6. This is because the derivative of r^6/6 is (1/6)*(r^6'), that is 1/6 times the derivative of r^6, which is 6*r^5 according to the known formula: r^n'=n*r^(n-1). The reason your classmate answered instantly is most likely because he has practised taking derivatives a lot, thus remembered the derivation formula. I refer you to my first bullet point: practice makes perfect! –  josh Sep 27 '11 at 0:27
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I agree with all the above, Math is just a bunch of rules and understanding how to put the rules together to solve the problem. Most rules have to be done in a certain order.

-The thing that makes these rules stick when you do the test is every problem you do write the base equation down. Even write the rule you use for each step in the margin or at the top of the page. This will help commit the rule to your memory so when you take the test you have already written down the "lookup" many times it will be more like hand memory.

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-1: Math is not "just a bunch of rules". See vitalik's answer; or stages 3 and 4 in Bill Michell's answer; or any math book with proofs; or here. –  Zev Chonoles Sep 27 '11 at 3:01
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@Zev: You're right. It's a non-empty set, together with a bunch of rules! –  The Chaz 2.0 Sep 27 '11 at 3:41
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@Jordan May I suggest that you stop solving problems altogether? I think that helps a lot. Problem solving might have its advantages late in the career but when beginning I find it to be completely counterproductive especially when it is too much. I think it is much more important to be be able to read the standard mathematics texts first - line by line - make sure you understand every line written in the text. Just doing a hell lot of problems is hardly going to teach any mathematics. It can help to get the subtle points across ONLY if you have first read the text thoroughly. By standard texts I mean things like - lets say Artin for algebra or Apostol for Calculus etc. (I am not sure as to what is your level)

OR

Many good students are known to take the approach of reading the text itself like a problem. They basically read the stuff written between the proofs and then they try to prove the theorems on their own. If they can't only then they might look at the "standard" argument. This can be an exciting way to do stuff unless speed of learning is a concern.

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Being a student, I don't think his teachers would be down with "May I suggest that you stop solving problems altogether?"... (see his previous questions, for instance) –  J. M. Sep 26 '11 at 20:58
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I must say, I have never heard anyone give the advice that one should stop solving problems in order to learn math effectively. Seems rather nonsensical to me. –  ItsNotObvious Sep 26 '11 at 21:07
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Judging by Jordan’s questions, the course is a pretty standard first-semester cookbook calculus course; proofs are very unlikely to be an issue. Part of the problem is weak algebra; another part seems to be failure to recognize common patterns. Stopping solving problems is very unlikely to help with either of these problems. –  Brian M. Scott Sep 26 '11 at 21:11
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Based on Brian M. Scott's comments and my own (limited) observations, I suggest that Jordan get a copy of the Schaum's Outline Series in College Algebra (a print version -- the digital version omits many of the worked examples according to amazon.com reviews) and, beginning with page 1, systematically work through all the solved and unsolved problems. amazon.com/dp/0071452273 –  Dave L. Renfro Sep 26 '11 at 21:36
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While the author's opinion might not be popular, based on my knowledge of the OP, there is some merit in it (when interpreted as a temporary way to get past roadblocks). Generally one needs an appropriate balance of said line-by-line reading of textbooks and problem-solving, lest one lose sight of the forest for the trees. –  Bill Dubuque Aug 31 '12 at 14:39
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