Is There a Better Way to Learn Math Efficiently?

[babysnatcher]

seems like the only way to accurately solve any problem is to go through every problem scenario. is there anyway to solve any problem without going through every little thing? like when the integrand is (1-x^2)^(1/2), i think it would take like a whole day for me to figure this one out without help. so I am just doing all problems i can and when i get stuck, i just ask for help and memorize the procedure and reapply it. i don't think memorizing all these procedures is the right way. i forget them like a month later. if this is the way to learn, then how do i memorize all these procedures? there's got to be a better way! is there?

 

[leehufford]

Maybe you could ask your professor to explain things in a way that you are more likely to remember them. That, with practice, practice, practice should help.

 

[AlephZero]

You won't succeed by trying to remember lots of unconnected "facts" about math. You need to see how things are related. For example your $\sqrt{1-x^2}$ problem should make you remember that $1 - \cos^2\theta = \sin^2\theta$.

At the level of a first calculus course, you can write down everything that you NEED to remember on one sheet of paper. Then you need lots of practice in using that small amount of information to solve problems.

 

[20Tauri]

When you're taught a strategy, say, trig substitution like AlephZero's suggesting you use here, you do a bunch of problems on it until you get familiar with situations where you use it. Then you learn other techniques, like u-substitution and integration by parts, and if you're still having trouble knowing when to use which, just compile a bunch of integration problems and work through them until you get better at identifying which technique to use. WolframAlpha's Show Steps option might be helpful if you get stuck. Additionally, there is kind of an order of integration techniques from "nice" to "only use if all else fails," so if you're not sure which to use, that might help you out. This is not an official thing, and everyone will vary as to which they like or dislike more. Mine personally goes u-sub, partial fractions, by parts, trig sub, although I think it's less that trig substitution is hard than that I've not had to use it in a while and have forgotten how it works.

Sometimes, you won't know the best way to solve a problem, and you'll just have to try everything until something works. You get better at it as you do more and more of a problem type, but I wouldn't worry too much about being "efficient." If they wanted efficiency, they'd run everything through Mathematica or another scientific computing program. What they want is to teach you problem-solving skills.

 

[homeomorphic]

(1-x^2)^(1/2) makes me see one of the sides of a right triangle, by association. I go from there. This isn't just a straight mnemonic trick, either. Once you draw the triangle I have in mind, you already have part of the solution.

Here's a more conceptually shallow, but useful example. The way I think of integration by parts, what I do is I "throw the derivative" from one function to the other and add a minus sign. There is another term in the equation for integrating by parts, but it can be ignored because it's generally easy to evaluate, so it is not part of the difficulty of solving the problem. So, when I am integrating f g', I throw the prime to the f and add a minus sign and I get - f' g. That's what really tells you how to use it. To get from g' to g, you integrate, and obviously, you differentiate to get from f to f'. So, even though this is purely formal symbol manipulation, you sort of feel like you're more involved somehow--you're throwing the derivative, rather than just using a formula that someone handed to you. Slightly less mechanical.

One thing I observed in my undergrad was that I forgot a lot of the equations in my physics class, but I remembered most of the pictures and things that were in my imagination.

The other part of it is what AlephZero said. You remember things that are connected to each other much better than separate facts. One thing triggers the other.

 

[babysnatcher]

I want efficientcy for myself. I don't mind sitting all day doing the problems but why dwell so long on something when I could be learning other things in the sections ahead. I want to be efficient so I could learn it all fast and properly. I also want to know how long I should dwell on something before asking for help. How do I know when I'm learning? I have a very good memory and I abuse intuition. Its hard to tell when I really know something or just memorized it and have taken the steps that make sense rather than use the proper step by step logic. I only use logic when I have no other choice but those problems rarely occur. I think the book I'm using is pretty watered down. Anyways... I just want to learn as much as possible in little time as possible. Right now I'm doing 1-99 odds in every section. When I get stuck for 15mins with no progress, I write down the problem number and ask someone later. Should I be doing this? Is it proper? Is it the best method? I've heard stories of people learning everything on their own. I would prefer that but not if its going to take too much time. Whatever is most efficient. I abuse intuition because its fast. Id rather just ask someone and memorize the procedure rather than spend a whole day on one problem. I understand the logic when it is explained but I just can't come up with these problem solving techniques for every case by case specific problem. Is math part logic and part mindless repetitive computation? Can I get by with just the logic?

 

[babysnatcher]

Oh... And People tell me to just keep doing what I'm doing and just learn "what's really going on" later but I want to know now. I have an easy teacher right now but I want to prepare for those teachers that constantly throw curve balls on the exams and then tell me "you have to learn the concept and be able to apply it to any problem." Its trippy. In my opinion, some of the stuff they want us to realize on the spot during the exam is a little too much. Like they want you to make specific connections that were never made during the homework. Same concept but different problem. Like using the concept in ways that weren't obvious. Idk what to do. All the knowledge is in my head but I just can't make the connections by myself when they throw a specific case by case problems on the exam. Like... Problems like #99 in a section. There is only one of that problem type so I can't practice my way to recognizing it. Then 8 or 16 sections later, I've already forgotten how I solved problem #99 from 8-16 sections ago and test is coming up on 2 chapters. Ah. Here is an example: section1= extrema, section2=mean value theorem, section3=increasing and decreasin functions, section4=concavity, section5= limits at infinity. We were to be test on all these sections so naturally I trained myself extensively throughout all the sections so I know how to sketch a graph. Now... This came up on the exam for this chapter "Two cyclist begin a race at 1pm at the same time. Both cyclist finish the race at 4:15pm at the same time. Prove their velocities were equal at some time during the race." The solution wasn't by mean value theorem. I forgot how she solved it but the point is there wasn't anyway to see this problem coming and get extensive practice in it. It was similar concept but different procedure than the problems in all these sections. Way different. I never seen the procedure she used but I understood everything. I just want to be able to solve anything. I can't always rely on extensive training because most of the problems in the book are straight forward or intermediate. There are few difficult problems in the book. And then if I don't get a difficult problem of the specific case that the teacher throws on the exam then I am just helpless. So what can I do to prepare for all these curveballs?

 

[chiro]

Hey babysnatcher and welcome to the forums.

I don't think I can give you the panacea, but in terms of understanding the best I way find is to forget the symbols, equations, identities and step back for a moment and ask 'what is going on here'.

Put it in a language that you are good with in terms of comprehension, analysis, and ultimately understanding. The symbols itself are just symbols if they carry no meaning, and the meaning is the most important thing.

This is not quite a straight-forward thing to do, but one thing you can do to a trained person who has spent a bit of their own time on a subject is to ask them to explain something in layman terms with respect to what a particular topic or subject is all about.

This is not a stupid question: far from it. It is the wisest question that will force someone to compress their understanding of something into a form that is simple, comprehensible by a novice or similar, and it forces them to use a language that is non-specific and without jargon.

If they can't actually do this, then they don't really know what they are teaching.

 

[babysnatcher]

Also. I'm looking for a better way to learn everything perfect because people speak so highly of learning the step by step logic, learning the concept and what's really going on. They speak of it as though that's all you need to get 100% on an exam with little to no trial n error practice for physics n math. And in this way you will be able to solve any problem with just basic practice problems. I don't see how this is possible. Confused. So confused

 

[DragonPetter]

Also. I'm looking for a better way to learn everything perfect because people speak so highly of learning the step by step logic, learning the concept and what's really going on. They speak of it as though that's all you need to get 100% on an exam with little to no trial n error practice for physics n math. And in this way you will be able to solve any problem with just basic practice problems. I don't see how this is possible. Confused. So confused

If it helps unconfuse you or reassure you, I had to do a lot of homework problems and examples to get good at my math courses, and I have to do more whenever I need to remember what I forgot. I can remember a lot of concepts and topics, but when it comes down to actually solving a problem out of a book, I often will need to look at some examples or work on one for a while before I remember how to do it. This actually happened to me recently when I was tutoring someone on parametric equations: I could tell them all about them and how they work, but, when we looked at a real problem, I needed to take a few minutes to remember how its done. You won't be able to do that in an exam, so its good to do the homework problems.

I actually find going back on old problems that I haven't studied in a long time and solving them on my own to be very insightful. When you are first learning the material and have deadlines for homework and exams, it can be really easy to miss the fine points or the important things since you're so focused on being able to solve as many problems as possible in as short a time as possible.

I don't know what its like for extraordinary students though. I'm sure we all need some exercises, but some need more than others.

I think no one can solve a problem without attempting examples, and the more you attempt the better you will be at solving those problems. As an analogy, no one can read the postulates of quantum mechanics and then derive the ladder function for energy states or anything else like that just from knowing the postulates. They actually need to do a lot of calculations and eventually come to the point of getting answers. Similarly, in electronics, knowing Maxwell's Equations will not get you very far to actually solving circuit analysis problems on your first attempt, even though, theoretically, that should be all you need to know. It seems that you are under the impression that knowing the basic rules of math should be enough to solve problems. I think, historically, a lot of those rules came after a person had figured out how to solve problems, not the other way around.