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Self study method for Math

#1
I want to study math, and cover what you'd expect to see in undergrad. However reopening up a math textbook quickly made me think that learning following the textbook method would be bad. It reminded me of language textbooks that attempt to teach you by listing grammar rules and force you to create truth tables to ultimately write very awkward useless sentences. While math is axiomatic, I am looking for a better method to learn than memorize axioms, mentally fill out truth tables, and run through proofs multiple times before putting out some sort of solution. After seeing the efficiency of a method that works for me such as Heisig's RTK, I wonder if there is a better manner to learn math than the one taught at school. And so I ask, among you forum members, does anyone have a clue on the existence of such a self-teaching method?
Edited: 2016-03-24, 1:24 pm
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#2
I was doing a khan academy math course for a while and it seemed pretty good. Not a whole new paradigm in math pedagogy, but it wasn't a textbook and the lecturer did a good job getting points across efficiently. I liked that they have a phone app so I could watch a lecture anywhere I had a few free minutes. With math, I believe you should probably do a lot of problems and derivations, so aside from the lecture or textbook or whatever, you need a lot of problems to work through.
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#3
Seconding khan academy, it's a really cool resource and it's done with love Smile
Edited: 2016-03-24, 2:30 pm
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#4
(2016-03-24, 2:08 pm)yogert909 Wrote: I was doing a khan academy math course for a while and it seemed pretty good.  Not a whole new paradigm in math pedagogy, but it wasn't a textbook and the lecturer did a good job getting points across efficiently.  I liked that they have a phone app so I could watch a lecture anywhere I had a few free minutes.  With math, I believe you should probably do a lot of problems and derivations, so aside from the lecture or textbook or whatever, you need a lot of problems to work through.

I've done some more research and found Barbara Oakley has some good ideas on the subject. She learned Russian to fluency in the army before going on to STEM. She advocates a high amount of practice without letting your mind get caught up in needing to know and understand everything. That understanding is not a conscious process but a side effect from practice magic. Well enough with me twisting her words here's the article in question, and google can provide more (she's got an entire book on the subject): http://nautil.us/issue/17/big-bangs/how-...nt-in-math

Any ways I've decided to go forth by skimming through the lesson and working through the problem set, while checking back on the lesson for clues when I'm stumped. Upon your recommendations I'll check out Khan academy though Salman Khan's ugly profile pic has put me off of it for so long. Thank you for the help, and best of luck to all your learning endeavors.
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#5
TD;LR: There are great book resources for self study. Don't try to do problem sets without understanding the concepts behind it as it the foundation for solving those particular problems. Patrick JMT ftw! Finally, keep track of your errors.


[[Long answer]]
Since you are on the pursuit of self-studying higher level math, I thought this Quora question would be perfect:

https://www.quora.com/Electrical-Enginee...o-about-it

I can't speak for beyond the "Absolute Beginner" category but I have worked through 1/4 of Serge Lang's "Basic Mathematics" to better prep myself for calculus as the usual college courses weren't doing it for me (took college algebra 3x before I passed, dropped trig). It is safe to say that I am not a math person based on my track record but I have grown a sort of appreciation for it.

I know that my foundation skills are lacking so a few things have helped me which were reading the book "How to Become a Straight-A Student"by Cal Newport, particularly the section on technical courses. But if you want a summary of it, here it is from a med school blogger:

Quote:Construct a Mega-Problem Set (Technical)

Problem set assignments are key, so make sure you keep all the ones you accumulate during the course for studying later.

Make corresponding piles for each type of problem set
Supplement with sample problems from notes
Match the lecture notes to the problem set that covers the same material
Copy sample problems from notes to blank paper (don’t need to copy steps or answer, just questions)
Label the blank sheet with the lecture date (so you know where the problems are from and where to find the answers)
Put this with the problem/lecture note set you created in the previous steps
Augment mega-problem sets with technical explanation questions that ask to explain the basics of the topic (ex: “Explain the general procedure for drawing a molecular structure, why this is useful, and what special cases must be kept in mind.”). This will reveal whether you understand what you’re studying or not by forcing you to write out the basics.
If practice exams are available, copy them and store them with your mega-problem sets
http://hexaneandheels.tumblr.com/post/58...t-studying

http://patrickjmt.com/ is a riduculously great resource for "no fuss" math lessons as they are short and concise. He has videos for Algebra, Calc, Differential Equations, etc.  I only watch Khan lessons when I want more in-depth explanations.

I like to do Patrict JMT lessons first as a primer before tackling textbook problem sets. But of course, I do it in a MPS (Mega-Problem Set) format where I answer technical explanation problems FIRST (E.g. What is the formula for the "Quadratic Function"?/ What is the "vertex"?...etc) before going to solve the real math problems (E.g. Find the x-intercepts and vertex of y = –x2 + 2x – 4.)

Here is an example from a page of my MPS (it's pretty sloppy): http://imgur.com/IGsYnou

I also keep a separate notebook of problems that I had trouble with and note what I should have done to solve it correctly.
Cover of notebook (terrible kanji writing xD ): http://imgur.com/ukmtfr3
A page in the notebook: http://imgur.com/yd713Rx

Sorry for the long post but I hope it has been of some help to you xD
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#6
Thanks for all the links RawrPk, patrickjmt in particular seems to be a great reference sheet.

Quote:Don't try to do problem sets without understanding the concepts behind it as it the foundation for solving those particular problems

I've used this approach before but it did not give me results (flunked linear algebra twice, third time 's the charm, eh?), but I'll definitely keep it in mind; something you do suggest is keeping track of problems that give you trouble and I see you keep track of errors in your notebook. How has that worked out?

Edit: I do need to get my hands on some of that fine Japanese stationary
Edited: 2016-03-24, 3:49 pm
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#7
Quote: http://patrickjmt.com/

Holy crap that's a lot of math videos! Too bad he doesn't have playlists for each subject (eg. trig, calc,...). Looks like I found out what i'll be doing once I get more comfortable with Japanese. So much to learn...
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#8
Lol yea Patrick JMT is a beast! My friend who was taking calc 3 at the time recommended the site to me during my 3rd attempt of college algebra. He does have playlists on his Youtube channel but it's not as organized as his website so I just look through there.

(2016-03-24, 3:47 pm)probablyathrowaway Wrote: Thanks for all the links RawrPk, patrickjmt in particular seems to be a great reference sheet.

Quote:Don't try to do problem sets without understanding the concepts behind it as it the foundation for solving those particular problems

I've used this approach before but it did not give me results (flunked linear algebra twice, third time 's the charm, eh?), but I'll definitely keep it in mind; something you do suggest is keeping track of problems that give you trouble and I see you keep track of errors in your notebook. How has that worked out?

Edit: I do need to get my hands on some of that fine Japanese stationary

3rd time was definitely the charm for me in terms of college algebra. 2nd attempt I was reckless trying to take a 5 week course :/

In terms of concepts, they are imo important because in my experience, it makes the solving process that much faster. I would rather use my time applying the concepts as I'm trying to solve the problem instead of getting stuck.

E.g: Does f(x) =x^3 have an inverse function?

Concepts to think about:
  • What is an inverse function? A function that contains the same points as the original function but x and y points are reversed; inverse fxn =(y,x) vs fxn = (x,y)
  • How do you determine whether a function is one-to-one? By using the horizontal line test
  • What is the horizontal line test? What can be determined by its results? The horizontal line test is a horizontal line drawn across the function. If the horizontal line touches the graph more than once = NOT one-to-one; if it passes only once = one-to-one.

If I know these concepts cold, all I need to do is draw the graph f(x) = x^3 and draw a horizontal line to solve my problem.

Source: http://www.coolmath.com/algebra/16-inver...-to-one-01

As for the notebook, it helped tremendously! I have a habit of doing the same mistakes on similar problems and the notebook has significantly lessened it. It's also a good way to find new ways to solve problems. I tried to find the site I got the idea from but no luck. Instead, I found a site that made an "Error Analysis Sheet" which essentially stated the same idea of keeping a notebook but with page templates.

As for the notebook, there is a local Japanese import good store in the local mall so I got it from there xD but I have also bought a 10 pack of Campus notebooks from Amazon. My math errors notebook is the generic brand of Campus.

If I were to buy more Campus notebooks, I'd get the pre-dotted lined ones next time (I bought the regular lined notebooks) as they are amazing for math/science Smile. You can check out the notebooks on JetPens but they are on sale for cheaper right now on Amazon.
Edited: 2016-03-24, 5:13 pm
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#9
There are short videos like khan academy's, there are recorded classes like MIT OCW's, there are online courses like EDx/coursera, and there are books (and pdfs ;d).

Depending on what kind of math you want to learn and with that purpose you may one or other. Books are not bad at all and actually I think that's the best to learn undergrad math you just need to find the book with the same objective as you.

Btw I'm a mathematician and I don't remember almost anything by memory neither does someone who is half decent in math, you just have to understand a handful of things and then you can deduce the rest. it's all about understanding Smile
Edited: 2016-03-24, 5:32 pm
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#10
Watch videos from Khan Academy (though I found their calculus videos somewhat lacking) or similar if you need help understanding the math, but the best way to learn math is to understand what you're learning, which means spending however long it takes to go through the derivations of formulae. There's a reason calculus classes start with limits and the limit definition of a derivative: understanding why you're doing something in math is fundamental to successfully applying it.

After understanding what it is you're doing, do practice problems. Do not simply go through other people's work! You will only understand and remember what and why you're doing something if you do it yourself. Only watch someone else go through a problem if you can't understand what's not working, and even then, you should redo the problem yourself or do another, nearly identical problem in order to ensure that you can do them yourself.

And yeah, there's not really any memorization required, as far as I've studied (multivariable calculus; I'm not a mathmatician), except for basic derivatives and integrals (and that damn unit circle), everything else is fairly easily derived.

Find a good old book. Good, because you'll be using it a lot; old, because you'll probably be able to buy it for a few dollars as opposed to a couple hundred.

I'm an engineering student (starting my junior year soon, so mostly just more physics applications, labs, and projects left), in case that's relevant. If you're ever learning physics, the above applies just the same, but with fewer practice problems (because there are only so many hour long problems you can do in a day...)

EDIT: Of course, after I wrote this post, I looked back and saw that you're much further in math studies than I am... Whatever, applies just the same: understanding --> practice --> better understanding.
That applies to every discipline based on logic; you simply can't learn well using techniques you don't understand.
Edited: 2016-03-24, 11:09 pm
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#11
I don't know if this will be useful for you or not, but one thing that has always helped me as a study skill for any subject really, but math in particular, is to look at the solution and then work backward until I understand it. In this regard, I think some teachers are terrible when it comes to not showing enough examples and then just throwing homework at you. The correct answer should not be what is graded, rather the thought process and steps shown to get the solution should be. That takes a lot of time to grade, though.

Along the same lines, Wolfram Alpha can show you step by step solutions if you are a premium member. Back when I took calc and linear algebra it was invaluable to my studies. You could (at the time) find a new trial account for premium on BugMeNot every week so you didn't even have to register.
Edited: 2016-03-25, 2:22 am
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#12
Quote:If I were to buy more Campus notebooks, I'd get the pre-dotted lined ones next time (I bought the regular lined notebooks) as they are amazing for math/science Smile
Pre-dotted lines is something I didn't know I needed Tongue

Quote:Of course, after I wrote this post, I looked back and saw that you're much further in math studies than I am...
You give me too much credit I've only completed first year calculus and a proofs course. I'll be looking around for some textbooks at thrift stores after the Easter weekend, something nice and cheap that I wouldn't mind drooling on, scribbling on, and treat like trash (broken windows theory anyone?). Older textbooks are also less likely to downplay the rigor needed in solving problems, and I enjoy their frankness. You're an engineering student? Have you checked out Factorio, great little game: http://imgur.com/a/xcYxk

Quote:Btw I'm a mathematician and I don't remember almost anything by memory neither does someone who is half decent in math, you just have to understand a handful of things and then you can deduce the rest. it's all about understanding.
That's comforting; I find  it charming to see mathematicians reconstruct a proof to a theorem rather than checking a reference. I'll keep that as a benchmark for learning.


Quote:Along the same lines, Wolfram Alpha can show you step by step solutions if you are a premium member. Back when I took calc and linear algebra it was invaluable to my studies. You could (at the time) find a new trial account for premium on BugMeNot every week so you didn't even have to register.
Yes! I used to be part of the BugMeNot Wolfram Alpha crew, if anything it was a great way to check that the method that led to the solution was correct. I'll been waiting for a Free as in Freedom (though I do appreciate the free drinks) alternative, though no one seems to be working on it.
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