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I love math, and I used to be very good at it. The correct answers came fast and intuitively. I never studied, and redid the demonstration live for the tests (sometimes inventing new ones). I was the one who answered the tricky questions in class (8 hours of math/week in high school)... You get the idea.

As such I used to have a lot of confidence in my math abilities, and didn't think twice about saying the first idea that came to mind when answering a question.

That was more than 10 years ago, and I (almost) haven't done any math since then. I've graduated in a scientific field that requires little of it (I prefer not to give details) and worked for some time.

Now I'm back at school (master of statistics) and I need to do math, once again. I make mistakes upon blunders with the same confidence I used to have when I was good, which is extremely embarrassing when it happens in class.

I feel like a tone deaf musician and an ataxic painter at the same time.

One factor that probably plays a role is that I've learnt math in my mother tongue, and I'm now using it English, but I wouldn't expect it to make such a difference.

I know that it will require practice and hard work, but I need direction.

Any help is welcome.

Kind regards,

-- Mathemastov

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Elsewhere: news.ycombinator.com/item?id=2423263 –  Charles Stewart Apr 8 '11 at 13:39

13 Answers 13

up vote 14 down vote accepted

I'd like to emphasize a remark that Eric made in a comment to his answer. Introspection is essential when learning mathematics - not only to analyze problem solving techniques - but also in many other ways. The web of mathematics is connected in many mysterious and marvelous ways. Spending a little effort attempting to discover these links can go a long way towards better understanding the essence of the matter. After you solve (or fail to solve) a problem you should spend some time trying to abstract a bit from the specific problem. Can it easily be generalized from a specific trick to a general method that you can add to your tools for later reuse? For example, if it's a problem about numbers can it be generalized to one about functions? If so, can the proof be simplified by appealing to properties unique to functions such as derivatives? For a simple example see here, or consider the trivial proof of the $\rm\:abc$ theorem for polynomials (vs. difficult conjecture for numbers), which proof exploits to the hilt the power of derivatives (viz. wronskians).

Eventually, with enough training, you will be able to effortlessly move back and forth between the general and the specific, and better recognize the essence of the matter when sizing up problems - the same way a chess grandmaster can evaluate a chessboard in a single glance. Don't be frustrated if this doesn't come easy - or if it has atrophied - because - just like chess - it takes much practice to remain proficient. Unlike vision, language, etc. these mental faculties were not programmed into our minds by evolution, so one must continually reprogram these faculties for other purposes - whether they be chess or mathematical reasoning (it's no accident that one frequently sees strong correlation between chess and math abilities - both depend strongly upon pattern-matching, e.g. see the famous studies by the psychologist de Groot: Thought and choice in chess).

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Thanks for your answer. Over the years, I've already explored a bit topology, abstract algebra and some category theory by myself, out of curiosity. I don't need to get back my ability to prove theorems, though, mostly calculus, linear algebra and the their illustrations in Euclidian spaces. Basic stuff, but I struggle. –  Mathemastov Feb 24 '11 at 21:37
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@Mat: If those topics were taught well, i.e. conceptually vs. rote, then it should come back fairly quickly, since you will already have the conceptual scaffolding in mind - which helps serve to generate the whole structure/theory. It's only natural to be very rusty after having not practiced for over a decade. –  Bill Dubuque Feb 24 '11 at 21:44
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Yep, I have the scaffolding, I had good teachers. I undestand my mistakes very well after the fact, although not always immediately, but I'm spontaneously dumb, there's no other word. Do you have any advice regarding exercises to get back in shape? –  Mathemastov Feb 24 '11 at 22:27
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@Mat: This forum has a large number of students who could probably point you at good resources if you ask a more focused question that will grab the attention of the appropriate folks, esp. mention calculus and linear algebra in the title, and that you are looking for resources for refreshing your knowledge. I seem to recall that some folks mentioned Schaum's outlines but I don't recall for which topics (the quality varies widely in this series - but it has been praised by some first-rate mathematicians, e.g. Rota) –  Bill Dubuque Feb 24 '11 at 23:11

I think the key to your problem is in your first paragraph. You say, "The correct answers came fast and intuitively. I never studied." This is the classic high school con that can lead one to doubt one's own abilities as soon as the going gets more challenging.

No matter what your abilities, to do worthwhile work in mathematics you will need to study and work hard. You say in your last paragraph that you know it will require practice and hard work, but I do not think you have fully taken this on board.

When you do, you will stop worrying about making mistakes. They are an essential part of the learning process and not something to beat yourself up about. Forget the belief, instilled in high school, that you should be able to come up with the correct answer straight away. Those questions were crafted to have short, snappy answers within the reach of rote learning.

Your mathematical career has now gone beyond this stage. So get down to study and make as many mistakes as you like on the path to a deeper understanding, and enjoy!

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I was a gifted, precocious child, praised for my wits and not my efforts. However, I've already been through university (up to grad school actually, even though I had to interrupt my PhD). I've been confronted to the chalenges you mention and know what hard work means. Whatever. I'm not worried about learning new stuff, but about what I've lost in intuition about stuff I used to know and apply without even thinking. I had mastered all the stuff I'm using right now. I still understand it, but I'm unable to spontaneously apply it correctly, which is at best frustrating. –  Mathemastov Feb 24 '11 at 21:16

One thing I remember realizing in high school was that I would often see something, or read something, and think to myself, "yes, that makes sense". But then a day later I wouldn't have a clue about it. That was my first introduction to the difference between what I later learned was "active understanding" as opposed to "passive understanding". It's something that is not obvious to anyone without a certain amount of introspection.

Another thing I learned (much later, in graduate school) was that if I wanted to learn something, it was a good idea to get at least three books on the subject. I always found that reading just one book, I would inevitably run across something that would stop me dead, completely unable to continue. But if I had one or two other books available, almost always there would be some alternative explanation that made sense to me and made it possible for me to go on.

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Yep, I've actually lost my active knowledge. Passive stuff is still ok, though. –  Mathemastov Feb 25 '11 at 20:27
    
Introspection + many references! +1 –  user2115 Jun 18 '11 at 16:42
    
What you’re saying (in your second paragraph) also has relevance to overcoming the language barrier, and so I’ve cross-posted this to the Esperanto-USA website. Here is the link: esperanto-usa.org/en/content/… –  Mike Jones Jan 10 '12 at 0:45

Interesting question. Not asking about mathematics as such, but about building up one's skills in a global environment suffering from a dearth of post-educational mathematical tuition for adults.

Mathematics taught in educational establishments tends, at least at the early stages of childhood, to focus on established formulae and already-solved problems which form the foundation of more complicated, less well-known mathematical works still being solved to this day. Other, higher levels of mathematics still have little to no common ground with what the establishment already taught you. They don't teach much about the Riemann Hypothesis and the significance of prime numbers in cryptography and national security in schools, for instance - and yet you see prime numbers around you every day.

Re-awakening your mathematical skills might take some time, because you may first have to define the /extent/ of your mathematical skills. If you have access to your old mathematics textbooks, dig them up. Look through them for the parts where you found a weakness in your methodology. Ask yourself why, for example, you may have found prime numbers, or factorials, or matrices, or calculus tiresome. Then go and delve into that tiresome topic.

Like exercising an old muscle you thought had gone to seed, you can exercise mental and cognitive faculties long since thought to have gone dormant and even atrophied through neglect. And you can only do it one way, just like working out: by going through that awful machine in the corner of the gym that makes your back hurt, rather than all the easy ones.

It will hurt, and it will hurt your brain to the point where you may virtually buy stock in paracetamol, but if you persist in your efforts, in the end you'll have an active mathematician's brain running in your head again.

And yes, to hear an indolent old wretch like me talk about exercise comes off as a supreme irony. However, only through the painful and rigorous exercise of your mathematical faculties can they spring back to life with renewed vigor.

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Perhaps of interest/relevance:

http://johnlawrenceaspden.blogspot.com/2011/01/effortless-superiority.html

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Thanks for this -- thought-provoking to see things from the point of view of one with such talent. –  Nick Wiggill Nov 11 '11 at 1:29

Revisit all of your High school books. It will be easier to grasp and remind you of many stuff you forgot.

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It would also be a fantastic way to lose all interest again... –  user89 Feb 23 at 16:49

Linear Algebra: Try MIT's OpenCourseware with the inimitable Gilbert Strang.

Calculus: Try Calculus Made Easy, available as a free and nicely typeset PDF.

I also second Khan Academy.

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Similar situation, except I didn't get as far originally - just a CS undergrad 13 years ago. Now trying to study machine learning (Stanford online class), and finding everything from Linear Algebra back to even simple derivatives and integration has atrophied immensely. All I can say is keep slogging away at it - if I compare how slow my brain felt three or four years ago to now, it's already night and day. It comes back, but it feels more like a gradual waking up again of the math parts of my brain than a sudden flash. Good luck! :-)

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I highly reccommend this book Concrete Mathematics: A Foundation for Computer Science. While the entire book is not that relevant, I think a few chapters (on Binomial Coefficients and Discrete Probability) can be helpful.

The book focuses on formula manipulation. It will sharpen your practical skills like coming up with proofs, simplifying expressions, etc. This book is suitable for independent study b/c there are exercises at different difficulty levels with explained answers. Give it a try!

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there are some great answers / advise here already ... what i would like to add is that it demonstrates, in my opinion the clear link between communication, speech, language patterns and numerical skills, that part of the brain that visualizes these patterns and helps you make sense of the answer ...

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I'm in my late 30s and still remember being crap at maths while at school (this might sound like cold comfort, but run with this a little) because it's like I said in a Yahoo Q&A some time ago now. a little each day will keep your brain sharp and active and you'll still be able to do a little of the advanced statistical mathematics. I used to be able to do long multiplications in my head (eg 1687x432 and come to the figure 728700, whereas the correct answer is 728784 as an example. manually calculating stuff like that isn't as easy these days) but I'd only be worried for you if you lost the ability to do the basics.

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I don't mean to echo @Bill and @Derek, but certainly the hard work of practice, and making many mistakes along the way is absolutely required to get back your mathematical skills.

I want to add to the discussion that you shouldn't underestimate the value of social learning. If you're having trouble with a proof (or even not having problems), grab a piece of chalk and work through the problem on the chalkboard with your fellow grad students. It's an exceptional way to learn. Having to vocalize your arguments really helps to formalize your thoughts.

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I think that having study groups, peer tutoring, watching youtube videos, iPhone applications, and just going through textbooks will serve you to increase your math skill strength.

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