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I minored in Math in college (B.S. Software Engineering), and went through advanced Calc, Differential Equations, etc. But about 6 months after I graduated I had lost all my formulas. Now, about 5 years later, I look at my class notes and have no idea what I was writing.

How do you re-hone your skills after years spent away from formal coursework? Are there resources you would recommend for daily puzzles, or decent refresher books, etc?

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by answering lots of questions on the math stackexchange! –  user58512 Jan 28 '13 at 18:11
Did you learn only formulas and skills in your math courses? If so, I'm not surprised that you forgot them. It seems lots of students do only that, because that's what's on tests and they're only there to get grades. Understanding of mathematics relies heavily on formulas and skills, but doesn't consist of formulas and skills. –  Michael Hardy Jan 28 '13 at 18:48
So I had a mix across various professors' classes. I believe the general problem solving strategies (algebraic logic, decomposition, etc) are preserved in some form or another, but more complex/specific things are not. For example, if asked, I could not perform a Laplace Transform, or even recall what it was really for. –  Gopherkhan Jan 29 '13 at 0:35
Laplace transforms were specifically one of the thing I had in mind when I wrote my answer below. A vague recollection that it made circuit analysis easy. But solving hw and test problems was the extent of it. Now, if you have an interest in math, you can study linear algebra and complex analysis leading to Fourier analysis on a "matherly" basis and see the real beauty of the theory and appreciate it as we never did before in engineering classes. –  Andrew Jan 29 '13 at 18:06
Incidentally, for me it seems getting a job at a Machine Learning company has helped the most. It's forced me to review a lot of old material, as well as take some Coursera courses to build my statistical base a bit. –  Gopherkhan Nov 26 '13 at 1:07

4 Answers 4

up vote 3 down vote accepted

Here are some recommendations to consider.

Reading Books

$\bullet$ What Is Mathematics? An Elementary Approach to Ideas and Methods Richard Courant (Author), Herbert Robbins (Author), Ian Stewart (Editor)

$\bullet$ How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library) G. Polya (Author)

$\bullet$ How to Think Like a Mathematician: A Companion to Undergraduate Mathematics Kevin Houston (Author)

Proof Skills

You certainly should learn and improve proof techniques using books I listed on: how to be good at proving?

Additionally, you could also look at:

Study Habits from Math Stack Exchange

$\bullet$ What are some good math study habits?



Free Resources


$\bullet$ Khan Academy.

Local College Libraries


Lastly, you can search for book recommendations on the MSE in the areas you had learned if those are the ones you want to re-learn.


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These are actually some awesome resources :) –  Gopherkhan Nov 26 '13 at 1:05

A five-year break is really not difficult to overcome. I experienced a lapse of 6 years between a year-long (intensive) course in calculus, and my pursuit of an undergraduate degree. I tested out of the normal calculus sequence, and enrolled in Advanced Calculus (a year long course in advanced calculus/analysis) simultaneously with courses in differential equations and a proof-based class in linear algebra.

I won't deny that I probably had to put more time than other students to successfully complete Adv. Calc and DE, but I was surprised how quickly the material I had previously learned came back, as needed. And I truly fell in love with math during the proof based class in linear algebra. Following that was abstract algebra, and the rest is history...

Your previous learning WILL come back, much more quickly than the time it took to learn in the first place. And the second time around, you will learn it in greater depth. "Use it, or you lose it": perhaps that's true to some extent. But if you simply "review it and re-use it", you'll find that what you thought you lost was just a bit buried and in need of dusting off the cobwebs...

As pointed out, beyond the more calculating-based classes in math, you will often rarely encounter some of the earlier material. If it weren't for having to teach lower level classes (calculus, diffeq, etc), most TAs and/or professors would struggle more with complicated integrals, e.g., than a student fresh out of the class.

Bottom-line: 5 years away from math is negligible. Review it, yes. Practice, surely. But, in fact, the added maturity you have gained can be used to your advantage in both reviewing and pursuing math at a deeper level of understanding.

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Nice resources: <br> MIT Open Course Ware. <br> Khan Academy. –  amWhy Jan 30 '13 at 0:57

Perhaps my experience will be of use to you. As Michael Hardy mentioned above, there is somewhat of a difference between formulae based math and understanding math. I had been out of the classroom for 40 years and even then never had a formal math education - mostly classes related to engineering.

I decided to give "pure math" a try. Needless to say it was nothing like what I studied. I was fortunate to find a free downloadable copy of lecture notes given by Fields Medal (math equivalent of the Nobel Prize) winner Vaughan Jones on Real Analysis. This can be considered the entry point to the beauty of pure math.

His presentation is extremely motivating, very clear, and self-contained. So you can start from the beginning without necessarily needing any prior courses. Of course, you have to apply yourself. But that should not be a problem since you are motivated.

What is particularly nice about this endeavor is that is doable, so if you just hang in there and cover a few pages at a time, you can really learn a lot, confident that you are being taught by a true master who's intention is for you to get it.

If you are interested, you can give it a try with this link:


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do you have a link for the field medal lectures? that seems very interesting –  pyCthon Jan 28 '13 at 19:41
@pyCthon The lectures I was referring to and linked to above were given to math students at Berkeley for their RA course. So while they were given by a FM winner, they were not in the auspices of the FM. For your convenience, here is a link to the FM website. Also on wikipedia you can see a list of past winners and check out their work fields.utoronto.ca/aboutus/jcfields/fields_medal.html. Regards, Andrew –  Andrew Jan 28 '13 at 20:20

A fair number of universities are offering M(ass) O(pen) O(nline) C(ourse)s. Google MOOC Mathematics. You may well find that a free course in first semester calculus brings back memories, and even with the lapse of use you are mentally in a better position to understand what that course is all about.

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