How do you retain your knowledge of complex mathematics over time?

[Yawzheek]

It's not a homework help question, but it's somewhat related, and doesn't have a better sub-forum: how do you guys/gals remember all the intricacies of mathematics so well that you can help people like me?

I ask this because, like many people, I've done it all before, but when I pour over the questions asked in homework help, I draw a blank. I practiced these things for many, MANY hours, and had it down well, but years later I see these same questions, and say, "I understand what's asked, but I forget how to find the answer."

This is making the assumption most of you have left college/university by now, and surely we have a much different background, but it amazes me you retain these things. My physics instructor, great man, was in charge of the mathematics faculty of my college, and while he could certainly explain basic algebra and trigonometry - the latter being a rare question for him - when I asked questions about things I assume are a bit obscure in mathematics (synthetic division, and whatnot, which I forget) he would say, "It has been so long, I forget how to do it." It's not that he didn't know mathematics, but he seemed to forget "lesser" mathematics, if that makes sense.

The fact that you guys/gals remember things I don't isn't surprising in any way - I'm not a mathematician or physicist by any stretch of the imagination - but I'm extremely interested in how you all manage to retain this knowledge, and I'll admit, I'm quite jealous.

 

[axmls]

You remember things that you use often. It just so happens that the homework helpers are either a) students themselves who have the material fresh in their minds, or b) professionals/academics/retirees who refresh their memory by helping people with their homework.

 

[jtbell]

Also, what you can't see behind your computer screen is the books that we have lying around that we often refer to in order to refresh our memories, and the Google searches that we often do, when answering questions.

We may not remember everything off the tops of our heads, but we do remember doing this stuff before, and we remember where to look it up (more or less).

 

[micromass]

Also, what you can't see behind your computer screen is the books that we have lying around that we often refer to in order to refresh our memories, and the Google searches that we often do, when answering questions.

We may not remember everything off the tops of our heads, but we do remember doing this stuff before, and we remember where to look it up (more or less).

Right. A lot of questions here appear over and over again, so after a while we know the answer to those. Other questions might be something that we might need to look up first.

Learning mathematics is not really about learning many facts. Sure, that's part of it. But it's more about learning the right methods and knowing where to look for answers.

 

[mathwonk]

well this browser just trashed my lengthy answer, but it is mostly the same as others, i.e. although I took my last coursework in 1977, I keep my hand in almost every day and have done so for some 45+ years, studying, learning, teaching, tutoring, writing research papers, posting here (over 7,000 times), and I still forget anything I haven't either reviewed lately or that is one of the few special or elementary topics I know absolutely cold after a lifetime of teaching it. In particular, when answering a question here I often brush up from one of my own books, I had over 300 math books on my shelf until last year when I narrowed down to about 150. (Just now I counted 10 calculus or advanced calculus books, 3 real analysis books, 5 complex analysis books, and 21 abstract algebra books; and that does not include the Riemann surface books, differential geometry or differential topology or complex manifolds, algebraic topology or algebraic geometry or elementary geometry or number theory or differential equations or category theory books.) Moreover most questions asked here can be anwered by googling them, which I also do. So we are all in the same boat as you. I.e. we remember more only if we repeat it more often. One possible difference is that for us answering math questions is fun and challenging and we like to do it, and we are willing to spend some time researching the answers in order to get the pleasure of helping and explaining something to someone.

In addition to these awshucks answers however I am going to try to add something that may be useful to you at least in the future, althoughn nothing can enhance your memory of things learned in the past.

namely the more thoroughly something is learned, at least in my case, the longer it is retained. E.g. topics that I learned in high school well, such as plane geometry facts, that I practiced for math competititons, over and over at the time, have never faded. E.g. the diagonal of a square is sqrt(2) times the side. and the root factor theorem, that for a polynomial, x=r is a root if and only if x-r is a factor. But those are easy things, and when it comes to calculus I still forget stuff. Of course I didn't really learn that stuff well as a youth. More advanced facts like the inductive definiion of the first countably many ordinals, I still recall since in college I memorized that stuff cold to make sure I got the A I needed for graduate admission.

But also stuff I taught over and over and even wrote up in notes and unpublished books, is easier to recall than other stuff. So this first tip is just to repeat it over and over, i.e. memory is related to amount of repetition.

Second, it is also related to how simple th concept is, or rather how simple you have gotten it down to in your mind. I always wrestle everything over and over until it becomes simple. Once I have sufficiently simplified something I have less to remember, i.e. juat the core idea, and that is more memorable.E.g. in differential geometry the definiton of "curvature" of surface is often given in some abstract terms using derivatives, connections on bundles, or whatever arcane concept. After years I have realized it is a simple relation between the radius and the circumference of a circle. I.e. roughly, a surface is flat if the circumference of a circle is 2π times the radius, and positively curved (like a sphere) if the circumference is smaller than that, and negatively curved (like a saddle) if the circumference is larger than that. That's basically all there is to it. And that is easy to remember just by visualizing the three basic surfaces mentioned (plane, sphere, saddle). Of course all concepts have simple intuitive versions like this, and more precise formal versions, but the precise versions are easier to learn and remember if you first, or also, learn the intuitive version and understand the connection between them.

It also helps to read the original master who discovered the subject. Gauss for instance explained simply how to measure curvature by the change in angle of a tangent or normal vector, as a point moves on the surface or curve, whereas nowadays even good writers on the topic apparently cannot say these simple words without mumbling about "fundamental forms". these words of course have no intrinsic meaning and just make my brain cloud over.

 

[gleem]

Well it has been almost 50 years since my last formal course and I'd be lucky if I could recall 10% of that knowledge and not necessarily 10 % of any subject either. However it does come back and relearning it is much easier. I look up almost all my responses to verify I on the right track except when I get cocky and and have to apologize for my mistakes.