What causes us to "click"?

[dekoyl]

What causes things in our brain to "click"? I'm sure everyone knows what I'm talking about. You'll be looking at a maths problem that you're stuck on and suddenly... *CLICK*. And then you realise it was so damn simple all along.

Oh, if only there was something that could manipulate this so we may "click faster".

I kid you not, after looking at a 2 mark question for spesh, on and off, for five hours, I FINALLY got it. Five. Hours!

[kamil9876]

I know what you mean, so frustrating 

e.g: just spent so much time on zzdfa's new problem and didn't study ;(

Happens to me all the time that such a simple solution to a problem only comes after so much time. Fortunately for us I hope real life isn't about solving 10 questions in an hour and more about real collaborative thinking over relatively more relaxed periods of time so hopefully good but not machine-like thinkers like us will somewhat thrive in such an environment. I heard about this phenomena from a few anecdotes from mathematicians.

but as far as studies go, i'll say go over not only the mechanics of the questions but also the mechanics of the thoughts, e.g: what you associate with what.. make general associations between concepts etc.
Also sometimes I leave a question that I don't get till last and while im working on another one it suddenly pops in. I remmeber my teacher advising that, "read all questions and then let your subconcious work on all of them simultaenously", in fact it sounds like a quote from famous french mathematician Poincare.

The worst is when you realise 5 hours after an exam all the mistakes you make, happened to me :/

[humph]

It's unavoidable, and gets worse the further you go along. Questions in high school maths usually won't take that long to click because they're usually very closely related to what you've been studying, so you know how to apply techniques. The further you get in uni maths, the less clear the path is to you when solving problems (at least in my experience). By the time you become a researcher, it's not unusual to spend months on problems - and like Kamil said, collaboration is one of the best ways to speed things up.

[Mao]

I go lie down on my bed, stare at the ceiling and try to visualize everything, and it start making sense soon. The key thing is lying down, it helps a lot [no idea why].

I also find it easier to think in the shower, and the glass screen tend to get scribbles on them (and sometimes graphs).

[TrueTears]

I go lie down on my bed, stare at the ceiling and try to visualize everything, and it start making sense soon. The key thing is lying down, it helps a lot [no idea why].

I also find it easier to think in the shower, and the glass screen tend to get scribbles on them (and sometimes graphs).

Yeap, exactly the same here, I remember back in the summer holidays when doing questions all day I even began to have dreams about certain questions ~_~

[QuantumJG]

It's unavoidable, and gets worse the further you go along. Questions in high school maths usually won't take that long to click because they're usually very closely related to what you've been studying, so you know how to apply techniques. The further you get in uni maths, the less clear the path is to you when solving problems (at least in my experience). By the time you become a researcher, it's not unusual to spend months on problems - and like Kamil said, collaboration is one of the best ways to speed things up.

These are some wise words.

I'm only doing first year maths and things still take around the same time to click. I'm guessing with next year it will take longer. The best way to speed up this process as what humph said was to collaborate because everyone thinks about something differently and you can explore more aspects. In uni the more open you are to other people's opinions and the more friends you have is what determines your success.

One of the things about maths I love is when a really hard problem suddenly clicks.

[humph]

It's unavoidable, and gets worse the further you go along. Questions in high school maths usually won't take that long to click because they're usually very closely related to what you've been studying, so you know how to apply techniques. The further you get in uni maths, the less clear the path is to you when solving problems (at least in my experience). By the time you become a researcher, it's not unusual to spend months on problems - and like Kamil said, collaboration is one of the best ways to speed things up.

Last night I finally solved a question on one of my assignments that I had been working on without success for the last two or three days (it's due tomorrow). You get a massive rush and feeling of awesomeness when you finally prove something that you've found really difficult  Annoying though cos it all came together in my head while I was cooking dinner and I couldn't rush off to write it all down without burning my food 

[Collin Li]

Jumping on a bed helps for one of my friends... weirdo

[Ahmad]

Poincare was a rather reflective person and often analysed his own thoughts. He described problem solving as four stages: preparation, incubation, revelation and verification, the idea being that your subconscious works in a combinatorial fashion trying all sorts of combinations of ideas, and when it thinks one is useful it's brought to your attention, and this is often how tough problems are cracked. You can read more about it here and here. Hadamard wrote the book: The Psychology of Invention in the Mathematical Field which discusses it further.

George Polya wanted to know the thought process behind how people solve problems, he did many investigations and came up with the book How to Solve It which has a list of problem solving heuristics (Terry Tao used this book while competing in olympiads). Moreover if you want to know how mathematicians think and learn Polya also has the book Mathematics and Plausible Reasoning which is a fascinating read.

[Ilovemathsmeth]

Thinking when you're about to fall asleep and in the shower are pretty good =D

[physics]

i think as we get older our brain kinda slows down so we click slower and slower

[kamil9876]

Quote from Mathematician G.H Hardy:

"No mathematician should ever allow him to forget that mathematics, more than any other art or science, is a young man's game. … Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work later; ... [but] I do not know of a single instance of a major mathematical advance initiated by a man past fifty. ... A mathematician may still be competent enough at sixty, but it is useless to expect him to have original ideas."

[zzdfa]

I go lie down on my bed, stare at the ceiling and try to visualize everything, and it start making sense soon. The key thing is lying down, it helps a lot [no idea why].

me too, i think it's because more blood into brain => more brainPower???

[/0]

I plod around the room. And when I'm home alone I talk to myself a lot while making hand gestures

[minilunchbox]

I plod around the room. And when I'm home alone I talk to myself a lot while making hand gestures

I do that regardless of where I am and who I'm around.