[DarkShikari]

Math is doable, and simply becomes something that you have to study for; you no longer get a freebie because you're smart.

I found it was the exact reverse. Many concepts, in my experience, could not be learned well how hard I studied. And yet people who did significantly less work than I did found them extremely easy. One example is natural deduction proofs; I slogged my way through them with extreme difficulty while some friends of mine did them nearly effortlessly by comparison.

Overall, at least my own experience suggests that studying and hard work can never fully replace innate ability.

[gxs]

There is a cheesy quote that says genius is 1% inspiration 99% perspiration. In college, I found this to be especially true, at least in the math department. While most people would like you to believe that they breeze through the subject, getting to know people in the department (undergrad) who were very successful, I learned that, while of course mathematically inclined, the amount of effort put forth was quite astonoshing. It was astonoshing to me at least, coming from a non-competitive high school. It is not unusual for someone on any given week to study 25+ hours for a single class. This was the case for some of the smartest people I met in the program. I found 25+ hours a week turned out to just not seem like a lot to someone who was a) very interested in the subject and b) used to spending that kind of time on things. I struggled to put 10 hours a week into it and would barely pass some classes. Now that I am older I notice it a lot more. Dedication is a common thread in all stories of success. From Einstein to the kids in my grade that made it. The ability to work hard and push through obstacles is a skill. Some people realize this at a young age and seem to have it very developed by adulthood. I find more and more that being smart is necessary but nowhere sufficient for success, be it in business or academia. All things being equal, I would weigh hardwork/diligence more than intelligence.

[scythe]

There's a funny thing about dedication in that it doesn't really feel like dedication if you're interested enough in what you're doing. I can recall stealing my father's old college calculus book out of the the top cabinet when I was in third grade and reading through it at night when I was supposed to be asleep, purely out of boredom with the math we were taught at that level. By then, I barely had any concept of a variable, and yet I somehow managed to teach myself a decent portion of calculus in the next three years.

These days, I struggle to apply that level of effort to anything I'm required to do.

[kajecounterhack]

I find that some concepts are slippery to some people, and others are slippery to others, but when you approach a slippery subject and an excellent teacher explains in the right way to you (or a bright peer), it becomes easier to understand. That combined with practice is good enough to fill you in and hard work does indeed compare with prodigy. I don't know if I believe in prodigies -- any piano prodigy I've known has played piano from a young age. Same with math. Practice is everything. Exercise your brain muscle and it gets bigger.

[jacquesm]

True, in piano though (as well as in string instruments) where finger dexterity is very important it really helps if you get this ingrained in your fine motor control at an early age. Simply put you can distinguish someone that has spent 5 years learning to play the piano from age 6 to 11 from someone that learned to do the same thing from age 30 to 35 with ease. Even if you were to stop at age 11 and restart when you're 30 you will probably still be ahead of the pack.

And then there is the 10,000 hour rule to consider:

http://www.gladwell.com/outliers/outliers_excerpt1.html

[cturner]

Very interesting. I've had experienced like this. So I guess it's some sort of learning strategy. Perhaps you blitz the field from certain approaches and are slower at others. Do you have any theories about what your strong approach is?

[DarkShikari]

In general I've found I'm best when I'm self-motivated, which really shouldn't be surprising to anyone on HN.

All the math courses I took up to Calculus BC were through the Stanford distance learning program (EPGY), which really helped enforce this. I also took three courses on C programming from them in 2nd-3rd grade; I was pretty terrible at the time but something from must have them stuck with me really well, because when I finally came back to C 10 years later it was nearly effortless.

Similarly, almost all the skills I would say I've learned really well in recent years (e.g. SIMD assembly) are things that I've learned on my own time outside of class.

Another related observation is that I tend to learn on an S curve ( http://en.wikipedia.org/wiki/Sigmoid_function ). For example, take the process of learning x86 SIMD assembly: it took me days to write my first assembly function (from scratch with no knowledge whatsoever other than an example for syntax and documentation pages). Then it took me mere hours to get acquainted with SIMD basics and start writing basic SIMD functions. In just a few more days I had written dozens of optimized SSE versions of MMX functions...

...and yet it probably took me a whole year after that to gain enough experience to feel comfortable writing extremely complicated functions from scratch.

In other words, initial learning is slow, then learning builds on itself to vastly accelerate speed, but then honing ones' abilities is slow again.