A new favorite quotation of mine about mathematics learning:
"Mathematics must be written into the mind, not read into it. 'No head for mathematics' nearly always means 'Will not use a pencil.'" Arthur Latham Baker, Elements of Solid Geometry (1894), page ix.
Hat tip to the Bay Area Math Circle for the quotation reference.
To put things in perspective, and to possibly invalidate the general application of my insight on the matter, Calculus was in fact the only course I passed my final semester of high school. I received a D shortly before I dropped out altogether. The D score was earned only after being the only student to ace the final, a task which was itself only possible after I proved the first fundamental theorem to myself (thanks to an especially verbose description of it in one of the exam questions) during the course of the test.
To this day I find that the actual solving of equations to be tedious and can only be interested in problems tenable to axiomatic and algorithmic approaches. Thats where all the fun is imho. Who cares about actually determining a number (or equation)? [the answer: all the smartest people do.]
I'm serious though, I'm learning with textbooks, etc.. and I'm a dropout so I have some of the basics already.
In my first year of a doctoral program, I could study from after lunch until next morning and I concentrated on getting the fundamentals right. The greatest book, the one where things began to click, was Rudin's "principles of mathematical analysis". Before I came to this book, I had wasted a lot of time finding my style of learning or, if you will, discovering my preferences. But if you already have the basics and are not going to waste as much time as I was trying to figure out what is what, you might pull it off much quicker.
Good luck! To me it was a cool time and I finally got rid of my math inferiority complex ;)
For me school has always been about discipline and dedication and not about skill. The school adapts itself to a lower difficulty so that more will pass and they compensate by making it more cumbersome so that the skilled, but not so disciplined, will get bored and not perform as well.
This was also my personal experience. I was pretty good at math but I never got any challanges so I just didn't do anything, but luckily I aced the tests and the only homeworks we got where to have complete the chapters. This translated to other courses where I was good, but I got bored and not motivated so I got a lot worse grades than I could've had if I was dedicated.
I managed to motivate myself later on to get good grades, other would call them great, but I know I could've had straight A grades if I just weren't so lazy and skipped a lot of hours.
Furthermore it doesn't help (doesn't show in grades) if you know a subject really well. Often the workaholics, or a bit nicer: those with more discipline than me, would just study like hell before a test and get a great score, where I wouldn't but I knew the subject well enough so I would get a great score too. I'm a bit upset with this because I doubt that they could remember anything from those tests but I felt I could remember them for months.
On one Calculus test I had, I didn't study or do any of the homework and pulled a ~90% score. When talking to someone who got a perfect 100%, he casually informed me that he forgot how to integrate between tests. I was shocked to hear this, but apparently it's the norm.
I couldn't have put it better, that's exactly what I feel too! It would be so cool if the educational system were designed to help under-disciplined people to perform to their best rather than weed out the ones who can't adapt.
Seriously, though, sometimes taking short breaks can cause you to be more productive than just slogging through. Not all trips to the internet are procrastination. Sometimes you really do need to just stop coding for a few minutes. Though I imagine there are better ways to recharge your mental batteries than HN. HN just makes you think too much sometimes.
I talk about the experience here:
BTW, it appears that fast thinkers are less likely to be creative people:
Some students have a "discipline" trait, they'll force themselves through boring drudgery to get the A, then never look at the material again, the A is all that matters. That's a waste of time.
True "discipline" comes from having an interest in the material. That interest has to be sparked by something, the material gives you a buzz, and working on it does not feel like work. That's not something I'd call self-discipline.
First has a lot to do with what is expected at school (wonder what the data looks for 3rd level education) and the rest is pretty obvious...
http://www.stevepavlina.com/blog/2005/12/rules-are-no-obstac... "Rules are no obstacles for committed people"
WTF? I thought Scribd was HTML5-in’ it up?
The trade-off comes in deciding whether you'd rather get good grades or spend time learning other things on your own. It comes down to how much you trust the educational system you're in to fully make use of your time and intelligence.
I think it's a bad thing that, in general, grades to not track IQ. Hard work is important, but only when the effort does work on your intelligence (similar to the physical definition of work). One can spend hours on a paper or a PowerPoint and have done nothing to further their intelligence; and yet these assignments are what students are typically graded on.
Hear Hear! And all learning is done on one's own, really.
IQ is, not to put too fine a point on it, bunk.
Grades are also bunk. They can never measure depth of knowledge, which is the thing that matters, intellectually speaking. Depth depends on connections formed with other areas of knowledge, which are unique to each student.
Grades can indicate the depth of knowledge, but with the disclaimer that both the arbitrating grader is trusted (in a very special sense), and that the method used to decide grades be both defensible and explicable to others.
To provide a concrete example. A paper on the first book of Hume's Treatise of Human Nature attempts to bring a new argument to bear on the consequences to modern epistemology incurred by Hume's concept of abstract ideas. The paper is written as a final in a graduate level philosophy class, and the instructor is known to be knowledgeable, intelligent, fair, and honest. The paper receives a B+. It is arguable that the mark is generally reflective of the student's understanding and ability at the time the student wrote the paper. (this example is taken from personal experience - I wrote this paper, and I feel that the grade was reflective of my knowledge on the topic)
But without the trustworthy instructor, the grade would be meaningless. The problem, if I were to speculate, is one of numbers - too many students, too few professors. A basic premise of security is that the more parties involved in a pact, the lest reliable the pact is, and if grades are to be a faithfully representation of a students ability, the reliability of the academic pact is paramount.
Sad to say, not everyone works that way. I'd say that more people are dependent upon others teaching them than anything else.
Agreed. I believe this is the most important part of education.