Although this may be just deflecting the blame, I do think that the educational system in the US by and large poorly serves gifted children. The underlying notion seems to be that gifted children are already ahead of the game and so don't deserve any special attention. Another piece of the puzzle is grading-as-certification rather than grading-as-progress report or some other scheme. Straight As can mask a lot of problems.
I've got a room mate who was put ahead a grade in middle?late-elementary school and missed the whole foundation of algebra. They did so because he got straight A's so obviously he's smart enough; he could just skip that. (They promised to give him a text book so he could learn on his own that summer, but never got around to giving it to him.)
Of course, he started struggling in school the next year. Now math is one of his worst subjects. They completely screwed him over because they were distracted by his grades.
Skipping a grade means giving up a full year of education. If that proposition doesn't give you pause, you probably don't have enough respect for the value of education. Even if you you get absolutely nothing out of that year aside from the soft skills you develop by sitting in a class with other kids and listening to lectures and grinding through homework (even stupid and boring homework), it's still vastly more valuable than the fleeting ego boost you get in exchange when you sacrifice it in order to skip a grade. Parents, please, this is one marshmallow test[1] you and your kids really don't want to fail.
[1]: http://en.wikipedia.org/wiki/Stanford_marshmallow_experiment
During elementary school, a friend and I got a hold of a Math review book meant for college-age folks who didn't really grok Math in High School. We enjoyed competing with each other in learning math, but it resulted in us going into 6th grade knowing everything we would be learning for the next three years. In Math alone. We were falling asleep in class, waking up at the end, and getting perfect-score-plus-bonus on every daily quiz.
We made the choice (not our parents, mind you) to try and skip a grade in Math alone. The school fought against us in this, and we eventually broke through, skipping two grades in math. Because our middle and high schools were next door to each other, we could walk across the street to take more advanced classes. The only downside was that I ran out of Math classes to take after my Sophomore year.
This isn't a brag. Far from it. All that this did was bring us into the level of math we were already prepared for, and which some private schools or wealthier schools already teach at that age-level, without skipping grades. I am grateful for the opportunity to not sit on my hands for two additional years, and to not be artificially disadvantaged by a system that moves too slow for the current standards of college preparation.
I have bumped up against the "Curse of the Gifted" in several places in life. I suspect I am hitting one right now in some aspects of how I produce software. But skipping two grades in lower-level Math was not one of them. In fact, it is now standard at my middle and high school, among the top students, to simply skip ahead at least one year in math. You end up with either more free time, more time for arts/personal pursuits, or more time to load up on AP sciences.
It is not a failed marshmallow test. It is a sign the curriculum is inadequate for the student's needs.
Maybe even in the more extreme case of a child skipping several grades altogether by himself, that's the best of a bad set of choices. But we shouldn't be offering only bad choices. At least in/near any decent size city, there should be a large enough population to create classes of a reasonable size of same age students at roughly the same level.
We actually ended up separated in our advanced math, and I was bullied heavily because I had to end up eating lunch with 8th graders while only being in 6th and 7th grade. It was socially sub-optimal and it sure wasn't rainbows and unicorns. However, I was already an outcast amongst my own grade level, so being an outcast amongst older kids wasn't much of a difference.
My point is that it was not intellectually hobbling, it did not set me up to feel a Curse of the Gifted in Math, nor did it teach me bad habits as the analogy with the marshmallow test would imply.
Why? There is great educational value in being in a class which has different levels.
If you are smarter: learning to help others, working together with people that are slower than you are, learning that usually intelligence is not one-dimensional and kids that are not very good at math might be great at writing…
If you are less smart the benefits are across the board. I remember reading a study about mixing kids from different incomes in the same school as opposed to wealth segregation (you might argue that smarts and wealth are not the same, but they are often highly correlated at least when starting out) and the conclusions were that there was no drawback for the rich kids, but huuuge advantages for the poorer.
the problem ended up being that i lost all my peers to study and eventually lost interest in the university. the result was that all my peers now have a ph.d and i struggled to get back in many years later to now write a master thesis...
I have absolutely no respect for the current educational system.
The grades kids usually skip are teaching 90% review every year, and skipping one just lets you get out of that worthless grind sooner. I've known multiple adults who skipped grades as kids, and all have said it was a good thing.
What can mess things up is throwing a kid into a group of older kids if they aren't socially prepared for it.
Stupid and boring homework is worthless. There's no point in defending it. Kids who have no assigned homework ever (in primary and secondary school [1]) can still grow up with a work ethic, a top college degree, and a great job. Heck, I know adults that were "unschooled" [2] as kids and who got exactly the jobs they wanted when they were ready.
Learning isn't something that should be restricted to school, hard or soft skills. You can develop way more soft skills interacting with others than you can being forced to sit still and not talk for hours at a time. What skill does the latter teach, exactly? I don't know why the constraints of school are celebrated when they should be condemned.
Can't agree more. -- someone who's happily skipped two grades at school (not in the US, mind you) and sometimes wishes he had skipped all twelve.
Fewer years of "writing/literature" could have weakened my writing skills, but I think that was primarily from my non-application due the incestuous lit-crit regurgitate-the-themes-the-teacher-says educational approach, and more of this certainly wouldn't have helped. I applied myself just fine when I was in technical classes in college and therefore had something real to write about.
Socially, I actually did better when being placed with the older kids. There was less Idiocracy-style jockeying for social position, or at least I just wasn't involved.
In science/math, most school years involved spending half of the year "reviewing" the previous year's material. My recollection of elementary arithmetic class is that you learn multiplication tables up to 6 one year, up to 10 the next, and up to 12 the next. For someone who gets it, will remember it, and just wants to move on, I'd say being held in these doldrums is more harmful than simply letting them go closer to their own pace.
Narratives are central to human identity. Particularly for young people who haven't yet developed the ability to reason in the abstract. If you want to create people who can follow, who can lead, who oppose tyranny, who oppose corruption, who believe in the ability to change the world, and who understand why they believe what they believe - narrative is the most powerful at your disposal.
Literature may be the only important subject at school. It teaches us about the use of the language we use in our every day interaction, and pushes the study of narratives about life. You guys who got bad literature teaching really missed out. You will have compensated for it somewhat through film and independent reading.
I value education, just not schooling, despite the attempts to equate the two.
>Parents, please, this is one marshmallow test you and your kids really don't want to fail.
Except that pushing yourself harder and earlier is analogous to waiting for the two marshmallows, not eating the one.
He started school a year ahead by being skipping Kindergarten. He progressively began testing out of math courses, and eventually took BC Calculus when he was in 9th grade. He then decided to skip tenth grade, but since he was already so far ahead in math, he really just had to test out of three courses -- English, Chemistry, and World History. He ended up graduating valedictorian, and he finished his undergrad in two years. He's now completing his masters in physics at Cambridge. He will likely do his Phd there as well.
Interestingly, he couldn't speak as a young child. But his father was able to teach him math, which he soaked up like a sponge. Most notably, his father instilled in him the importance of what my friend calls "the grind." He was taught that he couldn't be truly successful in mathematics unless he practiced hard, and this discipline has served him well to this day.
I think it hurt me socially a bit during high school, but I closed that gap very comfortably in college as well, so I really don't see any downside and think there was massive upside to completing my schooling 1 year early, essentially giving me one additional year of freedom as an adult.
In many ways, that's like getting an extra year of "life" (and specifically at age 21). What would you put as the fair value for that? For me, it's enormous.
My son goes to a Montessori school that has two different grades in each class, so you hear not just the stuff for your grade, but also for the next (or previous) one. I think that's a perfect environment for skipping grades, because you get to condense two years into one.
(A rough hack around not being able to skip a grade (if the institution is inflexible) is to supplement the school's curriculum with textbooks from other countries. China and Russia both have textbooks with harder problem sets for the same topics, which are great prep material if you've joined, say, a math team. What makes me sad is, that's their regular material for the same age group, not some super-advanced track.)
And gods forbid, the day comes when the kid who's held back by not skipping a grade goes to college and finds that their new peers had gotten more advanced material in their other schools while they themselves whiled away their years being inadequately challenged. And they're now hopelessly behind, or at least have to paddle that much harder just to keep up. There is no purpose to holding a kid back against their will. If my parents stopped me from skipping a grade, it would only have been punishment for doing nothing wrong. And I would have fought them tooth and nail.
I wanted to skip my senior year in high school as well, but my parents wouldn't let me. I have to say that I'm glad they didn't. I actually enjoyed that year a lot. One of my favorite classes was a humanities course that was rather unique. We had a team of teachers and would study the music, art, literature, and history from a period in time and gradually marched our way forward. It was great! Also, in college, the only humanities course that I took was science fiction, so if I hadn't taken that course, I would have missed out on a lot!
At least for me, skipping grades isn't about an "ego boost" it's about time. It's the same thing with AP courses--some people say that they aren't equal to college courses, but if you take them, pass, and the school accepts them, then it saves you an incredible amount of time.
In my case, I just studied at regular speed at school, and at home my older brother kept me up on what he learned about the topics that interested me (math & science). During regular school classes I enjoyed the ability to re-think about the stuff I already knew, and the extra time I didn't need to devote to them was time I did spend thinking and experimenting with other subjects like philosophy, writing, or computers and videogame development (I designed, drew, encoded and compressed the maps for our first two commercial games all by hand during class - but this was back in the 8 bit days).
I think the chance to experiment and pursue my interests without pressure or obligation was incredibly valuable.
The reasons skipping that grade was good: I avoided a poor teacher, whom I would not have gotten along well with. Part of the reason for me skipping a grade was that the disparity in reading levels was actually detrimental to the class, because of the diversion of the teacher's resources. Second reason is that I have an early birthday, so I was never tooo far behind my peers in the next grade, and in fact I felt much more comfortable in that situation and made friends better.
Just for an alternate perspective!
I suppose it can have an impact on socialization, but it still seems like a good trade if you gain a year of your life. I made the trade once the normal way and twice more by starting college early. Stuff was hard, but I can't imagine boring myself to death for three extra years in a town isolated from civilization would have been better.
But I was one of the oldest in my grade (so I was still sorta in the right age band), and it was the second grade (so our social structures weren't all that well-developed).
Education doesn't stop with schooling.
So, for calculus, I just got a good book and dug in. So, yes, first I did the chapters on analytic geometry, i.e., the conic sections -- hyperbolas, parabolas, ellipses -- and then one with calculus. For my next year, my brother was out of the good, private college, so I transferred in as a sophomore and hopeful math major. So, I started with their sophomore calculus. Did fine. Wrote my honors paper in math and got 800 on my GRE Math knowledge test.
But, right, I 'skipped' freshman calculus, never took it! Didn't get credit for it, either.
In grad school, there was an advanced course in linear algebra. I'd never had a course in linear algebra but still told the faculty that I didn't need that course. They said, with a patronizing smile, "Take it anyway.".
Okay. I blew away all the other students with by wide margins the highest scores on homework, tests, and midterm and final exams. Prof wrote, "Best performance in the class. Knows this material cold.". Yup. I told them so! Heck, I'd wanted to study optimal control theory, not waste time with linear algebra. How'd I do that? That is, essentially 'skip' a first, and, really, an advanced, second course in linear algebra? Easy? I already knew the material from independent study of a stack of books, including some of the best, especially Halmos, 'Finite Dimensional Vector Spaces'. But in total it was a big stack. How? Why? A long list of various projects in school and in my career in physics, signal processing, applied math, numerical analysis, multivariate statistics, and more along with some careful independent study.
Lesson: Independent study can work fine. Then can 'skip' about whatever you want, at least in math.
Really, guys, what the heck do you think a research prof or any researcher does? Actually know all that stuff just from sitting in classes in school? Heck no! Instead they learn, from texts, papers, seminars, etc. Actually, if need to know something and can get a good text, then usually can get good understanding of just what need from that text for less than 10% of the effort that would be required if had to learn the text well enough to get an A on all of it. Net, independent study is crucial.
I was gifted. I ran through years, absent-mindedly attending courses, handwaving tests and scoring incredible marks. The best thing that happened to me is getting into prépa[0]. They make you hit the wall on purpose. The jump from the previous years is purposely high. The only way to get through it is to snap yourself out of your gift/boredom/ego vicious circle and start to learn to discipline yourself. Only by surrendering your gift and unlearning what you know can you build a solid foundation and make your talent truly shine.
[0]: http://en.wikipedia.org/wiki/Classe_pr%C3%A9paratoire_aux_gr...
One of the most important thing I want to teach my 4 months old son, is the value of effort.
I am glad you shared your experience, I will keep it as one of the many tools I am gathering so that I can be able to teach what I haven't learned to my children.
I only managed to satisfy degree requirements by taking 'logic' (a philosophy class) instead of calculus. I was warned off trying this route by a few people though, as it was "hard". I'd been hobby programming for... 10 years at that point - there's little about 'logic' that could possibly be 'hard' in a logic 101 class, so I passed by doing something I was naturally good at. I simply couldn't afford (financially) taking calculus another 2-3 times to keep trying to pass.
It's still a sore point even now... 20+ years later, and a source of secret (perhaps not so secret now that I post this!) shame. :(
I'd never even had to try at anything (except music, perhaps) until well in to high school, at which point I'd mostly just given up. I scored in the top 2% of the ACT test (back when it was scored differently, in the 80s), and I'm not actually 100% sure I finished all the questions - I know I didn't study for it, nor was I particularly well rested. Yeah, as someone else wrote, schools are simply not oriented towards helping 'gifted' students, assuming that they're already 'ahead' of others, and seem to be left on their own - I know I was.
The prologue is great:
"Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks.
Some calculus-tricks are quite easy. Some are enormously difficult. The fools who write the textbooks of advanced mathematics (and they are mostly clever fools) seldom take the trouble to show you how easy the easy calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way.
Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can."
http://www.amazon.com/Calculus-Easy-Way-Douglas-Downing/prod...
As she described it "I needed to rewire my brain and music was the way to do that". From her perspective learning to play music helped her begin to understand how things "fit" together in her head. Her ability to memorize things was incredibly helpful (so she was initially a step-up in confidence), but as she played more she began to see how chords build on each other to form songs. She went from rote memorization to actually building a mental model of music.
In the end she became not only a fantastic musician but a MUCH better student.
I still didn't espouse the value of hard work until I've gotten out of college. I tried to cruise by like I always did, though it increasingly became untenable. What made me really understand working hard was actually through sports. There, I had no illusion of being superior, and it's established that you get better through practice. You get to see yourself getting better each time, and put yourself to the test. True iteration.
Then I took Calc 2. I don't think I've ever failed so hard in my life. ... Except freaking english...
College-level calculus burned me so much, I didn't had the algebra background necessary to grok it, much of it is memorizing heuristics and "tricks", and college professors (specially in my public uni) didn't had much patience either. I even did well in other math-related classes, but calculus was a chimera. Unfortunately, I fell so much behind the curriculum that I had to drop out of college to start working full-time.
Lately I've been picking up other math-related online classes, and I finally realized that I had never had real exposure to mathematics, and that high-school curriculum is something else. So, there's that too... we are mostly scammed into thinking we've learned something during high-school, and then life happens and you actually figure out you don't know shit. Hopefully I'm a curious person, so I'm always learning something by myself, mostly to broaden my knowledge in other topics, but to not let the gears grind too.
Relevant (an article I wrote): http://j2kun.svbtle.com/you-never-did-math-in-high-school
As an anecdote for you, since you're a mathematician: during school I had zero patience for math. I would often study just enough to get a passing grade. On the other hand, I enjoyed physics classes a lot, it involved more problem solving and holistic vision than memorization (since you can just memorize a couple formulas and derive everything else you need - e.g. the second and third equations of motion from the first). Maybe I would have enjoyed math classes more if it involved more intuition and less rote? I'm sure you'll come across kids like I've been during your voluntary lectures.
PS: I'll be following your blog.
EDIT: I just read your article, I think our math program is like you describe it should be.
I agree there is some baseline level of rote manipulation people can't get away with, but it's gone too far.
http://nces.ed.gov/fastfacts/display.asp?id=97
-- says that in 2009, just 16% of U.S. high school students took calculus. And a common complaint by college professors is that high school calculus courses teach rote procedures without explaining calculus in any depth -- the students know how to solve calculus problems, but they don't actually understand the subject.
When the AP test rolled around I was really struck. The problems were about the flow rate of water descending into a tank and what happens when you put a drain here with this rate of flow, etc. Completely different from our earlier problems which could be solved by rote memorization. And from my perspective - one of a student who knew a bunch of calculus "rules" but lacked understanding of how calculus really worked - this was incredibly difficult.
I got a 3, which is passing, but opted to take beginner-level engineer's calculus when I got to college anyway. Even if you "learn" calculus in high school, it does not teach the level of mathematical maturity required to understand the higher level manipulations. This kind of thought is the foundation for any good engineer, and I have no regrets about retaking the course.
I jumped to AP Calculus without taking the pre-requisite AP class everyone else had taken, and I thought I was so cool and smart. By the weeks end I no longer felt so cool and smart - I was completely lost.
As a side note, I can't stress enough how bad the education system is in Brazil (in the sense of lack of structure, motivation, etc.), even at college level. Almost everything I know is self-taught (including English). I sometimes hear complaints about public school in the USA, and there's a meme that americans are "dumb", but it seems miles ahead to what I had access to.
- Introduction to mathematical thinking [1]
- Introduction to Mathematical Philosophy [2]
- Machine Learning (actually a CS course, but involves linear algebra and some calculus) [3]
- Calculus: Single Variable [4]
[1] https://www.coursera.org/course/maththink
[2] https://www.coursera.org/course/mathphil
In my case, I actually gave up on maths in year 7 because I wanted to hang out with friends more than study. I consistently failed in high school despite being pretty promising in primary school ... but that's just 'cause I "wasn't trying", right?
Well, maybe so, but maths is a skill. You can't ignore it for 4 years and then pick it up in the same way you can with English or Japanese or Sociology.
So when it came to year 11 and 12, when I actually wanted to do well at maths, I couldn't, so I borderline failed high school maths.
A foible of the (Australian) state in which I went to year 11/12 meant that, despite not using maths towards my university entrance score, I was still allowed to use it as a pre-requisite to get into a university course.
I didn't go to uni straight away because I didn't really know what I wanted to do. At age 21 and on a whim, on the last day that you could mail your applications in, I picked Mechatronic Engineering from the handbook and sent in my application. Why? Because I thought it sounded cool (I used to love Robotech :) but I had no idea that Engineering was so maths heavy (that's how ignorant I was of the field).
First year I failed everything. The only thing I did well in was programming, so I took 2 years off and got a job as a programmer. In the last 6 months before I started uni again, I went through my Calculus text book from the beginning, going over my first year uni notes and assignments at the same time. Answering all the problems in the book, etc.
As it turns out, the things that were most nobbling my ability to do calculus was lack of understanding of algebra, trigonometry, geometry ... things like logs and graphs and functions, factorisation, all that stuff. Once I'd learned that, I discovered the rest wasn't so bad.
I re-did my first year subjects in summer school that year and passed (even got some Credits which is 65+!) and then managed to get a low distinction in one of my second year courses.
You never saw a guy so happy with his 75 in Vector Calculus :) all the engineering geeks were crying in their pillows because they got a 98 instead of a 100 but I was punching some serious air.
Getting past this personal myth that I was "just bad at maths" is one of the most important things I've ever done. It really taught me that, with persistent execution and incremental improvement, you can learn and do anything (within reason: ie. I probably can't learn to levitate although I'd love to be proven wrong :)
ESR "grandpa" was at the Wharton Computer Center when I was there. [1]
http://www.catb.org/~esr/resume.html
At Wharton, at the time, this was definitely true:
"When you were in college, did you ever meet bright kids who graduated top of their class in high-school and then floundered freshman year in college because they had never learned how to study? It's a common trap. A friend of mine calls it "the curse of the gifted""
I was lucky that I had to work hard in high school. I definitely noticed a bunch of kids that were actually depressed when they were surrounded by others equally gifted and got their first "b" grade. No adversity experience.
[1] APL was big back then. http://en.wikipedia.org/wiki/APL_%28programming_language%29
Pretty much high school was an obedience test. "Monkeys, if you do exactly what I tell you, when I tell you, I will rate you all 'Above Average'."
It's not something I've entirely conquered either, frustration leading to shut-down still happens from time to time.
I don't see the difference. In my mind, thinking about it logically is recognizing that one side has an excess and removing it from the other side. That's subtracting.
To you is it like looking at 5 * 4 and knowing it's 20 rather than thinking about 2 * 5 is 10 and two of those is 20?
I was the kind of rule-following kid who went through the algebra steps because you were supposed to rather than out of necessity. (Or so I like to think: I'm not good at math in my head because I always wrote it out or maybe I relied on paper since I couldn't do it in my head.) I recall rebelling against memorizing multiplication tables in grade ~4 because I thought my time was better spent leaning new stuff (or maybe understanding how to multiply faster -- wouldn't that be ironic!).
It's always interesting to hear how someone else's brain works. Thank you.
Switched to a more creative field, and circled back towards engineering-type jobs 10 years later. Happy (enough) with where I am now, but can't help but think things would have been better if laziness, hubris and structure of the school system didn't keep me from learning how to learn earlier.
I think in the end, the slap in the face came at the right time and made me more capable later. No matter how smart a person is, there are things they do not know and have to go learn. Learning how to learn it skill, and like all skills it takes practice.
In my Calculus class we weren't even allowed to use a calculator on the test. Actually, almost all my math classes through college wouldn't let us use calculators during tests.
AP Exams/Courses are offered in many subject areas and each have very strict guidelines (like which calculators to allow), which are dictated to the instructor by the College Board.
[1] http://en.wikipedia.org/wiki/AP_Calculus#Exam [2] http://en.wikipedia.org/wiki/College_Board
The test is designed so that you still need to have a fairly high-level understanding of calculus and how the functions work. They aren't testing your ability to crunch out 56/7, but your ability to understand that the rate of flow of a liquid out of an irregular trough changes as the fluid level drops, and that you can model each portion of the equation correctly. A calculator isn't going to help you know that you need to account for a changing volume, and that the volume is directly correlated to the outflow.
It'll help you get the right formula for the area of a circle or volume of a cylinder, but not the higher-level reasoning that is really important.
In my opinion, that's silly, as none of the 3 semesters of college calculus I took allowed the use of a calculator, but that's the test that was mentioned.
(I actually did the same thing on mine, but fortunately extra calculators were provided, possibly because of their past experiences with students pulling this kind of stunt).
Ideally, the gifted should have their own programs where they are forced to compete with other exceptional students the entire time they're in school, rather than follow a program meant to pander to the lowest common denominator students in society.
Also, I'm with you bro, fell into the same trap. It sucks.
I hit a wall within the first five minutes of programming. I know I will mistakes and I make up for that by making small, clear code commits and trying to make my code easy to read.
I'm a much better programmer than I am a physicist. I am also much happier working as a programmer than I was studying physics.
The nice thing about this is, some of those survive revisions and can then be line comments which explain the next few lines of code:
# Frob the Foo.womble, so that Bar gets its input
# in the format it expects
There is one on hand the "gifted" who runs on his own fumes with no other form of external effort. Much like @austennailred notes about his efforts in high school mathematics, this type of "gifted" often suffers from something similar to the Dunning-Kruger effect in the long run --> in other words, a sense of superiority can potentially drive you into believing you are better than you actually are.
This, I believe, can happen when people attempt to pursue things which do not particularly satisfy their innate curiosity or their genuine interests. Sure you can be great at math naturally, but your view of how good you are limited by the population size of those you compare yourself against can leave you damagingly unprepared when you realize that you actually aren't as great as you thought you were.
What makes the "curse of the gifted" the "blessing of the gifted" on the other hand, is when that moment where you realize your own fallibility seems to provide a moment of illumination rather than a sense of dejection.
For me, my first attempts at genuine college level discrete mathematics versus having used basic counting, permutations, probability, number theory principles in the past showed that computer science or at least the mathematics of computer science can be difficult. I had a similar issue in chemistry as well. The difference was even though I was always a strong math and science student, approaching college level chemistry made me just feel inferior and made me drop it. On the other hand, when faced with adversity in mathematics/CS I tended to take the difficulty as a challenge. Rather than drop it, realizing that perhaps there were others who could be better at mathematics, CS in this new environment drove me to pushing myself to a place of intense focus I had had before.
Now this is where I think passion crosses with the concept of the "curse of the gifted" to make it more of a blessing than a curse. When you love something and you know what it feels like to be best, realizing your own shortcomings can sometimes cause you to miss that feeling. Now in order to get back to a place where you "feel gifted" you have to be in love with the work to get there over "the idea of being gifted" and this I think can happen to those who are put in a position where the feeling of "being gifted" is compromised.
Anyway, just my view, first "longer" post I've written on HN but I thought austennailred had a great point and just wanted to comment.
Great piece by both Eric and Linus and crazy to think how 14 years later, we have Git.
Why? Because before his teachers, back to his age 7 or so, had carefully walked him through the music note by note and bar by bar and, likely then, the artistic expression. But, still he was darned gifted. So, a few years later, relearning violin, he was fine again for the rest of his life.
Being able to do arithmetic, algebra, and other mathematical operations in your head is fine. E.g., during WWII, John von Neumann, one of the best mathematicians of the 20th or any century, sometimes went to people working on war projects to see if he could help them out. At one stop, a guy was unsure about how to find a solution, so von Neumann suggested, "Let's try a local solution via infinite series and look at a few terms. So, von Neumann did the arithmetic in his head [he could take two eight digit numbers and multiply them together in his head]. When von Neumann left, the guy was near tears. Why? Because he had been up all night using a mechanical desk calculator to get just some of that arithmetic right!
I found in doing research in math (I've published such stuff), once get into the problem well then do most of the thinking for the actual research just between the ears and writing very little. Can even do some of the algebraic derivations just between the ears. The actual research, at least the way I do it, is heavily 'conceptual' where writing isn't very efficient anyway.
From what you wrote, I see nothing wrong with your abilities. If you are not handy with paper and pencil, then it is not too much to expect you to become handy with just some usage. So, try, and soon you will 'get it'. It sounds like your problem with calculus was mostly just a problem with paper and pencil -- so, 'get it' with paper and pencil. Unless are, say, another von Neumann, can't expect to do significant derivations in calculus, say, integration by parts or multiple integrals, without writing down the expressions step by step.
But, of course, for calculus, long ago some software efforts for integration worked out what expressions could be integrated in closed form and what ones could not and wrote software to do the integrations for all the ones that could. The results are in at least one of the old computer algebra packages! So, with that software, "Look, Ma, no pencil or paper!".
About your suspected faults, don't be the least discouraged about them! Some are just your imagination. Some can be easily enough corrected. Essentially all the rest can be circumvented.
In particular, it appears that you are concerned about some skills that were supposed to take you from a few days to at most a few months to get good enough with way back long before you shaved! Doing catch up on that material now should be a few days' walk in the park. E.g., in K-12, I didn't learn English grammar and punctuation very well. In college I was in math and physics where this gap was not very important, and otherwise I just wrote only sentences where I did understand the grammar and punctuation. Now I've long had a 12th grade English grammar book right at hand and look up details when I'm in doubt; long ago I got nearly everything I missed in K-12.
Back to working without writing, in plane geometry I refused to do the homework the way the teacher wanted, write out in full detail proofs of just three problems. Instead I worked all the non-trivial problems and then also the more challenging ones in the back of the book, often writing nothing, sometimes writing in the margin of the book, sometimes on scraps of paper, and only for a few of the most difficult problems in the book, actually writing out a complete proof. Worked fine! When the teacher wanted to see my homework, I showed her nothing, and she was torqued. When I came in second in the class on the state test, she was 'confused' since she thought that I didn't do any of the homework! Heck, I solved every non trivial problem in the book and, thus, likely did the most homework of anyone in the class. Also, I slept in class -- I learned from the book in a quiet room, not when she was making noise in front of the class! Being able to learn math just from a book proved crucial, the main way I got my Ph.D.!
Lesson: You don't always have to do it their way!
Thankfully I wasn't smart or gifted enough that I could ride it for long, but when it comes to math and problem-solving I rode it well into my high school years. I never learned to do algebra "by the book," because I didn't need to. Or maybe because I wasn't smart enough to.
The math teacher would teach "3x + 6 = 9." Basic algebraic problem-solving says you subtract the 6 from both sides, then divide by 3. So "3x = 3" then "x = 1." Easy. But I learned pretty early on that I could do it in my head. It was a little bit challenging, but then I wouldn't have to waste the time of writing it out, and I wasn't handicapped like all of those suckers who had to go through the motions no matter how simple the problem was. If the teacher wrote "x + 1 = 6" I didn't have to subtract 1 from each side, I just thought about it logically and knew the answer. Of course, the math got more complex, but I was good enough at doing it in my head that, at least for a long time, it never really mattered.
I thought it was because I just "got" math, and the other kids were on a lower level. But as the math grew in complexity, I fell behind. By the time we reached Calculus I was still doing most of it in my head, as I had never really learned to write it out on paper. And the complexity of the math outgrew my capacity to visualize. I showed up to my AP calculus test without a calculator, partially because I was forgetful and partly for fun, and it wasn't until I got my score back (a failing 2 of 5) that it finally hit me: I was actually behind. In school. I was cocky enough that this was a slap in the face.
I had to start from scratch, and I'm still not sure if I've made up for a lot of that. I ended up in more creative fields, mostly because I felt inferior to those who had learned the rules and not been cocky douchebags like I had been in the beginning.
This really sucks to write. I frequently wonder what could have been.