First of all, I don't think math/science is harder than history or art. I consider both Newton and Shakespeare to be geniuses; same with Bach and Leibniz. I think it's an established empirical fact (maybe not -- see NYT article) that to become an expert at anything, it will take you upwards of 10,000 hours -- be it playing violin, studying history, or doing number theory. So I don't think the effort is any differentiating factor.
You'd be surprised how often this happens. In my calculus classes it was virtually rampant. We'd get some material that was non-trivial to figure out on an exam. That's just bad teaching. Sure, some people will argue that it separates those a better grasp of the material from those with a worse one (maybe it does), but to me it seems unfair. Say I'm not a bright student but I understand the homework very well; the exam, however, uses a non-trivial combination of the elements found in the homework. I personally think that's bullshit. Why not go over the most mind-numbingly difficult problems in class? Often times, there are only a limited number of tricks that can trip you up; if professors would go over these, everyone would pass. But I guess we don't want that (why not?).
Consider an analogy: I decide to prepare for a running competition. My trainer shows me how to keep my heart-rate up, how to jog briskly, etc. All this is done on a treadmill on the low setting. Come competition day, it turns out that it's a 25k through the Australian outback. I'm not sure how anyone in their right mind would think the trainer did a good job if he knew exactly what I was getting into.
Exams like that are where the teaching function of school collides headlong with the (more important) sorting function of school. I found the phenomenon frustrating in engineering school, and even more frustrating in law school. They both prominently featured exams that looked nothing like what was taught in class. They both favor the kind of cynical person who ignores the larger ideas taught in class to focus on day 1 on the final exam, and also the socially competent who can get previous exams and the like from social networks. They also, completely incidentally, identify the few people who are so beyond the curve they could answer anything the professor might ask without studying, which lends them an air of legitimacy that perpetuates their existence.
You see it less in the humanities because the sorting function of school is so much less important than the teaching function. No lucrative jobs are riding on getting a 3.7 in art history versus a 3.4. (Everyone will be unemployed anyway...)
As an aside, I'm convinced grades are just an awful aspect of "education." They don't serve any purpose other than making it easier for employers to figure out who to hire, which I don't think is a legitimate function of school.
The irony here is that most R&D/engineering jobs do their own training. My mom, for example, works as a chem engineer/researcher -- every time they hire someone, they also train them. I think it's ironic that people bust their asses in school to get a 3.5+ and end up being trained like little robots anyway once they find their "dream job." Anyone with some elementary knowledge of chemistry could pass these training courses and get to work in a couple of weeks.
It just goes to show how disconnected academia is from real life.
> "the socially competent who can get previous exams and the like from social networks"
side note: at Cambridge, every past exam in every subject[1] is in the college library[2], and it's assumed that you'll photocopy them and work through the last 5 years or so for practice. This doesn't help much, of course, because you get a new crazily difficult exam every year[3].
[1] probably not going all the way back to 1209...
[2] i.e. 50-odd small libraries near the students' accommodation, mine was accessible 24 hours a day.
[3] unless it's Terentjev in TP1 (Theoretical Physics 1), who just set us a bunch of previous years questions, which I hadn't bothered to finish solving because they never reused questions. I'm not bitter, I promise...
> They don't serve any purpose other than making it easier for employers to figure out who to hire, which I don't think is a legitimate function of school.
Interesting point. At the same time it justifies the existence of schools since a lot of people go to school to get a better job.
>to become an expert at anything, it will take you upwards of 10,000 hours
Wrong. 10,000 hours will make a unskilled person moderately skilled, but if they lack the innate talent or intellectual capacity, it won't make them an expert. This sort of meritocracy where people just need to try harder is completely wrong, and dangerous to encourage.
That Time article leads with a mischaracterization of the debate. I haven't read Outliers, but I did read a bit of K. Anders Ericsson's original work. Ericsson qualified the 10,000 hours saying that it had to be a certain type of deliberative practice, and that there was a limit of about 4 hours per day which could be spent on it.
In other words, it has never been the case that 10,000 hours is all that's needed, only that that is a minimum number.
However, you reject here Ericsson's entire thesis, that being summarized in the Time article as "Ericsson doesn’t deny that genetic limitations, such as those on height and body size, can constrain expert performance in areas like athletics — and his research has shown this. However, he believes there is no good evidence so far that proves that genetic factors related to intelligence or other brain attributes matter when it comes to less physically driven pursuits."
You made a claim that "innate talent or intellectual capacity" is essential to being an expert. This is a widely held belief, first articulated (I'm told) in Vasari's "The Lives of the Artist" (1568). The debate is - where is the evidence which justifies this widely held belief?
You see a danger in promoting this "meritocracy" ideal. I have two objections to that. First, I think you are using the term incorrectly. Suppose we only have innate talents as part of our genetic nature. Under a meritocracy, those who are naturally smarter, etc. will get a better job or higher position than those who aren't. It doesn't make a difference if it was achieved through deliberative practice or innate nature. Instead, I think the better word is 'egalitarianism.'
Second, it reminds me too much of various view of "noble blood" and rusty arguments that women by nature can't do X, that black people by nature can't do Y, and that Swedes are by nature dumb squareheads. There's any number of these stereotypes once fervently held to be true, but which haven't stood up to the test of time.
Also, there shouldn't be a good argument that "people just need to try harder." Ericsson's view is that "deliberate practice requires effort and does not lead to immediate reward" (quoting from a summary by David Zach Hambrick). There are many good reasons to not become an expert on a topic, including: 1) the person could be interested in becoming an expert in another topic, 2) it can be hard to find the 10,000 hours needed for practice, especially if that time is spent making money needed for rent and food, and 3) other requirements may be missing; it's hard to be an expert surfer if one lives in Colorado, or downhill skier in Miami.
The path towards being an expert in anything is chaotic. I am one of the world experts in writing software for a certain subset of biochemistry. But had I gotten a different job just out of college, I would probably be in a completely different field. It's really hard for me to believe that I was born with this talent, given how much education and job-related experience I had through to get to this point.
Bravo for responding so clearly and accurately to such a pervasive and wrong belief. The best overview/summary I have read is The Genius in All of Us by David Shenk[1]. It would probably not add much to your understanding, but to anyone who hasn't looked extensively into these ideas, read this book.
Seriously, if you're reading right now, stop what you are doing, and read this book. It will change your understand of what you are capable of.
There are plenty of people that spent 10,000 hours studying math but due to mental limitations are far from experts. However, most people don't spend anywhere near that amount of time studying math unless there reasonably intelegent in the first place which suggests that the only validity to that number is the less gifted generally give up before wasting that much time.
What really makes math stand out is there are people who became recognized experts well before that magic 10,000 hour number. EX: Srinivasa Ramanujan http://en.wikipedia.org/wiki/Srinivasa_Ramanujan
Oh? He had almost no formal training, but that doesn't mean he didn't put a lot of work into it. How many hours do you think it took before he was a "recognized expert", or since this is Ramanujan, before he developed new theorems which would later be recognized as being that of an expert?
Quoting from the Wikipedia link: Ramanujan's introduction to formal mathematics began at age 10. He demonstrated a natural ability, and was given books on advanced trigonometry written by S. L. Loney that he mastered by the age of 12; he even discovered theorems of his own, and re-discovered Euler's identity independently. He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, Ramanujan had conducted his own mathematical research on Bernoulli numbers and the Euler–Mascheroni constant.
7 years at 4 hours per day = 10,000 hours.
Do you think he averaged less than two hours per day on math as a teenager? Based on what little I know about him, I don't think that's the case.
Also, your terminology is the core of the debate here. You say "the less gifted," but the debate is that there may be no "gift", but instead the dominate factor is the willingness of the person to go through a difficult learning method with delayed rewards, in order to become an expert.
I studied math, physics, and computer science as an undergraduate. I spent a lot of extra time learning and practicing software development, while I rarely did math and physics beyond what was needed for coursework. I believe most of my 'deliberate practice' went into CS. I'm now an expert in software development, especially as it relates to biomolecular structures. I firmly believe it is my interest in the topic and the lack of competition (meaning that it pays well) which led me here, and not some intrinsic gift.
It depends on how high you set the bar for 'expert'. If you mean 'doyen', then yes, 10k isn't going to do it. If you mean 'has a solid grasp of the ins and outs, avoids common pitfalls, and can carry an informed, intelligent conversation on the topic', then 10k will do it.
When you look at what many great accomplishments, you often find a person that had a drive to succeed in his field that bordered on an obsession. When you are truly so passionate about something that it becomes your entire life, I believe it is possible to appear as if you are a genius, or maybe to even become one. In fact, I think that true genius can be attributed more to hard work than anything else.
Look at people like Bobby Fischer, he spent an unhealthy amount of time playing / studying chess (supposedly like 18 hours a day), and it destroyed every other relationship in his life. I'm sure he was intelligent, but without developing an obsession with chess, he probably wouldn't have been much of a player.
There was also a kid whose father forced him to study every waking moment, according to an IQ test, he was a super-genius (I think it was over 200), but in reality he was just a poor kid with a fucked-up, abusive father. I remember reading about it in the news, but I can't seem to find the article.
Those were interesting articles, but there was a much more recent example. The kid in question was interviewed on talk shows and stuff. His mom divorced his father over the insane treatment of their child, and she gained custody.
I'm not sure how the article supports your argument. Looking at its conclusion, it looks like the "10,000-hour hypothesis" is an open question (at the very least).
Sports scientists have pointed out[1] that people who are innately good at a thing will do more of it, with less resistance to training, than people who aren't.
The 10,000 hours are done by people who enjoy a subject or activity enough to do it almost constantly. And that usually comes about because of early successes.
There are geniuses in every field, therefore every field is just as difficult intellectually? I don't follow your reasoning here. What sort of evidence would convince you that math/science are in fact harder for humans than history or art? (Edit: A somewhat different perspective I have is expressed by Feynman here: http://www.youtube.com/watch?v=NWjV0bNBPY4 The good men and women of a field are more important than the field, and you can probably find good men and women in every field if you look hard enough.)
As a teacher, I take strong issue with dvt's comment: "We'd get some material that was non-trivial to figure out on an exam. That's just bad teaching."
Exams are supposed to be non-trivial, if they are to test your understanding of the material. When I teach freshman calculus, I invariably get this kind of comments from students who aced math in high school because they had basically memorized all possible question patterns from the textbook. But did they understand it? More often than not, they hadn't, really. And when they get a question that doesn't fit a pattern they've seen before, they call it a "trick", when it's anything but.
I work hard at getting my students to understand that math is not about memorizing stuff but about understanding stuff. You have to know the basic concepts and techniques by heart, of course, same as any subject, but anything more is just icing (unless your brain works in such a way that memorizing patterns helps you understand general principles, in which case memorize away, but don't mistake the means for the end.
Many students tell me they don't understand why they got a failing mark on an exam because they did all the homework and/or put in tens of hours of study. They seem to think that these actions should somehow guarantee them a passing grade, and if it didn't, it's obviously because the exam was unfair.
Now let me be perfectly clear: I don't give hard exams. In fact, most of the questions I ask are downright easy, provided you understand the material. Here's an example: "Sketch the graph of a twice-differentiable function f(x) whose domain is the real numbers and which satisfies the following two conditions: f'(x) is negative for all x, and f''(x) always has the same sign as x." This was in fact a question in my calc 1 midterm last year.
Out of 60 students, 10 did not write anything. 10 drew something that was not the graph of a function. 10 drew a function that did not satisfy any of the requirements. 10 drew a decreasing function but got the concavity wrong somehow. 20 gave a correct answer. (This is all approximate, of course.) The average mark for this question was probably around 2/5.
Was this exam question harder than my homework problem sets? Absolutely not! It's just different. Here's an example of a homework question relating to the same material in a similar way: "A differentiable function f(x) is such that f'(x) never changes sign. What can be said about the number of zeros of f?" This is more difficult than the exam question because the step linking the sign of f' to the number of zeros of f (drawing a graph) is not explicitly suggested, and because the answer is "f has at most one zero" and not "f has exactly one zero".
You teach someone how to do X, lets assume this goes something like: Step 1, Step 2, Step 3, Step 4, done. You then teach someone how to do Y, this goes like: Step 5, Step 6, done. On the exam you ask someone to do Z. This follows from a nontrivial combination of Step 1, Step 3, Step 6, done. If anyone gets it right, don't flatter yourself. You didn't teach them how to do Z.
Either they have a sort of a priori intuition of the material (this is how I get by most of the time), they got lucky, or they had someone else teach them. Mathematicians (and other academics) feel the need to make their subjects so obtuse they seem insurmountable. Math is not hard - some guy saw an interesting behavior of a function and wanted to see what happens when he tries to differentiate it. Programming is not hard - some girl thought she could make her life easier by writing a program that writes other programs. This pretty much exemplifies all of human understanding. It's not much more than that.
Of course I'm not suggesting that complex analysis or the Dragon Book are trivial, all I'm saying is that they are not hard. But academics themselves often discourage people from pursuing science and math (numerous examples in this thread alone). We can blame the government, elementary schools, and parents all we want, but it's blatantly obvious that universities are broken. The fact that students are tested on material not covered in class (or nontrivial combinations of material covered in class) is inane.
That's called problem soloving. You see the problem, see that it is a combination of smaller problems, you solve them.
Lots of problem solving at school was teaching exactly that: how to transform a problem into the ones you can solve with step-by-step approach. This was true not only for math, but for physics and chemistry too.
Yes, but I would agree with dvt that there are professors who consider themselves "clever" for putting material on the exam that looks nothing like what showed up in lecture or in the homework.
Kinds of problems that can justify being on an exam are surely important enough to be in lecture or on the homework. Putting a special kind of problem on the exam that must be deconstructed before it can be transformed in a problem that showed up in the homework is a "trick".
People keep dodging my analogies, I've given two thus far. I guess one more won't hurt. This one isn't very good, but I hope you'll get the gist of it. You take an art class and you're taught the basics of painting -- color, contrast, texture, shading, etc. Your final exam is to reproduce the Mona Lisa (or pick any equally-daunting piece of art).
Of course da Vinci used the same principles of color and shading to paint the Mona Lisa, but the final exam does not seem to test the skills you were taught -- rather, it tests your innate ability to be a great painter. Undoubtedly, some people will get A's, some will get A-'s, and some will get B's. But if person X has some sort of innate talent that person Y does not have, X has a clear and distinct advantage on the exam -- an advantage that has nothing to do with the class and nothing to do with the teacher.
Consider another example: if a friend of mine asked me to "teach him how to program" I wouldn't give him the building blocks without the caveats -- one of the first things I'd do is tell him that off-by-one errors, for example, are a very common caveat in for loops.
And yet, I've taken programming courses in which this kind of trickery (CS professors love to fuck with you by giving retarded off-by-one puzzles) borders on immoral. I've had friends in said classes that had no experience with programming (unlike me) that received unsatisfactory grades because of this kind of incessant trickery. Thankfully, CS books are written magnitudes better than math books.
It's not obvious to me that the example you give is "downright easy, provided you understand the material". Now, I understand the material, and I find the example easy. But I can see that it involves a few little cognitive leaps which it may not be reasonable to expect a student to make. The straightforward solution, I think, involves considering x < 0 and x > 0 separately. How is a student supposed to know that that's a reasonable option? Is "when a question uses the phrase 'the same sign as x', try considering each sign separately" in the textbook? Are students supposed to figure that out somehow?
The question is whether exam material should seem substantially different from the course material.
Looking at your first question, I am hemming and hawing whether it is correct to say zero is positive or negative or both. I would say both, thus f''(x)=0 leads us towards a legitimate solution. Having not attended your course, it is possible the issue came up multiple times and this would cause no confusion. But it might confuse someone who has an otherwise excellent command of calculus.
But given the homework problem shown later, it is rather likely that those paying attention found your exam question perfectly fair.
"Was this exam question harder than my homework problem sets? Absolutely not! It's just different." The kind of differences matter. I do not see why the exam questions need to seem different in any non-trivial way. (I speak of a general principle. I do not hold an opinion about your particular questions.)
> f'(x) is negative for all x, and f''(x) always has the same sign as x.
Say, let f'(x) = -e^{-x^{2}}, and f''(x) = 2xe^{-x^{2}};
f(x) = \Int{f'(x)dx} = -0.5sqrt(pi)erf(x) + C, where erf is the error function(, OK, I cheated with Wolfram Alpha, and never worked out the integral part myself).
But the point is to draw a function with these properties. You just have to have a smooth curve that approaches 0 asymptotically and is concave up. No worries about graphing any particular function!
Either you got the description wrong or I'm especially rusty - in that case, f''(x) = e^(-x), which is positive even when x is negative, so it doesn't always have the same sign.
The quoted part is admittedly a bit awkward to read -- I exemplified exactly what I meant in my analogy. It's true that most good universities "sort" rather than "teach." My argument is that sorting is not what schools were invented for. We have Olympiads for that.
Plato's academy (and most of the ancient academies) were not built to "find out who was the best" at X or Y. They were built to a) teach current ideas and b) develop new ones.
The problem with post-scholastic (and I would argue even scholastic) thought is that we've mixed up the two roles. Grading (as we know it now) is a relatively new invention... probably invented around the late 18th century at Cambridge.
And yet these non-sorting educational institutions still produced Newton, Galileo, Augustine, and so many others.
But I do think that in math/engineering classes there is a disproportional amount of trickery. There are numerous memes that exemplify this: http://www.thefunnyblog.org/wp-content/uploads/2012/05/funny...
You'd be surprised how often this happens. In my calculus classes it was virtually rampant. We'd get some material that was non-trivial to figure out on an exam. That's just bad teaching. Sure, some people will argue that it separates those a better grasp of the material from those with a worse one (maybe it does), but to me it seems unfair. Say I'm not a bright student but I understand the homework very well; the exam, however, uses a non-trivial combination of the elements found in the homework. I personally think that's bullshit. Why not go over the most mind-numbingly difficult problems in class? Often times, there are only a limited number of tricks that can trip you up; if professors would go over these, everyone would pass. But I guess we don't want that (why not?).
Consider an analogy: I decide to prepare for a running competition. My trainer shows me how to keep my heart-rate up, how to jog briskly, etc. All this is done on a treadmill on the low setting. Come competition day, it turns out that it's a 25k through the Australian outback. I'm not sure how anyone in their right mind would think the trainer did a good job if he knew exactly what I was getting into.