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.
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.