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.
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".