Algebra Exams

Grading Algebra exams^* a few weeks back left me with the conclusion that my students don't understand functions.

For example, they are given that f(x) = sqrt(x) (i.e., "square root of x"; no square root symbols on this keyboard) and that g(x) = x + 1. When asked to find the domain of (f+g)(x), the two most popular answers are "all real numbers" and "sqrt(x)". The following are answers to the domain of the composition f(g(x)):

• all real numbers
• all real numbers except x = -1
• (sqrt(x))(x + 1)(x)
• {f < x < g}
• I don't know what I'm doing.

The last answer I can't really argue with. Should I give that a point?

Now, almost two weeks later, I gave the students a bonus quiz as an opportunity to make back some points. After having notice for some time that the quiz would cover the same material they bombed on the test, I'm not sure I see evidence that most students did any better on the quiz. In fact, I saw pretty much the same answers on the quiz.

I'm told that the failure rate for this course is somewhat high, so it seems my results are not atypical. But it leaves me wondering: What goes through the minds of my students? How are they studying? Do they think it's working?
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^*Yes, this is a somewhat math-y post. If you didn't take/don't remember algebra, you may not know why I find these answers funny.