# College Math Teaching

## March 5, 2014

### Comparing calculus exams…..and university students

Filed under: academia, calculus, pedagogy — Tags: — collegemathteaching @ 2:04 am

Some people tried to argue with me about calculus; they seemed to think that calculus at one institution is the same as at another one.

Hardly.

Not only can courses vary in terms of topic and difficulty, but so can exams…and the difference might be very subtle to those who are unfamiliar with giving and grading exams.

Here is one example: suppose you want to examine the students on the Mean Value Theorem. How might you do this?

1. State the Mean Value Theorem (yes, the bad students usually can’t even do this).

2. State and prove the Mean Value Theorem (prove using what?)

3. Let $0 < x < y < 1$. Show that there exists a $c$ between $x$ and $y$ so that $y^2 - x^2 = 2c(y-x)$.

4. Use the Mean Value Theorem to show that $|cos(x) - cos(y)| \leq |x - y |$

5. Show that for all real $x, y, |cos(x) - cos(y) | \leq | x-y|$

Not only do these questions vary in difficulty, they may or may not have been covered directly in class prior to the exam; that makes a big difference.