This question is targeted at teachers who taught both low and high level mathematics. I have a group of students that I'm currently teaching precalculus and they seem to be doing really well in all computational areas. For example, they can all do a problem like "Find the partial fraction decomposition of $\frac{3x}{x^2-4}$" or "Compute $\lim\limits_{x\to\infty}\frac{x^3+2e^x}{3-e^x}$". However, as soon as we hit abstract notions, they fall apart. We just reached Intermediate Value Theorem and they just can't seem to make the leap into abstraction. Consider a multiple choice question:
Let $f(x)$ be a continuous function and let $f(1) = 3$ and $f(3)=11$. Which of the below statements must be true?
- $f(x)$ is increasing on the interval $(1,3)$
- $f(2)=7$
- $f(x) = 5$ for some value of $x$
- $f(x)$ has a local maximum in the interval $(1,3)$
The majority of the students chose 1 and only a few chose the correct answer 3. We did 3 or 4 similar exercises and I explained each one (or so I thought). By the time we did a similar question #4, there was only a moderate improvement. Can someone suggest how to help students make a leap into the abstraction? How do you teach abstract concepts to students who have only encountered computational problems in their math careers?