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?

  1. $f(x)$ is increasing on the interval $(1,3)$
  2. $f(2)=7$
  3. $f(x) = 5$ for some value of $x$
  4. $f(x)$ 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?

  • $\begingroup$ Ask them to draw a picture that supports the answer given. $\endgroup$
    – Dan Fox
    Commented Jun 24, 2021 at 8:38

1 Answer 1


Here's one quick idea of an exercise that might shake students out of their narrow thinking.

  • Give the students three coordinate grids with (1,3) and (3,11) filled in with closed circles. (I'll use the numbers from your example, but I'd probably choose better numbers if I were doing this for real.)
  • In the first grid, ask the students to graph a continuous function on [1,3] that goes through the two points.
  • Then, in the second grid, ask them to graph a DIFFERENT continuous function that goes through the two points.
  • Then have them look over the four multiple choice options and see how many of the options are satisfied by both of their graphs.
  • In the third grid, graph a continuous function that goes through the two points such that only one of the four conditions is satisfied by all three of their graphs.
  • Have the students pair up, describe what they did, and then hold a class-wide conversation where people shared what their pairs talked about.
  • 2
    $\begingroup$ This is a really good answer. I will use this idea in other scenarios that require abstract thought. For example, I can already see how this can be used to illustrate the MVT - I could challenge the students to draw a continuous, not everywhere differentiable function that doesn't satisfy MVT, for example. I really appreciate your help. $\endgroup$ Commented Jun 19, 2021 at 1:31
  • $\begingroup$ @Misha Shklyar: (+1) I also thought this was a good answer (great, actually), especially the last part. In fact, I think it'd be better to practice examples like this by beginning with the students being in groups of two or three. $\endgroup$ Commented Jun 19, 2021 at 18:31

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