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I'm currently teaching Calculus I problem solving sessions for college freshmen. The problem is that the majority of students have studied the material in high school (what's a function, even/odd, limits, asymptotes...). What can I do in this situation to make those who're familiar with the material engaged in the problem solving sessions?

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    $\begingroup$ engaged in the problem solving sessions I think you already answer your question. Let them solve the problem while you watch them do it. $\endgroup$ – scaaahu Sep 18 '16 at 12:14
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    $\begingroup$ Why do you want to engage them? You are re-explaining something they already know. They should not be required to attend the lectures. $\endgroup$ – Federico Poloni Sep 18 '16 at 18:54
  • $\begingroup$ @Federico Poloni I totally agree, but it's the course instructor's decision that attendence is mandatory, which I can barely do anything about. $\endgroup$ – sars53 Sep 18 '16 at 18:58
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    $\begingroup$ If you're standing up in front of the class and solving problems on the board, stop doing that, because it's obviously not working. Offer students flexible possibilities for using the time well. For example, ask each student to write goals for the next session on an index card. Sort these cards into stacks, and find things they can do if they're in that stack. Email these things to them. Then at the next session, walk around the room and help them with what they're doing or check their work. $\endgroup$ – Ben Crowell Sep 19 '16 at 3:09
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One thing you might do is try to incorporate non-routine questions into your Calculus problem solving sessions. Knowing the requisite material is not enough to solve a [non-routine] problem (i.e., question for which the method of solution is unknown at the outset; used in contradistinction with exercise).

There is a lovely paper ("technical report") with five example problems that you might try out:

Selden AN, Selden JO, Hauk SH, Mason AL. (1999). Do calculus students eventually learn to solve non-routine problems? Department of Mathematics, Tennessee Technological University. Link.

Here are the five non-routine problems:

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The paper explores how effectively students did (or did not) use their resources (in the sense of Schoenfeld, 1985) when attempting these problems. Perhaps the ideas contained in the aforecited/aforelinked can help inform how you approach your own problem solving sessions.

To close, here is the final, non-original (Putnam 98/B2) problem foot-noted by the authors:

Given a point $(a, b)$ with $0< b< a$, determine the minimum perimeter of a triangle with one vertex at $(a, b)$, one on the $x$-axis, and one on the line $y=x$. You may assume that a triangle of minimum perimeter exists.

(I suggest trying all the problems yourself, first, before reading the solutions!)

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    $\begingroup$ Alright. I'll check the article out. I liked the idea that maybe I could incorporate this type of questions to the problem solving sessions. Thank you! $\endgroup$ – sars53 Dec 17 '16 at 20:49
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If your students are anything like my students in Calc I, I'd emphasize a different perspective on what they're doing. Many first year students have a mechanical/algorithmic understanding of the derivative and don't have a very strong geometric understanding. A favorite of mine is the inverse function theorem: if $f$ is differentiable and invertible near $b$, then $(f^{-1})'(b)=\frac{1}{f'(f^{-1}(b))}$. This has a great geometric justification. Sure, the strongest ones might be able to prove this algebraically using implicit differentiation, but it'd be nice for them to understand why it's "obvious" geometrically. (Also, problems involving $(f^{-1})'$ are also hard for them to interpret physically; that's another reason I emphasize this.)

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