- Show that: Show that if f(x) = sin(x^2) + e^(sin(x)) then f'(x) = 2xcos(x^2)+cos(x)*e^(sin(x)). This is an example of the "Show that A can be transformed by some relevant mathematical process into B" type of question. Such questions have the advantage that students at least know where they are going, and just have to find a way to get there. Also, the question itself provides immediate feedback on the accuracy of student work. Someone still has to ask the student to explain every step of their solution, but other than that the question is pretty self-contained.
- Spot my mistake: I claim that if f(x) = ln(x^3) + e^(x^2)*cos(x) then f'(x) = 3/x - 4*e^(4x)*sin(x). What is my mistake? This is an example of the "Critically examine to find a mistake and try to understand what the person making the mistake was thinking." Most of my students really struggled with this one because they did not recognizing the need for product rule. There were a lot of good discussions going on between students comparing their ideas of what might be wrong in the solution. To me, the benefits of this type of question are that they promote critical/logical meta-cognitive reasoning and discussion.
- Verify: Is it true that if f(x) = ln(sin(x^3)) then f'(x)=1/tan(x^3)? If not, explain what misunderstanding might have caused the mistake. This of course is very similar to the second type of question, but a little less scaffolded because the students are not told to expect a mistake. Once again, this gives rise to good discussions between students about whether the solution is correct, where potential mistakes are, and what misunderstandings/correct understandings give rise to this solution.
All-in-all, I see these questions as best being used in sequence, though depending on the difficulty/novelty of the topic one or all steps may be skipped. After these scaffolded questions students should be ready for the classic "solve this..." exercises. Because the three question types mentioned above have other benefits than just providing scaffolding, it makes sense to me to intermittently use them for variation and to encourage reasoning and discussion even once students are confident with the less scaffolded exercises.
Some questions that need to be answered: how do we ensure that students gradually let go of the scaffolding (their classmates, their books and notes)? How can scaffolding activities be combined/coexist with explorative investigation-driven open-ended work?