I have always found that in these kinds of situations learners respond in one of two ways:
either with excitement and curiosity or resentment and bitterness. Students who have come to my class from a previous course that emphasized rote memorization and procedures sometimes feel like they're "doing everything right" but still can't come up with the right answer- a huge source of frustration. Students who have been asked to reason on their own, look for patterns, and solve puzzles in previous courses often find these types of problems invigorating and exciting; they live for the "thrill" of discovering the right move to make.
One way I've tried to combat this and get everyone more on the "I can solve it!" page is to try to make these problems into puzzles whenever possible. I love the way this Time Magazine article discussed the human desire to problem solve:
Human brains have an extreme form of consciousness: they’re ravenous for new innovative solutions to problems in the world, ravenous for optimizing our lives, for building pyramids of knowledge.I have found that the more I try to celebrate the "cute" and elegant mathematical moves and help kids see the repeated reasoning that goes into these problems, the less resentful they become.
Here is a puzzle activity for converting polar and rectangular equations. Directions I give the students:
I have partners sort them into groups and try to put the "steps" in order. They then need to create a "booklet" (fold a piece of paper in half so there are 4 sections- 2 on the front, 2 on the back) with one problem on each "page", glue the steps down in order, and explain the reasoning between each step in their own words. I love the conversations I hear between partners, especially on a problem where they "just know" what the next line should be because it seems correct, but they need to think about how to justify it. This really helped my kids dig into the "why" of these sorts of problems.
