Generality vs. Specificity

We want our students to have problem solving methods that are general enough to work in any situation. If we assign a series of exercises that are too similar to each other, it becomes easy for students to lock onto the wrong pattern, or to use a ‘trick’ that works just frequently enough to seem worth the effort to learn it.

One thing I tried this year was to prompt students to make themselves aware of the spectrum from generality to specificity. What works for solving specifically this question? What general ideas apply to answering all of the problems on the page?

I used my randomized question generator to help create problems that worked this way. Here’s an example:

I only started a deliberate effort to prompt these conversations at the middle of the second semester. I wish I was doing it all year.