Corrinne Manogue at OSU is the source of this one:
You teach upper-level physics. Say, you want to teach your students about eigenvectors. You could
(A) Introduce the word "eigenvector" before or at the beginning of lecture, explaining what the term means and where it comes from. And then lecture on how to solve for eigenvectors, and then have students practice.
(B) Put students in groups: give each group a different (carefully chosen, of course) matrix and ask them to see if they can find any vectors that don't change direction when you multiply it by the matrix. Let them explore, remember how to perform matrix multiplication, encourage them to draw (not just do algebra), watch them develop an intuition for what each matrix is doing, and try guess and check, encourage them to use geometrical insight to rule out or hone in on solutions, let them struggle with whether there can be more than one solution. Then let them share their solutions with their peers. Point out important similarities and differences across problems and solutions strategies. Point out important things you'll need to bring up later. Then, then introduce the word "eigenvectors". Draw on the insights they have (and haven't) made and present the formal method for finding eigenvalues.
The argument for doing A could be this: "Students don't have any intuitions about eigenvectors and linear algebra. It's a weird word that's distracting. If I introduce the word before lecture, it will help them focus on the mathematical structure and methods I want to teach, rather than on the weird vocabulary."
The argument for doing B is this: "By drawing on what students do know and can do, you can quickly build up a set of intuitions that orient students to the concept of eigenvectors. Since, they are not likely to formally develop all the methods on their own, I can capitalize on what they end up doing to anchor the formal instruction to their own ideas and methods."
Anyway, what do you guys think?