*listening*to a clear and correct teacher explanation is even less effective than

*giving*an incorrect explanation to a computer avatar. And this year, I had the dramatic experience of walking three classes through a solution to a problem they had gotten wrong on a test, only to have virtually every student get the same problem wrong on a quiz the week later.

**time spent talking is time wasted**.

- Listening to
*anyone*talk is boring after a few minutes. How long can you sit quietly and listen to your best friend tell you about the crazy thing that happened on the way home from work? My voice talking about solving trig equations is certainly no more gripping than that. - Most teacher talk starts from where the teacher is and wants to go, not from what the students know and want to learn. Not only is this kind of talk emotionally irrelevant (see point 1), but it fails to address any underlying preconceptions or misconceptions--which is why a "clear explanation" so rarely is. To make matters worse, we math teachers often couch our teacher talk in vocabulary that the students only barely understand, so that the words themselves are mystifying.
- When I'm talking, my students are either ignoring me (point 1) or listening attentively and trying to take notes, but in neither case are they actually doing mathematics, which is the one activity that I can guarantee will produce learning gains.

- The message that talking is ineffective is counterintuitive. Math teachers are expected to be experts at doing math problems. So it's natural to think that explaining how to do a problem is the way to put this expertise to work. Students may believe this more deeply than teachers: I've had students this year complain that I no longer go over
*any*test questions, until I draw their attention to the fact*they*helped establish, that going over the solution didn't actually result in their learning any math. [For the record, I couched the issue as my problem/fault/responsibility rather than theirs.] - The intuition (that talking actually helps) is bolstered by our own experiences as math students: for the most part, my teachers talked at me, and here I am today. We have to remember that we are the
*survivors*of this method; but our survival is not proof the method worked. Someone lucky enough to be alive and healthy in London in 1667 would hardly credit the previous two years' outbreak of the Black Death as the reason for their vitality. - Teaching students to learn from their own talk is difficult; in my experience, the weakest students are also the ones who have the most trouble with the message that "the discussion is the lesson." (I have two hypotheses about this: first, that weaker students are rationally mistrustful of their own abilities, and overextrapolate to "anything
*I*do myself will not help me learn"; second, that the way they got to be weaker students is from being talked at by teachers, so that the weaker students are the ones who've been talked at the most, and who have the least experience with other ways of learning.) - Talking gives the illusion of speed. I can describe how to solve a problem in under three minutes, making time to go on to the next thing.
- As we've discussed, figuring out what to do
*besides*just explain-and-practice is extremely difficult: it requires planning, real-time data analysis and decisionmaking, and reflection and revision. So "teacher talk" is an easy default.