Sunday, February 27, 2011

Less Talk, More Math

Behind all this talk about teacher talk we've been dancing around a central idea that I think is worth stating and thinking about explicitly:  less teacher talk is more.  I think we've both come to believe that the animated Charlie Brown shows are pretty much on the money about the students' experience of listening to their teachers:

When I was in DC, the NSF showed me a slide that suggests that, as a learning activity, 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. 

The moral is simple: for the most part, time spent talking is time wasted.

Why is teacher talk so ineffective?  And why is it hard for us to shut up and let the students do the talking?

Addressing the first question:
  1. 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.
  2. 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.
  3. 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.
None of these problems are particularly surprising, and it's hard for me to imagine that most teachers aren't aware of them.  But we yak on.  Why?
  1. 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.]
  2. 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.
  3. 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.)
  4. 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.
  5. 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.
But if all the above are reasons for talking, I think there's an additional underlying cause:  being the center of attention is fun.  Part of being a teacher is having an odd kind of power, and power is intoxicating.  As with so much of the affective parts of teaching, I think it's important that we recognize our own complicity:  we talk, in part, because we get a charge out of talking.

And that's enough talk for now.  Next post: two of my favorite recent problems.

Sunday, February 20, 2011

What do we really think is important??

P.J. is right that we have to plan our questions, but then what happens when something occurs that we did not anticipate? It goes deeper than careful planning, at least in the usual sense of the phrase. We need to work hard to break our bad habits about questioning. We must develop habits of language and thought that enhance learning. We need to be clear about what our objectives are and pause before we ask--or answer--a question, and make sure that our response is consistent with the outcomes we desire.

With regard to PJ’s surprise solution, I think the moment in class when a student connected two ideas from totally different places and solved the problem ought to stand as one of the high points of any teacher’s career. At that moment, PJ’s student achieved a higher level of doing mathematics than merely understanding a well-known theorem. PJ’s student demonstrated, in the presence of the entire class, what it means to do mathematics. The student’s insight is more important than the theorem that was meant to be taught that day. A meta goal was reached. Exciting moments like that are rare and can only happen when students are encouraged to approach problems using their own intellect and intuition. If we could make those events happen every day, mathematics would be the most exciting class for every student in every school. It is important that PJ knew to set aside the lesson at hand and celebrate the insight of one of his students. But how does a teacher ensure that this sort of exciting moment happens?

I think we can establish an atmosphere in our classroom in which students will be willing to take chances and are not afraid to be wrong. I think we can structure our classes so students will try to think of clever solutions and will try to make connections. How can we do this? By asking problems that are rich enough that students can get started on them but will not necessarily see the end for a while, by celebrating many different ways to solve a problem, and by asking questions that encourage students to think for themselves. The first thing is to give them time to work and encourage collaboration. In other words: instead of asking if everyone understood, how about asking: Did anyone do the problem a different way? Does this solution remind anyone of another problem we have done? What made that student think of using the Power theorem?

Asking questions should be a means to stimulate further thought. If we want to know what students know, we can ask them to do an interesting problem, and then we can walk around and see if they can do it. We can observe their errors and let them sort out a solution among themselves. If a question does not take us further along in the investigation at hand, we don’t need to ask it. We can ask students: what relevant questions arise after this problem has been solved? What is the next question we might ask? Would this solution have worked if the coefficients had been irrational numbers? What if the point had not been on the circle? Did you need everything that was given to solve the problem?

And often, there should be no question—just a situation; the students’ first task is to determine what questions can be asked and answered.

Monday, February 14, 2011

We Have Seen the Enemy and He Is Us


Teaching my classes this week after reading your post, more than once I found myself staring into the cruel mirror of recognition: as much as I "know" the questions I shouldn't ask, I catch myself asking them, or almost asking them, more than I care to admit.

(And, for the record, they are terrible questions--so bad, in fact, they're not even worthy of the name, because as you point out, they don't even ask anything. So from now on, I'll call utterances like "Everybody get that?" or "Are there any questions?" nonquestions, as opposed to genuine questions like "So if that claim is true, what about ... ?" )

So why is that? Why is it so hard to ask genuine questions, and so easy to fall into the trap of asking nonquestions that don't accomplish anything?

Thinking about that this week, I've come up with two basic answers.

First, maintaining good questioning habits is hard. You have to:
  • Plan ahead of time what genuine questions you will ask and when you will ask them;
  • Either remember those questions or read them from a script;
  • As Kathleen suggested, get frequent feedback (from videotape or peer observations) that alerts you to poor questions when they happen, just like you would do to keep from backsliding on any bad habit.
Second, planning good questioning requires acknowledging the scarcity of two essential resources.

The first resource is time: as our friend Tom McDougal says, it's the teacher's most precious resource. If nothing else, the simple asking of a nonquestion and waiting for a nonresponse takes time. It sucks time out of genuine questions, in part because it creates ambiguity in every question, by forcing students to figure out whether the question is one to which the teacher really expects an answer, and thereby delaying or dampening students' responses.

The second resource is information: about what students know, think, understand, and can do. A genuine question allows a teacher to garner some information: about one student or, depending on questioning technique and the response mode (individual whiteboards or clickers, small group discussions, etc.) multiple students. A nonquestion wastes the opportunity to find out what students know, and as noted above, actually reduces the effectiveness of subsequent genuine questions by introducing ambiguity into the questioning framework.

For me, just reminding myself of the waste when I hear myself asking a nonquestion is good negative reinforcement. But obviously it's not enough, because I keep asking them, not often, but more than never. Your post and Kathleen's comment remind me that I need to get into others' classrooms more often, and I need to get them into my classroom more often--if only for this one thing.


Friday, February 4, 2011

Teacher talk

Everybody see that?
You want me to go over that again?
Did I go too fast for you?

Even without context, we recognize these as teacher phrases. These are things well-meaning teachers routinely say to students in an effort to be encouraging and positive about the lesson at hand. There are many more phrases such as these; I am sure you can think of some.

One of my mentors, David R. Johnson from Nicolet High School in suburban Milwaukee, wrote an article, “Every Minute Counts,” and a sequel, “Making Minutes Count Even More.” The articles deal with the nitty-gritty of teaching mathematics. Even the titles embrace an important idea: that good teachers make use of every minute. There is no time to be wasted. 

I bring up these articles now because David had considerable insight about these teacher phrases, and I would like to share some of his thoughts with you.

“Everybody see that?” This kind of question is not answerable by a student and teaches them to ignore my questions.

“You want me to go over that again?” They really didn’t want me to go over it the first time.

“Did I go too fast for you? “ No. Faster, faster. Let’s get this done.

“Here’s an easy one.” This comment could be one of the worst things we say. As a student: if I get it right, so what, it was easy; but what if I don’t? Then I know I can’t even do the easy ones.

When I first heard these comments from David, they hit me hard. I recognized the accuracy of his observations. I also recognized these remarks as things I said virtually every day. I tried to change my habits, but it was hard. Gradually, I realized that these bad habits were symptomatic of a larger problem with my teaching. I was still thinking of myself as the person who was explaining math so well that it would be clear to everyone. My classes were still teacher-centered. 

It took me a long time, and a lot of trial and error to change what was happening in my class so that students were working on authentic problems that taught them important ideas in a coherent way, while I observed and learned from them how they were thinking and what progress they had made.

The questions that David discusses are all about how well I am doing, not how well my students are doing. I look over these remarks, and it strikes me that all of them assume that it is my job as a teacher to explain, and it is the student’s job to listen and therefore learn. Those job descriptions highlight what is really wrong with these teacher comments. The answers to these questions, these comments, are really meant to reassure me that I am doing well in explaining, which is not the point at all. The real question, every minute of every class, is how well are each and every one of my students doing as they struggle to comprehend the new ideas I have confronted them with today. And the best way for me to measure comprehension is to walk around and listen to what they have to say to each other and look at what they write. Then it is still not my job to explain to them how math works. It is my job to ask interesting questions and then to direct the discussion students are having as they try to figure things out. 

A coda: I never would have figured all of this out by myself. We need each other: teachers need students, and teachers need other teachers so that we can all contemplate best models from every angle. I am a pretty good teacher, but only because of the wisdom that has been passed on to me by people like David Johnson.