What does a good discussion look like?

One of my favorite moments in the classroom is when students are thinking about some really interesting problem… perhaps they’ve even posed an extension of a problem in their textbook… and they are excitedly discussing it.  They build on one another’s ideas, they inevitably argue, there is a back-and-forth that continues until they’ve really gotten somewhere.  Occasionally I will step in to resolve a dispute or get the students to think more carefully about some misconception they’ve been running with, but for the most part it is the material itself that drives the discussion.

There is a tension, though, between letting the discussion flow naturally and between creating a balance of voices heard in the classroom.  When things get exciting, it is much harder, and perhaps not even the right thing, to let the students speak in turn.  Because there is often one person who has had the crucial idea, the other students’ comments tend to be directed at that person, who may then be speaking every other comment.  Because the discussion is heated and the people who’ve just spoken want to respond right away, there is also less “space” in the discussion for people who are not as in the thick of it to jump in.  I worry in these cases about quieter students, students who take a bit longer than others to formulate their ideas, more tentative students, and students who’ve simply missed some of the framing of the discussion and aren’t quite sure what we’re talking about.

Here are two strategies I’ve sometimes used to make these conversations more friendly to every student.  1) Go to a strict hand-raising system, in which the two or three most ardent students have to wait to bring their ideas forward while we hear from other people who have more tentative and perhaps less-formed opinions.  2) Go to group work for five minutes and let each group the chance to discuss the material, then report back, at which point multiple groups might have definitively solved the problem, or, if not, at least we can begin the discussion again with more students “on the same page.”

While I do sometimes use those strategies, #1 especially feels strange, as if I’m killing the momentum of the discussion.  At a private school we have the luxury of small classes, but there is still something that seems artificial about having a discussion with more than, say, three people at once.  What does it look like to have an open-ended discussion in which most students are involved, that at the same time builds on an idea and approaches a conclusion?

How long should I let kids struggle with a problem?

I’ve been thinking a lot about that question the last few weeks. I teach math to three different “ability groupings” of kids, and yet in many ways they have similar sets of reactions in grappling with a difficult problem.

When they first hear the problem, there generally is interest and excitement, especially if I have chosen a good problem. (By the way, by “problem”, I don’t mean a routine exercise, but rather something that requires them to think, to use what they have previously learned but in a different way.) Students generally start talking among themselves about what they might do, or they raise their hands to ask clarifying questions and to propose a plan of attack. At this point, things have a really good feel– students are engaged, and they are anticipating solving a thorny problem if they just make a sincere effort. And indeed, that is often how things go– students spend a few minutes trying an approach or two, make a mini-breakthrough, and then solve the problem.

But often, after 5-10 minutes of thinking, students find themselves at an apparent dead end.

Giuseppe’s Fingers

I assigned the following problem last week in my class:

Giuseppe likes to count on the fingers of his left hand, but in a peculiar way.  He starts by calling the thumb 1, the first finger 2, the middle finger 3, the ring finger 4, and the pinkie 5, and then he reverses direction, so the ring finger is 6, the middle finger is 7, the first finger is 8, the thumb is 9, and then he reverses again so that the first finger is 10, the middle finger is 11, and so on.

One day his parents surprise him by saying that if he can tell them some time that day what finger the number 1,234,567 would be, he can have a new sports car.  Giuseppe can only count so fast, so what should he do? 

Here’s how it went down:

 Some of my students realized instantly that counting up to 1,234,567 on their fingers wouldn’t be a very effective use of their time (or Giuseppe’s!).  Others counted up to about 180 until they “abandoned ship” on the brute force method.

Mice and Wine

This is one of my favorite problems:

You’re planning a huge party for tomorrow, which will include a toast exactly 24 hours from this moment.  You have 1000 bottles of wine, but one of them is contaminated with a slow-acting poison that will kill any living thing within 24 hours of being ingested.  You happen to have 10 altruistic mice on hand who have volunteered to test the poison.  How many bottles of wine can you safely serve at the toast?

I’ve given it to a number of classes, ranging in age and strength, and it’s produced wonderful discussions every time.  Here’s a reconstruction of how many of these have gone.

Right off, several students come up with the idea of splitting up the 1000 bottles evenly among the 10 mice.  When one of the mice dies, they explain, you would know that the poisoned bottle was among the 100 that it drank, and so the remaining 900 would be safe to serve.