One of the most common things that I do, and I figure most teachers do, is to ask students to try a few problems, and then address that work as a full class. For the purposes of this post, I’m going to narrow the focus: I’m less interested at the moment in a problem-solving or inquiry activity, but instead on humble practice that solidifies learning — problems similar to ones students have seen before, that most will solve successfully but some will struggle with, both as formative assessment and a teaching opportunity. To drill down even further, I’m not going to focus on times when the majority of students are making mistakes — I think these are more clear-cut, at least for where I am in my teaching at the moment. Instead, I’m thinking about times when the majority of students either got questions right or think that they got questions right.
My go-to moves come in three general styles:
Pick a high-leverage question or two and either question our way through it as a class, or analyze one or two examples of student work
Folks are doing well, and I feel like I can address misconceptions individually or in some other way — instead of taking the time to look at questions students were successful with, let’s just move on to another learning opportunity.
Give students answers to the problems, have them check the ones they got to, then move on.
I don’t love any of these. Questioning is one I often use, but when the class as a whole has been successful, it is hard to engage the group in looking at questions they have already completed, especially when they can sense that most kids can do it. Many times there is a unique strategy or particularly interesting problem to look at, but when there isn’t this seems like a bit of a hollow activity, and can take a bunch of time away from students actually doing math. Moving on is my favorite, but I worry that I send the message that the questions I asked the class to do are unimportant if we don’t address their work. And while giving answers makes that work more meaningful, it seems to reinforce answer-getting habits, and I’m skeptical it does a whole lot to promote learning.
So I asked Twitter over the weekend, and got a huge number of responses from some really thoughtful folks. Here’s what I learned:
Moving on is totally fine
If students are getting problems right, I should just tell them that, and say hey, we aren’t going to linger here, let’s move on to something more challenging. If I message the purpose behind it, and I don’t make a habit of giving students a cakewalk of a problem set on a daily basis, I shouldn’t feel any problem with doing it.
Giving answers ahead of time
Got this one from @k8nowak, and it’s one I’ve actually never used.
Kate talked about how he makes it routine from the beginning of the year that she will often give the answers to a set of problems to students ahead of time, so that they can see how they are doing. This also focuses student attention on the process, rather than the product — it underscores the emphasis on quality of work because students get feedback, and frees up teacher attention to what the students are thinking and how they are approaching the math, rather than telling them whether they are right or wrong.
Pose a similar, but different, problem
This one was from @hpicciotto, and is my favorite.
Henri proposed that, after students practice a set of problems that they feel confident with, but I know there may be some lingering misconceptions, that I pose a similar, but different, problem on the board for students to try. All students are actually doing math, rather than reciting math they have already done. It sends the message that the work students had been doing was important. And it exposes student misconceptions without having to coerce students into participating on a problem they’ve already tried.
These are small things, but the little things can make a huge difference, especially the small things that I do day after day.