What?? you might be thinking. What else could the teacher’s job be but to teach?
The teacher’s job is to ensure that students learn – all of them, we hope, though we know we will usually fall short.
In Japan, most (elementary) math lessons are designed as “teaching through problem solving” lessons (TtP). A teaching through problem solving lesson typically includes the following parts:
- introduce the problem
- explicitly pose the task for students
- students work on the task (5-10 minutes)
- share student ideas
- compare and discuss the ideas for the purpose of learning new mathematics
- summarize major points from the lesson
- student reflections
(There is sometimes overlap, and a back-and-forth between some of these, e.g. #4 & #5 may be combined.)
While students are working on the task (#3), the teacher walks around the room, monitoring their progress. Japanese educators have a term for this, kikkan shidō, or “[providing] guidance between the desks”. They recognize that there are different ways to do kikkan shidō, and it is often a subject of discussion in Lesson Study. During planning, for example, a team will usually discuss how – or whether – the teacher should respond to a student who exhibits a particular misconception; during the post-lesson discussion, there may be argument about whether the kikkan shidō was effective. And, it is considered a skill that new teachers need to develop.
Teachers who are inexperienced with TtP lessons often make an unfortunate error while doing kikkan shidō: they see a student who is struggling, or who has done something wrong, and they stop and help that student. After several minutes the teacher moves on, encounters another student who is having trouble, helps that student, and so on. Then, suddenly, time is up, and the lesson ends.
There are at least four important drawbacks to this type of kikkan shidō. First, as my description suggests, it uses up a lot of time. The teacher may never get around to all of the students, and other students who need help may never get it. Second, by addressing misconceptions privately rather than publicly, the teacher deprives other students of the opportunity to analyze those misconceptions and learn why they are incorrect. Any experienced teacher knows that certain misconceptions are very common, so when one student makes an error that stems from a common misconception, that offers an opportunity to “inoculate” other students against making the same error sometime later.
The third problem with tutoring students individually is that it conflicts with the whole premise of teaching through problem solving. You expect that some, or even all, of the students will have difficulty with the task; that’s why it’s called “problem solving” and not “practice.” Teaching through problem solving involves an expectation that students will have difficulty, but that the comparison and discussion phase will address their difficulties and that, by the end of the lesson, all (or almost all) of the students will have learned what they need to know.
And fourth, we want to help students learn to give viable arguments and to critique the reasoning of others, the third Standard for Mathematical Practice in the Common Core State Standards. To accomplish this, we need for students to share and discuss different, perhaps conflicting solutions. Students need to do the critiquing, not the teacher.
Of course, some errors are simply the result of sloppiness, or otherwise unrelated to the main learning goals of the lesson. So when the teacher sees an error while conducting kikkan shidō, he or she has to decide: should this be addressed privately or publicly? What should I say to this student? Do I expect that, by the end of the lesson, this student will understand what he or she has done wrong? This is a tricky decision, and an important part of lesson planning is anticipating different student responses, correct and incorrect, and deciding ahead of time how to handle them.
Caring teachers naturally feel drawn to help struggling students: they feel like it is their duty to help those students right now. To counteract that impulse, I say, bluntly:
It is not the teacher’s job to teach the students. It’s the teacher’s job to create a lesson that teaches the students.