In our discussions about teaching the cylinder problem, the conversation frequently turned to the importance of wait-time (Tobin, 1983a, 1983b). For us, wait-time includes providing time for students to generate their own solutions to problems, waiting for a student to find the words for an explanation, and listening patiently as students try to put their questions into words. In addition, we also realized the importance of giving ourselves wait-time -- time to weigh options before deciding what to do next.
In order to become proficient at self-directed inquiry, it helps for students to use other students as resources and to support each other through the frustrations of problem-solving. When we wait for several students to respond to a question, or when we accept different solutions to a problem before we respond to any one of them, we find that we are providing our students with the opportunity to evaluate and help each other with their responses.
We need to resist the temptation to answer all of our students' questions, or
to tell them what to do next. If we intervene too much, they may become
dependent on us and, more importantly, may not be able to cultivate the power
of their own thinking. Waiting after posing a question sends a message that students
can explore their ideas without help from the teacher, increasing their
confidence. In this clip, Susan Stein encourages one student to help another:
In this next clip, we see Susan Stein repeat a student's question in order to focus attention on the question asked and on the student's ownership of the question, rather than on herself [view clip].
A similar goal can be accomplished through a general review of the questions
posed by the class. In the following clip, we see Art Mabbott reminding
students of the questions they were trying to answer during class.
When we provide students with time to think, and enable them to face the moments of uncertainty in problem-solving, we show our confidence in their abilities. Given enough time, students can develop the confidence to begin posing their own questions, and can seek resources to find answers for themselves. Here, for example, four students in Susan Boone's class work together to determine the relation between a cylinder's radius and its volume.
In a supportive classroom students do learn to think together, helping each other to address their questions. In this clip from Susan Stein's class, we see one student correcting another as the teacher merely restates the proposed definition. This sort of experience provides students with evidence that thinking with others is an important tool in mathematics, and may give them confidence in their ability to guide each other.
In John McKinstry's class, students are also encouraged to share their thoughts [view clip]. At this point in the lesson, students are making predictions about which cylinder will hold more. Notice that John does not tell the students whether or not their ideas are correct.
"I don't get it."
While we began our conversation by focusing on the concept of wait-time, we are also talking about teachers not jumping in with an answer, but knowing when to make room for students to figure out what is needed. Another form of this becomes visible when a student says, "I don't get it." We could assume that the student does not understand anything about the problem, and might launch into a re-explanation of the concept or procedure. There are alternatives, however, that can do more to support students' problem-solving abilities. In an anecdote from a different point in the year, Susan Stein describes how she learned what was causing her student difficulty:
One of my seventh graders did not complete his homework several times during the first two weeks of school. The first time he said he didn't understand the question, the next time he said he forgot, and the third time he again said he "didn't get it." I was becoming concerned and finally pinned him down to meet with me.
Much can happen as a result of the mere expectation that the student has more to say. Whether we wait silently or ask questions, we create space for the student to figure out a way forward. We can ask for a description of what the problem asks. We can involve other students in explaining what they understand. We can ask a question like, "What part do you understand?" instead of, "What don't you understand?" We can help students articulate which parts of a problem make sense, in order to help them move from "I don't get it" to "I know this much, but I'm stuck on this part." When students start from what is already known, they are no longer totally lost.
Of course there are many important factors other than discourse strategies
that enable students to have confidence that they can work from what they
know, and can increasingly rely on themselves and each other. We would like
to mention classroom norms as one area that interests us, and as a topic we
hope is developed in the conversations that build from this paper.