What kinds of norms need to be established in the classroom in order to enable students to think about and share their confusions about mathematics? We find that it is important to state explicitly in class that respect for one another's ideas is mandatory. Few students will ask a question or risk attempting to solve a problem unless they feel they are safe (Charney, 1991).
In addition to establishing clear expectations for student behavior, teachers can model mathematical thinking that includes resolving their own confusion (Schoenfeld, 1987). This practice legitimizes such confusion, and identifies it as an important part of the process of problem-solving (Polya, 1945). In addition, the teacher can pose problems that allow for multiple paths to a solution.
In pointing out that there are different approaches to finding an answer, a teacher suggests that there is not just one right way to answer a problem. In this clip, Jon Basden demonstrates that there are several ways of finding different measurements for height and length that will result in an area of 216 square inches [view clip].
By encouraging students to take risks, a teacher both acknowledges their
feelings about publicly engaging math, and encourages them to engage so
that they can learn. In this clip, Art Mabbott asks his students to
"take risks" and answer a question.
By suggesting that students are able to revise their answers, a teacher implies that an answer does not have to be final, and that continuing to revisit it is useful practice. In this clip, Susan Stein asks her students whether anyone wishes to change his or her prediction. One of the students refers to another's answer, disagrees, and then explains her own thinking.
When students come to expect that there may be more than one answer, they place more value on finding different methods for solving the same problem, and are better able to analyze and work with problems (Redman, 1996). One of the obstacles for teachers in this regard is finding rich problems that encourage multiple approaches, and understanding the mathematics well enough to pursue the many dimensions that may become visible. Thus, nonroutine problems such as the cylinder problem play a major role in supporting discourse for the development of students' mathematical thinking.