Editor’s Section Commentary

In the three preceding studies, we see students of different ages engage in mathematical endeavor not merely for the purpose of filling a sheet of paper to be submitted to the teacher, but for the express purposes of articulating their individual| collective thinking to others and in ways already presupposed to be intelligible to an audience. Although in each case, this audience includes the teacher, it consists primarily of peers who are at approximately the same level in their mathematical development and at the same competency level with respect to talking mathematics.

From a traditional perspective on learning, there would little difference between what students do, say, and write when they address the teacher only—homework, test, examination—and when they address a wider audience. This is so because all a student is thought to do is empty out the contents of their minds, using language as a medium to carry the information from within themselves to without.