For more than a century, observers have been looking into classrooms and emerging with descriptions of how U.S. teachers teach.^{65} What is most striking in these observers’ reports is that the core of teaching—the way in which the teacher and students interact about the subject being taught—has changed very little over that time. The commonest form of teaching in U.S. schools has been called *recitation*.^{66} Recitation means that the teacher leads the class of students through the lesson material by asking questions that can be answered with brief responses, often one word. The teacher acknowledges and evaluates each response, usually as right or wrong, and asks the next question. The cycle of question, response, and acknowledgment continues, often at a quick pace, until the material for the day has been reviewed. New material is presented by the teacher through telling or demonstrating. After the recitation part of the lesson, the students often are asked to work independently on the day’s assignment, practicing skills that were demonstrated or reviewed earlier. U.S. readers will recognize this pattern from their own school experience because it has been popular in all parts of the country, for teaching all school subjects.

Although there are some differences in the way different subjects are taught,^{67} the description of recitation teaching is consistent with more recent descriptions of mathematics lessons. In the mid-1970s, the National Science Foundation funded a set of studies on classroom practice, including a national survey of teaching practices^{68} and a series of case studies.^{69} After observing a number of mathematics classrooms, one researcher said:

In all math classes I visited, the sequence of activities was the same. First, answers were given for the previous day’s assignment. The more difficult problems were worked by the teacher or a student at the chalkboard. A brief explanation, sometimes none at all, was given of the new material, and problems were assigned for the next day. The remainder of the class was devoted to working on the homework while the teacher moved about the room answering questions. The most noticeable thing about math classes was the repetition of this routine.^{70}

The findings for the full set of case studies are not easily summarized because there were some substantial differences between teachers, but a commissioned synthesis noted that the most common pattern in mathematics classrooms was “extensive teacher-directed explanation and questioning followed by student seatwork on paper-and-pencil assignments.”^{71}