When, If Ever, Should Mathematics Teachers Lecture?

Kurt Kreith

Early in my career as Principal Investigator for the Northern California Mathematics Project, I was invited to a meeting of the statewide CMP leadership. After some initial pleasantries, I was asked whether involvement in the NCMP had led me to reconsider my reliance on lectures in university classes? Though I knew the answer that was being sought, I decided to try to finesse the issue with a lighthearted reply, "No, I am still trying to give the perfect lecture."

At least one member of the CMP's Advisory Committee was not amused, and I vividly remember the stern and uncompromising lecture on the evils of lecturing that followed.

This was more than ten years ago, and the search for alternatives to the lecture format has since become the centerpiece of California's mathematics education reform movement. Professional development programs now stress "groups," "activities," "manipulatives," and "investigations" as providing more appropriate formats for mathematics instruction. By way of giving such alternatives official sanction, p.41 of the 1992 Framework states:

Teachers are facilitators of learning rather than imparters of information. They encourage and support students as they become active learners; establish a classroom climate in which students construct their own understanding of mathematics, providing rich environments for student investigation; and facilitate learning by asking questions that probe students' understanding and provoke their inquiry. In addition, teachers ask students to explain their thinking, help them hear and react to the ideas of others, and encourage them to make and refine mathematical generalizations.

It was not until the recent hearings by a Task Force appointed by California's Superintendent of Public Instruction's that this centerpiece of the mathematics education reform movement was challenged head-on. At these hearings a Palo Alto group named HOLD - Honest Open Logical Debate distributed a position paper that lists seven weaknesses of the Framework. The first of these refers to the above quote, stating:

It [the Framework] focuses totally on the constructivist model. As part of that, it insists on the teacher's role as facilitator only.
The constructivist model is unproved, and there is no (i.e., zero) experience with its broad and exclusive application to instruction.
The Framework totally ignores the proven fact that direct teaching is the most efficient method of transferring knowledge.

The existence of such opposite points of view suggests a need for serious efforts to find some common ground. In such a quest, I would expect that most proponents of lecturing would agree with the following:

There is little point in a teacher lecturing if, as a result, student learning does not take place.

I also believe that most proponents of the current reform movement would agree that:

If a child wants to know something that the teacher knows, that teacher may choose to tell the child - rather than insisting on all knowledge being "constructed" by a process distinct from "lecturing."

Given such a modest degree of flexibility at the two ends of the spectrum, it is seems useful to recall the pedagogical insights of an early visitor to our NCMP. Nicolas Rouche is a Belgian mathematician with deep concerns about ineffective instruction based on the lecture method. He stresses the need for teachers "to bring children to the epistemological threshold" before introducing new mathematical concepts or definitions. Activities, manipulatives or investigations provide one way (but not the only way!) of setting the stage for new learning.

However, once having brought children to this threshold, he argues that teachers have a further responsibility. It is to "synthesize and organize" such knowledge as they have now made accessible to their students.

In this context, we can think of many of the methods stressed by the current reform movement as efforts to "bring children to the epistemological threshold" for learning mathematics. However the subsequent steps, those of negotiating children over this thresholdand assuring that mathematical concepts are in fact synthesized and organized, may well involve something akin to "lecturing." For many teachers these steps involve interactive discussions with students as well as pure lectures. But whatever one calls it, the teacher's ability to verbalize mathematical ideas in a clear and coherent way remains crucial to the teaching process.

What has been de-emphasized in the current reform movement (perhaps to the point of accounting for some of its failings) is the need for teachers to organize and verbalize mathematical ideas. Countless in-service programs and presentations at professional meetings consist of descriptions of "activities," perhaps embellished with a few phrases indicating the area of mathematics in which "powerful learning" is expected to take place. Clear expositions of underlying mathematical ideas have become increasingly rare. This is unfortunate because, like any other professional skill, mathematical exposition needs to be encouraged and practiced if it is to be maintained.

By way of venturing answers to the question, "When, if ever, should mathematics teachers lecture?" let me describe three kinds of dialogues in which lecturing skills should be cultivated:

2. Professional dialogues in which a teacher describes to colleagues the larger mathematical context in which a particular activity can be useful. These dialogues should be carried out at a professional, rather than student, level.

3. Classroom dialogues in which the teacher precedes a classroom activity by setting the stage, and then follows that activity by synthesizing the mathematical ideas that this activity was chosen to convey. Classroom activities that do not lend themselves to such verbal interaction with students should be carefully scrutinized for their pedagogical value.

As California's mathematics education reform movement enters its second decade, it is certain to be subject to increased scrutiny and accountability. Attention to the mathematics that underlies its methodologies may be essential to maintaining credibility in the eyes of an increasingly wary and critical public.