by Glenda Lappan, NCTM President 1998-2000
NCTM News Bulletin, April 1999

The improvement of schooling in the United States remains a major topic of concern for politicians, school administrators, community leaders, and parents--and teachers continue to be the target, sometimes portrayed in a positive light but more often in a negative one. The cry for "high-quality teachers" is heard across the land. A more appropriate cry would be for "high-quality teaching." This does not mean that institutions of higher education do not need to improve their teacher education programs--they do; it does not mean that teachers should not know more mathematics--they should; it does not mean that teacher recruitment and retention are not important--they are.

All these efforts are absolutely necessary. But what they produce, at best, is a well-started teacher. We know that teaching is a lifelong learning experience, a continual quest to improve our knowledge of mathematics and our teaching practice. Improving one but not the other will not do the job. We collectively and individually need to be proactive in figuring out what we can do to improve our practice.

As members of NCTM, we know that we are the keys to what students learn in mathematics. We also know that the system to support us in our work frequently fails both our students and us. We know that we have little time to plan, to study, to evaluate, and to engage in professional work with our colleagues. Yet, to improve our practice, we must engage in exactly such study and reflection. We have no choice, and our system must support us in this way. There are no quick fixes or magic pills by which mathematics teaching and learning can be instantly improved, and to make matters more challenging, our own experiences can make us resistant to change.

Our own years of schooling in mathematics classes have built into us a model of teaching that is so much a part of our culture that it is virtually invisible to us. UCLA Professor Jim Stigler refers to this model as a teaching script, our American teaching script. For example, we do not even have to think about the lesson flow. We can operate on automatic pilot when it comes to structuring the lesson. We know how to get a class started, what happens in the middle, and how a class ends. The very invisibility of our long assimilation into mathematics teaching makes it extremely difficult for us to examine what we teach and how. And even if we can examine our deeply held notions about teaching mathematics, changing them is yet another order of magnitude in difficulty.

Given the challenges in making fundamental changes in how we teach, we could say, "Why bother? We have always taught mathematics this way, and it has worked for enough of our students." The stance of NCTM and the draft of NCTM's Principles and Standards for School Mathematics (PSSM) is that this is not good enough--every student deserves a high-quality mathematics program that sets high expectations for all. Undeniably, our current programs are not reaching our high expectations for many of our students, and we must address that.

PSSM sets goals for students that include thinking, reasoning, and developing the ability to tackle and solve unfamiliar mathematical problems with creativity, insight, inventiveness, and skill. PSSM recommends that students develop a deep understanding of important concepts and proficiency with important related skills across five major mathematical strands. This recommendation implies that our students deserve to learn more than the procedures of mathematics--they also need to make sense of mathematics. Going for such goals in the classroom requires a teaching script quite different from our traditional American one.

The typical American script--teacher explanation and demonstration followed by practice with problems of the same type--seldom challenges students to think deeply about ways to use what they already know to figure out something for themselves. Even the promising new curriculum materials that have more of a problem-centered focus to teaching mathematics can be problematic in practice. They are meant to engage students in making mathematical sense of problem situations that often require several steps to solve while using mathematics from several lessons. However, these kinds of materials offer a serious pedagogical challenge because that American teaching script by which we are so often guided simply does not work with such problems. If, following the script, the teacher demonstrates how to solve such a problem, the challenge is gone. If the teacher puts students into groups with no context setting, with no mathematical question or questions driving their work, students can flounder in unproductive ways for far too long. Writing a new script so that we can engage students with such problems is a challenge.

Perhaps the hardest part of this new script is summarizing at appropriate times to make sure the mathematics embedded in the problems is made very explicit for the students. Although the students have a role in this summarizing, the teacher has the greater responsibility. It is the teacher who knows where she or he is headed with a set of tasks for the students. Although the students may raise unexpected questions along the way, the teacher has to have a mathematical destination in mind and a plan to get there with the students.

Public support is growing for putting significant resources toward teaching. But change requires our commitment, time, and effort. We must continue to improve our knowledge and also our technique, rewriting our "scripts" when necessary. That effort, combined with stronger preservice teacher preparation, will ensure that we have both the teachers and the teaching we need to reach our goal. To paraphrase Donald Chambers of the University of Wisconsin--Madison, "Mathematics for all is the right goal at the right time. We just need to get the right teaching to get there."