by Glenda Lappan, NCTM President 1998-2000
NCTM News Bulletin, November 1998

One of the central components of a good mathematics program is good curriculum materials for the students. In my July/August President's Message, I argued that a teacher should not be expected to develop his or her own mathematics curriculum materials for students.

Given their usual responsibilities, teachers don't usually have the time it takes to develop coherent, connected, focused, and complete curriculum materials. But armed with high-quality materials, teachers themselves are an essential component of a successful mathematics program.

As I have worked with classroom teachers over the past decades, I have observed how good teachers work with curriculum materials to bring them to life for students. They do so by looking at each lesson through a number of different lenses. One way to think of these lenses is as a series of analytical questions teachers ask themselves.

Teachers start with planning for the lesson. Here the teacher analyzes the mathematical task that will comprise the students' classwork and asks, "What skill, mathematical reasoning, problem solving, and conceptual and procedural development will the task support? How does this task support my mathematics goals for the students?"

After the teacher decides that the task is worthwhile, the question becomes "What changes in the context would make the task more engaging to my students?" Here the challenge is to maximize the task's appeal to students without reducing its level of demand. All students are different. Yet the teacher must try to reach all students in the class. This leads to questions like "What are my core goals for the task? What do I want every student to understand? How will I engage the students so that they understand what is expected of them and so that the mathematics is clear? What mathematics extension questions will push the more interested students? What supporting questions will help students who are struggling but not reduce the cognitive demand of the task?"

After the mathematical potential of the task is clear, the focus of analysis turns to setting the stage for students' success with the task. Questions include "What tools and resources do the students need to tackle the task? What skills, processes, ways of thinking, and so forth, would contribute to students' success with the task?" These tools and resources might be intellectual, such as prior knowledge. They might be physical--students might need graph paper, calculators, measuring devices, or other physical tools to help model the task and their thinking about the task. The resources needed might even be human--"Is this task an appropriate challenge for student pairs or groups?"

After the students have made progress on the task, the most important part of the lesson, the summary, occurs. In preparing to summarize the lesson, good teachers think about what will help students make the mathematics used and invented more explicit. "What questions can I ask to encourage students to reflect on their thinking? How can I help organize and talk about students' responses to help them make connections between the mathematics of the task and their growing mathematical skills, understandings, and ways of thinking?"

Both during the lesson and after it has been taught, teachers think about assessing and evaluating students' performance. "What can I ask to assess what sense my students have made of the task? How can I report students' progress to students, parents, and others? How do I use the assessment to hold students accountable and to help students understand both what is expected of them and what constitutes good work in mathematics?"

But for good teachers, assessment does not stop with the students. The curriculum itself and the learning environment need to be examined on a regular basis. Teachers ask, "How coherent, connected, and powerful is the sequence of tasks I have used to promote understanding of the core mathematics? Do I have the 'right stuff' for my students to chew on? What does the assessment of my students and of the curriculum tell me about what I need to do next?"

And finally, an important part of examining the learning environment is asking about the climate in the classroom. "Are students respectful of one another? Are they learning to work together, to help others and to seek help when they need it, and to reach for and meet high expectations?"

This sort of thinking and planning is already intuitive to many teachers. Others of us can be helped by purposefully asking ourselves such questions as we teach. Just as good questions help students learn mathematics, good questions help us learn more about being effective teachers.