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

Builders often sink building supports into bedrock, the solid rock that underlies the earth's surface, to provide the stability on which towering structures can be built. In many ways, we as mathematics teachers need to do the same thing--ensure that our mathematics programs are built on the rock solid commitment to excellence and to providing support for students to learn.

Does this mean that every student will learn the same things from each experience we provide? No. The complex job of a teacher is to ensure that every student learns the core ideas and proceeds as far as possible. But with diverse students who have different interests and ways of learning, the question becomes "how?" Here are some ideas:

• Connect to students' interests. One of our long-term challenges is that many students see no way in which mathematics is relevant to their lives. They have little experience with problems that are genuinely interesting to them and that give rise to substantive mathematics. We can change that. Perhaps not every day, but often enough that students come to see mathematics as useful and relevant.
• Develop both the logical and the experimental aspects of mathematics. For far too long, we have overemphasized the routines of mathematics without allowing students to experience the joy of trying out new situations and developing their own wonderful ideas. Does this mean that understanding the language and structure of mathematics is unimportant? Absolutely not. It means that we need a healthy blending of students' own ideas and experiments with a teacher's careful guidance in making the mathematics more explicit--helping students abstract, generalize, prove, and apply the mathematics embedded in situations.
• Create a tool-rich environment for learning mathematics. Tools for making sense of and doing mathematics include cubes, tiles, grid paper, measuring devices, shape sets, and so forth--tools that allow students to explore and to represent their own ideas in mathematics. Tools also include computer and graphing technologies. Our responsibility as teachers is to instill in our students the mental skills and decision-making capabilities that allow them to use tools of all kinds with insight and understanding.
• Understand how students are making sense of mathematics. Teachers can make a real difference in students' learning by paying attention to how well students understand the mathematics being studied. Rather than only use tests to assign a grade, we need to see tests and everyday work as information we can use to strengthen our teaching as well as feedback through which students can monitor their own learning. Looked at in this way, tests are an important part of learning.

To help bring these ideas to life, consider the following problem recently tackled by students of a Kentucky middle school teacher: Take a standard-sized sheet of paper and tape the short edges together to form the lateral surface of a cylinder. Take a second sheet of the same-sized paper and cut it in half with a cut parallel to the longer edges. Tape the short ends of both pieces together to form the lateral surface of a new cylinder whose height is half the original cylinder's height and whose circumference is twice as long.

Continue by cutting a same-sized sheet in fourths parallel to its long edges, then tape the four long pieces together to form the lateral surface of a new cylinder whose height is half that of the previous one and whose circumference is double. Continue to make cylinders in this way, each time cutting the height in half and making the circumference twice as long. Set the cylindrical surfaces on a table. How do the volumes of these cylinders compare?

The students became immediately intrigued with cutting and building the cylinders. Then the teacher helped them to nest the cylinders on a table, tallest in the center. After guessing that they would all hold the same amount because the changes in the heights and circumferences countered each other, they filled the center cylinder with popcorn kernels. The teacher carefully removed the center cylinder and let the kernels spread out into the next-sized cylinder and then the next. The students were completely captivated and amazed. Instead of holding the same amount, it looked as if in each succeeding cylinder the kernels reached only half as high. This means that the second cylinder held twice as much as the first; the third cylinder held four times as much as the first cylinder; the fourth cylinder held eight times as much as the first, and so on.

Now the search for why became a goal of the students, not an assignment of the teacher. Each student learned something about cylinders, volume, formulas, and exponential growth in the context of a problem they found exciting. They used a variety of methods and tools in their search for patterns.

Activities such as these pursue substantive mathematical goals by making the problem real for all and engaging to students, who naturally hold a variety of interests and perspectives. They also help us establish in the students the firm bedrock for building towers of mathematics knowledge for years to come. It's well worth our efforts to find ways of accommodating every student. Their future depends on it.