n this chapter we explore three learner-centered classrooms. While the focus of the activity is on the student, the teacher establishes the lesson context and provides the tools and structure to complete the activity.

 Students enter school with estimation skills, aware of their approximate height or age, for example. By building lessons around this prior knowledge, teachers can help students develop a personal understanding of mathematics. In real life we estimate all the time--for example, when determining the number of hours to reach a destination or when figuring out how much money to leave for a tip. In this activity for a fifth grade classroom4, the teacher breaks the class into several small groups and introduces three estimation exercises. First, students are given a cluster of ten dots. They must estimate several other clusters as "fewer than ten," "more than ten," or "about ten." Students discuss "good" estimates--how close the estimate must be to the exact number--and then emphasize that in some situations, an estimate is just as good as the exact count. Students estimate the number of candies in a jar and pencils in a box, documenting how they arrived at their estimate. Next, students choose strategies to respond to the problem "What is the sum of 243 + 479?" One group adds hundreds and tens to produce an approximate sum of 700. Another estimates a sum of less than 750 by rounding 479 up to 500 and 243 up to 250. Finally, students estimate the dimensions of classroom objects. To calculate the height of the door, one group places their tallest member against the doorjamb. He knows that he is five feet tall and reaches slightly more than halfway to the top of the door, so the door is about nine feet. One girl, measuring the teacher's desk, recalls reading that a child's hand is about five inches. Her group decides that two "hands" equal a foot and estimates the desk length to be four and one half feet. These estimation exercises encourage numerical flexibility, mastery of a certain level of mathematical computation, and reflection about spatial and mathematical concepts. Such skills are especially important when using calculators or computers, since students must have an estimate available for comparison with the calculator's answer. By posing problems that demand reflection and self-generated meaning, teachers can help students build their own understanding and gain a better sense of what the numbers represent. In the above example, students draw upon their pool of knowledge (the boy as five feet tall, the size of a hand) to create a way to solve the problem. Let's examine what the learners and teacher did in this activity. First, the estimation exercise provides students the opportunity to use their prior knowledge to solve real world problems. Second, students accessed a variety of tools (clusters of dots, candies in a jar, and classroom objects) to construct their own understanding of the mathematical principles involved. Finally, by posing problems that demand activity, reflection and self-generated meaning, the teacher helped students build their own understanding and gain a better sense of what the numbers represent.