adaptive reasoning and conceptual understanding. He also knows how much he has to do to make sure that the productive work the students are doing comes together at the end. He has found this way of working valuable. He is sensitive to the critical role that he plays during the lesson, even though it seems that the students are doing a large amount of the talking and the work.

We have been looking at upper elementary and middle school classrooms. In the last sample lesson (Box 9–5), a fourth teacher, Ms. Kaye, is attempting to develop her first-grade students’ understanding of subtraction as it is used to compare quantities. She wants the students to find and consider their own ways of making comparisons of two-digit whole numbers in which the larger number has the smaller digit in the ones place.

Box 9–5 Ms. Kaye—Teaching First Graders About Comparing Prices

Ms. Kaye gives her first-grade class a problem that involves comparing prices on a menu. She reads the following problem several times and writes the numbers on the overhead projector:

At Wu’s Dairy a single ice cream cone costs 59¢. A double costs 85¢. How much more does a double dip cost than a single dip?

The children eagerly set to work on the problem at their desks. A number of tools— including counters of various kinds, plastic coins, and base-10 blocks—are available in the corner of the room. While the children work, Ms. Kaye talks with individual children about their solutions.

Ms. Kaye stops at Kurt’s desk and asks him what he is doing. He explains that he is trying to find out how much more 85 is than 59 and proceeds to make 59 with base-10 blocks. Ms. Kaye asks him what he is going to do next. Without answering, Kurt makes 85, again with the blocks. Once more Ms. Kaye asks him what he is going to do next. Staring at the blocks, Kurt does not respond. Ms. Kaye asks what he is trying to figure out. “How much bigger 85 is than 59,” he murmurs. He does not seem know how to proceed. Ms. Kaye focuses his attention on the base-10 blocks and asks whether they could help him figure it out. Saying that he wants find out how much more there is in the 85 set of blocks than the 59 set, Kurt proceeds to match the two sets, pairing block for block. He trades in a rod (a 10) from the 85 set for 10 ones to make possible the matching of the 5 ones and the 9 ones. After the matching is complete, Kurt counts the blocks left unmatched and gets

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