We worked in a fifth grade classroom of 22 students, a fourth grade classroom of 20 students, and a combination class that included 19 third and 10 fourth graders. The three heterogeneous classrooms were selected because of the teachers' experience and expertise with problem-based instruction. The teachers had successfully established social norms to encourage students to exchange points of view with respect to their mathematical thinking. All three teachers had worked collaboratively with the university-based research team over a period of three years in a grant-supported teacher leadership program. In the program they studied constructivist theories of learning mathematics and collaboratively designed curriculum and practiced instructional strategies to be consistent with those theories.
The instructional intervention was conducted in February and March, when students were accustomed to the classroom routines and to problem-based instruction. We were in each classroom over a period of five or six consecutive days. Written assessment tasks and four 90-minute lessons were presented.
Written Digit Correspondence Assessment Tasks
We designed tasks that could be administered to the whole class, rather than in individual interviews. Each student received a picture of 35 objects. Aided by an overhead projector transparency of the picture, the researcher elicited a consensus that the number of objects in the picture was 35. The researcher then wrote "35" on the transparency and said "Thirty-five stands for the 35 beans (or squares)." She then circled the "5" in one color and asked the students to do the same. She then asked, "What does this part of 35 have to do with how many beans are in the picture? Write down what you think and color the picture to show what you mean. After allowing for response time, she circled the "3" with another color, asked students to do the same and asked "How about this part? What does THIS part have to do with how many beans are in the picture?" For the preassessment, the 35 objects in the picture were arranged in a rectangular five-by- seven array. A second version, picturing 35 objects in an ungrouped collection, was administered at the close of the instructional period, and again in June which was three months after the instruction.
Each lesson began with a problem-solving task to set the stage for the digit-correspondence (experimental) task, which was to decide what the parts of the number had to do with how many objects are in a collection. The stage-setting tasks were designed to reflect typical intermediate-grades curriculum (topics included area, multiplication, and division), and to provide entry for all students. Manipulative materials and/or drawings were part of all the tasks. A set of detailed lesson descriptions including samples of student work is available from the authors.
144 Squares. Students were asked to decide, in groups, whether or not three gridded paper shapes were the same amount of paper (area). The rectangular shapes were 12cm x 12cm (144), 13cm x 11cm (143), and a shape 6cm x 24 cm with one square centimeter cut off each corner (140). Students reached consensus that the yellow square was the largest, with an area of 144 square centimeters. In the digit-correspondence task, students were asked "what does this part of 144 (circling each individual digit beginning with the 4 in the units place, then the tens digit and finally the "1") have to do with how many square centimeters are in the yellow shape?" We provided each group a transparency picture of the 12 x 12 square for preparing their presentation to the whole class.
124 Cubes. In this lesson we used a "factory" context of filling orders for cubes. Base ten blocks were available as models for cubes "prepackaged" in sets of ten and one hundred. We asked, "How many ways can you fill an order for 124 cubes?" Making a list was modeled as a problem-solving strategy. For the digit-correspondence task, each group was assigned one of the non-standard ways to fill the order (e.g., seven long packages and 54 individual cubes) and asked to "decide which blocks would fill the order for each of the three parts of the number" (digits).
26 Wheels. Students were asked to determine how many toy wheels were contained in a clear plastic bag, based on the following two clues: "There are enough for six cars. There are two left over." After students reached a consensus that there would be 26 wheels we asked what each part of 26 (the "6" and then the "2") had to do with "how many you have." A diagram of the six cars (each with its four wheels) and the remaining two wheels was provided each student as they worked individually and a transparency version was provided to each group.
62 Wheels. "If each car has four wheels, how many cars can be fitted with 62 wheels?" After arriving at a consensus of 15 cars (with two left over), we asked, What does each part (the "2" and the "6") of 62 had to do with how many wheels you have? Students made their own drawings.
Typically, a member of the research team presented the task, researchers and the classroom teacher circulated among groups during the cooperative-group work time, and the classroom teacher led the whole-class discussion while groups presented their results on overhead transparencies. Teachers used questions and comments to focus attention on differences and similarities among the ideas presented, and often asked students to elaborate by showing with the picture. Special care was taken to provide neither any direct instruction about the "tens and ones" meanings of the digits nor any judgments about the correctness of the ideas presented.
Transcripts of the lessons were based on note-taking by a trained observer and audio recordings. All individual written work and group work, which was usually in the form of overhead projector transparencies, were collected for analysis.
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