## Place-Value: Problem-Solving and Written Assessment Using Digit-Correspondence Tasks

### Results and Discussion

We sorted the individual written assessment papers into categories of similar responses and developed a descriptive rubric of nine distinct categories. Reading the 71 preassessment papers was discouraging. Twelve students failed to respond to the questions, and 32 invented meanings that gave no hint of the "3" representing 30. One student gave the response that the "5" meant five squares and the "3" stood for three squares. All the other students attempted to account for the whole collection of 35 squares; "rows of five" and "counting by threes" (even accounting for the remainder) were common responses. Fourteen students used the language of tens and ones in their written responses, but we could not be sure they were talking about the collection of squares in the picture or simply describing the names they had learned for the "coloms" (sic). Eight students wrote responses that strongly suggested that they might understand the meanings of the digits, but included no pictorial evidence of a 30 and 5 partitioning. Only five students gave truly convincing written and pictorial evidence of understanding.

With only one or two students in each classroom demonstrating understanding at the beginning of the instruction, we were concerned that there would be insufficient numbers of knowledgeable peers for the social-interaction design to produce changes in student conceptions. However, students found the lessons engaging and were soon immersed in making sense out of the digits. They examined many ideas, and lively debate often occurred as students exchanged points of view about the meaning of the individual digits. The task that elicited the most heated debate was "26 Wheels." One viewpoint was that the "2" stood for twenty wheels (usually in five cars), and that the "6" stood for the remaining six wheels. Other students were equally adamant that the "2" stood for the two wheels left over and the "6" stood for the six cars or the wheels on the six cars.

The responses on the postassessments were generally both more correct and more expansive than those on the preassessment. In the delayed postassessment, 23 students described the "3" in 35 as representing not simply 30, but also as three sets of ten; 10 of the 23 partitioned the accompanying picture into sets of ten while the remaining 13 partitioned it into 30 and 5. An additional 29 students wrote that the digits represented five squares and 30 squares; 15 of these included pictures. We concluded that there might be three reasons for the improvement. One is that students constructed meanings for the individual digits in a multidigit numeral that were more consistent with our place-value numeration system than those they held before the instructional intervention. Another is that they became better at expressing their mathematical thinking after the experience of talking and writing about their ideas, and hearing and seeing other students' ideas. Finally, because relatively few students used the pictures to show what they meant in the preassessment, we chose to prompt the use of coloring the pictures more assertively in administering the postassessments.

Some small groups seemed to get stymied with a single incorrect interpretation because it was the viewpoint of a student in their group who was a respected leader in the classroom. Students in these groups might have benefited more had we changed the groups so that they could have experienced a more fluid exchanges of ideas. Although we were constrained to present the lessons on consecutive days, the lessons might have been more effective if spaced across the school year, because teachers change the composition of the collaborative groups every few weeks.

To evaluate changes in student thinking about the digits, we compared the preassessments with the postassessments in terms of success. All responses that related the "5" in 35 to a set of five objects and the "3" in 35 to the remaining set of 30 objects or three sets of ten objects were defined as "successful." On the preassessment, the work of only 13 of the 71 students (18%) demonstrated that they knew that the "3" represented thirty of the objects; these 13 were also successful on both the immediate and delayed postassessments. On the immediate postassessment, 45 additional students (63%) were successful, falling to 41 (58%) on the delayed postassessment. Among those who were initially unsuccessful, 65% of the third graders, 76% of the fourth graders, and 63% of the fifth graders were successful on the delayed postassessment. Two were absent and 15 (21%) remained unsuccessful on the postassessments.

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