The purpose of this study was to examine the effects of researcher-developed lessons on students' understanding of two- and three-digit numeration. Digit-correspondence tasks, often used for individual interview assessment of place value understanding, were adapted to be used as problem-solving tasks. The tasks were presented to three classes, grades 3-5. Students were given ample opportunities, in cooperative groups and as a whole class, to discuss and exchange points of view. In the selected classrooms the social norms established by the teacher encouraged such exchanges. A scoring rubric was developed for a whole-class, digit-correspondence task requiring individual written responses. Only 18% were successful on the preassessment. Of the 58 students initially unsuccessful, 71% were successful after the instructional intervention as measured by a delayed postassessment.
In digit-correspondence tasks, students are asked to construct meaning for the individual digits in a multidigit numeral by matching the digits to quantities in a collection of objects. As measured by such tasks, even in the fourth and fifth grades no more than half the students demonstrate an understanding that the "5" in "25" represents five of the objects and the "2" the remaining 20 objects (Kamii, 1982; Ross, 1986; Ross, 1990).
Constance Kamii has argued that young students' developing understanding of place value is eroded by traditional algorithmic instruction in addition and subtraction, where individual digits are all treated as "ones" (Kamii & Lewis, 1993). Significant gains in conceptual understanding of place value have been demonstrated among first and second grade children participating in full-year studies where students are encouraged to invent their own methods for multidigit addition and subtraction (cf. Fuson & Smith, 1994; Hiebert & Wearne, 1992; Kamii, 1989).
In this study we examined the learning among older students who in prior grades had experienced traditional algorithmic instruction for multidigit addition and subtraction. In earlier work we had individually interviewed numerous students using digit-correspondence tasks, and had often wondered how children would react if they heard the ideas of other students and had an opportunity to react. We designed this study with two questions in mind. What would students learn from their peers if they share their thinking about the meanings of the digits in digit-correspondence tasks? What could be learned about student thinking by examining their individual responses to a written whole-class assessment instead of by using individual interviews.
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