Longitudinal Study

A Longitudinal Study of Invention and Understanding in Children's Multidigit Addition and Subtraction*
Journal for Research in Mathematics Education, Vol. 29, No. 1, pp. 3-20 January 1998

Thomas P. Carpenter, University of Wisconsin-Madison Megan L. Franke, University of California at Los Angeles Victoria R. Jacobs, California State University-San Marcos Elizabeth Fennema, University of Wisconsin-Madison Susan B. Empson, University of Texas at Austin

This 3-year longitudinal study investigated the development of 82 children's understanding of multidigit number concepts and operations in Grades 1-3. Students were individually interviewed 5 times on a variety of tasks involving base-ten number concepts and addition and subtraction problems.
For example, most procedures for adding or subtracting multidigit numbers essentially reduce the calculation to a series of sums or differences of single-digit numbers (or corresponding multiples of 10, 100, etc.). There are, however, some fundamental differences between standard algorithms and the strategies that children construct to solve multidigit problems. Standard algorithms have evolved over centuries for efficient, accurate calculation. For the most part, these algorithms are quite far removed from their conceptual underpinnings. Invented strategies, on the other hand, generally are derived directly from the underlying multidigit concepts. For example, with standard addition and subtraction algorithms, numerals are aligned so that the ones, tens, hundreds, and larger digits can be added in columns. But in the addition of columns, no reference is made to the fact that the addition involves the same unit (ones, tens, hundreds, etc.); one simply adds numbers in a column. Most invented strategies, on the other hand, specifically label the units being combined. For example, in the above addition examples, the invented strategies are based on adding 30 and 20 or 3 tens and 2 tens rather than adding two numbers that appear in the same column.1 Sequence of Development of Major Concepts and Procedures Longitudinal
The interviewer then pointed to the numeral 7 on the card and asked the students to show with the chips what that part meant. Then the interviewer pointed to the numeral 1 and asked what that meant. The rest of the base-ten-number-concept tasks were set in the context of word problems involving groups of 10. In one problem, students were asked to find the total number of pieces of gum in four packs of gum with 10 sticks of gum in each pack. In another problem, students were asked how many groups of 10 could be made from an initial set of 36. the final problem involved making teams of 10 each from a group of 241 children. Each problem was coded as demonstrating base-ten knowledge if students responded immediately on the basis of their knowledge of tens or counted by 10 without using counters or base-ten materials. The bundles task was used in the first two interviews, the multiplication word problem in the first three, and the division problem involving 241 children in the last two. The other two tasks were used in all five interviews. In the first two interviews, students were classified as demonstrating knowledge of base-ten number concepts if three out of four of their responses were coded as demonstrating base-ten knowledge; in the last three interviews the criterion was two out of three. Addition and