Enhancing children's conceptual understanding of mathematics through Chartworld software.(Report)

Abstract. Two randomized control/treatment experiments tested the effectiveness of Chartworld software on enhancing mathematical knowledge of 3rd-graders. In Experiment 1, 196 children were randomly assigned to receive either Chartworld or textbook instruction on number operations. ANCOVAs indicated significant group differences in favor of Chartworld in the magnitude of posttest performance while controlling for pretested ability (p

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Chartworld is a flexible computer program that allows children to create a wide variety of patterns. There is a mathematical reason for each pattern, and one of the strengths of the program is that children can explore the patterns at any stage of understanding, and can learn mathematics as they explore. With Chartworld, children can easily create models showing: multiples, the commutative property of multiplication, perfect squares, division as the inverse of multiplication, division with remainder, factors, prime numbers, divisibility tests, and the Sieve of Eratosthenes.

Chartworld emphasizes conceptual reasoning based on number sense, in addition to the more traditional focus on rote memorization. The development of Chartworld was influenced by the work of Seymour Papert (1980), who developed Logo, a highly popular computer language during the 1980s. Logo proved to be simple enough for kindergartners to draw shapes and yet powerful enough for college students to write sophisticated programs to solve problems in differential geometry (Abelson & diSessa, 1981). Despite all of the potential benefits, Logo has not gained widespread acceptance in the public school classrooms.

In an effort to build upon Logo's considerable strengths and to correct its weaknesses, Andrea diSessa (2000; diSessa & Abelson, 1986) developed Boxer, a programming language in which text editing, graphics, and database capability are all integrated with programming. Chartworld is written in Boxer, which has proved to be flexible enough for students to write programs on a wide range of topics (diSessa 2000; diSessa & Abelson, 1986). Crucially, it is very easy to modify any program in response to requests from teachers and students.

This research incorporates a fully randomized controlled experimental design. Theoretical and empirical evidence converge to support the conclusion that poor performance in working with numbers may be largely traced to separation between conceptual understandings of properties about numbers and performing operations with numbers, such as performing arithmetic. Conceptual knowledge can be defined as the awareness of what mathematical symbols mean, and the ability to represent relations among numbers in multiple ways.

Bransford, Brown, and Cocking (2000) and Kilpatrick, Swafford, and Findell (2001) reviewed the major cognitive science perspectives that provide a framework on which any information-processing model of mathematics skills should be based. Consistent with these reports, this study on Chartworld also assumes that it is necessary for children to coordinate conceptual knowledge in order to understand the underlying meanings of arithmetical operations with numbers. Thus, conceptual knowledge reveals the logical structure of any math domain. The ability to understand the meaningfulness of numbers, operations, and applications is often cited as one of the defining characteristics of the emerging construct in the mathematical cognition literature referred to as "number sense" (Berch, 2005). Conceptual understandings about whole numbers are important for children to be able to apply their understandings toward solving procedural and word problem tasks (Hecht 2002, 2006; Hecht, Close, & Santisi, 2003; Hecht, Vagi, & Torgesen, 2007). Children's ability to represent quantitative information enables them to see the meaning behind number operations. This, in turn, enables accurate representation of arithmetical relations in word problems.