‘‘Concrete’’ Computer Manipulatives in Mathematics Education

The use of ‘‘concrete manipulatives’’ in mathematics education is supported by research and often accepted as a sine qua non of ‘‘reform’’ approaches. This article reviews the research on the use of manipulatives and critiques common notions regarding concrete manipulatives. It presents a reformulation of the definition of concrete as used in educational psychology and educational research and provides a rationale of how, based on that reformulation, computer manipulatives may be pedagogically efficacious. The article presents 7 hypothesized, interrelated affordances of manipulatives and briefly reviews evidence for their empirical validity. KEYWORDS—mathematic education; child development; manipulatives; cognition; abstract; technology; computers; concrete models The notion of ‘‘concrete,’’ from concrete manipulatives to pedagogical sequences such as ‘‘concrete to abstract,’’ is embedded in educational theories, research, and practice, especially in mathematics education (see Kaminski, Sloutsky, & Heckler, 2009; Martin, Lukong, & Reaves, 2007; McNeil & Uttal, 2009). Like many widely accepted notions that have a good deal of truth behind them, this one has become nearly immune from critical reflection. In this article, we consider research on the use of mathematics manipulatives—for example, ‘‘colored counters, miscellaneous ‘junk’ items, patterning material, blocks of various colors, shapes and sizes, linking cubes, and base-ten blocks’’ (English, 2004, pp. 205–206)—in early and elementary education and offer a critique of common notions concerning concrete manipulatives and concrete ideas. From a reformulation of these notions, we reconsider the role computer manipulatives may play in helping students learn mathematics. EARLY RESEARCH ON MANIPULATIVES Early research on mathematics learning with manipulatives supported the notion that students who use manipulatives in their mathematics classes usually outperform those who do not (Driscoll, 1983; Greabell, 1978; Johnson, 2000; Lamon & Huber, 1971; Raphael & Wahlstrom, 1989; Sowell, 1989; Suydam, 1986). The studies also showed an increase in scores on retention and problem-solving tests. Not all the early research was confirmative, however. Fennema (1972), for example, found that on a test of transfer, students who did not use Cuisenaire (colored) rods to learn multiplication as repeated addition outperformed students randomly assigned to an instructional intervention that used manipulatives. Other researchers reported a lack of connection between students’ representations; for example, students did not connect the arithmetic they performed with manipulatives to their paper-and-pencil computations (Resnick & Omanson, 1987; Thompson & Thompson, 1990). Taken together, this early research, supported by subsequent studies (e.g., Gagatsis, 2003; Martin et al., 2007; Uttal, O’Doherty, Newland, Hand, & DeLoache, 2009), suggests that instruction should begin ‘‘concretely’’ but also that This paper was based on work supported in part by the National Science Foundation under Grant ESI-9730804 to D. H. Clements and J. Sarama ‘‘Building Blocks—Foundations for Mathematical Thinking, Pre-Kindergarten to Grade 2: Research-Based Materials Development’’ and in small part by the Institute of Educational Sciences (U.S. Department of Education, under the Interagency Educational Research Initiative, or IERI, a collaboration of the IES, NSF, and NICHHD) under Grant R305K05157 to D. H. Clements, J. Sarama, and J. Lee, ‘‘Scaling up TRIAD: Teaching Early Mathematics for Understanding With Trajectories and Technologies.’’ Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding agencies. Some sections reflect previous work (Clements, 1999). Correspondence concerning this article should be addressed to Julie Sarama, University at Buffalo, State University of New York, 505 Baldy Hall (North Campus), Buffalo, NY 14260; e-mail: [email protected]. a 2009, Copyright the Author(s) Journal Compilation a 2009, Society for Research in Child Development Volume 3, Number 3, Pages 145–150 CHILD DEVELOPMENT PERSPECTIVES