Computer-Based Dynamic Assessment of Multidigit Multiplication

Contemporary special education research, perhaps more than other areas of educational inquiry, is more concerned with variance than with central tendency of populations. For this reason, special educators and psychologists interested in disability often express dissatisfaction with standardized measures of achievement and intelligence, especially when such measures are intended to support decisions pertaining to special education eligibility, classification, and instructional intervention. Over the past 3 decades, research on cognition and information processing has raised hopes that alternative approaches to assessment can be developed that are more oriented toward understanding and depicting variance in emerging competence than in detecting deviance. For this purpose, an array of cognitive and cognitive-behavioral approaches to assessment have been developed (Frederiksen & White, 1990; Lidz, 1987). These approaches have common focus on the dynamics of assessment processes, rather than on the static summaries of performance typically generated by psychometric approaches. Also, they are centrally concerned with describing the individual learner's change and susceptibility to instruction over the course of problem-solving, rather than with "fixing" the relative location of that individual in a distribution of which he or she is a part. However, the concept of a "dynamic" assessment is oriented toward changes in intraindividual processes that produce observable changes in qualities of performance. Dynamic assessment also gauges social-instructional processes-the directions, the hints, the explanations, the models-that influence individuals in such a way that these processes are not apart from, but rather part of, the performance being assessed. Principles of dynamic assessment fundamentally reject the proposition that any given test performance (i.e., behavioral sample) can unambiguously portray or reveal children's capacity to learn and achieve. Instead, even very different approaches share the belief that attempts to modify children's academic performance is both the method and motive of a dynamic assessment process. Contingent instruction, in this view, cannot logically be separated from educational assessment. PROBLEMS WITH PREVIOUS WORK Despite striking similarities in underlying concepts, various approaches to dynamic assessment appear to differ significantly in emphasis, if not substance (Hasselbring, 1987; Joyce & Wolking, 1987; Lajoie & Lesgold, 1992; Minick, 1987; Rothman & Semmel, 1990). Because most current assessment relies on complex theories to derive clinical guidelines, standards, and practices, classroom teachers and other practitioners find dynamic assessment particularly difficult to learn and use effectively and successfully. Moreover, surprisingly few studies exist to illustrate how one should conduct or evaluate dynamic assessment of performance on those academic tasks especially relevant to students with mild disabilities-or their teachers. DYNAMATH PROJECT Over the past 3 years, we have developed and field-tested a computer-based dynamic assessment of multidigit, multiplication facts for secondary-school-level students with mild disabilities (Semmel, Gerber, & Semmel, 1992). This program (DynaMath) assesses student performance on problems having one or two multipliers and two or three multiplicands (i.e., the Domain). Figure 1 shows a schematic of a generalized multidigit multiplication problem as it is represented within the program. (The dynamic assessment program described here was developed under the name "DynaMath." The program has no connection with the magazine Scholastic Dynamath, and future versions of the program will be called DynoMath.) At the outset of assessment, students complete, on a standard keyboard, a computer-based preassessment of all 100 single-digit combinations. The program uses the results to create a "problem space" for each student in this domain, in which the difficulty of retrieving single-digit products is uniquely scaled for each student. …