Using Time Pressure to Promote Mathematical Fluency
Time pressure helps students practice efficient strategies. We report strong effects from using games to promote fluency in mathematics.
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Encouraging students to share and compare solution methods is a key component of reform efforts in mathematics, and comparison is emerging as a fundamental learning mechanism. To experimentally evaluate the effects of comparison for mathematics learning, the authors randomly assigned 70 seventh-grade students to learn about algebra equation solving by either (a) comparing and contrasting alternative solution methods or (b) reflecting on the same solution methods one at a time. At posttest, students in the compare group had made greater gains in procedural knowledge and flexibility and comparable gains in conceptual knowledge. These findings suggest potential mechanisms behind the benefits of comparing contrasting solutions and ways to support effective comparison in the classroom. Current educational reforms in mathematics advocate that the teacher act more as a facilitator, encouraging students to share and compare their own thinking and problem-solving methods with other students (Hiebert & Carpenter, 1992; National Council of Teachers of Mathematics, 1991, 2000). Despite an abundance of descriptive research suggesting the promise of this approach, ex-perimental studies that demonstrate its benefits are largely absent. In this study, we experimentally evaluated a potentially pivotal component of this instructional approach that is supported by basic research in cognitive science: the value of students comparing multiple examples. We used a unique design that allowed for random assignment to condition within intact classrooms to in-crease both internal and external validity. Seventh-grade students learned about algebra equation solving by either (a) comparing and contrasting alternative solution methods or (b) reflecting on the alternative solution methods one at a time. The findings have theoretical implications for when and why contrasting examples facilitate learning and practical implications for how to support effective comparison in the classroom.
The choice/no-choice method provides a means of obtaining unbiased estimates of the performance characteristics of strategies. The three experiments in the study illustrate the method's usefulness for testing predictions of alternative models of strategy choice. The experiments focused on 20- and 70-year-olds' choices among mental calculation, use of a calculator, and use of pencil and paper as strategies for solving multidigit multiplication problems. As predicted by the Adaptive Strategy Choice Model (ASCM), (a) differences in the speed and accuracy yielded by the strategies were the strongest predictors of the frequency with which each strategy was chosen on a given problem; (b) features of problems exerted an additional independent influence; and (c) having a choice resulted in better performance than not having one. These results held true for both older and younger adults. Potential extensions of the choice/no-choice method and of ASCM are discussed.
Consistent individual differences were found in first graders' strategy choices in addition, subtraction, and reading (word identification). Differences were present along 2 dimensions: knowledge of problems and stringency of thresholds for stating retrieved answers. Cluster analyses indicated that children could be classified into 3 groups: good students, not-so-good students, and perfectionists. Perfectionists were children who had good knowledge of problems and set very high thresholds for stating retrieved answers, good students also had good knowledge of problems but set lower thresholds, and not-so-good students had less good knowledge of problems and set low thresholds. Differences among the 3 groups were evident on measures not included in the cluster analysis as well as measures that were. Further, the groups differed in standardized achievement test performance 4 months after the experiment in ways consistent with the experimental analysis. The pattern of individual differences was similar in 2 experiments with different samples of children and problems and different methods for assessing strategy use. The results illustrated how detailed cognitive models can contribute to understanding of individual differences.