Abstract:

There is a vast body of example-based literature consistently showing that studying multiple examples is more effective than one example to promote learning because the comparison evoked by comparing multiple examples is generally good for learning. Not all comparisons, however, may equally be effective. The effectiveness of comparing multiple examples depends on the variability of the examples being compared and the prior knowledge of students who compare the examples, which are still unsolved questions and need further research. The current study experimentally examined the effects of different comparisons on the learning of multistep linear equation solving and their interaction effects with students’ prior knowledge. According to previous studies, there are two critical aspects for learning how to solve linear equation: problem type and solution method. There are three types of multistep linear equation and each type of equation can be solved by two types of solution methods. Three comparison conditions were designed accordingly to teach equation solving. We employed a pretest-intervention-posttest design. During the 3-day intervention, 186 seventh-grade students were randomly assigned to learn equation solving by comparing variation of the two critical aspects of problem type and solution method (abbreviated as comparing type and method; n = 60), by comparing variation of problem type (abbreviated as comparing type; n = 65), or by comparing variation of solution method (abbreviated as comparing method; n = 61). To investigate the role of prior equation solving knowledge, we categorized students as using shortcut or not using shortcut to solve equations at pretest, and we tested for a Prior Knowledge × Condition interaction. Students’ procedural knowledge, flexibility knowledge, and conceptual knowledge were assessed at pretest and posttest to evaluate the effectiveness of different comparisons. Results showed that the effectiveness of different comparisons in learning to solve equations was moderated by students’ prior knowledge. Comparing the two critical aspects of problem type and solution method (comparing type and method) led to more flexibility knowledge and conceptual knowledge than comparing only one of the two critical aspects (comparing type and comparing method) for students who did not use shortcuts at pretest. The different effectiveness of conditions, however, was absent for students who used shortcuts at pretest; those students did not show any preference for any condition. Students who have different levels of prior knowledge may perceive different aspects as critical for their learning and thus benefit differently from the same instruction. Students with higher prior knowledge should perceive fewer critical aspects than those with lower prior knowledge and are more likely to benefit from any instruction. Multiple examples should focus on aspects and features that are critical for student learning, and should be designed with controlled variation to ensure that students discern their critical aspects first separately and then simultaneously. Separate variation of each critical aspect prepares students to compare examples that are different in all critical aspects; experiencing simultaneous variation of all critical aspects is important for completely understanding the concept.

摘要：

186名初一年级学生通过对比三种不同变异类型的样例学习解一元一次方程，他们对比方程问题类型和解法这两个关键特征的变异、只对比问题类型的变异、或者只对比解法的变异。结果显示样例关键特征不同变异类型的学习效果受到学生先前知识的影响：对于在前测中没有使用简便方法的学生，变异类型和解法两个关键特征比只变异类型或解法其中一个关键特征更有利于他们学习变通性知识和概念性知识；而对于在前测中有使用简便方法的学生，不同变异类型的效果没有显著差别。多重样例变异性的设计需要提供机会让学生充分对比学习每个关键特征。