ABSTRACT An ongoing debate concerns whether novel mathematical concepts are better learned using contextualised or decontextualised representations. A barrier to resolving this debate, and therefore to progress in the discipline, has been the paucity of validated methods of measuring students’ understanding of mathematical concepts. We developed an innovative and efficient method for measuring, in experimental settings, students’ understanding of any mathematical concept using comparative judgement. We demonstrate the method by applying it to the comparison of learning outcomes from two teaching conditions. Participants (260 15–16 year olds across six schools) were introduced to differential calculus using contextualised or decontextualised representations. We then assessed participants’ comparative conceptual understanding of derivatives. We found evidence that contextualised and decontextualised representations were equally effective at promoting student learning in this context. The assessment method yielded valid and reliable results, suggesting that it offers a robust and efficient approach for the problem of assessing conceptual understanding in experimental or other comparative settings.

中文翻译：

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使用情境化和非情境化表征进行教学：通过比较判断技术检验微积分的案例

摘要 正在进行的辩论涉及是否使用上下文化或去上下文化的表示可以更好地学习新的数学概念。解决这场争论并因此推动学科进步的一个障碍是缺乏有效的方法来衡量学生对数学概念的理解。我们开发了一种创新且有效的方法，用于在实验环境中使用比较判断来衡量学生对任何数学概念的理解。我们通过将其应用于两种教学条件下学习结果的比较来演示该方法。参与者（来自六所学校的 260 名 15-16 岁的学生）被引入使用情境化或非情境化表征的微分学。然后，我们评估了参与者对衍生品的比较概念理解。我们发现有证据表明，情境化和去情境化的表征在促进学生在这种情况下的学习方面同样有效。该评估方法产生了有效和可靠的结果，表明它为评估实验或其他比较环境中的概念理解问题提供了一种稳健有效的方法。