Mickey Kevin W, McClelland James L
Department of Psychology, Stanford University, Stanford, CA, United States.
Front Psychol. 2025 Jun 3;16:1507670. doi: 10.3389/fpsyg.2025.1507670. eCollection 2025.
Mathematics relies on formal systems of rules that can be treated in isolation or grounded in a conceptual system that provides meaning for the relationships the rules express. Here, we show how the conceptual system provided by the unit circle, a visuospatial structure that provides a meaning for formal expressions in the domain of trigonometry, supports a generalizable understanding of trigonometric relationships, allowing for transfer beyond relationships explicitly taught. We examined the utility of the unit circle in our first study, in which we presented trigonometric identity problems to undergraduates ( = 50) who had prior coursework in pre-calculus trigonometry. Students reported using the unit circle to solve these problems more often than other approaches, and those who reported using the circle solved more problems correctly. Using other students from the same population, we then manipulated the systems they used by presenting a refresher lesson, using either formal rules or rules grounded in relationships on the unit circle ( = 35 in each group). Students in both conditions improved on taught problems, but only students in the grounded condition showed improvement on held-out transfer problems. Using findings from a third study further exploring the grounded condition ( = 64 participants), we found evidence that the circle supported transfer in two ways: by providing a procedure that could be used to solve both taught and transfer problems without rules and by allowing students to appreciate rules as capturing relationships between meaningful quantities, facilitating their application and extension. This project served as the starting place for the development of a curriculum that supports reliance on the unit circle and led to robust learning and retention of trigonometric relationships for most students with sufficient relevant prior knowledge, as described in Part II of this article.
数学依赖于形式化的规则系统，这些规则系统可以孤立地处理，也可以基于一个概念系统，该概念系统为规则所表达的关系赋予意义。在这里，我们展示了单位圆所提供的概念系统，一种为三角学领域的形式表达式赋予意义的视觉空间结构，如何支持对三角关系的可推广理解，从而实现超越明确教授的关系的迁移。在我们的第一项研究中，我们考察了单位圆的效用，在这项研究中，我们向有预微积分三角学先修课程的本科生（(n = 50)）呈现三角恒等式问题。学生报告说，他们比使用其他方法更频繁地使用单位圆来解决这些问题，而且那些报告使用单位圆的学生正确解决的问题更多。然后，我们使用来自同一群体的其他学生，通过呈现复习课程来操纵他们使用的系统，复习课程要么使用形式规则，要么使用基于单位圆上关系的规则（每组(n = 35)）。两种条件下的学生在已教授的问题上都有进步，但只有基于单位圆关系条件下的学生在留存的迁移问题上有进步。利用第三项研究的结果进一步探索基于单位圆关系的条件（(n = 64)名参与者），我们发现有证据表明单位圆以两种方式支持迁移：通过提供一种可用于在不使用规则的情况下解决已教授问题和迁移问题的程序，以及通过让学生认识到规则捕捉了有意义数量之间的关系，促进他们对规则的应用和扩展。如本文第二部分所述，该项目是开发一门支持依赖单位圆的课程的起点，并导致大多数有足够相关先验知识的学生对三角关系进行了稳健的学习和记忆。
