Grounding mathematics in an integrated conceptual structure, part I: experimental evidence that grounded rules support transfer that formal rules do not.

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.