Grounding mathematics in an integrated conceptual structure, part II: intervention study demonstrating robust...

Mathematical reasoning systems are often treated as systems for manipulating formal expressions according to structure sensitive rules. However, these expressions typically reference objects, properties, and relationships in a target domain in which they have meaning. One case in point is precalculus trigonometry, a crucial part of high-school preparation for university level mathematics. In work reported in Part I of this 2-part publication, we found evidence that reliance on the unit circle, a visuospatial structure that provides meaning for formal expressions in trigonometry, provides an integrated conceptual framework that supports successful mastery of foundational trigonometric relationships that are often very difficult for students to learn. Importantly, however, although coverage of the unit circle is standard in classrooms and textbooks, many students fail to rely on it and so fail to benefit from the conceptual model it provides. Here, we consider some of the reasons why mastery of this system may be challenging and some of the pitfalls in the ways these relationships are often taught. We then describe a set of principles we used to guide the development of a tutorial curriculum aimed at addressing these challenges and pitfalls. After refinement, our curriculum was successful in allowing many high-school and community college students to learn to solve trigonometric identity problems students often fail to master in typical classroom settings and to retain what they had learned over a 2-3 week delay. Our findings demonstrate the value of developing structured, conceptually grounded, and research-motivated teaching materials to allow students to gain mastery of mathematical systems they might otherwise fail to learn.