A Unique Experience Learning Calculus: Integrating Variation Theory with Problem-Based Learning

The paper proposes a pedagogical approach to teaching and learning calculus differentiation formulas that synthesizes the principles of variation theory (VT) and bianshi in a problem-based learning (PBL) format. Unlike traditional approaches that view formulas procedurally, the paper adapts Steinbring’s (1989) distinction between “concept” and “symbol,” abstracting differentiation calculus formulas as “concept” (i.e., the meaning of the formula) and “symbol” (i.e., procedural knowledge about how to apply the formula). The paper then aligns this distinction with VT and bianshi pedagogies. While VT emphasizes more static elements of conceptual knowledge (e.g., highlighting the contrast between conceptual and non-conceptual features of the object of learning), bianshi broadens the concept of variation, offering more dynamic principles of variation through procedural variation (e.g., via the process of problem solving) (Gu et al., 2004). Combining VT and bianshi into a single pedagogical application yields an eight-step approach to teaching and learning calculus differentiation formulas.