Discovering mathematics can be likened to piecing together a puzzle. You gather fragments of mathematical knowledge from various sources, gradually filling in the gaps. However, the relationship between individual components and the overall picture differs in mathematics compared to puzzles.
In a puzzle, individual pieces can be unclear, but the complete image is readily comprehensible. On the contrary, when diving into a mathematical text, the core components, when found, are typically quite natural and easily understood individually. However, when intertwined, they form an abstract mathematical entity with definitions, statements, and proofs. This systematic presentation promotes rigor but can obscure simplicity.
In many teaching scenarios, instructors tend to present the final, precise, but somewhat opaque sequence of mathematical steps. We advocate a different approach – one that emphasizes discovery-based teaching. With this method, you convey mathematics in a manner that echoes the process of its discovery: you first explain the natural and usually simple core ideas and then adeptly fit them into a deductive framework. The challenge lies in maintaining clarity and oversight, a skill honed through decades of practice.
Through this approach, students gain insights into how mathematics operates. They internalize this mathematical method, a critical skill for their future professional lives.
