Visualizing Logical Thinking using Homotopy A new learning method to survive in dynamically changing cyberworlds

For logical thinking learners, it is important to learn problem solving formally and intuitively. Homotopy which is modern mathematics combines algebra and geometry together. Homotopy gives the most abstract level of explanation or the most basic invariant which is the most required characteristics for logical thinking. The homotopy lifting property gives good explanation of inductive reasoning. The homotopy extension property, which is dual to the homotopy lifting property, provides a good tool for deductive reasoning. As homotopy combining algebra with geometry, it provides intuitive explanation with informative figures. This paper shows how puzzle-based learning is explained mathematically and intuitively using these properties.