Emerging from problem solving to modeling through dimensional exchange of the respective illustrative diagrams

Our teaching experience tells that students may feel easier to grasp mathematical concepts and ideas through illustrative examples that directly confront their previous mathematical knowledge with what is going to be learnt. Illustration usually appears in the form of rough sketch or diagram with quick calculation. It aims to hint a mathematical idea. Effective manipulation of the appearance of such illustrative diagrams may enable teachers to deliver an effective lesson. Such manipulation not only includes how to label, present, and interconnect various objects in the diagrams, but also skillfulness of changing the dimensions of structure of the shown figures or diagrams, such as projecting a three dimensional picture to a two dimensional one, or vice versa. We discuss on how this skill of dimension exchange changes the way of teaching that may lead to a hint or clue for modelers in solving modeling tasks in a real life situation.