Mathematics as presented in textbooks, journals, and lectures is primarily constituted by a formal language of demonstration that is distinct from the informal heuristic language of investigation that gave shape to and informed what became the polished presentation. The fundamental problem is with regard to the learning of mathematics, which research locates historically as a challenging outcome for a considerable portion of US students. The findings while undesired are understandable as students are presented with a language of demonstration with regard to learning mathematics, yet are in effect asked to draw upon a language of investigation for doing mathematics. The focus of this paper is to make evident the necessity for providing a mathematics classroom and textbook experience where the formal demonstration includes the informal heuristic decisions that were essential in informing and shaping the final presentation.