Role of representations

In mathematics, knowing how to say the same thing in many different ways is a powerful asset; at the same time, jumping to algebra first, while an "elegant" approach, can hide details and fail to truly illuminate the "discovery" of the final product; thus, I must say that external representations done wrong can actually increase the cognitive load on the student. While I may seem to be slightly disagreeing with the article, both of those points are actually supported by the same Russian experiment on teaching trigonometric proofs seems to assert as much.


Speaking of jumping to the abstract first, a case in point would be starting a group theory course in university by dropping the four group multiplication properties out of thin air. When group theory was the hottest frontier back in the 19th century, the notations and concepts were nowhere near as slender as today; it took several generations until Cayley came along and cleaned things up.


When the authors discuss Hiebert's insights on developing competencies in working with mathematical symbols, it would appear that after the initial connection between the symbols and the underlying objects are made, pupils shall learn how to manipulate symbols, and then that is that. It is tempting to assert that understanding how the objects are manipulated underneath the said symbols cannot be neglected.






Assuming that "iconic" includes other kinds of multimedia in addition to pictures in static frames like ye olde portraits, I do not think anything in particular was "missed" per se.