Reflection - Role of Representation(s) in Developing Mathematical Understanding

I am convinced by the author's discussion about how the development of mathematical understanding begins through various representations but I believe that it is limiting in the way we develop mathematical skills overall. Personally, I believe that once one develops the representations discussed, they must also be able to communicate it and that requires understanding mathematics from a language-present perspective. By this, I mean that once a student is able to visualize a mathematical concept through the representations discussed, then they should be able to reason and manipulate these concepts as though they were words in a sentence.

One example I have is how we simplify equations by collecting like-terms. Fractions can be thought more abstractly as objects than the "parts of a circle" discussed and so, one quarter plus one quarter is two quarters. This further generalizes to higher level concepts like polynomials where "one ex-squared plus three ex-squared is four ex-squared." Once a student understands this, they can break equation, or "math sentence" down into smaller parts and rearrange them as though they were rewriting an english sentence to have a different sentence structure.