Writing mathematics is an important activity of the mathematician. In trying to write one's mathematics, one finds ways to balance intuition and rigor and to efficiently communicate concepts and ideas along with the results of calculation.
In requiring students to write mathematics in full sentences, we often require that they provide justification for what they do. We get a more complete picture of what the students are thinking, and check blind and mindless computation.
I have recently told my students that mathematical notation is a very powerful shorthand that one must earn the use of by demonstrating that one knows why one is doing what one is doing.
Of course, in reality we all compute when working on a problem: we all make formal substitutions according to rules appropriate to a symbolic system. We are not always mindful, in language, of the validity of these substitutions as we are doing them. Arguably, a different part of our brain is engaged when we are carrying out computations than when we are writing them up. In short, we all (I imagine rather often) behave like formalists.
A more serious objection to writing while one works is that mathematical notation is a very powerful shorthand. Using symbolic substitutions provides a great deal of leverage that writing everything down misses. Just think of basic arithmetic with place value!!!
Of course, attempts to reduce mathematics to formal logic a la Whitehead and Russell surround the issue.
Question: Has there been any study of the dichotomy between the mental processes of writing mathematics and calculation?
Writing and calculating just feel so different, I wonder if someone has studied this.