This writing is characterized by fairly formal rules of writing. This is the type of writing that is typically graded in mathematics classes. Solution writing is the write-up of solutions for problems. Formality requirements vary. Proof writing requires the use of a formal set of rules which varies in rigorousness from situation to situation.
Presentations require students to prepare writing for oral, and possibly visual, presentation.
Quick writing assignments can be collected or discussed in class. These are designed to get instructors an assessment of how well the students are learning and what they are thinking. Discuss the most difficult homework problem. Insiders and Outsiders, Academic Connections , p.1-5.