The challenge of incorporating more writing into your syllabus is that if you add more assignments you will have to read and respond to all those papers.  Though your students might eventually write better with more opportunity to practice and more guidance from you, instructors are for the most part already too busy to take on the extra work. I suggest it is possible to ask your students to write more, as long as you adapt your grading style. 

Writing to Learn
Students presumably learn through reading and class discussion of course materials, and then show you what they’ve learned by producing a paper or research project. But you can also use informal, non-graded assignments to allow students to show themselves what they’ve taken in from lectures, discussion, and readings, especially at a point where they may not have fully mastered the material. The idea is that you are looking for understanding of concepts, rather than polished work. Such work will not be assessed for grammar and correctness. Effortful but faulty answers are also acceptable. Your grading may simply consist of putting a check mark on it to note that it was handed in.  

Rather than putting comments on paper after paper, you can teach course concepts and writing by featuring one or two exemplary papers. You can project them for the whole class and discuss why these examples are worthwhile. You can also use informal writing-to-learn assignments to show how ideas might be developed further for a formal paper. You can show where unsubstantiated assertions would need to be supported with evidence from the text, or where transitions might be needed to connect several ideas. 

Giving an unannounced writing assignment can work to salvage a class where discussion has sputtered and everyone is looking down at the table. Likewise in discussions where some speakers are very heated and others are apparently afraid to venture an opinion, asking the whole class to write for ten minutes gives everyone a chance to commit to an opinion. You can use a writing-to-learn assignment as homework, and start the next class discussion by requiring students to read their essays. Used in this way, there’s really no need to grade or comment on the writing, or in some cases even to collect it. The writing-to-learn assignment should strengthen the student’s connection between critical thinking and writing, allow everyone to participate in class discussion, if only silently, and force students to be responsible for their own learning by having them produce their thoughts on paper. 

Strengthening the Writing Process
Though many instructors do assign formal papers with graduated deadlines for proposals, drafts, and finished products, students still often have difficulty breaking their writing into steps and truly revising. When students are given informal writing assignments with a very short deadline, they are forced to produce rough drafts.  If you then use the drafts to show where revision should happen, then students begin to see the difference between rough and polished work. 

You can also emphasize the difference between rough and polished work with your approach to grading. Demand excellence of finished work—work that has been thought through, revised, well documented, properly formatted, edited, and proofread. Don’t penalize students for stumbling, disorganized, horribly spelled and perhaps conceptually faulty rough work, as long as it is prepared thoughtfully. This work is only a step towards mastering the material.