On the student page, each of the course's tasks should be displayed, along with the current status of that the task. Per Greg, the term for this status is a "completed task." If a task is not completed, then it will appear blank.
The student should not be able to mark a task as completed.
An instructor viewing a student's page should be able change the status of the task. For version 1, these tasks will be completed by entering plain text. According to a wireframe, perhaps a status color should be specified by the instructor as well.