I do not use traditional methods for going over homework in the mathematics classes I teach. The accepted wisdom for many college mathematics instructors is that one spends ten to fifteen minutes each class period answering questions from the homework. This ``question answering'' is usually the instructor doing an exercise on the board, ``explaining'' each step.2 What typically happens is that students, very devotedly, copy down exactly what the instructor writes on the board. The more anxious a student is about the material the more likely it is that learning gears are not engaged while this collective scribing takes place. Students become collectors, convincing themselves that having the solution to an exercise is equivalent to understanding the solution process. Further witness to this perception on the part of students - that possession of solutions suffices - is epitomized in the following remarks by a student in her mathematical autobiography (see §1.3.2 for more on my work with mathematical autobiography),
One aspect of geometry that I will forever dread was the truth tables. I had the hardest time learning how to prove that a square is a rectangle ...I don't think I ever want to cross truth tables again ...I'm planning on becoming an elementary teacher, but if I were to ever change over to high school, I would like to teach Algebra or geometry (please note, crossing truth tables as a teacher wouldn't be that bad because I will have the teacher's book).[Hauk, 2005, p.43]Relying on the ``authority'' of the textbook is a common theme in school mathematics culture (among both teachers and students).
Part of my role as teacher is to help students see that possessing a solution bears little relation to understanding a solution method. Students may memorize the actions but still have little or no understanding of the processes associated with a concept [Breidenbach, Dubinsky, Hawks, & Nichols, 1992]. Undergraduate students tend to equate seeing with knowing. Among the ways to challenge the seeing=knowing view, I have found each of the following effective: (1) make a solutions manual available to students;, (2) Have students create their own solutions manual.
In the first case, students quickly realize that possessing/seeing the answers does not equate to facility with mathematical concepts. Allowing students a solutions manual also answers the most common questions students have about many exercises. In the second case, when a student is responsible for creating her own solutions manual, she has a reason for participating in mathematical conversation with classmates. Students can see each other as resources. Since they generally have difficulty in communicating mathematically, discussion elicted by one student asking another, ``How did you do number 39?'' may be rife with false starts and struggles on both sides in the effort to communicate about the mathematics. Nonetheless, the practice at mathematically sensible verbalization is valuable to all participants in the conversation [Schoenfeld, 1992; Sfard, 1991].
In either case, at the beginning of each undergraduate class section I come in and put an Advanced Organizer on the board consisting of a list of the topics and activities for the class meeting. If the course is a lower division, general education course, the first order of business is to allow students to voice comments and concerns about their work, the class, or anything listed for the day's activities. I have found that making a habit of soliciting students' views, especially in general education courses where mathematics anxiety is frequently high, makes for a relaxed and cooperative classroom atmosphere. In a more advanced course for mathematics majors, however, I find it more conducive to an engaging classroom atmosphere if the first item on the Advanced Organizer for the first few weeks of the course is the presentation (by myself or a student volunteer) of a solution to a homework exercise exemplifying the actions and processes the students might be expected to understand. I do this at the beginning of the term because I want students accustomed to the seeing=knowing notion to be comfortable. As the semester progresses, the students take over this part of the class meeting, often filling the board with work before I arrive in the room. In the graduate courses I have taught I rarely write on the board at all, leaving that up to the graduate student participants in the courses. Most of the time I run graduate courses using the European tutorial model, much like an American seminar but with a great deal more writing and email communication outside of class.