This thread is for the person that is interested in seeing how I teach a day to day class. Each link discusses something about my teaching. The first few links are samples of days that I teach, and the last link contains a discussion about one curious nature of the course I taught and the sorts of rich student driven mathematical conversations we had. This last contains some thoughts on what were some factors that inhibit conversations.
The main thing I am trying to get across in this link is how I teach the class with a goal in mind of producing conversations and creating student motivated thoughts. The first day of class is set to create an atmosphere where the students start thinking about how the class can discuss high school mathematics (fractions) in the context of advanced mathematics (equivalence relations), but it is also designed to give them some feeling for me as a person. I am a mathematician, but I am not someone that knows everything. Rather, mathematicians (like myself) investigate mathematics, and sometimes we don't know the answers, but we enjoy investigating the mathematics.
The second day is the first day of in class group work, where the students can start gaining confidence of speaking out in small groups. Moreover, they ideally should begin to see the big idea of mathematics being one of investigating and answering questions. Thus, they begin small (looking for one digit in a decimal expansion) and then move through the homework to bigger and more interesting things.
The cubic discussion (which encompasses many classes) is one of the places where we see my more traditional platonic discussion/lecture style. I let the students lead me at times, but at other times I will lead them through ideas. This is seen in the cubic classes when I have the students tell me how to derive the quadratic formula, but then I lay out the idea behind the algebra in the cubic solution.
The day the teacher hid selection is a full-fledged artifact from the main portfolio. I chose this piece because that day was the culmination of creating a community of mathematicians in the classroom. On that day, the class decided whether an answer was correct without me. The students have thus taken more and more of the responsibility of seeing whether what we do is correct. It is really after this point that several students start contradicting homework questions on taking mathematics to the high school. This is something I need them to do, because it is the conflict over these ideas that produces students (and thus teachers) that think and argue about what material can be brought down to the students and what material can influence their teaching.
Finally, the conversations piece tells of the success of getting these students to start asking questions about deep mathematical topics. A feature of this particular class that was extremely exciting.
For me, I try and teach in a way that follows Lampert's discussion of being a dance instructor (Lampert, 1990). Some of the time, the students watch me show them how to dance, some of the time they practice dancing with each other, and some of the time, they get a chance to dance with me. Unlike Lampert, however, much of the dancing with each other and dancing with me can happen out of class during office hours or in project meetings. Thus, while the above classes show a diversity of what I do, they do not show it in the percentages that I carry out. Probably 75% of the classes I teach are Socratic lecture format, 20% are group work, and the remaining 5% are like the day the teacher hid.
Thus, the final element I want the viewer of this portion of the portfolio to understand is that the ideal of student conversations does not rest on avoiding the traditional Socratic lecture method in mathematics.