Center for Teaching and Learning
Focused Inquiry Groups (FIGs) - BSI
Reading Apprenticeship - Members - Ming Ho
Reading Apprenticeship Faculty Inquiry Group
2009 - 2010
Linear algebra, a sophomore level math course, is a class in which many students begin writing frequent formal math proof for the first time. Many students have difficulty in proof writing. Since this is an advanced course, I want the students to get a sense of how mathematician discuss problems, formulate solutions, and revise their work for presentation. I didn’t feel that individual RA techniques were the right tools for my student, but the ideas of cognitive apprenticeship, the basis of RA, were very appealing. As Brown, Collins, & Duguid describe it, "Cognitive apprenticeship methods try to enculturate students into authentic practices through activity and social interaction in a way similar to that evident in craft apprenticeship." Therefore, I moved my office hour to Friday mornings during the same time as the MW class and in the same classroom to get students to do some collaborative mathematics.
During office hour, everything has to be written and discussed at the board for all to see. Sometimes students have particular questions, but most of the time, students work together on homework problems at the board. In any case, a shared workspace over which participants discuss problems and formulate solutions is how professional mathematicians collaborate, and this authentic setting has allowed me to engage in elements of cognitive apprenticeship: Modeling, coaching (which includes scaffolding), exploration, articulation, and reflection.
Implicit in modeling under cognitive apprenticeship is cognitive modeling, in which thoughts and reasons for the process accompany the tasks being demonstrated. While I do that as a part of the lecture, I encourage my students to replicate that process as they do the problem, so they can model for each other. Meanwhile, I coach them by filling in incomplete explanation and highlight important points that link ideas together.
As students write their solution on the board, we can all see what’s going on. I encourage them to talk through (articulate) the process, as I have done. I work with them to edit their solution to follow cultural conventions of the field and to polish up their proof for ease of reading. Being in a more social atmosphere, sometimes a student would ask a “what-if” question and I would encourage them to go down that path. Students appear more likely to explore under guidance. Often outside of class, a student would say that he had an idea, but no work was done to follow through. Being able to monitor them as they explore provides authentic teaching moments on proof-writing and proof techniques, which mostly seem to go over students’ heads when they are not the one actively engaged in problem solving. At the end of their work while the problem context is most immediate to them, I can help them reflect on the similarities between what they have just completed and what they have seen before, extracting out common structures.
Because I only have four students in the class, the students get very personal attention during the workshop as they work. With a larger class, I may have to split students into small working groups and have each group working in front of the board. I am not sure really how to evaluate the time my students spent with me on these Fridays, as compared to what might have been if I didn’t do this. However, one sign that students find these sessions help is that all but one student come to it. (The one student works Friday mornings.) One of the students lives in Berkeley and comes on Fridays even though he has no other classes.
Brown, J. S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher, 18(1), 32-42.