In a subject such as high school mathematics, whose >primary value is in application, how can we balance the use of discussion to >promote learning and written exams to test ability to apply the learned >concepts?
In kindergarten, everyone is used to group work as something normal; a part of every day school work. As we get older, group work is often seen as some sort of lesser form of learning. If teachers are going to be "nice", they "let" their students work in groups. It seems to me this is the wrong way for this type of learning to progress. Group work is a way of balancing discussion with written exams for a fuller understanding of concepts as well as application; let me explain.
Working in groups is a great way to consolidate knowledge. When we have to explain our own understanding to someone else, we become more firm in our own understanding. Moreover, if we are having trouble with a certain type of problem, it can help to work through that problem with someone who has recently worked through it, or is working through problems of a similar type; a similar level of difficulty. This can provide a sort of scaffolding for further learning. Working in groups can get at this type of concept consolidation to which Hall was referring by.
How can we set this up in classrooms? Give student written exams on which they can work in groups. This promotes discussion of the ideas that are being tested, but it also encourages students to be ready to put their ideas into an application in a problem. One way to make this more enticing to students is by perhaps having one of the group members take notes and write a sort of journal on the process and what the group learned. Then this write up would be a part of the grade for the test as well. This way we focus on the MEANS of undestanding as well as the actual right or wrong-ness of a problem.