The arguments about problem solving versus drill and speed have arisen again. I think what Lucille says is important (see below). In addition to the _completion_ that she talks about I would like to put a word in for _connection_. I have just finished reading Children and Number by Martin Hughes. This gave me cause to worry about what I was teaching in Math for Elem Ed this summer and to think about classes I have taught in the past.
I always had the idea that if I interspersed drill with activities that (I hope) lead to understanding then the students will make the connection themselves. And even if they don't make the connection themselves, then surely they listen to my five minute lectures where I point out the connection. I am wrong. Connections are very difficult for students to "make." It is difficult for them to "see" that manipulating Dienes blocks has any connection to the algorithm for multiplying two two-digit numbers. Even if they have been "shown" the connection.
In other words, no matter what focus we choose, we must make our students understand that mathematics is not composed of individual tasks that stand apart, but fibers that become threads that become cloth that becomes ... . The problem is that today we work on a fiber that will be part of the sleeve and tomorrow we work on a fiber that becomes part of the pocket. It is extremely difficult for students to see this big picture and even more so if they don't make the smaller (obviously) connections.
>...Yes, keep the focus on understanding and >meaning. Help people to see that they can construct mathematics. Teach so >they are in control of the material they are learning. But then carry that >understanding forward to some form of completion, with competence, accuracy, >efficiency and rigor. With the necessary shift in focus toward problem >solving and constructivism some (myself included) have lost sight of the >necessity to carry the job to completion. Where that line of completion is >at this time I am not quite so sure. I'm still thinking about this. > >Lucille L. Peterson
--------------------- Murphy Waggoner Department of Mathematics Simpson College 701 North C Street Indianola, IA 50125 firstname.lastname@example.org ---------------------