Rather than teaching about problem solving - somewhat of a separate strand of the mathematics curriculum - this chapter is about the teaching that should go on almost every day in a constructivist classroom. It is based on the thesis that the best way to promote active reflective thought and student involvement in mathematics, and hence understanding, is through solving problems. Furthermore, the proposition is that virtually all mathematics can be developed in a problem-solving manner. Problem solving objectives - process, metacognitive, affective - still remain. However, these are to be developed as we teach the regular mathematics curriculum with problems.
In effect, then, this is the chapter that explains to teachers how to teach and how to develop those problems that are most useful to them. This chapter implements the constructivist view of Chapter 3.
Teaching Through Problem Solving
In their 1996 article, Hiebert and his colleagues propose the following principle: "Students should be allowed to make the subject problematic. … Allowing the subject to be problematic means allowing students to wonder why things are, to inquire, to search for solutions, and to resolve incongruities. It means that both curriculum and instruction should begin with problems, dilemmas, and questions for students" (Hiebert, et al, 1996, p. 12, and T25 and 26)
This is the idea that is at the heart of this chapter and which I hope you will find echoed throughout the book. Quite simply, we need to teach mathematics in a problem-based manner. This completely reverses the notion of teaching mathematics that most teachers and your students will have. It is the exact opposite of teaching by telling. The teacher does not do the explaining.
The Before, During, After Model
Whether it is a 10 minute activity, a full-period problem, or a multi-day task, a simple way to think about organizing the instruction is around these three parts: before, during, and after. For each of these lesson parts, the text lists separately agendas that must be met and corresponding teaching actions that can be used to meet these agendas. (T- 32 to 38) For most teachers, the most difficult part is the during section. Here teachers must let go of their control and their role of being the font of all knowledge.
The most frequently neglected section is the after portion of the lesson. In order to make teaching with problem-based tasks effective, there must be a significant period in which students share and discuss their results and justify them to the class (not to the teacher). When different results and strategies occur, there is opportunity for rich discourse and learning. More "blue dots" will be activated.
One of my main agendas is to convince teachers that problem-based tasks are not all that difficult to develop. If you let them believe that the only way they can find these tasks is to search through journals and resource books, it is likely that they will never completely adopt this approach to instruction. While there are many excellent sources of good tasks, and these should certainly become part of a teacher's professional library, the easiest place to begin is with their basal textbook. In this chapter you will find two examples of converting traditional textbook lessons into problematic lessons or tasks (pp.49, 50). I have found that once teachers get the idea of teaching through problem solving they are quite able to convert these lessons with little difficulty.
With inservice teachers, I have found that it is first important to focus on "big ideas," or to establish a unit perspective concerning the content they have to teach. Every chapter in Section 2 begins with a list of Big Ideas that I hope will help. Preservice teachers have not yet wrestled with traditional textbooks or the objective lists found in many school districts that tend to atomize content into isolated behavioral objectives that get in the way of the performance task approach I am suggesting.
Developing a Community of Learners
Many teachers believe that good discussions in the after portion of a lesson will simply happen. In fact, students must be taught how to behave as a community of learners. Although social norms for classroom discussions are often taught in general methods classes, considerable research in this area has been done by mathematics educators. Based on this research, suggestions for developing a community of learners is found on p. 47 with a listing on T
Teaching About Problem Solving
This section of the chapter (pp. 57 to the end) reflects the older notion of problem solving as a strand of the curriculum. Here you will find outlined (also on a T-43) the general goals of problem solving. Rather than teach to these goals via the process problems popular in the 1980s and early 1990s, I believe that these goals can be achieved as problem-based tasks are used to teach the regular curriculum. I personally will be happy if my students can understand the importance of the objectives of problem solving as outlined pages 57 - 59.