Rutgers/Lucent ALLIES IN TEACHING MATHEMATICS AND TECHNOLOGY Grant Using technology not simply to do things better, but to do better things. Thinking About Teaching Mathematics: Questions for Guiding Problem Solving (adapted from J. Mason, Thinking Mathematically, Addison-Wesley, 1982) Background By definition, a problem presents a situation and a question for which you do not initially know an answer; often, even a process to develop an answer may not be immediately evident. Thus, solving problems is an explorative process rather than an algorithmic process. Even if you know a set of general strategies, you may often be stuck while probing your way through solving a rich or complex problem. One heuristic model is to think of the process of solving problems as having three loosely defined stages or phases: entry, attack and review. The boundaries between these stages are usually blurred. Entry The entry stage includes understanding the problem. This often requires "playing" with the situation - prodding and exploring - to expose the subtleties or complexities of the setting and elements of the problem situation. Attack The attack stage includes getting to the solution, or realizing the need to cycle back to get a better understanding of the problem. It is in this stage that people often get stuck and feel most frustrated. Usually, an 'attack' involves more than a simplistic approach like "devise and carry out a plan". Often, especially with a rich problem, a plan emerges slowly and haltingly, and the solution becomes evident almost as a quick surprise ending. Frequently, the plan becomes clear only when you examine the completed process as a cogent sequence. Review The review stage is where reflection, and therefore the greatest learning, takes place. This should be more than just checking with an authority to see if the answer is 'right'. This stage provides the opportunity to understand the process, and know that the solution seems reasonable. Beyond that, reviewing provides the opportunity to think about alternative strategies and approaches, and their relative 'elegance'. During this stage we should also review what was learned about mathematical connections, about problem solving approaches, and about how we experience the process Š the "thrill of victory and the agony of defeat". Some questions for guiding people in the ENTRY phase: What was given? What do you know? What other information do you need? How would knowing that help? In how many different ways could the situation be interpreted? How does choosing any particular interpretation help me move forward? How can you organize what you know - will a diagram, list, table, or graph help? What are you asked to find? Will a wild guess or reasoned estimate help? If there are many answers, can you start with a simple answer? How would you check an answer if you had one? HOW DO YOU START? What could you do first? Can you simplify the problem by working on a part of it, or by ignoring one or more of the conditions? Can you find something/anything to do? Some questions for guiding people in the ATTACK phase: Can you make a conjecture? Can you confirm the conjecture? If it is true, what does that tell you about the problem? If it is false, can you modify it to make it true? Can you create some examples and look for a pattern? IF YOU ARE STUCK: Have you reviewed what you are trying to find? Have you reviewed what you know? How else could you use what you know? How else could you organize what you know? What other connections can you imagine between what you know and what you are trying to find? Have you considered such strategies as: Work backwards? Make a model? Solve an easier related problem? Use an algebraic representation? Use a process of elimination, or guess-check-revise? Have you tried a changing your point of view and looking at the situation in a completely different way? What have you tried that has NOT worked? Why didnÕt it work? Would trying it again produce a different result? Some questions for guiding people in the REVIEW phase: How do you know that you have a solution? Could you convince a friend that it is correct? Can you find the solution in a different way? Is one way simpler or easier to present than another? What did you learn about mathematics while solving this problem? Can you make another problem related to this problem that would be engaging and worth trying to solve? What did you learn about problem solving while solving this problem? What did you learn about yourself while solving this problem? Return to Agenda THE MATH FORUM: Creating community, developing resources, constructing knowledge...