Mathematical Problem Solving

From the mathematical, or subject-matter, point of view, one sees mathematical problems as defining the discipline of mathematics. Mathematics is not simply the famous problems that great mathematicians have worked on; "all mathematics is created in the process of formulating and solving problems". Schoolchildren seldom see much of this face of mathematics; for them, the construction scaffolding has been taken away, and all that remains is the completed edifice through which they are guided.

Although some mathematics educators might question the impact that computer simulation models of cognitive processes have had on our thinking about the teaching of mathematical problem solving, one of the healthy lessons these models have taught us is that people do not live by processes alone. Studies of expert problem solvers and computer simulation models have shown that the solution of a complex problem requires (1) a rich store of organized knowledge about the content domain, (2) a set of procedures for representing and transforming the problem, and (3) a control system to guide the selection of knowledge and procedures. It is easy to underestimate the deep knowledge of mathematics and extensive experience in solving problems that underlie proficiency in mathematical problem solving. On the other hand, it is equally easy to underestimate the sophistication of the control processes used by experts to monitor and direct their problem-solving activity.

Stolurow's argument was that the computer permits researchers to construct predictive and descriptive models that can be used to make explicit various elements of teaching and their relationships. The parallel argument with respect to problem solving might suggest that from the various computer simulations of problem-solving behavior one can get some guidance for problem-solving instruction, and that is the direction some researchers have pointed. Whether mastering the model is superior to modeling the master in mathematical problem solving remains a largely unexplored question.

The various perspectives on teaching mathematical problem solving that have been advocated in recent years can be put, with some oversimplification, into five categories: osmosis, memorization, imitation, cooperation, and reflection. The names are meant to reflect the primary emphasis of the approach; most programs of problem-solving instruction combine features of several categories.

Some bottom-up approaches to instruction in mathematical problem solving are built upon task analyses that decompose the solution of a problem into atomic procedures, each of which is then taught. Essentially, an algorithm is developed that will handle a class of problems, and students are "programmed" to follow the algorithm to obtain a solution. Such approaches can be effective within narrow limits, but they cannot be used with problems to which the algorithm does not apply, and students often have trouble recognizing when the algorithm is applicable.