Mathematical Problem Solving
Our third example is Schoenfeld’s (1983, 1985) method for teaching mathematical problem solving to college students. Like the other two, this method is based on a new analysis of the knowledge and processes required for expertise, where expertise is understood as the ability to carry out complex problem-solving tasks. And like the other two, this method incorporates the basic elements of a cognitive apprenticeship, using the methods of modeling, coaching, and fading and of encouraging student reflection on their own problem-solving processes. In addition, Schoenfeld’s work introduces some new concerns, leading the way toward articulation of a more general framework for the development and evaluation of ideal learning environments.
One distinction between novices and experts in mathematics is that experts employ heuristic methods, usually acquired tacitly through long experience, to facilitate their problem solving. To teach these methods directly, Schoenfeld formulated a set of heuristic strategies, derived from the problem-solving heuristics of Polya (1945). These heuristic strategies consist of rules of thumb for how to approach a give problem. One such heuristic specifies how to distinguish special cases in solving math problems: for example, for series problems in which there is an integer parameter in the problem statement, one should try the cases n=1,2,3,4, and try to make an induction on those cases; for geometry problems, one should first examine cases with minimal complexity, such as regular polygons and right triangles. Schoenfeld taught a number of these heuristics and how to apply them in different kinds of math problems. In his experiments, Schoenfeld found that learning these strategies significantly increased students’ problem-solving abilities.
But as he studied students’ problem solving further, he became aware of other critical factors affecting their skill, in particular what he calls control strategies. In Schoenfeld’s analysis, control strategies are concerned with executive decisions, such as generating alternative courses of action, evaluating which will get you closer to a solution, evaluating which you are most likely to be able to carry out, considering what heuristics might apply, evaluating whether you are making progress toward a solution, and so on. Schoenfeld found that it was critical to teach control strategies, as well as heuristics.
As with the reading and writing examples, explicit teaching of these elements of expert practice yields a fundamentally new understanding of the domain for students. To students, learning mathematics had meant learning a set of mathematical operations and methods. Schoenfeld’s method is teaching students that doing mathematics consists not only in applying problem-solving procedures but in reasoning about and managing problems using heuristics and control strategies.
Schoenfeld’s teaching employs the elements of modeling, coaching, scaffolding, and fading in a variety of activities designed to highlight different aspects of the cognitive processes and knowledge structures required for expertise. For example, as a way of introducing new heuristics, he models their selection and use in solving problems for which they are particularly relevant. In this way, he exhibits the thinking processes (heuristics and control strategies) that go on in expert problem solving but focuses student observation on the use and management of specific heuristics. The example in the sidebar provides a protocol from one such modeling.
Next, he gives the class problems to solve that lend themselves to the use of the heuristics he has introduced. During this collective problem solving, he acts as a moderator, soliciting heuristics and solution techniques from the students while modeling the various control strategies for making judgments about how best to proceed. The division of labor has several effects. First, he turns over some of the problem-solving process to students by having them generate alternative courses of action but provides a major support or scaffolding by managing the decisions about which course to pursue, when to change course, etc. Second, significantly, he no longer models the entire expert problem-solving process but a portion of it. In this way, he shifts the focus from the application or use of specific heuristics to the application or use of control strategies in managing those heuristics.
Like Scardamalia and Bereiter, Schoenfleld employs a third kind of modeling that is designed to change students’ assumptions about the nature of expert problem solving. He challenges students to find difficult problems and at the beginning of each class offers to try to solve one of their problems. Occasionally, the problems are hard enough that the students see him flounder in the face of real difficulties. During these sessions, he models for students not only the use of heuristics and control strategies but the fact that one’s strategies sometimes fail. In contrast, textbook solutions and classroom demonstrations generally illustrate only the successful solution path, not the search space that contains all of the dead-end attempts. Such solutions reveal neither the exploration in searching for a good method nor the necessary evaluation of the exploration. Seeing how experts deal with problems that are difficult for them is critical to students’ developing a belief in their own capabilities. Even experts stumble, flounder, and abandon their search for a solution until another time. Witnessing these struggles helps students realize that thrashing is neither unique to them nor a sign of incompetence.
In addition to class demonstrations and collective problem solving, Schoenfeld has students participate in small-group problem-solving sessions. During these sessions, Schoenfeld acts as a “consultant” to make sure that the groups are proceeding in a reasonable fashion. Typically he asks three questions: What are they doing, why are they doing it, and how will success in what they are doing help them find a solution to the problem? Asking these questions serves two purposes: First, it encourages the students to reflect on their activities, thus promoting the development of general self-monitoring and diagnostic skills; second, it encourages them to articulate the reasoning behind their choices as they exercise control strategies. Gradually, the students, in anticipating his questioning, come to ask the questions of themselves, thus gaining control over reflective and metacognitive processes in their problem solving. In these sessions, then, he is fading relative to both helping students generate heuristics and, ultimately, to exercising control over the process. In this way, they gradually gain control over the entire problem-solving process.