Challenges of Teaching Problem Solving

Challenges of Teaching Problem Solving - Although Polya presented the inquiry-based framework for teaching problem solving more than 50 years ago, there has yet to be widespread implementation of his ideas in U.S. classrooms. This suggests that there are a number of challenges to making this shift in mathematics teaching.

Teaching nonroutine problem solving is difficult. True problem solving is as demanding on the teacher as it is on the students. The art of teaching mathematical problem solving is best mastered over a long period of time (Thompson, 1989). Teaching problem solving is difficult, writes Schoenfeld (1992). Teachers:

• Must perceive the implications of students' different approaches, whether they may be fruitful and, if not, what might make them so.
• Must decide when to intervene, and what suggestions will help the students whill leaving the solution essentially in their hands, and carry this through for each student.
• Will at times be in the position of not knowing; to work well without knowing all the answers requires experience, confidence, and self-awareness.

Burkhardt (1988, as cited in Schoenfeld, 1992) states even more succinctly that teaching problem solving is difficult for teachers mathematically, pedagogically, and personally. Teachers must have the mathematical expertise to understand the different approaches that students might take to a problem and how promising those approaches will be. Many elementary teachers are trained as generalists and often do not have the strong mathematical background required to teach from a problem-solving approach.

Pedagogically, teachers must make complex decisions about the level of difficulty of the problems assigned, when to give help, and how to give assistance that supports students’ success while ensuring that they retain ownership of their solution strategies. Personally, teachers will sometimes find themselves in the uncomfortable position of not knowing the solution. Letting go of the “expert” role teachers have traditionally played requires experience, confidence, and self-awareness. Often, teachers are asked to teach mathematics they never encountered in school and in a way that differs from how they were taught. For these reasons, teachers may need additional training in mathematical content and theory, as well as in methods for teaching problem solving.

Nonroutine problems are difficult for students. Nonroutine, open-ended problems are often, by their nature, difficult for many students. Shannon and Zawojewski (1995) conducted a ministudy that demonstrated the difficulty presenting problem-solving tasks without providing hints and procedural steps poses to students. In the study, two groups of students were presented with similar tasks. In one task, “Supermarket Carts,” students were given a scale drawing of 12 shopping carts nested together and asked to create a rule to determine the length of storage space needed for any number of carts and the number of carts that would fit into a given space. This was essentially all the direction given.

A second group of students was assigned the task “Shopping Carts,” which included several prompts or subproblems to help guide them toward a solution. Students were asked to find the length of one shopping cart, find how much a cart sticks out when the carts are nested, find the total length of 20 carts, and find how many carts could fit into a 10-meter space. Then they were asked to find the two formulas that were asked for in the Supermarket Carts task.

The researchers reported that students attempting the Supermarket Carts task had difficulty knowing how to get started. Only a few students successfully derived the formulas required. On the other hand, none of the students working on the Shopping Carts task had any difficulty getting started, and all but one group successfully derived the requested formulas. The authors conclude that, “the sense of students’ having to struggle was greater in Supermarket Carts than in Shopping Carts.” Watching their students struggle in frustration is often very difficult for teachers. Knowing when to give hints and how much help to give requires striking a delicate balance that comes with experience and knowing students’ capabilities.

Teachers are concerned about content coverage. The TIMSS research characterized the U.S. curriculum as “a mile wide and an inch deep” compared to the mathematics curriculum in other countries (Peak, 1996, 1997; Takahira, et al, 1998). Teachers in the U.S. are generally expected to cover large areas of content each year. Yet solving challenging, nonroutine problems takes time. Often a single problem can occupy a class for a whole period or more. Therefore, it’s essential that content and skills be integrated within the context of problem solving. By selecting rich, engaging, and worthwhile tasks, teachers can ensure that time is well-spent.

Textbooks present few nonroutine problems . Although they are improving, many textbooks do not provide an adequate number of nonroutine problems from which teachers can choose. Many teachers are not comfortable straying from the scope and sequence the textbook provides, but they must develop the confidence to search out and develop other materials to supplement their texts.

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