• Interpretations of the term “problem solving” vary considerably, ranging from the solution of standard word problems in texts to the solution of non routine problems. In turn, the interpretation used by an educational researcher directly impacts the research experiment undertaken, the results, the conclusions, and any curricular implications (Fuson, 1992c).
• Problem posing is an important component of problem solving and is fundamental to any mathematical activity (Brown and Walter, 1983, 1993).
• Teachers need assistance in the selection and posing of quality mathematics problems to students. The primary constraints are the mathematics content, the modes of presentation, the expected modes of interaction, and the potential solutions (concrete and low verbal). Researchers suggest this helpful set of problem-selection criteria:
1. The problem should be mathematically significant.
2. The context of the problem should involve real objects or obvious simulations of real objects.
3. The problem situation should capture the student’s interest because of the nature of the problem materials, the problem situation itself, the varied transformations the child can impose on the materials, or because of some combination of these factors.
4. The problem should require and enable the student to make moves, transformations, or modifications with or in the materials.
5. Whenever possible, problems should be chosen that offer opportunities for different levels of solutions.
6. Whatever situation is chosen as the particular vehicle for the problems, it should be possible to create other situations that have the same mathematical structure (i.e., the problem should have many physical embodiments).
7. Finally, students should be convinced that they can solve the problem and should know when they have a solution for it.Most of these criteria apply or are appropriate to the full grade scale, K–12 (Nelson and Sawada, 1975).
• A problem needs two attributes if it is to enhance student understanding of mathematics. First, a problem needs the potential to create a learning environment that encourages students to discuss their thinking about the mathematical structures and underlying computational procedures within the problem’s solution. Second, a problem needs the potential to lead student investigations into unknown yet important areas in mathematics (Lampert, 1991).
• Algebra students improve their problem solving performance when they are taught a Polya-type process for solving problems, i.e., understanding the problem, devising a plan of attack, generating a solution, and checking the solution (Lee, 1978; Bassler et al., 1975).
• In “conceptually rich” problem situations, the “poor” problem solvers tended to use general problem solving heuristics such as working backwards or means-ends analysis, while the “good” problem solvers tended to use “powerful content-related processes” (Larkin et al., 1980; Lesh, 1985).
• Mathematics teachers can help students use problem solving heuristics effectively by asking them to focus first on the structural features of a problem rather than its surface-level features (English and Halford, 1995; Gholson et al., 1990).
• Teachers’ emphasis on specific problem solving heuristics (e.g., drawing a diagram, constructing a chart, working backwards) as an integral part of instruction does significantly impact their students’ problem solving performance. Students who received such instruction made more effective use of these problem solving behaviors in new situations when compared to students not receiving such instruction (Vos, 1976; Suydam, 1987).
• Explicit discussions of the use of heuristics provide the greatest gains in problem solving performance, based on an extensive meta-analysis of 487 research studies on problem solving. However, the benefits of these discussions seems to be deferred until students are in the middle grades, with the greatest effects being
realized at the high school level. As to specific heuristics, the most important are the drawing of diagrams, representing a problem situation with manipulative objects, and the translation of word situations to their representative symbolic situations (Hembree, 1992).
• Mathematics teachers who help students improve as problem solvers tend to ask frequent questions and use problem resources other than the mathematics textbook. Less successful teachers tend to demonstrate procedures and use problems taken only from the students’ textbook (Suydam, 1987).
• Young children often make errors when solving mathematical problems because they focus on or are distracted by irrelevant aspects of a problem situation (Stevenson, 1975). This error tendency decreases as students pass through the higher grades, yet the spatial-numerical distracters (i.e., extraneous numbers or diagrams) remain the most troublesome over all grade levels (Bana and Nelson, 1978).
• In their extensive review of research on the problem solving approaches of novices and experts, the National Research Council (1985) concluded that the success of the problem solving process hinges on the problem solvers’ representation of the problem. Students with less ability tend to represent problems using only the surface features of the problem, while those students with more ability represent problems using the abstracted, deeper-level features of the problem. The recognition of important features within a problem is directly related to the “completeness and coherence” of each problem solver’s knowledge organization.
• Young students (Grades 1–3) rely primarily on a trial-and-error strategy when faced with a mathematics problem. This tendency decreases as the students enter the higher grades (Grades 6–12). Also, the older students benefit more from their observed “errors” after a “trial” when formulating a better strategy or new “trial” (Lester, 1975).
• While solving mathematical problems, students adapt and extend their existing understandings by both connecting new information to their current knowledge and constructing new relationships within their knowledge structure (Silver and Marshall, 1990).
• Students solving a mathematics problem in small groups use cognitive behaviors and processes that are essentially similar to those of expert mathematical problem solvers (Artz and Armour-Thomas, 1992).
• Solution setup (e.g., organizing data into a table, grouping data into sets, formulating a representative algebraic equation) is the most difficult of the stages in the problem solving process (Kulm and Days, 1979).
• Problem solving ability develops slowly over a long period of time, perhaps because the numerous skills and understandings develop at different rates. A key element in the development process is multiple, continuous experiences in solving problems in varying contexts and at different levels of complexity (Kantowski, 1981).
• Results from the Mathematical Problem Solving Project suggest that willingness to take risks, perseverance, and self-confidence are the three most important influences on a student’s problem solving performance (Webb et al., 1977).
• A reasonable amount of tension and discomfort improves the problem solving performance of students, with the subsequent release of the tension after the solution of problem serving as a motivation. If students do not develop tension, the problem is either an exercise or they are “generally unwilling to attack the problem
in a serious way” (Bloom and Broder, 1950; McLeod, 1985).
• The elements of tension and relaxation are key motivational parts of the dynamics of the problem solving process and help explain why students tend not to “look back” once a problem is solved. Once students perceive “that the problem solution (adequate or inadequate) is complete, relaxation occurs and there is no
more energy available to address the problem” (Bloom and Broder, 1950).
• Students tend to speed-read through a problem and immediately begin to manipulate the numbers involved in some fashion (often irrationally). Mathematics teachers need to encourage students to use “slow-down” mechanisms that can help them concentrate on understanding the problem, its context, and what is being
asked (Kantowski, 1981).
• A “vital” part of students’ problem solving activity is metacognition, which includes both the awareness of their cognitive processes and the regulation of these processes (Lester, 1985).
• Students can solve most one-step problems but have extreme difficulty trying to solve nonstandard problems, problems requiring multi-steps, or problems with extraneous information. Teachers must avoid introducing students to techniques that work for one-step problems but do not generalize to multi-step problems, such as the association of “key” words with particular operations (Carpenter et al., 1981).
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