Motivation 

To summarize the research on the teaching of problem solving–how people apply their knowledge to new situations, reason about scenarios for which they have incomplete or uncertain information, and solve novel problems.

Problem Solving Definitions

Problem solving: a cognitive process that is used to transform a given state into a goal state when a problem does not have an obvious solution, often used interchangeably with thinking and reasoning. Problem solving can be academic, such as solving an unfamiliar arithmetic word problem, or non-academic, such as how get 3/4 of 2/3 of a cup of cottage cheese.

Types of problems (well-defined vs. ill-defined): Well-defined problems have clearly specified given (problem) states, goal (solution) states, and problem-solving spaces (i.e., the relevant information required to solve the problem and the rules/logic/operators that connection different bits of information). For example, an arithmetic problem, no matter how complex, is well-defined. In ill-defined problems, the given state, goal state, or problem-solving space might be unclear. For example, writing an essay or designing a sustainable building are ill-defined problems. The knowledge of the problem solver does not determine whether problems are well- or ill-defined.

Types of problems (routine vs. nonroutine): Unlike well- and ill-defined problems, the knowledge of the problem solver can determine whether problems are routine or nonroutine. For routine problems, problem solvers have a practiced procedure for solving the problem. For nonroutine problems, problem solvers have to identify a new path through the problem solving space to reach the goal state. The ability to solve nonroutine problems is relevant to the ability to transfer knowledge, i.e., use established knowledge to solve novel problems.

Cognitive Processes: Problem solving involves representing (i.e., defining given state, goal state, and problem-solving space), planning/monitoring (i.e., devising a method for solving the problem and evaluating its efficacy), executing (i.e., carrying out method), and self-regulating (i.e., monitoring progress and modifying method if appropriate). Most instruction focuses on execution/procedures, but problem solvers need to learn effective strategies for all cognitive processes.

Instruction that Promotes Problem Solving

Seven instructional methods for improving problem solving

1. Load-reducing methods – these methods aim to reduce the cognitive load required to solve the problem. Two examples are automaticity methods, in which the learner automates simple skills before tackling complex skills, and constraint removal methods, in which the learner does not need to use some undeveloped skills.
2. Structure-based methods – these methods help learners to select, organize, and integrate information required for cognitive processes, especially abstract processes. For example, using manipulatives (objects in early math instruction or labs in science instruction) can help learners process abstract information through interaction with concrete objects.
3. Schema-based methods – these methods provide or activate an existing schema to help learners process new information. Three examples are advanced organizers, which help learners understand the relationships among information, pretraining, which tells learners what the components of the system are before they learn about them, and cueing, which alerts learners to prior relevant knowledge.
4. Generative methods – these methods require learners to make connections between existing knowledge and new information. Four examples are elaborative methods, which asks learners to make these connections, note-taking, which ask learners to summarize and synthesize main points, self-explanation, which asks learners to explain to themselves why something works or a procedural step was taken, and questioning methods, which asks learners to create assessment questions for the information that they are learning.
5. Guided discovery methods – these methods provide various levels of guidance, depending on the skill being learned or the prior knowledge of the learner, for student-driven learning rather than instructor-driven direct instruction. These methods are designed to allow the learner to extend prior knowledge by discovering concepts and principles.
6. Modeling methods – these methods provide learners with models of problem solving for learners to mimic while solving novel problems. Two examples are example methods, which give learners the solutions to typical problems in the field to use as a model for solving other problems, and apprenticeship methods, which pairs learners with an experienced person so that the learner can observe the experienced person and, after completing scaffolding tasks, the learner can become independent.
7. Teaching thinking skills – this approach is different than the others in that it teaches generalizable problem solving strategies that apply to many kinds of problems rather than teaching a specific problem solving procedure in a way that transfers to solving novel problems of the same type. They have had various levels of success.

Why this is important

From these methods, Mayer and Wittrock identified and recommended general principles for problem-solving instruction (p. 299):

“Domain-specific principle: Rather than attempting to teach general problem-solving heuristics, it is better to teach problem solving skills within specific disciplines.

Near-transfer principle: Rather than expecting problem-solving skill to be applicable to a wide range of problems, it is better to expect that problem solving skills will be largely restricted with respect to their range of applicability.

Knowledge integration principle: Rather than focusing mainly on teaching of facts and procedures or on teaching concepts and strategies, it is better to integrate teaching of all these kinds of knowledge within guided problem solving tasks.”

These principles, based on decades of work in educational psychology and related fields, are important to keep in mind when teaching problem solving. If you want to go against these principles, then you’d better have a good justification for why.

Mayer, R. E., & Wittrock, M. C. (2006). Problem solving. In P. A. Alexander & P. H. Winne (Eds.), Handbook of Educational Psychology, 2nd edition (pp. 287-304). Mahwah, NJ: Lawrence Erlbaum Associates.

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