problem

A problem is a situation that presents a gap between a current state and a desired goal, requiring action to resolve. It involves an obstacle, uncertainty, or constraint that makes achieving the goal nontrivial. Problems can be perceived or real, routine or novel, personal or collective, and may be addressed through reasoning, experimentation, collaboration, or a combination of methods.

Etymology and usage: the word problem comes from the Greek problēma, meaning “a thing put forward,” from proballein, “to throw forward,” through Latin and into English. In everyday language, a problem is any situation calling for a solution or decision, while in formal fields it often denotes a precisely defined challenge with specific inputs and outputs.

Types and contexts: problems are categorized in various ways. Practical problems concern real-world tasks and constraints, such as repairing a device or meeting a deadline. Theoretical problems aim to understand principles or systems. Mathematical problems seek proofs, calculations, or constructions, and may be well-defined (with a clear goal and constraints) or ill-defined (requiring framing). In computer science, problems may be optimization, decision, or search tasks, sometimes with varying levels of computational difficulty.

Problem solving: addressing a problem typically follows a process that may be formal or informal. Key stages include recognizing and defining the problem, generating possible strategies, selecting and implementing a plan, and evaluating results. Common approaches include algorithmic methods, heuristics, trial and error, and collaborative reasoning. Reflecting on outcomes can lead to improved strategies for future problems.

Impact and pedagogy: problems drive inquiry and learning across disciplines. Problem-based learning, design thinking, and critical thinking curricula use real-world problems to develop analytical and collaborative skills, emphasizing the choice and justification of solutions rather than a single correct answer.