An education concept defining a learning, teaching, or organizational practice used to develop knowledge and skills. It applies when prerequisites are met and produces observable changes in learner performance or organizational capability. It does not ensure outcomes without appropriate implementation, feedback, and adjustment based on evidence. It materially affects learning and equity by shaping access, quality, and effectiveness of educational experiences. The concept is generally stable, though standards, tools, and research evidence evolve over time.

Principle

Problem solving combines domain knowledge, heuristic strategies, model building, and metacognitive regulation; success depends on iterating between representation, computation, and justification.

Demonstration

Students encounter a rate problem about mixing solutions, translate the context into ratios and equations, test a proposed approach, revise their model after a counterexample, and justify the final answer in writing.

Misapplication

Teaching problem solving as merely applying memorized algorithms to superficially novel questions or presenting only step lists without fostering model building and reflection undermines transfer.

Consequence

Authentic problem solving cultivates transferable reasoning, adaptive strategy use, and the capacity to tackle non-routine tasks across content domains.

Reversal

A reversal is limiting practice to routine exercises with known solution paths, which builds speed on familiar items but fails to develop strategies for new or ill-formed problems.

Boundary

Pertains to non-routine tasks that require interpretation and strategy; excludes routine procedural practice, drill for isolated facts, or tasks whose solution depends only on applying a single memorized step.

Semantic Tension

Tension exists between problem solving as open-ended exploration and as structured skill practice; educators must weigh guidance versus independence to support learning without denying challenge.

Synthesis

Problem Solving in mathematics is the cyclic activity of modeling unfamiliar situations, applying and adapting strategies, checking and revising work, and producing justified solutions that extend beyond rote application.